deploy: a84e784

Accelerated Imaging Methods.html

@@ -74,7 +74,7 @@
     <script>DOCUMENTATION_OPTIONS.pagename = 'Accelerated Imaging Methods';</script>
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-    <link rel="next" title="MRI Notation and Terminology" href="MRI%20Notation.html" />
+    <link rel="next" title="Key MRI Concepts and Equations" href="Key%20MRI%20Concepts.html" />
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@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>
@@ -416,6 +420,7 @@ <h2> Contents </h2>
                 <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="#general-formulation-of-mri-reconstruction">General Formulation of MRI Reconstruction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#partial-fourier-imaging">Partial Fourier Imaging</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parallel-imaging">Parallel Imaging</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#image-space-methods">Image-space Methods</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#auto-calibrated-k-space-methods">Auto-calibrated K-space Methods</a></li>
@@ -517,6 +522,14 @@ <h2>General Formulation of MRI Reconstruction<a class="headerlink" href="#genera
 <div class="math notranslate nohighlight">
 \[ \hat{x} = \mathbf{F^{H}} y \]</div>
 </section>
+<section id="partial-fourier-imaging">
+<h2>Partial Fourier Imaging<a class="headerlink" href="#partial-fourier-imaging" title="Permalink to this heading">#</a></h2>
+<p>Partial Fourier imaging, also known as fractional NEX or partial k-space, utilizes the conjugate symmetry property of the Fourier Transform.  This property states that the Fourier Transform of a real-valued function has conjugate symmetry.  If our image, <span class="math notranslate nohighlight">\(m(\vec{r})\)</span>, is real-valued, then the k-space data satisfies</p>
+<div class="math notranslate nohighlight">
+\[ M(\vec{k}) = M(-\vec{k})^*\]</div>
+<p>In this case, only half of k-space is required, and the other half can be filled in by this property.</p>
+<p>In practice, there is typically low spatial frequency phase components in the image so the initial assumption of a real-valued image is violated.  However, the data acquisition can still be accelerated by acquiring slightly more than half of k-space, and the fully sampled center of k-space can provide sufficient information about the low spatial frequency phase to still allow for accelerationg based on this conjugate symmetry property.</p>
+</section>
 <section id="parallel-imaging">
 <h2>Parallel Imaging<a class="headerlink" href="#parallel-imaging" title="Permalink to this heading">#</a></h2>
 <p>For parallel imaging (PI), we need to consider the coil sensitivity profiles, <span class="math notranslate nohighlight">\(\mathbf{C}_q\)</span>, for each RF coil into encoding matrix along with a Fourier Transform encoding matrix, <span class="math notranslate nohighlight">\(\mathbf{F}\)</span>, as well as a k-space sub-sampling operator, <span class="math notranslate nohighlight">\(\mathbf{S}\)</span>, for the measurements from each RF coil, <span class="math notranslate nohighlight">\(\mathbf{y}_q\)</span>:</p>
@@ -697,11 +710,11 @@ <h3>Reference<a class="headerlink" href="#reference" title="Permalink to this he
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-       href="MRI%20Notation.html"
+       href="Key%20MRI%20Concepts.html"
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-        <p class="prev-next-title">MRI Notation and Terminology</p>
+        <p class="prev-next-title">Key MRI Concepts and Equations</p>
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@@ -722,6 +735,7 @@ <h3>Reference<a class="headerlink" href="#reference" title="Permalink to this he
     <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="#general-formulation-of-mri-reconstruction">General Formulation of MRI Reconstruction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#partial-fourier-imaging">Partial Fourier Imaging</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parallel-imaging">Parallel Imaging</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#image-space-methods">Image-space Methods</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#auto-calibrated-k-space-methods">Auto-calibrated K-space Methods</a></li>

Artifacts.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>

FOV and Resolution.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>
@@ -527,7 +531,10 @@ <h2>Spatial Resolution<a class="headerlink" href="#spatial-resolution" title="Pe
 <p>The spatial resolution is determined by the maximum sample extent in k-space, with the simple relationship based on the maximum k-space sample locations, <span class="math notranslate nohighlight">\(k_{max}\)</span> as:</p>
 <div class="math notranslate nohighlight">
 \[ \delta = \frac{1}{2 k_{max}}\]</div>
-<p>For example, <span class="math notranslate nohighlight">\(\delta_x = \frac{1}{2 k_{x,max}}\)</span></p>
+<p>For example, <span class="math notranslate nohighlight">\(\delta_x = \frac{1}{2 k_{x,max}}\)</span>.</p>
+<p>For symmetric sampling in k-space, this can also be defined based on the width of the k-space sampling, <span class="math notranslate nohighlight">\(W_k = 2 k_{max}\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[ \delta = \frac{1}{W_k}\]</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">rectangular</span> <span class="nb">object</span> <span class="n">to</span> <span class="n">demonstrate</span> <span class="n">resolution</span>

Fast Imaging Pulse Sequences.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>

Fun with RF Pulses.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>

Gradient and Spin Echoes.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>

Image Reconstruction.html

@@ -214,6 +214,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>

Introduction.html

@@ -211,6 +211,10 @@
 <li class="toctree-l1"><a class="reference internal" href="Fast%20Imaging%20Pulse%20Sequences.html">Fast Imaging Pulse Sequences</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Accelerated%20Imaging%20Methods.html">Accelerated Imaging Methods</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Summary</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="Key%20MRI%20Concepts.html">Key MRI Concepts and Equations</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Reference Material</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="MRI%20Notation.html">MRI Notation and Terminology</a></li>
@@ -485,6 +489,8 @@ <h2>Acknowledgements<a class="headerlink" href="#acknowledgements" title="Permal
 </div>
 <div class="toctree-wrapper compound">
 </div>
+<div class="toctree-wrapper compound">
+</div>
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