deploy: 3214c8e

Image Reconstruction.html

@@ -450,6 +450,10 @@ <h2> Contents </h2>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 </div>

MR Physics - Bloch Equation.html

@@ -460,6 +460,10 @@ <h1>Bloch Equation<a class="headerlink" href="#bloch-equation" title="Permalink
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 </div>

MRI Signal Equation.html

@@ -453,6 +453,10 @@ <h2> Contents </h2>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 </div>

Magnetic Fields and RF Coils.html

@@ -471,6 +471,10 @@ <h1>Magnetic fields in MRI<a class="headerlink" href="#magnetic-fields-in-mri" t
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 </div>

Signal to Noise Ratio.html

@@ -476,7 +476,8 @@ <h2>SNR Dependencies<a class="headerlink" href="#snr-dependencies" title="Permal
 <h3>Voxel Volume<a class="headerlink" href="#voxel-volume" title="Permalink to this heading">#</a></h3>
 <p>The MRI signal is an integral over a volume of the transverse magnetization.  Therefore there is a linear dependency on the MRI signal and the volume of magnetization being examined.  So we have</p>
 <div class="math notranslate nohighlight">
-\[SNR \propto \mathrm{Voxel\ Volume} = \Delta x\ \Delta y\ \Delta z\]</div>
+\[SNR \propto \mathrm{Voxel\ Volume} = \delta x\ \delta y\ \delta z\]</div>
+<p>where <span class="math notranslate nohighlight">\(\delta x, \delta y, \delta z\)</span> are the voxel dimensions in x, y, and z.</p>
 </section>
 <section id="data-acquisition-time">
 <h3>Data Acquisition Time<a class="headerlink" href="#data-acquisition-time" title="Permalink to this heading">#</a></h3>

Spectral-Spatial RF Pulses.html

@@ -478,6 +478,10 @@ <h2>Spectral-Spatial RF Pulse Design<a class="headerlink" href="#spectral-spatia
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>warning: trying to compile MEX file from /home/runner/work/MRI-education-resources/MRI-education-resources/RF Pulses/rf_tools/mex_files/abrx.c

Spin Physics.html

@@ -632,6 +632,10 @@ <h3>Relaxation Equations<a class="headerlink" href="#relaxation-equations" title
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>loading image
 loading signal
+error: graphics_toolkit: qt toolkit is not available
+error: called from
+    graphics_toolkit at line 88 column 5
+    startup at line 43 column 5
 </pre></div>
 </div>
 </div>

_sources/FOV and Resolution.ipynb

@@ -14,25 +14,17 @@
     "    * Describe what determines the image FOV and resolution\n",
     "1. Manipulate MRI sequence parameters to improve performance\n",
     "    * Manipulate the gradients and k-space sampling to change the FOV and resolution\n",
-    "\n",
-    "## Field-of-View (FOV)\n",
-    "\n",
-    "The FOV is determined by the sample spacing in k-space, with the simple relationship based on the sample spacing, $\\Delta k$ as:\n",
-    "\n",
-    "$$ FOV = \\frac{1}{\\Delta k}$$\n",
-    "\n",
-    "\n",
-    "## Spatial Resolution\n",
-    "\n",
-    "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, $k_{max}$ as:\n",
-    "\n",
-    "$$ \\Delta = \\frac{1}{2 k_{max}}$$\n"
+    "\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "octave"
+    }
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -49,6 +41,143 @@
     "startup"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Field-of-View (FOV)\n",
+    "\n",
+    "The FOV is determined by the sample spacing in k-space, with the simple relationship based on the sample spacing, $\\Delta k$ as:\n",
+    "\n",
+    "$$ FOV = \\frac{1}{\\Delta k}$$\n",
+    "\n",
+    "For example, $FOV_{x} = \\frac{1}{\\Delta k_x}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "octave"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "% rectangular object to demonstrate FOV\n",
+    "\n",
+    "N = 256;\n",
+    "kx = [-N/2:N/2-1]/N;\n",
+    "N_rect = N/4;\n",
+    "kdata = sinc(kx *N_rect).' * sinc(kx *N_rect);\n",
+    "\n",
+    "rect_recon = ifft2c(kdata);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon), [0 max(abs(rect_recon(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('Full FOV')\n",
+    "\n",
+    "kdata2 = kdata;\n",
+    "kdata2(1:2:end,:) = 0;\n",
+    "kdata2(:,1:2:end) = 0;\n",
+    "rect_recon2 = ifft2c(kdata2);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata2)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon2), [0 max(abs(rect_recon2(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('Doubled \\Delta k, Half FOV')\n",
+    "\n",
+    "kdata3 = kdata;\n",
+    "kdata2(1:3:end,:) = 0;\n",
+    "kdata2(2:3:end,:) = 0;\n",
+    "kdata2(:,1:3:end) = 0;\n",
+    "kdata2(:,2:3:end) = 0;\n",
+    "rect_recon3 = ifft2c(kdata3);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata3)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon3), [0 max(abs(rect_recon3(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('Tripled \\Delta k, One Third FOV')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Spatial Resolution\n",
+    "\n",
+    "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, $k_{max}$ as:\n",
+    "\n",
+    "$$ \\delta = \\frac{1}{2 k_{max}}$$\n",
+    "\n",
+    "For example, $\\delta_x = \\frac{1}{2 k_{x,max}}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "octave"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "% rectangular object to demonstrate resolution\n",
+    "\n",
+    "N = 256;\n",
+    "kx = [-N/2:N/2-1]/N;\n",
+    "N_rect = N/2;\n",
+    "kdata = sinc(kx *N_rect).' * sinc(kx *N_rect);\n",
+    "\n",
+    "rect_recon = ifft2c(kdata);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon), [0 max(abs(rect_recon(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('Full Resolution')\n",
+    "\n",
+    "kdata2 = kdata;\n",
+    "kdata2([[1:N/4],[3*N/4+1:N]],:) = 0;\n",
+    "kdata2(:,[[1:N/4],[3*N/4+1:N]]) = 0;\n",
+    "rect_recon2 = ifft2c(kdata2);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata2)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon2), [0 max(abs(rect_recon2(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('Half k_{max}, Double voxel size')\n",
+    "\n",
+    "kdata4 = kdata;\n",
+    "kdata4([[1:3*N/8],5*N/8+1:N]],:) = 0;\n",
+    "kdata4(:,[[1:3*N/8],5*N/8+1:N]]) = 0;\n",
+    "rect_recon4 = ifft2c(kdata4);\n",
+    "\n",
+    "subplot(221)\n",
+    "imagesc((abs(kdata4)), [0 (1)])\n",
+    "colormap(gray), axis equal tight off\n",
+    "subplot(222)\n",
+    "imagesc(abs(rect_recon4), [0 max(abs(rect_recon4(:)))])\n",
+    "colormap(gray), axis equal tight off\n",
+    "title('1/4 k_{max}, 4x voxel size')\n",
+    "\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

_sources/Signal to Noise Ratio.ipynb

@@ -29,7 +29,9 @@
     "\n",
     "The MRI signal is an integral over a volume of the transverse magnetization.  Therefore there is a linear dependency on the MRI signal and the volume of magnetization being examined.  So we have\n",
     "\n",
-    "$$SNR \\propto \\mathrm{Voxel\\ Volume} = \\Delta x\\ \\Delta y\\ \\Delta z$$\n",
+    "$$SNR \\propto \\mathrm{Voxel\\ Volume} = \\delta x\\ \\delta y\\ \\delta z$$\n",
+    "\n",
+    "where $\\delta x, \\delta y, \\delta z$ are the voxel dimensions in x, y, and z.\n",
     "\n",
     "### Data Acquisition Time\n",
     "\n",