From 84672ba096f97f7c4c9ff45b2d98e8f66e1f669e Mon Sep 17 00:00:00 2001
From: agentmess <agentmess@users.noreply.github.com>
Date: Thu, 17 Oct 2024 23:37:29 +0000
Subject: [PATCH] deploy: f5fbcc4e5ce57178b20b47cdec276d87bac7adcd

---
 Artifacts.html                                |   7 +
 Fast Imaging Pulse Sequences.html             |  56 ++++
 Gradient and Spin Echoes.html                 |   4 +-
 MRI Contrast.html                             |  20 +-
 MRI Signal Equation.html                      |  12 +-
 Magnetic Fields and RF Coils.html             |   2 +-
 RF Pulses.html                                |   4 +-
 Spatial Encoding.html                         | 274 ++++++++++--------
 Spectral-Spatial RF Pulses.html               |   2 +-
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 _sources/Artifacts.ipynb                      |  10 +
 _sources/FOV and Resolution.ipynb             |   2 +-
 _sources/Fast Imaging Pulse Sequences.ipynb   |  47 ++-
 _sources/Gradient and Spin Echoes.ipynb       |   4 +-
 _sources/MRI Contrast.ipynb                   |  42 +--
 _sources/MRI Signal Equation.ipynb            |   6 +-
 _sources/Magnetic Fields and RF Coils.ipynb   |   2 +-
 _sources/RF Pulses.ipynb                      |   4 +-
 _sources/Spatial Encoding.ipynb               | 101 ++++++-
 reports/Spatial Encoding.err.log              |  59 ++++
 searchindex.js                                |   2 +-
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diff --git a/Artifacts.html b/Artifacts.html
index 08f7f7c..d7a91f2 100644
--- a/Artifacts.html
+++ b/Artifacts.html
@@ -413,6 +413,7 @@ <h2> Contents </h2>
             <nav aria-label="Page">
                 <ul class="visible nav section-nav flex-column">
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#artifact-comparison">Artifact Comparison</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations-of-artifacts">Simulations of Artifacts</a></li>
 </ul>
@@ -450,6 +451,11 @@ <h2>Learning Goals<a class="headerlink" href="#learning-goals" title="Permalink
 <li><p>Identify artifacts and how to mitigate them</p></li>
 </ol>
 </section>
+<section id="introduction">
+<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading">#</a></h2>
+<p>Many of the artifacts that occur in MRI can be understood and analyzed using the k-space perspective.  In particular, they can be understood as the k-space data being modified by some function.  This is described mathematically in
+<span class="xref myst">MRI Signal Equation and K-psace - K-space Data Weighting</span></p>
+</section>
 <section id="artifact-comparison">
 <h2>Artifact Comparison<a class="headerlink" href="#artifact-comparison" title="Permalink to this heading">#</a></h2>
 <table class="table">
@@ -755,6 +761,7 @@ <h2>Simulations of Artifacts<a class="headerlink" href="#simulations-of-artifact
   <nav class="bd-toc-nav page-toc">
     <ul class="visible nav section-nav flex-column">
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#artifact-comparison">Artifact Comparison</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations-of-artifacts">Simulations of Artifacts</a></li>
 </ul>
diff --git a/Fast Imaging Pulse Sequences.html b/Fast Imaging Pulse Sequences.html
index 65a0e73..de63f6a 100644
--- a/Fast Imaging Pulse Sequences.html	
+++ b/Fast Imaging Pulse Sequences.html	
@@ -415,6 +415,12 @@ <h2> Contents </h2>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#echo-planar-imaging-epi">Echo-planar Imaging (EPI)</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fast-turbo-spin-echo-fse-tse">Fast/Turbo Spin-echo (FSE/TSE)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fast-gradient-echo-sequences">Fast gradient-echo sequences</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#spoiler-or-crusher-gradients">Spoiler or Crusher gradients</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#rf-spoiling">RF spoiling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#types-of-fast-gradient-echo-sequences">Types of fast gradient-echo sequences</a></li>
+</ul>
+</li>
 </ul>
             </nav>
         </div>
@@ -442,18 +448,21 @@ <h2>Learning Goals<a class="headerlink" href="#learning-goals" title="Permalink
 <ul class="simple">
 <li><p>Describe how EPI works</p></li>
 <li><p>Describe how FSE/TSE sequences work</p></li>
+<li><p>Describe how fast gradient-echo sequences work</p></li>
 </ul>
 </li>
 <li><p>Manipulate MRI sequence parameters to improve performance</p>
 <ul class="simple">
 <li><p>Understand how and when to accelerate with EPI</p></li>
 <li><p>Understand how and when to accelerate with FSE/TSE</p></li>
+<li><p>Understand how contrast changes in fast gradient-echo sequences</p></li>
 </ul>
 </li>
 <li><p>Identify artifacts and how to mitigate them</p>
 <ul class="simple">
 <li><p>Identify EPI artifacts including distortion and T2*</p></li>
 <li><p>Identify FSE/TSE artifacts include T2 blurring</p></li>
+<li><p>Identify fast gradient-echo artifacts such as banding in bSSFP</p></li>
 </ul>
 </li>
 </ol>
@@ -466,6 +475,47 @@ <h2>Echo-planar Imaging (EPI)<a class="headerlink" href="#echo-planar-imaging-ep
 <h2>Fast/Turbo Spin-echo (FSE/TSE)<a class="headerlink" href="#fast-turbo-spin-echo-fse-tse" title="Permalink to this heading">#</a></h2>
 <p>These pulse sequences use multiple spin-echo refocusing pulses after a single exictation pulse.  This enables multiple k-space lines to be acquired during the multiple spin-echoes that are formed.  This technique was originally called Rapid Acquisition with Relaxation Enhancement (RARE), and is known on various MRI scanners as fast spin-echo (FSE, GE Healthcare), turbo spin-echo (TSE, Siemens Healthineers), or turbo field-echo (TFE, Philips Healthcare).</p>
 </section>
+<section id="fast-gradient-echo-sequences">
+<h2>Fast gradient-echo sequences<a class="headerlink" href="#fast-gradient-echo-sequences" title="Permalink to this heading">#</a></h2>
+<p>The basic gradient-echo sequence is typically a spoiled gradient-echo sequence.  The sequence is called spoiled because the transverse magnetization is spoiled by a spoiler gradient before the next RF pulse.</p>
+<section id="spoiler-or-crusher-gradients">
+<h3>Spoiler or Crusher gradients<a class="headerlink" href="#spoiler-or-crusher-gradients" title="Permalink to this heading">#</a></h3>
+<p>A large, unbalanced gradient will create dephasing of the transverse magnetization across the imaging voxels.  This effectively eliminates the signal.  However, the net magnetization is not truly eliminated, and this can be refocused by gradients.</p>
+</section>
+<section id="rf-spoiling">
+<h3>RF spoiling<a class="headerlink" href="#rf-spoiling" title="Permalink to this heading">#</a></h3>
+<p>The RF axis of rotation is changed every TR.  This reduces the chances of magnetization from previous TRs to become coherently excited.  Specific RF spoiling schemes, such as quadratic phase incrementation, is required for this to be effective.</p>
+</section>
+<section id="types-of-fast-gradient-echo-sequences">
+<h3>Types of fast gradient-echo sequences<a class="headerlink" href="#types-of-fast-gradient-echo-sequences" title="Permalink to this heading">#</a></h3>
+<p>Spoiled gradient-echo (SPGR)/fast low-angle shot (FLASH)</p>
+<ul class="simple">
+<li><p>Aims to full spoil transverse magnetization every TR</p></li>
+<li><p>Both gradient and RF spoiling are used</p></li>
+<li><p>Pure T1 weighted contrast</p></li>
+</ul>
+<p>Gradient-recalled acquisition in the steady state (GRASS)/fast imaging with steady-state precession (FISP)</p>
+<ul class="simple">
+<li><p>Allows for residual transverse magnetization to be used in the next TR</p></li>
+<li><p>Refouces frequency and phase encoding gradients</p></li>
+<li><p>RF spoiling used</p></li>
+<li><p>Increases T2* contrast</p></li>
+</ul>
+<p>Steady-state free precession (SSFP)/time-reversed fast imaging with steady-state precession (PSIF)</p>
+<ul class="simple">
+<li><p>Use gradients to refocus signals from previous TRs</p></li>
+<li><p>Gradient spoiling but refocused in later TRs</p></li>
+<li><p>Creates T2/T1 contrast</p></li>
+</ul>
+<p>Balanced SSFP/TrueFISP</p>
+<ul class="simple">
+<li><p>Balanced gradients every TR</p></li>
+<li><p>No spoiling, all magnetization preserved each TR</p></li>
+<li><p>Creates T2/T1 contrast</p></li>
+<li><p>High SNR efficiency</p></li>
+</ul>
+</section>
+</section>
 </section>
 
     <script type="text/x-thebe-config">
@@ -534,6 +584,12 @@ <h2>Fast/Turbo Spin-echo (FSE/TSE)<a class="headerlink" href="#fast-turbo-spin-e
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#echo-planar-imaging-epi">Echo-planar Imaging (EPI)</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fast-turbo-spin-echo-fse-tse">Fast/Turbo Spin-echo (FSE/TSE)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fast-gradient-echo-sequences">Fast gradient-echo sequences</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#spoiler-or-crusher-gradients">Spoiler or Crusher gradients</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#rf-spoiling">RF spoiling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#types-of-fast-gradient-echo-sequences">Types of fast gradient-echo sequences</a></li>
+</ul>
+</li>
 </ul>
   </nav></div>
 
diff --git a/Gradient and Spin Echoes.html b/Gradient and Spin Echoes.html
index f82fb4a..ecca049 100644
--- a/Gradient and Spin Echoes.html	
+++ b/Gradient and Spin Echoes.html	
@@ -487,8 +487,8 @@ <h2>Gradient Echo Pulse Sequence (GE or GRE)<a class="headerlink" href="#gradien
 <h2>Off-resonance<a class="headerlink" href="#off-resonance" title="Permalink to this heading">#</a></h2>
 <p>Ideally, the magnetic field is uniform in the absense of any applied gradients.  However, in practice there are unavoidable variations in the magnetic field that lead to changes in resonance frequency.  These magnetic field variations are referred to as “off-resonance”, and can be represented as</p>
 <div class="math notranslate nohighlight">
-\[\Delta f_r(\vec{r})\]</div>
-<p>This typically includes imperfections in the main <span class="math notranslate nohighlight">\(B_0\)</span> magnetic as well as subject-specific changes in the magnetic field that are largely due to magnetic susceptibility effects.</p>
+\[\Delta f_r(\vec{r}) = \bar \gamma $\Delta B_0(\vec{r})$ \]</div>
+<p>These are due imperfections in the main <span class="math notranslate nohighlight">\(B_0\)</span> magnet as well as subject-specific changes in the magnetic field that are largely due to magnetic susceptibility effects.</p>
 </section>
 <section id="effects-of-off-resonance-and-t2">
 <h2>Effects of Off-resonance and T2*<a class="headerlink" href="#effects-of-off-resonance-and-t2" title="Permalink to this heading">#</a></h2>
diff --git a/MRI Contrast.html b/MRI Contrast.html
index 27d521b..7561cfc 100644
--- a/MRI Contrast.html	
+++ b/MRI Contrast.html	
@@ -686,6 +686,13 @@ <h3>90-degree flip angles<a class="headerlink" href="#degree-flip-angles" title=
 <div class="math notranslate nohighlight">
 \[S \propto M_0 (1- \exp(-TR/T_1) )\]</div>
 <p>Illustrated in the first example below.  This shows that the magnetization reaches steady state in the 2nd TR.</p>
+</section>
+<section id="id3">
+<h3>&lt; 90-degree flip angles<a class="headerlink" href="#id3" title="Permalink to this heading">#</a></h3>
+<p>For T1-weighting, it is generally more efficient in terms of signal acqruied per time to use &lt; 90-degree flip angles every TR, in which case</p>
+<div class="math notranslate nohighlight">
+\[S \propto M_0 \sin(\theta) \frac{1- \exp(-TR/T_1)}{1- \cos(\theta) \exp(-TR/T_1)}\]</div>
+<p>Illustrated in this second example below.  This shows that the magnetization can take many TRs to reach steady state.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% Signal evolution between TRs with 90-degree pulses</span>
@@ -774,13 +781,6 @@ <h3>90-degree flip angles<a class="headerlink" href="#degree-flip-angles" title=
 <img alt="_images/eda58af1383601bd31799acf1da6673f51aedf43bb9071f27aa042e63808fbbb.png" src="_images/eda58af1383601bd31799acf1da6673f51aedf43bb9071f27aa042e63808fbbb.png" />
 </div>
 </div>
-</section>
-<section id="id3">
-<h3>&lt; 90-degree flip angles<a class="headerlink" href="#id3" title="Permalink to this heading">#</a></h3>
-<p>For T1-weighting, it is generally more efficient in terms of signal acqruied per time to use &lt; 90-degree flip angles every TR, in which case</p>
-<div class="math notranslate nohighlight">
-\[S \propto M_0 \sin(\theta) \frac{1- \exp(-TR/T_1)}{1- \cos(\theta) \exp(-TR/T_1)}\]</div>
-<p>Illustrated in this second example below.  This shows that the magnetization can take many TRs to reach steady state.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% Signal evolution between TRs with &lt;90-degree pulses</span>
@@ -1144,6 +1144,9 @@ <h2>Simple Contrast Phantom<a class="headerlink" href="#simple-contrast-phantom"
 <div class="highlight-Octave notranslate"><div class="highlight"><pre><span></span><span class="n">signal_gre</span> <span class="p">=</span> <span class="n">MRsignal_spoiled_gradient_echo</span><span class="p">(</span><span class="n">flip</span><span class="p">,</span> <span class="n">TE</span><span class="p">,</span> <span class="n">TR</span><span class="p">,</span> <span class="n">M0</span><span class="p">,</span> <span class="n">T1</span><span class="p">,</span> <span class="n">T2</span><span class="p">)</span>
 </pre></div>
 </div>
+<section id="challenge-yourself">
+<h3>Challenge Yourself!<a class="headerlink" href="#challenge-yourself" title="Permalink to this heading">#</a></h3>
+<p>Based on the images and TE/TR parameters below, can you estimate T1 and T2 relaxation times of the 3 objects?</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% contrast_phantom</span>
@@ -1201,9 +1204,6 @@ <h2>Simple Contrast Phantom<a class="headerlink" href="#simple-contrast-phantom"
 <img alt="_images/03b97f10f51c0541981866f210657bdf0512e1458495d474287dff422cf2540f.png" src="_images/03b97f10f51c0541981866f210657bdf0512e1458495d474287dff422cf2540f.png" />
 </div>
 </div>
-<section id="challenge-yourself">
-<h3>Challenge Yourself!<a class="headerlink" href="#challenge-yourself" title="Permalink to this heading">#</a></h3>
-<p>Based on the images and TE/TR parameters above, can you estimate T1 and T2 relaxation times of the 3 objects?</p>
 </section>
 </section>
 </section>
diff --git a/MRI Signal Equation.html b/MRI Signal Equation.html
index d6c735d..d81e740 100644
--- a/MRI Signal Equation.html	
+++ b/MRI Signal Equation.html	
@@ -578,8 +578,8 @@ <h2>Relaxation during signal acquisition<a class="headerlink" href="#relaxation-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" src="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" />
 <img alt="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" src="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" />
+<img alt="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" src="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" />
 </div>
 </div>
 <p>Thus the reconstructed image will corrupted by a convolution (denoted by <span class="math notranslate nohighlight">\(*\)</span>) based on the k-space amplitude weighting</p>
@@ -625,8 +625,8 @@ <h2>Relaxation during signal acquisition<a class="headerlink" href="#relaxation-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" src="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" />
 <img alt="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" src="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" />
+<img alt="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" src="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" />
 </div>
 </div>
 <p>In the above plots, the height of the main peak in the center represents the expected SNR, including losses due to blurring, while the signal amplitude outside of the main peak represents blurring that will occur.   These show that the blurring and signal loss from <span class="math notranslate nohighlight">\(T_2^*\)</span> gets worse as the relaxation time is shorter, the blurring it is much worse for EPI (in phase encoding direction) versus Cartesian trajectories.</p>
@@ -636,8 +636,8 @@ <h2>Off-resonance and Chemical Shift<a class="headerlink" href="#off-resonance-a
 <p>Off-resonance and chemical shift lead to changes in the precession frequency of the magnetization.  We can add this onto the idealized signal equation.</p>
 <p>Here we separately define:</p>
 <ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(\Delta f_r(\vec{r})\)</span> magnetic field inhomogeneity (macroscopic), including main field imperfections, and magnetic susceptibility effects</p></li>
-<li><p><span class="math notranslate nohighlight">\(\Delta f_{cs}\)</span> frequency changes due to chemical shift, so depends on the chemical species being examined (could sum or index signal over this dimension)</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Delta f_r(\vec{r}) = \bar \gamma \)</span>\Delta B_0(\vec{r})$$ - magnetic field inhomogeneities, including main field imperfections, and magnetic susceptibility effects</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Delta f_{cs}\)</span> - frequency changes due to chemical shift, so depends on the chemical species being examined (could sum or index signal over this dimension)</p></li>
 </ul>
 <!-- $$s(t) = \int m(\vec{r})\ e^{-i 2 \pi \Delta f_{cs} t} e^{-i 2 \pi \Delta f_r(\vec{r}) t} e^{-i 2 \pi \vec{k}(t) \cdot \vec{r}} \  d\vec{r}$$ -->
 <p>For simplicity, combine all off-resonance and/or chemical shift into a single frequnecy shift term, <span class="math notranslate nohighlight">\(\Delta f(\vec{r}) = \Delta f_{cs} + \Delta f_r(\vec{r})\)</span> as</p>
@@ -697,8 +697,8 @@ <h2>Off-resonance and Chemical Shift<a class="headerlink" href="#off-resonance-a
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" src="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" />
 <img alt="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" src="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" />
+<img alt="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" src="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" />
 </div>
 </div>
 <p>For frequency shift, the main peak of the convolution kernels is shifted frfom the origin.  This will result in a shift in the reconstructed image.  The shift is much larger for EPI and is in the phase encoding instead of the frequency encoding direction.  (The residual side lobes are due to sinc interpolation effects, similar to Gibbs ringing.)</p>
@@ -722,7 +722,7 @@ <h2>K-space Data Weighting<a class="headerlink" href="#k-space-data-weighting" t
 \[\hat{M}(\vec{k}) = M(\vec{k}) W(\vec{k}) \]</div>
 <div class="math notranslate nohighlight">
 \[\hat{m}(\vec{r}) = m(\vec{r}) * \mathcal{F}^{-1} \{ W(\vec{k}) \} = m(\vec{r}) * w(\vec{r})\]</div>
-<p>Where <span class="math notranslate nohighlight">\(w(\vec{r})\)</span> is the inverse Fourier Transform of the weighting function.Thus the effects can be boiled down to a convolution operation in the resulting reconstructed image.</p>
+<p>Where <span class="math notranslate nohighlight">\(w(\vec{r})\)</span> is the inverse Fourier Transform of the weighting function.  This is also known as a point spread function (PSF) or impulse response function.  Thus the effects can be boiled down to a convolution operation in the resulting reconstructed image.</p>
 <p>When the frequency shift and relaxation rates vary across space, it is helpful to decompose the image into components, <span class="math notranslate nohighlight">\(m_i\)</span> that experience the same weighting (e.g. same frequency shift, or same <span class="math notranslate nohighlight">\(T_2^*\)</span>) <span class="math notranslate nohighlight">\(W_i\)</span>.</p>
 <div class="math notranslate nohighlight">
 \[ m(\vec{r}) = \sum_i m_i(\vec{r}) \]</div>
diff --git a/Magnetic Fields and RF Coils.html b/Magnetic Fields and RF Coils.html
index a2900a7..3d4230d 100644
--- a/Magnetic Fields and RF Coils.html	
+++ b/Magnetic Fields and RF Coils.html	
@@ -513,7 +513,7 @@ <h2>Magnetic Fields<a class="headerlink" href="#magnetic-fields" title="Permalin
 <td><p>-</p></td>
 </tr>
 <tr class="row-even"><td><p>Magnetic Field Gradients</p></td>
-<td><p><span class="math notranslate nohighlight">\(\vec{G}(t)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\vec\)</span>\Delta B_0(\vec{r})<span class="math notranslate nohighlight">\( {G}(t)\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(z\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(\approx 1\)</span> kHz</p></td>
 <td><p><span class="math notranslate nohighlight">\(\approx 10\)</span> mT</p></td>
diff --git a/RF Pulses.html b/RF Pulses.html
index 76fd99f..ccf6490 100644
--- a/RF Pulses.html	
+++ b/RF Pulses.html	
@@ -506,9 +506,9 @@ <h3>Slice Thickness<a class="headerlink" href="#slice-thickness" title="Permalin
 </section>
 <section id="slice-shifting">
 <h3>Slice Shifting<a class="headerlink" href="#slice-shifting" title="Permalink to this heading">#</a></h3>
-<p>The slice location can be shifted by a distance <span class="math notranslate nohighlight">\(\delta z\)</span> through shifting the frequency of the RF pulse by <span class="math notranslate nohighlight">\(\delta f\)</span>:</p>
+<p>The slice location can be shifted by a distance <span class="math notranslate nohighlight">\(z_{off}\)</span> through shifting the frequency of the RF pulse by <span class="math notranslate nohighlight">\(f_{off}\)</span>:</p>
 <div class="math notranslate nohighlight">
-\[ \delta f = \frac{\gamma}{2\pi} G_{Z,SS} \  \delta z \]</div>
+\[ f_{off} = \frac{\gamma}{2\pi} G_{Z,SS} \  z_{off} \]</div>
 </section>
 <section id="specific-absorption-rate">
 <h3>Specific Absorption Rate<a class="headerlink" href="#specific-absorption-rate" title="Permalink to this heading">#</a></h3>
diff --git a/Spatial Encoding.html b/Spatial Encoding.html
index d818f64..19a3370 100644
--- a/Spatial Encoding.html	
+++ b/Spatial Encoding.html	
@@ -419,7 +419,7 @@ <h2> Contents </h2>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#k-space">k-space</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mri-signal">MRI signal</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-mri-data-to-images">From MRI data to images</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cartesian-k-space-frequency-and-phase-encoding-2dft-imaging">Cartesian K-space (Frequency and Phase encoding, 2DFT Imaging)</a><ul class="nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cartesian-k-space-frequency-and-phase-encoding-or-ft-imaging">Cartesian K-space (Frequency and Phase encoding, or FT Imaging)</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#frequency-encoding">Frequency Encoding</a></li>
 </ul>
 </li>
@@ -463,7 +463,7 @@ <h2>Effects of Magnetic Field Gradients<a class="headerlink" href="#effects-of-m
 \[\Delta B_z = \vec{G}\cdot\vec{r}\]</div>
 <p>which leads to linear variation in the resonance frequency as a function of position:</p>
 <div class="math notranslate nohighlight">
-\[ \omega = \gamma \vec{G}\cdot\vec{r} \]</div>
+\[ f = \bar \gamma \ \vec{G}\cdot\vec{r} \]</div>
 <p>This forms the basis for spatial encoding.</p>
 <p>For constant gradients, <span class="math notranslate nohighlight">\(\vec{G}(t) = \vec{G}_0\)</span>, this variation in frequency can be described by (neglecting relaxation)</p>
 <div class="math notranslate nohighlight">
@@ -520,10 +520,10 @@ <h2>From MRI data to images<a class="headerlink" href="#from-mri-data-to-images"
 </ol>
 <p>We have now have the incredible result that we used magnetic field gradients, the k-space framework, and appropriate acquisition and ordering of the MR signal to create an <strong>IMAGE</strong>!!</p>
 </section>
-<section id="cartesian-k-space-frequency-and-phase-encoding-2dft-imaging">
-<h2>Cartesian K-space (Frequency and Phase encoding, 2DFT Imaging)<a class="headerlink" href="#cartesian-k-space-frequency-and-phase-encoding-2dft-imaging" title="Permalink to this heading">#</a></h2>
+<section id="cartesian-k-space-frequency-and-phase-encoding-or-ft-imaging">
+<h2>Cartesian K-space (Frequency and Phase encoding, or FT Imaging)<a class="headerlink" href="#cartesian-k-space-frequency-and-phase-encoding-or-ft-imaging" title="Permalink to this heading">#</a></h2>
 <p>By far the most common way to perform spatial encoding is using a Cartesian k-space trajectory, which the name refers to the fact that data is sampled on a regularly spaced grid in k-space.
-This is also known 2DFT imaging, since we reconstruct images with an inverse 2D Fourier Transform.</p>
+This is also known FT imaging, since we reconstruct images with a 2D Fourier Transform.</p>
 <p>It consists of 2 types of encoding:</p>
 <ol class="arabic simple">
 <li><p>Frequency encoding - one dimension is encoded using a constant gradient applied during data acquisition</p></li>
@@ -533,66 +533,54 @@ <h2>Cartesian K-space (Frequency and Phase encoding, 2DFT Imaging)<a class="head
 <h3>Frequency Encoding<a class="headerlink" href="#frequency-encoding" title="Permalink to this heading">#</a></h3>
 <p>Typically 1 dimension of the object is encoded using “frequency encoding”.  This means that, after RF excitation, a magnetic field gradient is turned on and the signal is read out.  The frequencies present in the signal correspond to given spatial locations.  By convention, this is applied in the x-direction (but in practice can be rotated to any direction);</p>
 <div class="math notranslate nohighlight">
-\[\omega = \gamma G_{xr} x\]</div>
+\[f = \bar \gamma G_{xr} x\]</div>
 <p>where <span class="math notranslate nohighlight">\(G_{xr}\)</span> is the readout gradient amplitude.  Therefore, the position is proportional to the frequency:</p>
 <div class="math notranslate nohighlight">
-\[ x = \frac{\omega}{\gamma G_{xr}}\]</div>
+\[ x = \frac{f}{\bar \gamma G_{xr}}\]</div>
 <p>From frequency encoding data alone, a 1D image can be reconstructed with an inverse Fourier Transform</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% simulate frequency encoding</span>
-<span class="n">N</span> <span class="p">=</span> <span class="mi">8</span><span class="p">;</span>
-<span class="n">Mxy</span><span class="p">=</span> <span class="nb">ones</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span> <span class="n">simulate</span> <span class="n">frequency</span> <span class="n">encoding</span>
+<span class="n">N</span> <span class="o">=</span> <span class="mi">8</span><span class="p">;</span>
+<span class="n">Mxy</span><span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span>
 
-<span class="n">x</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">];</span>
-<span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">]</span> <span class="p">=</span> <span class="nb">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">);</span>
-<span class="n">Splot</span> <span class="p">=</span> <span class="mf">0.5</span><span class="p">;</span>
+<span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">];</span>
+<span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">);</span>
+<span class="n">Splot</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">;</span>
 
-<span class="n">kFE</span> <span class="p">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">;</span>
-<span class="n">dt</span> <span class="p">=</span> <span class="mf">0.1</span><span class="p">;</span>
-<span class="n">Tfe</span> <span class="p">=</span> <span class="mi">2</span><span class="p">;</span>
+<span class="n">kFE</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">;</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">;</span>
+<span class="n">Tfe</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span>
 
-<span class="n">GAMMA</span> <span class="p">=</span> <span class="mf">42.58</span><span class="p">;</span>
+<span class="n">GAMMA</span> <span class="o">=</span> <span class="mf">42.58</span><span class="p">;</span>
 
 
-<span class="k">for</span> <span class="n">tfe</span> <span class="p">=</span> <span class="nb">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span><span class="o">*</span><span class="n">Tfe</span>
-    <span class="n">phase_x</span> <span class="p">=</span> <span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">*</span><span class="n">kFE</span><span class="o">*</span><span class="n">x</span> <span class="o">*</span><span class="n">tfe</span><span class="o">/</span><span class="n">Tfe</span><span class="p">;</span>
+<span class="k">for</span> <span class="n">tfe</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span><span class="o">*</span><span class="n">Tfe</span>
+    <span class="n">phase_x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">kFE</span><span class="o">*</span><span class="n">x</span> <span class="o">*</span><span class="n">tfe</span><span class="o">/</span><span class="n">Tfe</span><span class="p">;</span>
         
-    <span class="n">Mxy_FE</span> <span class="p">=</span> <span class="n">Mxy</span> <span class="o">.*</span> <span class="nb">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">phase_x</span><span class="p">);</span>
+    <span class="n">Mxy_FE</span> <span class="o">=</span> <span class="n">Mxy</span> <span class="o">.*</span> <span class="n">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">phase_x</span><span class="p">);</span>
         
-    <span class="nb">figure</span>
-    <span class="nb">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">),</span> <span class="nb">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">:</span><span class="n">dt</span><span class="p">:</span><span class="n">Tfe</span><span class="o">+</span><span class="n">dt</span><span class="p">],</span>  <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="nb">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Tfe</span><span class="o">/</span><span class="n">dt</span><span class="p">)</span><span class="o">*</span><span class="n">kFE</span><span class="o">/</span><span class="p">(</span><span class="n">Tfe</span><span class="o">/</span><span class="n">dt</span><span class="p">),</span> <span class="mi">0</span><span class="p">],</span> <span class="n">tfe</span><span class="o">*</span><span class="nb">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.05</span> <span class="mf">0.05</span><span class="p">]),</span> <span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;G_X&#39;</span><span class="p">)</span>
-    <span class="nb">title</span><span class="p">(</span><span class="s">&#39;Frequency Encoding&#39;</span><span class="p">)</span>
-    <span class="nb">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
-    <span class="nb">quiver</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="nb">real</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">y</span><span class="o">-</span><span class="nb">imag</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="nb">real</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">),</span> <span class="nb">imag</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">),</span> <span class="n">Splot</span><span class="p">)</span>
-    <span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-    <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
+    <span class="n">figure</span>
+    <span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">),</span> <span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">:</span><span class="n">dt</span><span class="p">:</span><span class="n">Tfe</span><span class="o">+</span><span class="n">dt</span><span class="p">],</span>  <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Tfe</span><span class="o">/</span><span class="n">dt</span><span class="p">)</span><span class="o">*</span><span class="n">kFE</span><span class="o">/</span><span class="p">(</span><span class="n">Tfe</span><span class="o">/</span><span class="n">dt</span><span class="p">),</span> <span class="mi">0</span><span class="p">],</span> <span class="n">tfe</span><span class="o">*</span><span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.05</span> <span class="mf">0.05</span><span class="p">]),</span> <span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;G_X&#39;</span><span class="p">)</span>
+    <span class="n">title</span><span class="p">(</span><span class="s1">&#39;Frequency Encoding&#39;</span><span class="p">)</span>
+    <span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+    <span class="n">quiver</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">real</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">y</span><span class="o">-</span><span class="n">imag</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">real</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">),</span> <span class="n">imag</span><span class="p">(</span><span class="n">Mxy_FE</span><span class="p">),</span> <span class="n">Splot</span><span class="p">)</span>
+    <span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">),</span> <span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
     <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
-    <span class="nb">title</span><span class="p">(</span><span class="s">&#39;M_{XY}&#39;</span><span class="p">)</span>
-    <span class="nb">drawnow</span>
-<span class="k">end</span>
+    <span class="n">title</span><span class="p">(</span><span class="s1">&#39;M_</span><span class="si">{XY}</span><span class="s1">&#39;</span><span class="p">)</span>
+    <span class="n">drawnow</span>
+<span class="n">end</span>
 </pre></div>
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>warning: using the gnuplot graphics toolkit is discouraged
-
-The gnuplot graphics toolkit is not actively maintained and has a number
-of limitations that are ulikely to be fixed.  Communication with gnuplot
-uses a one-directional pipe and limited information is passed back to the
-Octave interpreter so most changes made interactively in the plot window
-will not be reflected in the graphics properties managed by Octave.  For
-example, if the plot window is closed with a mouse click, Octave will not
-be notified and will not update it&#39;s internal list of open figure windows.
-We recommend using the qt toolkit instead.
+<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span>  <span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">5</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">];</span>
+             <span class="o">^</span>
+<span class="ne">SyntaxError</span>: invalid syntax
 </pre></div>
 </div>
-<img alt="_images/633fd62680141f067fd0027944b5856b7f36825f6b9614c7852ef6dc936df389.png" src="_images/633fd62680141f067fd0027944b5856b7f36825f6b9614c7852ef6dc936df389.png" />
-<img alt="_images/eacaff314d6ca100a1babd6711c2a3ebefa7a000b233cd854414a16ddfcf76f8.png" src="_images/eacaff314d6ca100a1babd6711c2a3ebefa7a000b233cd854414a16ddfcf76f8.png" />
-<img alt="_images/aa622fa30e442abad38f8adbf513e8f8afc703d56338cb1620ca3acd0ebe30ab.png" src="_images/aa622fa30e442abad38f8adbf513e8f8afc703d56338cb1620ca3acd0ebe30ab.png" />
-<img alt="_images/94f0bb94ef66078a0702a9a60652e1592422b13023ec17d8caf206975519be25.png" src="_images/94f0bb94ef66078a0702a9a60652e1592422b13023ec17d8caf206975519be25.png" />
-<img alt="_images/ee9681136c56b2b7a3abc0065df5a7c4465b4d0c3564bb93bcbfcdb805e76a81.png" src="_images/ee9681136c56b2b7a3abc0065df5a7c4465b4d0c3564bb93bcbfcdb805e76a81.png" />
-<img alt="_images/77cfcd054449f55c4465bccad72f41e206eb0861f80630d348f6a70d7eb54681.png" src="_images/77cfcd054449f55c4465bccad72f41e206eb0861f80630d348f6a70d7eb54681.png" />
 </div>
 </div>
 </section>
@@ -605,56 +593,56 @@ <h2>Phase Encoding<a class="headerlink" href="#phase-encoding" title="Permalink
 <p>These additional dimensions are fully encoded by repeating this pulsed gradient with different amplitudes.  This is equivalent to taking different samples of a frequency encoding gradient.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% 2D object</span>
-<span class="c">% simulate frequencies + phase encoding</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span> <span class="mi">2</span><span class="n">D</span> <span class="nb">object</span>
+<span class="o">%</span> <span class="n">simulate</span> <span class="n">frequencies</span> <span class="o">+</span> <span class="n">phase</span> <span class="n">encoding</span>
 
-<span class="n">N</span> <span class="p">=</span> <span class="mi">8</span><span class="p">;</span>
-<span class="n">Mxy</span><span class="p">=</span> <span class="nb">ones</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span>
+<span class="n">N</span> <span class="o">=</span> <span class="mi">8</span><span class="p">;</span>
+<span class="n">Mxy</span><span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span>
 
-<span class="n">x</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">];</span>
-<span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">]</span> <span class="p">=</span> <span class="nb">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">);</span>
-<span class="n">Splot</span> <span class="p">=</span> <span class="mf">0.5</span><span class="p">;</span>
+<span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">];</span>
+<span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">);</span>
+<span class="n">Splot</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">;</span>
 
-<span class="n">kPE</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">;</span> <span class="c">%</span>
-<span class="n">dt</span> <span class="p">=</span> <span class="mf">0.1</span><span class="p">;</span>
-<span class="n">Tpe</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
+<span class="n">kPE</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">;</span> <span class="o">%</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">;</span>
+<span class="n">Tpe</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>
 
-<span class="n">GAMMA</span> <span class="p">=</span> <span class="mf">42.58</span><span class="p">;</span>
+<span class="n">GAMMA</span> <span class="o">=</span> <span class="mf">42.58</span><span class="p">;</span>
 
-<span class="k">for</span> <span class="n">Ipe</span> <span class="p">=</span> <span class="mi">1</span><span class="p">:</span><span class="nb">length</span><span class="p">(</span><span class="n">kPE</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">Ipe</span> <span class="o">=</span> <span class="mi">1</span><span class="p">:</span><span class="n">length</span><span class="p">(</span><span class="n">kPE</span><span class="p">)</span>
     
-    <span class="k">for</span> <span class="n">tpe</span> <span class="p">=</span> <span class="n">Tpe</span> <span class="c">%[0:dt:Tpe]</span>
-        <span class="n">phase_y</span> <span class="p">=</span> <span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">*</span><span class="n">kPE</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span><span class="o">*</span><span class="n">y</span> <span class="o">*</span><span class="n">tpe</span><span class="o">/</span><span class="n">Tpe</span><span class="p">;</span>
+    <span class="k">for</span> <span class="n">tpe</span> <span class="o">=</span> <span class="n">Tpe</span> <span class="o">%</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="n">dt</span><span class="p">:</span><span class="n">Tpe</span><span class="p">]</span>
+        <span class="n">phase_y</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">kPE</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span><span class="o">*</span><span class="n">y</span> <span class="o">*</span><span class="n">tpe</span><span class="o">/</span><span class="n">Tpe</span><span class="p">;</span>
         
-        <span class="n">Mxy_PE</span> <span class="p">=</span> <span class="n">Mxy</span> <span class="o">.*</span> <span class="nb">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">phase_y</span><span class="p">);</span>
+        <span class="n">Mxy_PE</span> <span class="o">=</span> <span class="n">Mxy</span> <span class="o">.*</span> <span class="n">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">phase_y</span><span class="p">);</span>
         
-        <span class="nb">figure</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span>
+        <span class="n">figure</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span>
         
-        <span class="nb">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
-        <span class="nb">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">:</span><span class="n">dt</span><span class="p">:</span><span class="n">Tpe</span><span class="o">+</span><span class="n">dt</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="nb">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Tpe</span><span class="o">/</span><span class="n">dt</span><span class="p">)</span><span class="o">*</span><span class="n">kPE</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">Tpe</span><span class="o">/</span><span class="n">dt</span><span class="p">),</span> <span class="mi">0</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.05</span> <span class="mf">0.05</span><span class="p">]),</span> <span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;G_Y&#39;</span><span class="p">)</span> 
-        <span class="nb">title</span><span class="p">([</span><span class="s">&#39;Phase encoding, Step &#39;</span> <span class="n">int2str</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)])</span>
-        <span class="nb">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
-        <span class="nb">quiver</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="nb">real</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">y</span><span class="o">-</span><span class="nb">imag</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="nb">real</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">),</span> <span class="nb">imag</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">),</span> <span class="n">Splot</span><span class="p">)</span>
-        <span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-        <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
+        <span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+        <span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">:</span><span class="n">dt</span><span class="p">:</span><span class="n">Tpe</span><span class="o">+</span><span class="n">dt</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Tpe</span><span class="o">/</span><span class="n">dt</span><span class="p">)</span><span class="o">*</span><span class="n">kPE</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">Tpe</span><span class="o">/</span><span class="n">dt</span><span class="p">),</span> <span class="mi">0</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.05</span> <span class="mf">0.05</span><span class="p">]),</span> <span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;G_Y&#39;</span><span class="p">)</span> 
+        <span class="n">title</span><span class="p">([</span><span class="s1">&#39;Phase encoding, Step &#39;</span> <span class="n">int2str</span><span class="p">(</span><span class="n">Ipe</span><span class="p">)])</span>
+        <span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+        <span class="n">quiver</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">real</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">y</span><span class="o">-</span><span class="n">imag</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">)</span><span class="o">*</span><span class="n">Splot</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">real</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">),</span> <span class="n">imag</span><span class="p">(</span><span class="n">Mxy_PE</span><span class="p">),</span> <span class="n">Splot</span><span class="p">)</span>
+        <span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">),</span> <span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+        <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
         <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
-        <span class="nb">title</span><span class="p">(</span><span class="s">&#39;M_{XY}&#39;</span><span class="p">)</span>
-        <span class="nb">drawnow</span>
-    <span class="k">end</span>
-<span class="k">end</span>
+        <span class="n">title</span><span class="p">(</span><span class="s1">&#39;M_</span><span class="si">{XY}</span><span class="s1">&#39;</span><span class="p">)</span>
+        <span class="n">drawnow</span>
+    <span class="n">end</span>
+<span class="n">end</span>
 </pre></div>
 </div>
 </div>
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+<img alt="_images/6b498db82f8fd884a4ee33ce2be5e3f46d68600f324d095ee38180f4610835f8.png" src="_images/6b498db82f8fd884a4ee33ce2be5e3f46d68600f324d095ee38180f4610835f8.png" />
+<img alt="_images/b2ebaf1ffc7beaa5180e2c06110c556ecdf7ae48c942e0bb7f7822ff236d55b1.png" src="_images/b2ebaf1ffc7beaa5180e2c06110c556ecdf7ae48c942e0bb7f7822ff236d55b1.png" />
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 </section>
@@ -681,58 +669,100 @@ <h3>K-space trajectories<a class="headerlink" href="#id1" title="Permalink to th
 <p>The k-space pattern during a MRI experiment is referred to as the k-space “trajectory”.  The most common are Cartesian trajectories, in which parallel lines of k-space are covered to sample a 2D (or 3D) grid.  K-space trajectories with other patterns, such as radial lines, spirals, rastered lines (echo-planar trajectories), or blades can also be used.</p>
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-<div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">%  plot k-space trajectories</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span>  <span class="n">plot</span> <span class="mi">2</span><span class="n">D</span> <span class="n">k</span><span class="o">-</span><span class="n">space</span> <span class="n">trajectories</span>
 
-<span class="c">% Cartesian</span>
-<span class="n">N</span> <span class="p">=</span> <span class="mi">8</span><span class="p">;</span>
+<span class="o">%</span> <span class="n">Cartesian</span>
+<span class="n">N</span> <span class="o">=</span> <span class="mi">8</span><span class="p">;</span>
 
-<span class="n">k</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">;</span>
-<span class="p">[</span><span class="n">ky</span><span class="p">,</span><span class="n">kx</span><span class="p">]</span> <span class="p">=</span> <span class="nb">meshgrid</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">);</span>
+<span class="n">k</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">;</span>
+<span class="p">[</span><span class="n">ky</span><span class="p">,</span><span class="n">kx</span><span class="p">]</span> <span class="o">=</span> <span class="n">meshgrid</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">);</span>
 
-<span class="nb">figure</span>
-<span class="nb">plot</span><span class="p">(</span><span class="n">kx</span><span class="p">,</span> <span class="n">ky</span><span class="p">,</span> <span class="s">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">])</span>
-<span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;k_x&#39;</span><span class="p">),</span><span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;k_y&#39;</span><span class="p">)</span>
-<span class="nb">title</span><span class="p">(</span><span class="s">&#39;Cartesian trajectory&#39;</span><span class="p">)</span>
+<span class="n">figure</span>
+<span class="n">plot</span><span class="p">(</span><span class="n">kx</span><span class="p">,</span> <span class="n">ky</span><span class="p">,</span> <span class="s1">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Cartesian trajectory&#39;</span><span class="p">)</span>
 
-<span class="c">% echo-planar</span>
-<span class="n">kx_ep</span> <span class="p">=</span> <span class="n">kx</span><span class="p">;</span> <span class="n">kx_ep</span><span class="p">(:,</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="k">end</span><span class="p">)</span> <span class="p">=</span> <span class="n">kx_ep</span><span class="p">(</span><span class="k">end</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="k">end</span><span class="p">);</span>
-<span class="n">kx_ep</span> <span class="p">=</span> <span class="n">kx_ep</span><span class="p">(:);</span>
+<span class="o">%</span> <span class="n">echo</span><span class="o">-</span><span class="n">planar</span>
+<span class="n">kx_ep</span> <span class="o">=</span> <span class="n">kx</span><span class="p">;</span> <span class="n">kx_ep</span><span class="p">(:,</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="n">kx_ep</span><span class="p">(</span><span class="n">end</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">end</span><span class="p">);</span>
+<span class="n">kx_ep</span> <span class="o">=</span> <span class="n">kx_ep</span><span class="p">(:);</span>
 
-<span class="n">ky_ep</span> <span class="p">=</span> <span class="n">ky</span><span class="p">(:);</span>
+<span class="n">ky_ep</span> <span class="o">=</span> <span class="n">ky</span><span class="p">(:);</span>
 
-<span class="nb">figure</span>
-<span class="nb">plot</span><span class="p">(</span><span class="n">kx_ep</span><span class="p">,</span> <span class="n">ky_ep</span><span class="p">,</span> <span class="s">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">])</span>
-<span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;k_x&#39;</span><span class="p">),</span><span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;k_y&#39;</span><span class="p">)</span>
-<span class="nb">title</span><span class="p">(</span><span class="s">&#39;Echo-Planar trajectory&#39;</span><span class="p">)</span>
+<span class="n">figure</span>
+<span class="n">plot</span><span class="p">(</span><span class="n">kx_ep</span><span class="p">,</span> <span class="n">ky_ep</span><span class="p">,</span> <span class="s1">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Echo-Planar trajectory&#39;</span><span class="p">)</span>
 
-<span class="c">% radial</span>
+<span class="o">%</span> <span class="n">radial</span>
 
-<span class="n">k_theta</span> <span class="p">=</span> <span class="nb">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">N</span><span class="p">));</span>
-<span class="n">k_radial</span> <span class="p">=</span> <span class="n">k</span><span class="p">.</span><span class="o">&#39;</span> <span class="o">*</span> <span class="n">k_theta</span><span class="p">;</span>
+<span class="n">k_theta</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">N</span><span class="p">));</span>
+<span class="n">k_radial</span> <span class="o">=</span> <span class="n">k</span><span class="o">.</span><span class="s1">&#39; * k_theta;</span>
 
-<span class="nb">figure</span>
-<span class="nb">plot</span><span class="p">(</span><span class="nb">real</span><span class="p">(</span><span class="n">k_radial</span><span class="p">),</span> <span class="nb">imag</span><span class="p">(</span><span class="n">k_radial</span><span class="p">),</span> <span class="s">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">])</span>
-<span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;k_x&#39;</span><span class="p">),</span><span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;k_y&#39;</span><span class="p">)</span>
-<span class="nb">title</span><span class="p">(</span><span class="s">&#39;Radial trajectory&#39;</span><span class="p">)</span>
+<span class="n">figure</span>
+<span class="n">plot</span><span class="p">(</span><span class="n">real</span><span class="p">(</span><span class="n">k_radial</span><span class="p">),</span> <span class="n">imag</span><span class="p">(</span><span class="n">k_radial</span><span class="p">),</span> <span class="s1">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Radial trajectory&#39;</span><span class="p">)</span>
 
 
-<span class="c">% spiral</span>
-<span class="n">n</span> <span class="p">=</span> <span class="nb">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">201</span><span class="p">);</span>
-<span class="n">Nturns</span> <span class="p">=</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">;</span>
-<span class="n">k_spiral</span> <span class="p">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">.*</span><span class="nb">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">*</span><span class="n">Nturns</span><span class="o">*</span><span class="n">n</span><span class="p">);</span>
+<span class="o">%</span> <span class="n">spiral</span>
+<span class="n">n</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">201</span><span class="p">);</span>
+<span class="n">Nturns</span> <span class="o">=</span> <span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">;</span>
+<span class="n">k_spiral</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">.*</span><span class="n">exp</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">Nturns</span><span class="o">*</span><span class="n">n</span><span class="p">);</span>
 
-<span class="nb">figure</span>
-<span class="nb">plot</span><span class="p">(</span><span class="nb">real</span><span class="p">(</span><span class="n">k_spiral</span><span class="p">),</span> <span class="nb">imag</span><span class="p">(</span><span class="n">k_spiral</span><span class="p">),</span> <span class="s">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="nb">xlim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="p">.</span><span class="mi">6</span> <span class="p">.</span><span class="mi">6</span><span class="p">])</span>
-<span class="nb">xlabel</span><span class="p">(</span><span class="s">&#39;k_x&#39;</span><span class="p">),</span><span class="nb">ylabel</span><span class="p">(</span><span class="s">&#39;k_y&#39;</span><span class="p">)</span>
-<span class="nb">title</span><span class="p">(</span><span class="s">&#39;Spiral trajectory&#39;</span><span class="p">)</span>
+<span class="n">figure</span>
+<span class="n">plot</span><span class="p">(</span><span class="n">real</span><span class="p">(</span><span class="n">k_spiral</span><span class="p">),</span> <span class="n">imag</span><span class="p">(</span><span class="n">k_spiral</span><span class="p">),</span> <span class="s1">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Spiral trajectory&#39;</span><span class="p">)</span>
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+<img alt="_images/5b39e1ebc881f649c67f5ac15ce8c8a75e4a4c90e457944d4fcdebecb9765e0e.png" src="_images/5b39e1ebc881f649c67f5ac15ce8c8a75e4a4c90e457944d4fcdebecb9765e0e.png" />
+<img alt="_images/7fb4dad9c315d2e6180b1051ed906c6aa4d468982a0a6b51fb2c3ce32f3b84b0.png" src="_images/7fb4dad9c315d2e6180b1051ed906c6aa4d468982a0a6b51fb2c3ce32f3b84b0.png" />
+<img alt="_images/682a80458c6d6462569bebe71790fb3febb4b4c02786b73b1137a08da5e5bd7d.png" src="_images/682a80458c6d6462569bebe71790fb3febb4b4c02786b73b1137a08da5e5bd7d.png" />
+<img alt="_images/2149bec492121a4c0c21ee11bde6587912c06f6bb9a9c159cbccaa6653da593c.png" src="_images/2149bec492121a4c0c21ee11bde6587912c06f6bb9a9c159cbccaa6653da593c.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span>  <span class="n">plot</span> <span class="mi">3</span><span class="n">D</span> <span class="n">k</span><span class="o">-</span><span class="n">space</span> <span class="n">trajectories</span>
+
+<span class="o">%</span> <span class="n">Cartesian</span>
+<span class="n">N</span> <span class="o">=</span> <span class="mi">8</span><span class="p">;</span>
+
+<span class="n">k</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">;</span>
+<span class="p">[</span><span class="n">ky</span><span class="p">,</span><span class="n">kx</span><span class="p">,</span><span class="n">kz</span><span class="p">]</span> <span class="o">=</span> <span class="n">meshgrid</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">);</span>
+
+
+<span class="n">figure</span>
+<span class="n">plot3</span><span class="p">(</span><span class="n">reshape</span><span class="p">(</span><span class="n">kx</span><span class="p">,</span> <span class="p">[</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span> <span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="p">]),</span> <span class="n">reshape</span><span class="p">(</span><span class="n">ky</span><span class="p">,</span> <span class="p">[</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span> <span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="p">]),</span> <span class="n">reshape</span><span class="p">(</span><span class="n">kz</span><span class="p">,</span> <span class="p">[</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span> <span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="p">]),</span> <span class="s1">&#39;LineWidth&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
+<span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">zlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">),</span> <span class="n">zlabel</span><span class="p">(</span><span class="s1">&#39;k_z&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Cartesian trajectory&#39;</span><span class="p">)</span>
+
+<span class="o">%</span> <span class="n">radial</span>
+
+<span class="n">N_radial</span> <span class="o">=</span> <span class="n">N</span><span class="o">^</span><span class="mi">3</span><span class="p">;</span>
+
+<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="p">:</span><span class="n">N_radial</span><span class="p">;</span>
+
+<span class="n">kz</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="n">N_radial</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">N_radial</span><span class="p">;</span>
+<span class="n">kx</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">N_radial</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span> <span class="o">.*</span> <span class="n">asin</span><span class="p">(</span><span class="n">kz</span><span class="p">))</span> <span class="o">.*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">kz</span><span class="o">.^</span><span class="mi">2</span><span class="p">);</span>
+<span class="n">ky</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">N_radial</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span> <span class="o">.*</span> <span class="n">asin</span><span class="p">(</span><span class="n">kz</span><span class="p">))</span> <span class="o">.*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">kz</span><span class="o">.^</span><span class="mi">2</span><span class="p">);</span>
+
+<span class="n">figure</span>
+<span class="n">plot3</span><span class="p">([</span><span class="n">zeros</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_radial</span><span class="p">);</span> <span class="n">kx</span><span class="p">],</span> <span class="p">[</span><span class="n">zeros</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_radial</span><span class="p">);</span> <span class="n">ky</span><span class="p">],</span> <span class="p">[</span><span class="n">zeros</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_radial</span><span class="p">);</span> <span class="n">kz</span><span class="p">])</span>
+<span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">]),</span> <span class="n">zlim</span><span class="p">([</span><span class="o">-</span><span class="mf">.6</span> <span class="mf">.6</span><span class="p">])</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;k_x&#39;</span><span class="p">),</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;k_y&#39;</span><span class="p">),</span> <span class="n">zlabel</span><span class="p">(</span><span class="s1">&#39;k_z&#39;</span><span class="p">)</span>
+<span class="n">title</span><span class="p">(</span><span class="s1">&#39;Radial trajectory&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="_images/5b39e1ebc881f649c67f5ac15ce8c8a75e4a4c90e457944d4fcdebecb9765e0e.png" src="_images/5b39e1ebc881f649c67f5ac15ce8c8a75e4a4c90e457944d4fcdebecb9765e0e.png" />
+<img alt="_images/7fb4dad9c315d2e6180b1051ed906c6aa4d468982a0a6b51fb2c3ce32f3b84b0.png" src="_images/7fb4dad9c315d2e6180b1051ed906c6aa4d468982a0a6b51fb2c3ce32f3b84b0.png" />
+<img alt="_images/682a80458c6d6462569bebe71790fb3febb4b4c02786b73b1137a08da5e5bd7d.png" src="_images/682a80458c6d6462569bebe71790fb3febb4b4c02786b73b1137a08da5e5bd7d.png" />
+<img alt="_images/2149bec492121a4c0c21ee11bde6587912c06f6bb9a9c159cbccaa6653da593c.png" src="_images/2149bec492121a4c0c21ee11bde6587912c06f6bb9a9c159cbccaa6653da593c.png" />
 </div>
 </div>
 <p>These movies illustrate the phase accumulation during non-Cartesian trajectories</p>
@@ -752,16 +782,16 @@ <h3>K-space trajectories<a class="headerlink" href="#id1" title="Permalink to th
         },
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+    <script>kernelName = 'python3'</script>
 
                 </article>
               
@@ -811,7 +841,7 @@ <h3>K-space trajectories<a class="headerlink" href="#id1" title="Permalink to th
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#k-space">k-space</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mri-signal">MRI signal</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-mri-data-to-images">From MRI data to images</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cartesian-k-space-frequency-and-phase-encoding-2dft-imaging">Cartesian K-space (Frequency and Phase encoding, 2DFT Imaging)</a><ul class="nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cartesian-k-space-frequency-and-phase-encoding-or-ft-imaging">Cartesian K-space (Frequency and Phase encoding, or FT Imaging)</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#frequency-encoding">Frequency Encoding</a></li>
 </ul>
 </li>
diff --git a/Spectral-Spatial RF Pulses.html b/Spectral-Spatial RF Pulses.html
index df58b2e..852a7f9 100644
--- a/Spectral-Spatial RF Pulses.html	
+++ b/Spectral-Spatial RF Pulses.html	
@@ -690,8 +690,8 @@ <h2>Spectral-Spatial RF Pulse Design<a class="headerlink" href="#spectral-spatia
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>ans = -11.589
 </pre></div>
 </div>
-<img alt="_images/72387277d34c4cc95f56a18b92a17de6c14b3d7aab6aff85e9a8659fcccb9387.png" src="_images/72387277d34c4cc95f56a18b92a17de6c14b3d7aab6aff85e9a8659fcccb9387.png" />
 <img alt="_images/fc82530ab4317581e65ff0523a04d4453463988ae99daed57ce74a13127a6732.png" src="_images/fc82530ab4317581e65ff0523a04d4453463988ae99daed57ce74a13127a6732.png" />
+<img alt="_images/72387277d34c4cc95f56a18b92a17de6c14b3d7aab6aff85e9a8659fcccb9387.png" src="_images/72387277d34c4cc95f56a18b92a17de6c14b3d7aab6aff85e9a8659fcccb9387.png" />
 </div>
 </div>
 <p>The spectral pulse, <span class="math notranslate nohighlight">\(H_f(k_f)\)</span>, creates the overall envelope, and is applied as a weighting for repeated versions of the spatial pulse, <span class="math notranslate nohighlight">\(H_z(k_z)\)</span>.  This creates the excitation k-space profile, <span class="math notranslate nohighlight">\(B_1(k_f, k_z)\)</span> shown above, which results in the spectral-spatial profile shown below:</p>
diff --git a/_images/011e2056fd0c8fbb0da3c4e71a79966c2eee21e53b758099e3d05cd0014d4d84.png b/_images/011e2056fd0c8fbb0da3c4e71a79966c2eee21e53b758099e3d05cd0014d4d84.png
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diff --git a/_sources/Artifacts.ipynb b/_sources/Artifacts.ipynb
index 33d6ca5..745d97c 100644
--- a/_sources/Artifacts.ipynb
+++ b/_sources/Artifacts.ipynb
@@ -37,6 +37,16 @@
     "1. Identify artifacts and how to mitigate them\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Introduction\n",
+    "\n",
+    "Many of the artifacts that occur in MRI can be understood and analyzed using the k-space perspective.  In particular, they can be understood as the k-space data being modified by some function.  This is described mathematically in \n",
+    "[MRI Signal Equation and K-psace - K-space Data Weighting](MRI%20Signal%20Equation.ipynb#k-space-data-weighting)"
+   ]
+  },
   {
    "attachments": {},
    "cell_type": "markdown",
diff --git a/_sources/FOV and Resolution.ipynb b/_sources/FOV and Resolution.ipynb
index bed46db..54ad93b 100644
--- a/_sources/FOV and Resolution.ipynb	
+++ b/_sources/FOV and Resolution.ipynb	
@@ -200,7 +200,7 @@
    ],
    "mimetype": "text/x-octave",
    "name": "python",
-   "version": "3.11.6"
+   "version": "3.11.9"
   }
  },
  "nbformat": 4,
diff --git a/_sources/Fast Imaging Pulse Sequences.ipynb b/_sources/Fast Imaging Pulse Sequences.ipynb
index 158d6f9..829900c 100644
--- a/_sources/Fast Imaging Pulse Sequences.ipynb	
+++ b/_sources/Fast Imaging Pulse Sequences.ipynb	
@@ -23,12 +23,15 @@
     "1. Understand the most popular pulse sequences and their acronyms\n",
     "    * Describe how EPI works\n",
     "    * Describe how FSE/TSE sequences work\n",
+    "    * Describe how fast gradient-echo sequences work\n",
     "1. Manipulate MRI sequence parameters to improve performance\n",
     "    * Understand how and when to accelerate with EPI\n",
     "    * Understand how and when to accelerate with FSE/TSE\n",
+    "    * Understand how contrast changes in fast gradient-echo sequences\n",
     "1. Identify artifacts and how to mitigate them\n",
     "    * Identify EPI artifacts including distortion and T2*\n",
-    "    * Identify FSE/TSE artifacts include T2 blurring\n"
+    "    * Identify FSE/TSE artifacts include T2 blurring\n",
+    "    * Identify fast gradient-echo artifacts such as banding in bSSFP"
    ]
   },
   {
@@ -49,6 +52,48 @@
     "These pulse sequences use multiple spin-echo refocusing pulses after a single exictation pulse.  This enables multiple k-space lines to be acquired during the multiple spin-echoes that are formed.  This technique was originally called Rapid Acquisition with Relaxation Enhancement (RARE), and is known on various MRI scanners as fast spin-echo (FSE, GE Healthcare), turbo spin-echo (TSE, Siemens Healthineers), or turbo field-echo (TFE, Philips Healthcare)."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Fast gradient-echo sequences\n",
+    "\n",
+    "The basic gradient-echo sequence is typically a spoiled gradient-echo sequence.  The sequence is called spoiled because the transverse magnetization is spoiled by a spoiler gradient before the next RF pulse.\n",
+    "\n",
+    "### Spoiler or Crusher gradients\n",
+    "\n",
+    "A large, unbalanced gradient will create dephasing of the transverse magnetization across the imaging voxels.  This effectively eliminates the signal.  However, the net magnetization is not truly eliminated, and this can be refocused by gradients.\n",
+    "\n",
+    "### RF spoiling\n",
+    "\n",
+    "The RF axis of rotation is changed every TR.  This reduces the chances of magnetization from previous TRs to become coherently excited.  Specific RF spoiling schemes, such as quadratic phase incrementation, is required for this to be effective.\n",
+    "\n",
+    "### Types of fast gradient-echo sequences\n",
+    "\n",
+    "Spoiled gradient-echo (SPGR)/fast low-angle shot (FLASH)\n",
+    "* Aims to full spoil transverse magnetization every TR\n",
+    "* Both gradient and RF spoiling are used\n",
+    "* Pure T1 weighted contrast\n",
+    "\n",
+    "Gradient-recalled acquisition in the steady state (GRASS)/fast imaging with steady-state precession (FISP)\n",
+    "* Allows for residual transverse magnetization to be used in the next TR\n",
+    "* Refouces frequency and phase encoding gradients\n",
+    "* RF spoiling used\n",
+    "* Increases T2* contrast\n",
+    "\n",
+    "\n",
+    "Steady-state free precession (SSFP)/time-reversed fast imaging with steady-state precession (PSIF)\n",
+    "* Use gradients to refocus signals from previous TRs\n",
+    "* Gradient spoiling but refocused in later TRs\n",
+    "* Creates T2/T1 contrast\n",
+    "\n",
+    "Balanced SSFP/TrueFISP\n",
+    "* Balanced gradients every TR\n",
+    "* No spoiling, all magnetization preserved each TR\n",
+    "* Creates T2/T1 contrast\n",
+    "* High SNR efficiency"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
diff --git a/_sources/Gradient and Spin Echoes.ipynb b/_sources/Gradient and Spin Echoes.ipynb
index fcfdc16..db29c25 100644
--- a/_sources/Gradient and Spin Echoes.ipynb	
+++ b/_sources/Gradient and Spin Echoes.ipynb	
@@ -73,9 +73,9 @@
     "\n",
     "Ideally, the magnetic field is uniform in the absense of any applied gradients.  However, in practice there are unavoidable variations in the magnetic field that lead to changes in resonance frequency.  These magnetic field variations are referred to as \"off-resonance\", and can be represented as\n",
     "\n",
-    "$$\\Delta f_r(\\vec{r})$$\n",
+    "$$\\Delta f_r(\\vec{r}) = \\bar \\gamma $\\Delta B_0(\\vec{r})$ $$\n",
     "\n",
-    "This typically includes imperfections in the main $B_0$ magnetic as well as subject-specific changes in the magnetic field that are largely due to magnetic susceptibility effects.\n",
+    "These are due imperfections in the main $B_0$ magnet as well as subject-specific changes in the magnetic field that are largely due to magnetic susceptibility effects.\n",
     "\n",
     "## Effects of Off-resonance and T2*\n",
     "\n",
diff --git a/_sources/MRI Contrast.ipynb b/_sources/MRI Contrast.ipynb
index 2f8a8ab..387a2d7 100644
--- a/_sources/MRI Contrast.ipynb	
+++ b/_sources/MRI Contrast.ipynb	
@@ -207,7 +207,17 @@
     "\n",
     "$$S \\propto M_0 (1- \\exp(-TR/T_1) )$$\n",
     "\n",
-    "Illustrated in the first example below.  This shows that the magnetization reaches steady state in the 2nd TR.\n"
+    "Illustrated in the first example below.  This shows that the magnetization reaches steady state in the 2nd TR.\n",
+    "\n",
+    "\n",
+    "### < 90-degree flip angles\n",
+    "\n",
+    "For T1-weighting, it is generally more efficient in terms of signal acqruied per time to use < 90-degree flip angles every TR, in which case \n",
+    "\n",
+    "$$S \\propto M_0 \\sin(\\theta) \\frac{1- \\exp(-TR/T_1)}{1- \\cos(\\theta) \\exp(-TR/T_1)}$$\n",
+    "\n",
+    "Illustrated in this second example below.  This shows that the magnetization can take many TRs to reach steady state.\n",
+    "\n"
    ]
   },
   {
@@ -327,21 +337,6 @@
     "colorbar\n"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "\n",
-    "### < 90-degree flip angles\n",
-    "\n",
-    "For T1-weighting, it is generally more efficient in terms of signal acqruied per time to use < 90-degree flip angles every TR, in which case \n",
-    "\n",
-    "$$S \\propto M_0 \\sin(\\theta) \\frac{1- \\exp(-TR/T_1)}{1- \\cos(\\theta) \\exp(-TR/T_1)}$$\n",
-    "\n",
-    "Illustrated in this second example below.  This shows that the magnetization can take many TRs to reach steady state.\n",
-    "\n"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": 13,
@@ -831,7 +826,11 @@
     "\n",
     "```Octave\n",
     "signal_gre = MRsignal_spoiled_gradient_echo(flip, TE, TR, M0, T1, T2)\n",
-    "```"
+    "```\n",
+    "\n",
+    "### Challenge Yourself!\n",
+    "\n",
+    "Based on the images and TE/TR parameters below, can you estimate T1 and T2 relaxation times of the 3 objects?"
    ]
   },
   {
@@ -928,15 +927,6 @@
     "end"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Challenge Yourself!\n",
-    "\n",
-    "Based on the images and TE/TR parameters above, can you estimate T1 and T2 relaxation times of the 3 objects?"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
diff --git a/_sources/MRI Signal Equation.ipynb b/_sources/MRI Signal Equation.ipynb
index 8778e97..a1e206e 100644
--- a/_sources/MRI Signal Equation.ipynb	
+++ b/_sources/MRI Signal Equation.ipynb	
@@ -271,8 +271,8 @@
     "Off-resonance and chemical shift lead to changes in the precession frequency of the magnetization.  We can add this onto the idealized signal equation.\n",
     "\n",
     "Here we separately define:\n",
-    "* $\\Delta f_r(\\vec{r})$ magnetic field inhomogeneity (macroscopic), including main field imperfections, and magnetic susceptibility effects\n",
-    "* $\\Delta f_{cs}$ frequency changes due to chemical shift, so depends on the chemical species being examined (could sum or index signal over this dimension)\n",
+    "* $\\Delta f_r(\\vec{r}) = \\bar \\gamma $\\Delta B_0(\\vec{r})$$ - magnetic field inhomogeneities, including main field imperfections, and magnetic susceptibility effects\n",
+    "* $\\Delta f_{cs}$ - frequency changes due to chemical shift, so depends on the chemical species being examined (could sum or index signal over this dimension)\n",
     "\n",
     "\n",
     "<!-- $$s(t) = \\int m(\\vec{r})\\ e^{-i 2 \\pi \\Delta f_{cs} t} e^{-i 2 \\pi \\Delta f_r(\\vec{r}) t} e^{-i 2 \\pi \\vec{k}(t) \\cdot \\vec{r}} \\  d\\vec{r}$$ -->\n",
@@ -401,7 +401,7 @@
     "\n",
     "$$\\hat{m}(\\vec{r}) = m(\\vec{r}) * \\mathcal{F}^{-1} \\{ W(\\vec{k}) \\} = m(\\vec{r}) * w(\\vec{r})$$\n",
     "\n",
-    "Where $w(\\vec{r})$ is the inverse Fourier Transform of the weighting function.Thus the effects can be boiled down to a convolution operation in the resulting reconstructed image.\n",
+    "Where $w(\\vec{r})$ is the inverse Fourier Transform of the weighting function.  This is also known as a point spread function (PSF) or impulse response function.  Thus the effects can be boiled down to a convolution operation in the resulting reconstructed image.\n",
     "\n",
     "When the frequency shift and relaxation rates vary across space, it is helpful to decompose the image into components, $m_i$ that experience the same weighting (e.g. same frequency shift, or same $T_2^*$) $W_i$.  \n",
     "\n",
diff --git a/_sources/Magnetic Fields and RF Coils.ipynb b/_sources/Magnetic Fields and RF Coils.ipynb
index 3705e99..0a5265d 100644
--- a/_sources/Magnetic Fields and RF Coils.ipynb	
+++ b/_sources/Magnetic Fields and RF Coils.ipynb	
@@ -56,7 +56,7 @@
     "| --- | --- | --- | --- | --- | --- |\n",
     "| Main Field | $B_0$ | $z$ | 0 Hz | $\\approx 1$ T | Polarization |\n",
     "| Main Field Inhomogeneities | $\\Delta B_0(\\vec{r})$ | $z$ | 0 Hz | $\\approx 1-10$ ppm | - |\n",
-    "| Magnetic Field Gradients | $\\vec{G}(t)$ | $z$ | $\\approx 1$ kHz | $\\approx 10$ mT | Spatial Encoding |\n",
+    "| Magnetic Field Gradients | $\\vec$\\Delta B_0(\\vec{r})$ {G}(t)$ | $z$ | $\\approx 1$ kHz | $\\approx 10$ mT | Spatial Encoding |\n",
     "| Transmit RF Coils | $B_1^+(\\vec{r},t)$ | $x,y$ | $\\approx 100$ MHz | $\\approx 10 \\mu T$ | Excitation |\n",
     "| Receive RF Coils | $B_1^-(\\vec{r},t)$ | $x,y$ | $\\approx 100$ MHz | $\\approx 1 \\mu T$ | Reception |\n",
     "| Net Magnetization | $\\vec{M}(\\vec{r},t)$ | $z,y,z$ | $\\approx 100$ MHz | $\\approx 1 \\mu \\mathrm{T}$ | Signal Source |\n",
diff --git a/_sources/RF Pulses.ipynb b/_sources/RF Pulses.ipynb
index 0d4e635..8d1e818 100644
--- a/_sources/RF Pulses.ipynb	
+++ b/_sources/RF Pulses.ipynb	
@@ -66,9 +66,9 @@
     "\n",
     "### Slice Shifting\n",
     "\n",
-    "The slice location can be shifted by a distance $\\delta z$ through shifting the frequency of the RF pulse by $\\delta f$:\n",
+    "The slice location can be shifted by a distance $z_{off}$ through shifting the frequency of the RF pulse by $f_{off}$:\n",
     "\n",
-    "$$ \\delta f = \\frac{\\gamma}{2\\pi} G_{Z,SS} \\  \\delta z $$\n",
+    "$$ f_{off} = \\frac{\\gamma}{2\\pi} G_{Z,SS} \\  z_{off} $$\n",
     "\n",
     "### Specific Absorption Rate\n",
     "\n",
diff --git a/_sources/Spatial Encoding.ipynb b/_sources/Spatial Encoding.ipynb
index ce62282..4ae70a4 100644
--- a/_sources/Spatial Encoding.ipynb	
+++ b/_sources/Spatial Encoding.ipynb	
@@ -29,7 +29,7 @@
     "\n",
     "which leads to linear variation in the resonance frequency as a function of position:\n",
     "\n",
-    "$$ \\omega = \\gamma \\vec{G}\\cdot\\vec{r} $$\n",
+    "$$ f = \\bar \\gamma \\ \\vec{G}\\cdot\\vec{r} $$\n",
     "\n",
     "This forms the basis for spatial encoding.\n",
     "\n",
@@ -106,10 +106,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Cartesian K-space (Frequency and Phase encoding, 2DFT Imaging)\n",
+    "## Cartesian K-space (Frequency and Phase encoding, or FT Imaging)\n",
     "\n",
     "By far the most common way to perform spatial encoding is using a Cartesian k-space trajectory, which the name refers to the fact that data is sampled on a regularly spaced grid in k-space.\n",
-    "This is also known 2DFT imaging, since we reconstruct images with an inverse 2D Fourier Transform.\n",
+    "This is also known FT imaging, since we reconstruct images with a 2D Fourier Transform.\n",
     "\n",
     "It consists of 2 types of encoding:\n",
     "1. Frequency encoding - one dimension is encoded using a constant gradient applied during data acquisition\n",
@@ -119,10 +119,10 @@
     "\n",
     "Typically 1 dimension of the object is encoded using \"frequency encoding\".  This means that, after RF excitation, a magnetic field gradient is turned on and the signal is read out.  The frequencies present in the signal correspond to given spatial locations.  By convention, this is applied in the x-direction (but in practice can be rotated to any direction);\n",
     "\n",
-    "$$\\omega = \\gamma G_{xr} x$$\n",
+    "$$f = \\bar \\gamma G_{xr} x$$\n",
     "where $G_{xr}$ is the readout gradient amplitude.  Therefore, the position is proportional to the frequency:\n",
     "\n",
-    "$$ x = \\frac{\\omega}{\\gamma G_{xr}}$$\n",
+    "$$ x = \\frac{f}{\\bar \\gamma G_{xr}}$$\n",
     "\n",
     "From frequency encoding data alone, a 1D image can be reconstructed with an inverse Fourier Transform"
    ]
@@ -471,7 +471,7 @@
     }
    ],
    "source": [
-    "%  plot k-space trajectories\n",
+    "%  plot 2D k-space trajectories\n",
     "\n",
     "% Cartesian\n",
     "N = 8;\n",
@@ -518,6 +518,85 @@
     "\n"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "%  plot 3D k-space trajectories\n",
+    "\n",
+    "% Cartesian\n",
+    "N = 8;\n",
+    "\n",
+    "k = [-N/2:N/2]/N;\n",
+    "[ky,kx,kz] = meshgrid(k,k,k);\n",
+    "\n",
+    "\n",
+    "figure\n",
+    "plot3(reshape(kx, [N+1 (N+1)^2]), reshape(ky, [N+1 (N+1)^2]), reshape(kz, [N+1 (N+1)^2]), 'LineWidth',10)\n",
+    "xlim([-.6 .6]), ylim([-.6 .6]), zlim([-.6 .6])\n",
+    "xlabel('k_x'),ylabel('k_y'), zlabel('k_z')\n",
+    "title('Cartesian trajectory')\n",
+    "\n",
+    "% radial\n",
+    "\n",
+    "N_radial = N^3;\n",
+    "\n",
+    "n = 1:N_radial;\n",
+    "\n",
+    "kz = (2*n - N_radial - 1) / N_radial;\n",
+    "kx = cos(sqrt(N_radial*pi) .* asin(kz)) .* sqrt(1-kz.^2);\n",
+    "ky = sin(sqrt(N_radial*pi) .* asin(kz)) .* sqrt(1-kz.^2);\n",
+    "\n",
+    "figure\n",
+    "plot3([zeros(1,N_radial); kx], [zeros(1,N_radial); ky], [zeros(1,N_radial); kz])\n",
+    "xlim([-.6 .6]), ylim([-.6 .6]), zlim([-.6 .6])\n",
+    "xlabel('k_x'),ylabel('k_y'), zlabel('k_z')\n",
+    "title('Radial trajectory')"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -541,9 +620,9 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Octave",
-   "language": "octave",
-   "name": "octave"
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
   },
   "language_info": {
    "file_extension": ".m",
@@ -562,8 +641,8 @@
     }
    ],
    "mimetype": "text/x-octave",
-   "name": "octave",
-   "version": "5.1.0"
+   "name": "python",
+   "version": "3.11.9"
   }
  },
  "nbformat": 4,
diff --git a/reports/Spatial Encoding.err.log b/reports/Spatial Encoding.err.log
new file mode 100644
index 0000000..8e62f4a
--- /dev/null
+++ b/reports/Spatial Encoding.err.log	
@@ -0,0 +1,59 @@
+Traceback (most recent call last):
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution
+    executenb(
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/nbclient/client.py", line 1305, in execute
+    return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped
+    return loop.run_until_complete(inner)
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
+    return future.result()
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/nbclient/client.py", line 705, in async_execute
+    await self.async_execute_cell(
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/nbclient/client.py", line 1058, in async_execute_cell
+    await self._check_raise_for_error(cell, cell_index, exec_reply)
+  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/nbclient/client.py", line 914, in _check_raise_for_error
+    raise CellExecutionError.from_cell_and_msg(cell, exec_reply_content)
+nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
+------------------
+% simulate frequency encoding
+N = 8;
+Mxy= ones(N+1,N+1);
+
+x = [-N/2:N/2];
+[x,y] = meshgrid(x,x);
+Splot = 0.5;
+
+kFE = 1/2;
+dt = 0.1;
+Tfe = 2;
+
+GAMMA = 42.58;
+
+
+for tfe = linspace(0,1,6)*Tfe
+    phase_x = 2*pi*kFE*x *tfe/Tfe;
+        
+    Mxy_FE = Mxy .* exp(i*phase_x);
+        
+    figure
+    subplot(121), plot([0:dt:Tfe+dt],  [0, ones(1,Tfe/dt)*kFE/(Tfe/dt), 0], tfe*ones(1,2), [-0.05 0.05]), ylabel('G_X')
+    title('Frequency Encoding')
+    subplot(122)
+    quiver(x-real(Mxy_FE)*Splot/2,y-imag(Mxy_FE)*Splot/2,real(Mxy_FE), imag(Mxy_FE), Splot)
+    xlabel('x'), ylabel('y')
+    xlim([-N/2-1, N/2+1])
+    ylim([-N/2-1, N/2+1])
+    title('M_{XY}')
+    drawnow
+end
+
+
+------------------
+
+
+  Cell In[1], line 5
+    x = [-N/2:N/2];
+             ^
+SyntaxError: invalid syntax
+
+
diff --git a/searchindex.js b/searchindex.js
index ddb383d..6cfa524 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"docnames": ["Accelerated Imaging Methods", "Artifacts", "FOV and Resolution", "Fast Imaging Pulse Sequences", "Fun with RF Pulses", "Gradient and Spin Echoes", "Image Reconstruction", "Introduction", "MR Physics - Bloch Equation", "MRI Contrast", "MRI Math Concepts", "MRI Notation", "MRI Signal Equation", "MRI System", "Magnetic Fields and RF Coils", "Pulse Sequence", "Questions", "RF Pulses", "Signal to Noise Ratio", "Software_Resources", "Spatial Encoding", "Spectral-Spatial RF Pulses", "Spin Physics", "build_notes"], "filenames": ["Accelerated Imaging Methods.ipynb", "Artifacts.ipynb", "FOV and Resolution.ipynb", "Fast Imaging Pulse Sequences.ipynb", "Fun with RF Pulses.ipynb", "Gradient and Spin Echoes.ipynb", "Image Reconstruction.ipynb", "Introduction.md", "MR Physics - Bloch Equation.ipynb", "MRI Contrast.ipynb", "MRI Math Concepts.ipynb", "MRI Notation.ipynb", "MRI Signal Equation.ipynb", "MRI System.ipynb", "Magnetic Fields and RF Coils.ipynb", "Pulse Sequence.ipynb", "Questions.md", "RF Pulses.ipynb", "Signal to Noise Ratio.ipynb", "Software_Resources.md", "Spatial Encoding.ipynb", "Spectral-Spatial RF Pulses.ipynb", "Spin Physics.ipynb", "build_notes.md"], "titles": ["Accelerated Imaging Methods", "Artifacts", "Field of View (FOV) and Resolution", "Fast Imaging Pulse Sequences (Under Construction)", "Fun with RF Pulses?", "Gradient and Spin Echo Pulse Sequences", "Image Reconstruction", "Introduction to Principles of MRI", "Bloch Equation", "MRI Contrast", "MRI Math Concepts", "MRI Notation and Terminology", "The MRI Signal Equation and K-space", "The MRI System", "Magnetic fields in MRI", "The Pulse Sequence", "MRI Questions", "RF Pulses", "Signal to Noise Ratio (SNR)", "MRI Software Resources", "Spatial Encoding", "Spectral-Spatial RF Pulses", "Spin Physics", "&lt;no title&gt;"], "terms": {"scan": [0, 3, 6, 7, 13, 14, 16, 18], "can": [0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22], "appli": [0, 4, 5, 6, 8, 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