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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 1, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "# Load packages \n", | ||
| "from PrepFunctions import * " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 2, | ||
| "metadata": {}, | ||
| "outputs": [ | ||
| { | ||
| "name": "stdout", | ||
| "output_type": "stream", | ||
| "text": [ | ||
| "Using matplotlib backend: MacOSX\n" | ||
| ] | ||
| } | ||
| ], | ||
| "source": [ | ||
| "%matplotlib" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# How magnetic waves are used to construct non-invasive internal anatomic and physiologic images \n", | ||
| "\n", | ||
| "Mira Liu 9/09/2022\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# How the way we hear sounds is like MRI:\n", | ||
| "\n", | ||
| "### - When plucking a string, one can hear a note. \n", | ||
| "### - That note comes from oscillation at a particular frequency. \n", | ||
| "### - We then hear the sum of the frequencies, that's a chord.\n", | ||
| "\n", | ||
| "# BUT when we hear a chord, we don't just hear that one sum: \n", | ||
| "### - We hear the individual notes (waves) that are making up that chord! \n", | ||
| "### - Our brain can deconstruct a chord and hear the individual notes of the chord \n", | ||
| "### - Being able to \"hear\" individual frequencyes from a sum is how MRI works! \n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code (click the code below and press \"shift\" and \"return\") to see the C major chord as the individual notes, and their sum at the bottom is the sound that reaches our ears:</span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 3, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "# Three sound waves \n", | ||
| "\n", | ||
| "x = np.linspace(0, .1, 1000) #from 0 to .1 seconds, sampling rate of .1/1000\n", | ||
| "\n", | ||
| "C_wave = np.sin(2*np.pi*261*x) #frequency of middle C 261 Hz\n", | ||
| "E_wave = np.sin(2*np.pi*329*x) #frequency of E 329 Hz\n", | ||
| "G_wave = np.sin(2*np.pi*392*x) #frequency of G 392 Hz\n", | ||
| "\n", | ||
| "ShowThreeWaves(x,C_wave,E_wave,G_wave)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Fourier Transform: \n", | ||
| "### - Our brain is performing a \"fourier transform\", where signal is decomposed into the waves of varying frequencies. \n", | ||
| "### - This \"decomposition\" is called a fourier transform or \"FT\"\n", | ||
| "### - One wave with one frequency returns a line at the frequency it is oscillating at. \n", | ||
| "### - A chord, the sum of multiple waves, will show lines at the frequencies of the notes the chord is made of!\n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code to see how as we increase the frequency of the wave (higher pitch from C to E to G) it shifts frequency in fourier space, and how a chord is made of those three individual notes!</span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 4, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "# Load Data\n", | ||
| "with open('data/Fig1_1dChordWaves.pickle','rb') as handle: \n", | ||
| " wave1,ft_sin1,wave2,ft_sin2,wave3,ft_sin3,wavesum,ft_sinsum = pickle.load(handle)\n", | ||
| "\n", | ||
| "# Three sound waves and their fourier transform\n", | ||
| "x = np.linspace(0, .05, 10001) #from 0 to 1, sampling rate of 1/1000ft_freq = np.linspace(-500,500,1000) #kmax = 1/2Deltax, Delta kx = 1/FOVx\n", | ||
| "ft_freq = np.linspace(-100000,100000,10001) #kmax = 1/2Deltax, Delta kx = 1/FOVx\n", | ||
| "\n", | ||
| "ShowFTOfWaves(x,ft_freq,wave1,wave2,wave3,ft_sin1,ft_sin2,ft_sin3,wavesum,ft_sinsum)\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Hearing sound waves make sense, but what about in images?\n", | ||
| "### - Make those 1D sound waves two dimensional (2D), they're planar waves now\n", | ||
| "### - this means they will be two dimensional frequency space\n", | ||
| "### - that doesn't make much sense intuitively, does it... so let's take a look:\n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code to see that the 2D planar wave of a single frequency, and its 2D fourier transform! Like before, it's just a point at the wave's frequency! </span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 15, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "with open('data/Fig2_2dWaves.pickle','rb') as handle: \n", | ||
| " img,kspace = pickle.load(handle)\n", | ||
| " \n", | ||
| "x = np.linspace(0, 1, 100)\n", | ||
| "ft_freq = np.linspace(-50,50,100)\n", | ||
| "\n", | ||
| "ShowFTof2Dwaves(x,ft_freq,img,kspace)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# What does 2D wave mean?\n", | ||
| "\n", | ||
| "### - A 1D wave doesn't have any direction, it only has frequency. (this can be hard to understand, but if you think about it, even for a 1D wave to travel in a direction, that direction has to be a direction in 2D space!)\n", | ||
| "\n", | ||
| "### - A 2D wave has both a frequency AND a direction in which it is going!\n", | ||
| "### - This means that the direction of the frequency must be included in the Fourier Transform.\n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code to see rotating planar waves means rotation of the fourier transform!</span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 16, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "with open('data/Fig3_Rotating.pickle','rb') as handle: \n", | ||
| " image_rotating,kspace_rotating = pickle.load(handle)\n", | ||
| "\n", | ||
| "fig, (ax1, ax2) = pl.subplots(1,2)\n", | ||
| "for j in range(np.shape(image_rotating)[2]):\n", | ||
| " ax1.imshow(image_rotating[:,:,j],cmap = 'gray',extent = [0,1,0,1])\n", | ||
| " ax1.set_title('Rotating\\nPlanar Wave')\n", | ||
| " ax2.imshow(np.abs(kspace_rotating[:,:,j]),cmap = 'gray',extent = [-50,50,-50,50])\n", | ||
| " ax2.set_title('Corresponding FT\\n\"k-space\"')\n", | ||
| "\n", | ||
| " pl.pause(.5)\n", | ||
| " " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# What about the 2D frequency though? \n", | ||
| "\n", | ||
| "## Remember the 1D chord notes? Just like that...\n", | ||
| "### - frequency is from the distance of the point from the center\n", | ||
| "### - the direction of the wave is the angle of the delta function from the center! \n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code to see a wave that is increasing in frequency while rotating. We can see that the fourier transform is showing the points are moving away from the center and rotating as well!</span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 10, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "with open('data/Fig4_RotatingIncreasing.pickle','rb') as handle: \n", | ||
| " image_rotateincrease,kspace_rotateincrease = pickle.load(handle)\n", | ||
| "fig, (ax1, ax2) = pl.subplots(1,2)\n", | ||
| "for j in range(np.shape(image_rotating)[2]):\n", | ||
| " ax1.imshow(image_rotateincrease[:,:,j],cmap = 'gray',extent = [0,1,0,1])\n", | ||
| " ax1.set_title('Rotating & Increasing Hz\\nPlanar Wave')\n", | ||
| " ax2.imshow(np.abs(kspace_rotateincrease[:,:,j]),cmap = 'gray',extent = [-50,50,-50,50])\n", | ||
| " ax2.set_title('Corresponding FT\\n\"k-space\"')\n", | ||
| "\n", | ||
| " pl.pause(.5)\n", | ||
| " " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# From this MRI can make images! \n", | ||
| "\n", | ||
| "### - Just as a chord is formed by summing the 1D waves of individual notes, an image is formed by summing the 2D waves of individual frequencies! \n", | ||
| "\n", | ||
| "### - MRIs can use magnetic fields to measure the frequency, direction, and phase of the 2D wavepatterns of tissue. \n", | ||
| "### - By imaging this we can get the individual contribution of each type of wave, and from this we can get images of biologic anatomy and physiology. \n", | ||
| "\n", | ||
| "## <span style=\"font-weight:bold;color:blue\">Run the following code to see an example of an MRI brain scan and how it is formed from summing 2D waves</span>\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 17, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "name = 'Coronal' #Sagittal, Axial\n", | ||
| "\n", | ||
| "fov_img = 256\n", | ||
| "\n", | ||
| "with open('data/'+name + '_superposition.pickle','rb') as handle1:\n", | ||
| " stack_image = pickle.load(handle1)\n", | ||
| "with open('data/'+ name +'_superposition_masks.pickle','rb') as handle2:\n", | ||
| " stack_waves = pickle.load(handle2)\n", | ||
| "with open('data/'+ name + '_superposition_kspace.pickle','rb') as handle3:\n", | ||
| " stack_kspace = pickle.load(handle3)\n", | ||
| "with open('data/'+ name + '_FullImages.pickle','rb') as handle4:\n", | ||
| " img,kspace = pickle.load(handle4)\n", | ||
| " \n", | ||
| " \n", | ||
| "fig, (ax1, ax2, ax3,ax4) = pl.subplots(1,4)\n", | ||
| "ax4.imshow(img,cmap = 'gray')\n", | ||
| "ax4.set_title('Final Image')\n", | ||
| "for i in np.arange(0,np.shape(stack_image)[2]-65):\n", | ||
| " ax1.imshow(stack_kspace[:,:,i],vmin = 0,vmax = 8000,extent = [-.5, .5, -.5, .5],cmap = 'gray')\n", | ||
| " ax1.set_title('k-space')\n", | ||
| " ax2.imshow(stack_waves[:,:,i],cmap = 'gray')\n", | ||
| " ax2.set_title('planar\\nwaves')\n", | ||
| " ax3.imshow(stack_image[:,:,i],cmap = 'gray')\n", | ||
| " ax3.set_title('Image\\nForming')\n", | ||
| " pl.pause(.1)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# See what the MRI measures, and the image we form from it: \n", | ||
| "\n", | ||
| "<span style=\"font-weight:bold;color:blue\">Run the following code to see an example of an MRI brain scan and how it is formed from summing 2D waves</span>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 14, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "fig, ((ax1,ax2), (ax3,ax4)) = pl.subplots(2,2)\n", | ||
| "\n", | ||
| "# MRI image\n", | ||
| "ax1.imshow(np.abs(kspace),vmin = 0,vmax = 8000,extent = [-.5, .5, -.5, .5],cmap = 'gray')\n", | ||
| "ax1.set_title('k-space collected \\nwith MRI')\n", | ||
| "ax2.imshow(img,cmap = 'gray')\n", | ||
| "ax2.set_title('Image from MRI')\n", | ||
| "\n", | ||
| "# Chord\n", | ||
| "x = np.linspace(0, .05, 10001) #from 0 to 1, sampling rate of 1/1000ft_freq = np.linspace(-500,500,1000) #kmax = 1/2Deltax, Delta kx = 1/FOVx\n", | ||
| "ft_freq = np.linspace(-100000,100000,10001) #kmax = 1/2Deltax, Delta kx = 1/FOVx\n", | ||
| "\n", | ||
| "ax4.plot(x,wavesum)\n", | ||
| "ax4.set_title('C Chord\\n(What our ears receive)')\n", | ||
| "ax4.set_xlabel('Time (s)')\n", | ||
| "ax3.plot(ft_freq,abs(ft_sinsum))\n", | ||
| "ax3.set_title('FT of C Chord\\n(What our brains process)')\n", | ||
| "ax3.set_xlim([0,500])\n", | ||
| "ax3.vlines(x=261,ymin =0,ymax=5000,color = 'red',label = ' C')\n", | ||
| "ax3.vlines(x=320,ymin =0,ymax=4000,color = 'purple', label = 'E')\n", | ||
| "ax3.vlines(x=400,ymin =0,ymax=4000,color = 'green',label = 'G')\n", | ||
| "ax3.set_xlabel('Hz (1/s)')\n", | ||
| "ax3.legend()\n", | ||
| "fig.tight_layout()\n", | ||
| "pl.show()\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3", | ||
| "language": "python", | ||
| "name": "python3" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.7.4" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 2 | ||
| } |
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