{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You can read an overview of this Numerical Linear Algebra course in [this blog post](http://www.fast.ai/2017/07/17/num-lin-alg/). The course was originally taught in the [University of San Francisco MS in Analytics](https://www.usfca.edu/arts-sciences/graduate-programs/analytics) graduate program. Course lecture videos are [available on YouTube](https://www.youtube.com/playlist?list=PLtmWHNX-gukIc92m1K0P6bIOnZb-mg0hY) (note that the notebook numbers and video numbers do not line up, since some notebooks took longer than 1 video to cover).\n",
"\n",
"You can ask questions about the course on [our fast.ai forums](http://forums.fast.ai/c/lin-alg)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 4. Compressed Sensing of CT Scans with Robust Regression "
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"## Broadcasting"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The term **broadcasting** describes how arrays with different shapes are treated during arithmetic operations. The term broadcasting was first used by Numpy, although is now used in other libraries such as [Tensorflow](https://www.tensorflow.org/performance/xla/broadcasting) and Matlab; the rules can vary by library.\n",
"\n",
"From the [Numpy Documentation](https://docs.scipy.org/doc/numpy-1.10.0/user/basics.broadcasting.html):\n",
"\n",
" Broadcasting provides a means of vectorizing array operations so that looping \n",
" occurs in C instead of Python. It does this without making needless copies of data \n",
" and usually leads to efficient algorithm implementations."
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The simplest example of broadcasting occurs when multiplying an array by a scalar."
]
},
{
"cell_type": "code",
"execution_count": 718,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 2., 4., 6.])"
]
},
"execution_count": 718,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([1.0, 2.0, 3.0])\n",
"b = 2.0\n",
"a * b"
]
},
{
"cell_type": "code",
"execution_count": 128,
"metadata": {
"hidden": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1 2 3] (3,)\n"
]
}
],
"source": [
"v=np.array([1,2,3])\n",
"print(v, v.shape)"
]
},
{
"cell_type": "code",
"execution_count": 129,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[1, 2, 3],\n",
" [2, 4, 6],\n",
" [3, 6, 9]]), (3, 3))"
]
},
"execution_count": 129,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m=np.array([v,v*2,v*3]); m, m.shape"
]
},
{
"cell_type": "code",
"execution_count": 133,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"n = np.array([m*1, m*5])"
]
},
{
"cell_type": "code",
"execution_count": 134,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[[ 1, 2, 3],\n",
" [ 2, 4, 6],\n",
" [ 3, 6, 9]],\n",
"\n",
" [[ 5, 10, 15],\n",
" [10, 20, 30],\n",
" [15, 30, 45]]])"
]
},
"execution_count": 134,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n"
]
},
{
"cell_type": "code",
"execution_count": 136,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"((2, 3, 3), (3, 3))"
]
},
"execution_count": 136,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n.shape, m.shape"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"We can use broadcasting to **add** a matrix and an array:"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 2, 4, 6],\n",
" [ 3, 6, 9],\n",
" [ 4, 8, 12]])"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m+v"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Notice what happens if we transpose the array:"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[1],\n",
" [2],\n",
" [3]]), (3, 1))"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1=np.expand_dims(v,-1); v1, v1.shape"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 2, 3, 4],\n",
" [ 4, 6, 8],\n",
" [ 6, 9, 12]])"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m+v1"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"#### General Numpy Broadcasting Rules"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"When operating on two arrays, NumPy compares their shapes element-wise. It starts with the **trailing dimensions**, and works its way forward. Two dimensions are **compatible** when\n",
"\n",
"- they are equal, or\n",
"- one of them is 1"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Arrays do not need to have the same number of dimensions. For example, if you have a $256 \\times 256 \\times 3$ array of RGB values, and you want to scale each color in the image by a different value, you can multiply the image by a one-dimensional array with 3 values. Lining up the sizes of the trailing axes of these arrays according to the broadcast rules, shows that they are compatible:\n",
"\n",
" Image (3d array): 256 x 256 x 3\n",
" Scale (1d array): 3\n",
" Result (3d array): 256 x 256 x 3"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"#### Review"
]
},
{
"cell_type": "code",
"execution_count": 165,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"v = np.array([1,2,3,4])\n",
"m = np.array([v,v*2,v*3])\n",
"A = np.array([5*m, -1*m])"
]
},
{
"cell_type": "code",
"execution_count": 166,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"((4,), (3, 4), (2, 3, 4))"
]
},
"execution_count": 166,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v.shape, m.shape, A.shape"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Will the following operations work?"
]
},
{
"cell_type": "code",
"execution_count": 159,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[[ 5, 10, 15],\n",
" [10, 20, 30],\n",
" [15, 30, 45]],\n",
"\n",
" [[-1, -2, -3],\n",
" [-2, -4, -6],\n",
" [-3, -6, -9]]])"
]
},
"execution_count": 159,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 158,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[[ 6, 12, 18],\n",
" [11, 22, 33],\n",
" [16, 32, 48]],\n",
"\n",
" [[ 0, 0, 0],\n",
" [-1, -2, -3],\n",
" [-2, -4, -6]]])"
]
},
"execution_count": 158,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A + v"
]
},
{
"cell_type": "code",
"execution_count": 167,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[[ 5, 10, 15, 20],\n",
" [ 10, 20, 30, 40],\n",
" [ 15, 30, 45, 60]],\n",
"\n",
" [[ -1, -2, -3, -4],\n",
" [ -2, -4, -6, -8],\n",
" [ -3, -6, -9, -12]]])"
]
},
"execution_count": 167,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 168,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"(4, 3, 2)"
]
},
"execution_count": 168,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T.shape"
]
},
{
"cell_type": "code",
"execution_count": 169,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[[ 5, -1],\n",
" [ 10, -2],\n",
" [ 15, -3]],\n",
"\n",
" [[ 10, -2],\n",
" [ 20, -4],\n",
" [ 30, -6]],\n",
"\n",
" [[ 15, -3],\n",
" [ 30, -6],\n",
" [ 45, -9]],\n",
"\n",
" [[ 20, -4],\n",
" [ 40, -8],\n",
" [ 60, -12]]])"
]
},
"execution_count": 169,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"### Sparse Matrices (in Scipy)"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"A matrix with lots of zeros is called **sparse** (the opposite of sparse is **dense**). For sparse matrices, you can save a lot of memory by only storing the non-zero values.\n",
"\n",
"![\"floating](\"images/sparse.png\")
\n",
"\n",
"Another example of a large, sparse matrix:\n",
"\n",
"![\"floating](\"images/Finite_element_sparse_matrix.png\")
\n",
"[Source](https://commons.wikimedia.org/w/index.php?curid=2245335)\n",
"\n",
"There are the most common sparse storage formats:\n",
"- coordinate-wise (scipy calls COO)\n",
"- compressed sparse row (CSR)\n",
"- compressed sparse column (CSC)\n",
"\n",
"Let's walk through [these examples](http://www.mathcs.emory.edu/~cheung/Courses/561/Syllabus/3-C/sparse.html)\n",
"\n",
"There are actually [many more formats](http://www.cs.colostate.edu/~mcrob/toolbox/c++/sparseMatrix/sparse_matrix_compression.html) as well.\n",
"\n",
"A class of matrices (e.g, diagonal) is generally called sparse if the number of non-zero elements is proportional to the number of rows (or columns) instead of being proportional to the product rows x columns.\n",
"\n",
"**Scipy Implementation**\n",
"\n",
"From the [Scipy Sparse Matrix Documentation](https://docs.scipy.org/doc/scipy-0.18.1/reference/sparse.html)\n",
"\n",
"- To construct a matrix efficiently, use either dok_matrix or lil_matrix. The lil_matrix class supports basic slicing and fancy indexing with a similar syntax to NumPy arrays. As illustrated below, the COO format may also be used to efficiently construct matrices\n",
"- To perform manipulations such as multiplication or inversion, first convert the matrix to either CSC or CSR format.\n",
"- All conversions among the CSR, CSC, and COO formats are efficient, linear-time operations."
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"## Today: CT scans"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"[\"Can Maths really save your life? Of course it can!!\"](https://plus.maths.org/content/saving-lives-mathematics-tomography) (lovely article)\n",
"\n",
"![\"Computed](\"images/xray.png\")
\n",
"\n",
"(CAT and CT scan refer to the [same procedure](http://blog.cincinnatichildrens.org/radiology/whats-the-difference-between-a-cat-scan-and-a-ct-scan/). CT scan is the more modern term)\n",
"\n",
"This lesson is based off the Scikit-Learn example [Compressive sensing: tomography reconstruction with L1 prior (Lasso)](http://scikit-learn.org/stable/auto_examples/applications/plot_tomography_l1_reconstruction.html)"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true,
"hidden": true
},
"source": [
"#### Our goal today"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Take the readings from a CT scan and construct what the original looks like.\n",
"\n",
"![\"Projections\"](\"images/lesson4.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"For each x-ray (at a particular position and particular angle), we get a single measurement. We need to construct the original picture just from these measurements. Also, we don't want the patient to experience a ton of radiation, so we are gathering less data than the area of the picture.\n",
"\n",
"![\"Projections\"](\"images/data_xray.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true,
"hidden": true
},
"source": [
"### Review"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"In the previous lesson, we used Robust PCA for background removal of a surveillance video. We saw that this could be written as the optimization problem:\n",
"\n",
"$$ minimize\\; \\lVert L \\rVert_* + \\lambda\\lVert S \\rVert_1 \\\\ subject\\;to\\; L + S = M$$\n",
"\n",
"**Question**: Do you remember what is special about the L1 norm?"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true,
"hidden": true
},
"source": [
"#### Today"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"We will see that:\n",
"\n",
"![\"Computed](\"images/sklearn_ct.png\")
\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Resources:\n",
"[Compressed Sensing](https://people.csail.mit.edu/indyk/princeton.pdf)\n",
"\n",
"![\"Computed](\"images/ct_1.png\")
\n",
"\n",
"[Source](https://www.fields.utoronto.ca/programs/scientific/10-11/medimaging/presentations/Plenary_Sidky.pdf)"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true,
"hidden": true
},
"source": [
"### Imports"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np, matplotlib.pyplot as plt, math\n",
"from scipy import ndimage, sparse"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"np.set_printoptions(suppress=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Generate Data"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"### Intro"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true,
"hidden": true
},
"source": [
"We will use generated data today (not real CT scans). There is some interesting numpy and linear algebra involved in generating the data, and we will return to that later.\n",
"\n",
"Code is from this Scikit-Learn example [Compressive sensing: tomography reconstruction with L1 prior (Lasso)](http://scikit-learn.org/stable/auto_examples/applications/plot_tomography_l1_reconstruction.html)"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"### Generate pictures"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"def generate_synthetic_data():\n",
" rs = np.random.RandomState(0)\n",
" n_pts = 36\n",
" x, y = np.ogrid[0:l, 0:l]\n",
" mask_outer = (x - l / 2) ** 2 + (y - l / 2) ** 2 < (l / 2) ** 2\n",
" mx,my = rs.randint(0, l, (2,n_pts))\n",
" mask = np.zeros((l, l))\n",
" mask[mx,my] = 1\n",
" mask = ndimage.gaussian_filter(mask, sigma=l / n_pts)\n",
" res = (mask > mask.mean()) & mask_outer\n",
" return res ^ ndimage.binary_erosion(res)"
]
},
{
"cell_type": "code",
"execution_count": 208,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"l = 128\n",
"data = generate_synthetic_data()"
]
},
{
"cell_type": "code",
"execution_count": 209,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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ngHsAquqJJA8DTwIvAx+uqr/rqXZJWliGsKqxj2N22zpyYtnnGWpd0houVtWpo27kERSS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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(5,5))\n",
"plt.imshow(data, cmap=plt.cm.gray);"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true,
"hidden": true
},
"source": [
"#### What generate_synthetic_data() is doing"
]
},
{
"cell_type": "code",
"execution_count": 155,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"l=8; n_pts=5\n",
"rs = np.random.RandomState(0)"
]
},
{
"cell_type": "code",
"execution_count": 156,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[0],\n",
" [1],\n",
" [2],\n",
" [3],\n",
" [4],\n",
" [5],\n",
" [6],\n",
" [7]]), array([[0, 1, 2, 3, 4, 5, 6, 7]]))"
]
},
"execution_count": 156,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x, y = np.ogrid[0:l, 0:l]; x,y"
]
},
{
"cell_type": "code",
"execution_count": 170,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 0, 1, 2, 3, 4, 5, 6, 7],\n",
" [ 1, 2, 3, 4, 5, 6, 7, 8],\n",
" [ 2, 3, 4, 5, 6, 7, 8, 9],\n",
" [ 3, 4, 5, 6, 7, 8, 9, 10],\n",
" [ 4, 5, 6, 7, 8, 9, 10, 11],\n",
" [ 5, 6, 7, 8, 9, 10, 11, 12],\n",
" [ 6, 7, 8, 9, 10, 11, 12, 13],\n",
" [ 7, 8, 9, 10, 11, 12, 13, 14]])"
]
},
"execution_count": 170,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x + y"
]
},
{
"cell_type": "code",
"execution_count": 157,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 16.],\n",
" [ 9.],\n",
" [ 4.],\n",
" [ 1.],\n",
" [ 0.],\n",
" [ 1.],\n",
" [ 4.],\n",
" [ 9.]])"
]
},
"execution_count": 157,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(x - l/2) ** 2 "
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 32., 25., 20., 17., 16., 17., 20., 25.],\n",
" [ 25., 18., 13., 10., 9., 10., 13., 18.],\n",
" [ 20., 13., 8., 5., 4., 5., 8., 13.],\n",
" [ 17., 10., 5., 2., 1., 2., 5., 10.],\n",
" [ 16., 9., 4., 1., 0., 1., 4., 9.],\n",
" [ 17., 10., 5., 2., 1., 2., 5., 10.],\n",
" [ 20., 13., 8., 5., 4., 5., 8., 13.],\n",
" [ 25., 18., 13., 10., 9., 10., 13., 18.]])"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(x - l/2) ** 2 + (y - l/2) ** 2"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[False, False, False, False, False, False, False, False],\n",
" [False, False, True, True, True, True, True, False],\n",
" [False, True, True, True, True, True, True, True],\n",
" [False, True, True, True, True, True, True, True],\n",
" [False, True, True, True, True, True, True, True],\n",
" [False, True, True, True, True, True, True, True],\n",
" [False, True, True, True, True, True, True, True],\n",
" [False, False, True, True, True, True, True, False]], dtype=bool)"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mask_outer = (x - l/2) ** 2 + (y - l/2) ** 2 < (l/2) ** 2; mask_outer"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAD8CAYAAABaQGkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ngQOF/Rf3BIPnGnrH8wAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(mask_outer, cmap='gray')"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 0., 1., 0., 0., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 0., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 0., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 1., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 1., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 0., 0., 0., 0., 1.],\n",
" [ 0., 0., 0., 0., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 1., 0., 0., 0., 0.]])"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mask = np.zeros((l, l))\n",
"mx,my = rs.randint(0, l, (2,n_pts))\n",
"mask[mx,my] = 1; mask"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAD8CAYAAABaQGkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ntAAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(mask, cmap='gray')"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"mask = ndimage.gaussian_filter(mask, sigma=l / n_pts)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(mask, cmap='gray')"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAD8CAYAAABaQGkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nFsEhqI8yLQAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"res = np.logical_and(mask > mask.mean(), mask_outer)\n",
"plt.imshow(res, cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"hidden": true,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAD8CAYAAABaQGkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(ndimage.binary_erosion(res), cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {
"hidden": true,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAD8CAYAAABaQGkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(res ^ ndimage.binary_erosion(res), cmap='gray');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Generate Projections"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### Code"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"def _weights(x, dx=1, orig=0):\n",
" x = np.ravel(x)\n",
" floor_x = np.floor((x - orig) / dx)\n",
" alpha = (x - orig - floor_x * dx) / dx\n",
" return np.hstack((floor_x, floor_x + 1)), np.hstack((1 - alpha, alpha))\n",
"\n",
"\n",
"def _generate_center_coordinates(l_x):\n",
" X, Y = np.mgrid[:l_x, :l_x].astype(np.float64)\n",
" center = l_x / 2.\n",
" X += 0.5 - center\n",
" Y += 0.5 - center\n",
" return X, Y"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"def build_projection_operator(l_x, n_dir):\n",
" X, Y = _generate_center_coordinates(l_x)\n",
" angles = np.linspace(0, np.pi, n_dir, endpoint=False)\n",
" data_inds, weights, camera_inds = [], [], []\n",
" data_unravel_indices = np.arange(l_x ** 2)\n",
" data_unravel_indices = np.hstack((data_unravel_indices,\n",
" data_unravel_indices))\n",
" for i, angle in enumerate(angles):\n",
" Xrot = np.cos(angle) * X - np.sin(angle) * Y\n",
" inds, w = _weights(Xrot, dx=1, orig=X.min())\n",
" mask = (inds >= 0) & (inds < l_x)\n",
" weights += list(w[mask])\n",
" camera_inds += list(inds[mask] + i * l_x)\n",
" data_inds += list(data_unravel_indices[mask])\n",
" proj_operator = sparse.coo_matrix((weights, (camera_inds, data_inds)))\n",
" return proj_operator"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### Projection operator"
]
},
{
"cell_type": "code",
"execution_count": 210,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"l = 128"
]
},
{
"cell_type": "code",
"execution_count": 211,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"proj_operator = build_projection_operator(l, l // 7)"
]
},
{
"cell_type": "code",
"execution_count": 212,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"<2304x16384 sparse matrix of type ''\n",
"\twith 555378 stored elements in COOrdinate format>"
]
},
"execution_count": 212,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"proj_operator"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"dimensions: angles (l//7), positions (l), image for each (l x l)"
]
},
{
"cell_type": "code",
"execution_count": 213,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"proj_t = np.reshape(proj_operator.todense().A, (l//7,l,l,l))"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The first coordinate refers to the angle of the line, and the second coordinate refers to the location of the line."
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The lines for the angle indexed with 3:"
]
},
{
"cell_type": "code",
"execution_count": 214,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[3,0], cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 215,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[3,1], cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 216,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[3,2], cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 217,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[3,40], cmap='gray');"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Other lines at vertical location 40:"
]
},
{
"cell_type": "code",
"execution_count": 218,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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+SPB48gJgKdADqCd4JPlBM9dRpSCSpUGDBgHB3AyNK1I9+eSTZ/uIZnMW6Qh69eqVGs+w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\nP+kAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[4,40], cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 219,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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AOoUczJo1C4i6guYvOGk+xXjo0KHE6hLJhToFEQmoU2inyy+/PHWKsfm7KGbMmMEf/vCH\nJMsSyRuFQivOP/98IP1x5vHjx6fOJqxatSqpskQKRsMHEQmoU2hB88eXmzuEH/zgB0A0kXjy5Mmk\nyhIpOHUKIhJQp5Bh2rRpQHQ1Yl1dHQDDhw8H0l+NJtLRqVMQkUCn7xQ+/elPp04x9u7dG4AFCxaw\nadOmJMsSSUynC4WysjIgPYE4ZcqU1P3mtRVEOjMNH0Qk0GoomNmDZnbUzHZmbLvXzF40s7+Z2Xoz\n65Xx3BIz229me83s6kIVno1FixZx4MABDhw4wPHjxzl+/DiDBg1i+fLl6hJEYm3pFH4BTDhl21PA\nCHe/EHgJWAJgZsOBqcDH49esMLMueatWRAqu1TkFd99qZgNP2Za55vk24Mb4/iRgrbv/C6g3s/3A\naOCZvFTbTpMnTwbSX4dWX1/P2LFjAXjhhReSKEmk5OVjovFrwK/j+/2IQqLZq/G2ornwwgtTE4eD\nBw8G0t98tH79+mKWInJGyikUzGwpcBJYncVr5wBzcjm+iORf1qFgZjOBicA4T69n/xpwXsZu/eNt\n/8bdq4Hq+L2yXor+7LPPBtJDhG984xup+zfccEO2byvSaWV1StLMJgC3Ade5+zsZT20EpppZdzOr\nBIYA+i5zkTNIq52Cma0BLgc+amavAncSnW3oDjxlZgDb3H2uu+8ys3XAbqJhxbfc/b1CFT9v3rxU\nV/Dwww8DcMEFF3D06NFCHVKkw7N0559gEW0cPlx9dXTZQ3MQHDt2LDWJuG3bthZfJyIAbHf3Ua3t\npCsaRSRQ8p99aD6tePfddzN69Ggg/bmF1avbfdJDRFqhTkFEAiXXKZx1VpRTzfMGS5cuBaILkKZP\nn55YXSKdhToFEQmUVKcwc+bMVIewZcsWACorKwFoaGhIrC6RzqQkQuFjH/sYK1eupGvXrsycORNI\nh4KIFJeGDyISKJWLl5qAt4E3kq4F+CiqI5PqCJ3JdQxw97LWdiqJUAAws7q2XG2lOlSH6ihsHRo+\niEhAoSAigVIKheqkC4ipjpDqCHX4OkpmTkFESkMpdQoiUgJKIhTMbEK8TsR+M1tcxOOeZ2ZbzGy3\nme0ys1vj7X3M7Ckz2xf/7F2EWrqY2V/N7PEEa+hlZg/Ha3rsMbPPJFTHwvj/x04zW2NmHy5WHS2s\nc9LisQu1zkmS660kHgrxuhDLgWuA4cC0eP2IYjgJLHL34cAY4FvxsRcDte4+BKiNHxfarcCejMdJ\n1HA/sMndhwEXxfUUtQ4z6wcsAEa5+wigC9FaIsWq4xf8+zonpz12gdc5OV0dxVlvxd0TvQGfAX6X\n8XgJsCShWjYAVwJ7gYp4WwWwt8DH7U/0y3YF8Hi8rdg1fASoJ55nythe7Dr6AYeAPkSX4T8OXFXM\nOoCBwM7W/gxO/V0Ffgd8plB1nPLc9cDqQtSReKdA+pegWdHXigCIF7wZCTwLlLt7Y/zUYaC8wIe/\nj+iLcN/P2FbsGiqBJmBVPIz5uZmdXew63P014EfAK0AjcMyjxYeK/eeRqaVjJ/m7+zXgyULUUQqh\nkDgz6wk8Anzb3f+R+ZxH0VuwUzRmNhE46u7bW9qn0DXEugKXAD9z95FEl50HLXox6ojH65OIQqov\ncLaZzSh2HS1J8tjNcllvpS1KIRTavFZEIZhZN6JAWO3uj8abj5hZRfx8BVDIr4f+LHCdmR0E1gJX\nmNmvilwDRP+6vOruz8aPHyYKiWLXMR6od/cmdz8BPApcmkAdmVo6dtF/dzPWW5keB1Te6yiFUHgO\nGGJmlWb2IaIJk43FOLBF30//ALDH3X+c8dRGoCq+X0U011AQ7r7E3fu7+0Ci//an3X1GMWuI6zgM\nHDKzofGmcURf1V/UOoiGDWPMrEf8/2cc0YRnsevI1NKxi7rOSdHWWynkpFE7JlSuJZpNfRlYWsTj\njiVqBf8GPB/frgXOJZr42wf8HuhTpHouJz3RWPQagIuBuvjP4zGgd0J13A28COwEHiJaY6QodQBr\niOYyThB1T7M+6NjA0vj3di9wTYHr2E80d9D8u7qyEHXoikYRCZTC8EFESohCQUQCCgURCSgURCSg\nUBCRgEJBRAIKBREJKBREJPD/edYlu+Qk+WgAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[15,40], cmap='gray');"
]
},
{
"cell_type": "code",
"execution_count": 220,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAD8CAYAAAB+fLH0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nkgqGgqSCoSCpYChIKvw/rtmVt2+NXmYAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(proj_t[17,40], cmap='gray');"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### Intersection between x-rays and data"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Next, we want to see how the line intersects with our data. Remember, this is what the data looks like:"
]
},
{
"cell_type": "code",
"execution_count": 221,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAATsAAAEyCAYAAACF03cPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAACURJREFUeJzt3dtO60oWhtHQ6vd/ZfpiCYmOHONDlatm/WNc7r2A4JDp\nz+ev7+/vF8Dq/jP6BQA8wbADIhh2QATDDohg2AERDDsggmEHRDDsgAiGHRDhv6NfwOv1en19fbmM\nA7jk+/v768i/U3ZABMMOiGDYAREMOyCCYQdEMOyACIYdEMGwAyIYdkAEww6IYNgBEQw7IMIUNwLg\nuk+Pwvz6OnRtdDdHH9E5+nWSQ9kBEZRdIVu19KmMfv7tU+X0/tqO/twzvxPcoeyACMqugKcr7ajf\nVXb1tW193ay/L7UpOyCCspvQ1f1fwGfKDoig7CZQreR671P7+b723dGSsgMiKLuBlAs8R9kBEZTd\nYtQibFN2QATDDohg2AERDDsgggMUA72fPPv+34+odkIyjKLsgAjKbgLvNXb0luZbX/uErSJt+Tqc\nPkMPyg6IoOxu6rHPTNFAe8oOiKDsLvq0Xylpf9Pv39EDd5idsgMiKLsTeh19XMHVI8qWI09RdkAE\nZXfAmf1wbin+T+rvzbyUHRBB2e1IrzP4rfrnQdkBEQw7IILN2A3Vcx1aev88VP18KDsggrL7peoa\nC55UtfCUHRBB2b3qraGgtyOXRlYrPGUHRFB2jVVZy0EaZQdEiC47FQb3Vdl3p+yACNFl12KN5CHV\n8M/shafsgAjRZffjrzXS3i3GZ1t7wWi9H6J+lbIDIii7X7bWSL//O6TYekxm9c+BsgMiGHZABJux\nL5ut0MNsm8LKDogQU3ZnTh+ZYS1EX2r+uJYH7kZ+tpQdEGHZsruzFpr9sheO+1T0Thw/78wy++tr\nR1B2QITlyq5ljSm8us6+Z3v/zvu/rdryUHZAhOXKjmw9Kkzhr0HZARGWKbuea91Zb1nDP0+9Lwqv\nNmUHRDDsgAiGHRBhmX125Bm178y+u5qUHRDBsAMiGHZABMMOiOAAxQmz3WYaOE7ZARGWKbunTwdw\n+gG01/PzpOyACMuU3SifHkby/v+Bz96LrkfhKTsgwnJl1/Kxb1d+7juP7Otn9K237Le978llp+yA\nCMuV3Y9ZHpW39zqUAGzrUe3KDoiwbNm986i8dY2+smX0vkOOUXZAhJiy2/PkUbXRFUI/3tu5KTsg\ngmEHRLAZ+4uTROvzHq6l5a4BZQdEUHYsSeHxTtkBEZQdS1N4/FB2QARlBwzzZHkrOyCCstvQc23j\nYvEx7Lur7+57qOyACMoOGlHt1z1R3soOiKDsduw9JvHomscDd+Zi310uZQdEUHYHbK399x7g89fX\nMp5qz6PsgAiGHRDBZuxFNlnWYBdFPXu7IPYoOyCCsoM3iu15bgQA0IhhB0Qw7IAI9tkBwzx52Z6y\nAyIYdkAEww6IYJ8d03EFAz0oOyCCshvo7LV9P1Yqmpa3WDrzteRRdkAEZfewFg9lWeHBLi3Or9q7\nY8kTy+Vqmb9edd+3ypQdEMGwAyLYjH1Iy82r39/DU7L6+7S5emeZ9/ieFT35tDdlB0RQdp31XmNV\new5qpeXR87V++p4rHHyalbIDIkxZdlcO6VsL0srISk7dH/vEFoqyAyJMVXZ3pvpsT2dPWitDKz0L\nT9kBEaYquzve1wCjymrUz612VHY2Mx4FTX5Prz4Ie4+yAyIsU3bAeo7U7NH6U3ZAhGXL7un9HYn7\nVWbkfeATZQdEWLbsmFPyEUbGUnZABMMOiGAz9iabY+N4uhhnKDsggrJjiBaXA7WsuBlvrTTL61iF\nsgMiKDuauFpoqoWnKDsgwrJlZ39HPy2Pgrq10v+bcXmsQtkBEZYtO9pLq+W/jhh7SHYtyg6IsEzZ\nzfbAnVGqXWg/4/lt74480LrV9+T/uS07wElTld2ds+qtKfuZtbhGszz6OfI3d3ZOKDsgwlRl96PS\nGnPWfWSzvi7Yc6bozm4JKjsggmEHRJhyM5Z2bM5SwRN/n8oOiKDsgGGe3OJQdkAEZRfCvjtmMuLv\nUNkBEZQdf1KF3NXiRh13//6UHRBB2YXZusRmllJTkLXtXbY1w3up7IAI5cvu6EXAM6xZqtu78LrH\nA6u9t3OrdsNcZQdEKFd2V9cmvWskydZy61Fh3ts5Vd2nquyACCXKrsWRw70aqbaGaqH1g27OVliP\nfXy9fw7Pa3nWgLIDIhh2QIQSm7Gsw4nDdVV/z5QdECG67FpWhmKZi/eDd8oOiFCi7FqfJgHMrcfn\nXNkBEUqUXSX2FcGclB0QoVzZVSmnKq8TZtLz86LsgAjlyq4ahccqqv8tKzsgQnTZPbmGmu1BN6Nf\nR+/lUbU+Uj3xfik7IELZsqu6/2D01SDVlhfz+fRApDN/UyMe1qPsgAiGHRCh7GbsWTM+gerJTfFZ\nN1+3Nuu3/t+eGd/bBO/L+OgT5ra+9gnKDohQvuyOPj0+ZU3/aTlU+P2vlkKF3y3B7O+DsgMilC+7\nH7OvVXoZcQj/KSv9Loyn7IAIy5RdFS2OHI6+1AsqUnZABGX3kFnPc9viKCgrUnZABGXXWYWic+UC\nCZQdEEHZFdTiNlEtjuhufV2FkiWTsgMiKLtOFA7MRdkBEQw7IILN2DC9N6+rPhuE9Sk7IIJhB0Qw\n7IAIhh0QwbADIhh2QATDDojgPLuC3JYdzlN2QARl14krCWAuyg6IoOw6a1F4LR+E/f567n4/qELZ\nARGU3Y6jjxR8vf6uo62iavW9r2hxa3eoRNkBEQw7IILN2B1nDi4c3RRM2VR06g2zUXZABGV3wJHT\nNZTMNsuFWSg7IIKyO+HI6RpVSubpk4qrLBfWpeyACMruoqqlUu31QivKDoig7G66cxnYk2YpuqpF\nTH3KDoig7Bp5L5TZSw/SKDsggrLrZJZ9UfaNwT/KDohg2AERDDsggmEHRHCAAor6dHqTg1HblB0Q\nQdnxKKfCXPfXieqW6T5lB0RQdjA5++baUHZABGW3uFluqTT656/AsrtH2QERlB1dvO9nUiXnqeG2\nlB0QQdmF2Lt9/NVy2DvvS43cZxm2peyACMouzFYtXL2FvPKgEmUHRFB2KDQiKDsggmEHRDDsgAiG\nHRDBsAMiGHZABMMOiGDYARG+rl4qBFCJsgMiGHZABMMOiGDYAREMOyCCYQdEMOyACIYdEMGwAyIY\ndkAEww6IYNgBEQw7IIJhB0Qw7IAIhh0QwbADIhh2QATDDohg2AERDDsggmEHRDDsgAj/A3R0ficj\nNo8tAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(5,5))\n",
"plt.imshow(data, cmap=plt.cm.gray)\n",
"plt.axis('off')\n",
"plt.savefig(\"images/data.png\")"
]
},
{
"cell_type": "code",
"execution_count": 222,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"proj = proj_operator @ data.ravel()[:, np.newaxis]"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"An x-ray at angle 17, location 40 passing through the data:"
]
},
{
"cell_type": "code",
"execution_count": 223,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAATsAAAEyCAYAAACF03cPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIIX/ATGQOHfWpmE5AAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(5,5))\n",
"plt.imshow(data + proj_t[17,40], cmap=plt.cm.gray)\n",
"plt.axis('off')\n",
"plt.savefig(\"images/data_xray.png\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Where they intersect:"
]
},
{
"cell_type": "code",
"execution_count": 224,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAD8CAYAAAB+fLH0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nrkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"both = data + proj_t[17,40]\n",
"plt.imshow((both > 1.1).astype(int), cmap=plt.cm.gray);"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The intensity of an x-ray at angle 17, location 40 passing through the data:"
]
},
{
"cell_type": "code",
"execution_count": 225,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"6.4384498372605989"
]
},
"execution_count": 225,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.resize(proj, (l//7,l))[17,40]"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The intensity of an x-ray at angle 3, location 14 passing through the data:"
]
},
{
"cell_type": "code",
"execution_count": 226,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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c6sOJwFxgMnAy8EEzu7JyHQ/mmfcR3r/QzAbMbKDZGEQkeXGqD7OAF9z9ZQAzuw+4AHjJ\nzLrcfdDMuoCD1d7s7iuBleF7qxYcMqQonamUeaSnKHdzxrkk+SIww8yONTMDZgLbgdXAgnCdBcAD\n8UIUkSzFaVPYYGb3AE8BR4DfE5z5jwPuNrOrgD3AvCQCrVSrL7fOZtXHI4h7XOq9N0XHv35FvNM3\n1tUHd18GLBu2+B2CrCFRjRw8DTU+pNrM3PWu38zrw/fTycd+NEVuCFaPRhGJcvfcHwRXKKo+ent7\nvbe3d8TX63nE2UYS+0/jkVZMSW+3qMcv70dax6XGdgfq+T4qUxCRCN0lKRHqPZmuVji2HVEoxLnO\nrh6C8ej4tR5VH0QkovCZgnrTidSWZCamTEFEIlQo1GnRokUNjzYt0opUKIhIROHbFEqKcqmsKHGI\nlCTd3tYyhUKR6DKbNKsVPjuqPohIREdkCmmVynlcLs3iDJPk2azIZ8RWp3kfRCQTLZUpFLU+lkXj\nY9H+Zomn2UFwshirwsJbl3PV6BiNtb4geQ6yksaXN+/enEX84LajDEa12uTu02utpOqDiES0ZKZQ\nMlLJWoSzajMxFPHvGU5jNLY0ZQoi0riWamgcrh3OSK1W/26FGCWeli4UiqKZL7auJkhRqfogIhHK\nFGIoytlejX+SJGUKIhJRM1Mws9uAy4GD7n5GuGw8cBfQDewG5rn7a+FrS4CrgL8C17r72lQiz1ER\nMgR1KJK01JMp/ByYPWzZYqDf3acA/eFzzGwqMB84PXzPj81sTGLRikjqamYK7v6YmXUPWzwXuDj8\nfRWwHrgpXH6nu78DvGBmO4HzgN8lE277aPY+jmazlGpzSipjkGqabWic4O6D4e8HgAnh7xOBJyrW\n2xcuaxt534cgkrbYVx/c3ZvppmxmC4GFcfcvIslqtlB4ycy63H3QzLqAg+Hy/cApFetNCpcdxd1X\nAiuh+XsfOkmSKb/m0pDRNHtJcjWwIPx9AfBAxfL5ZnaMmU0GpgBPxgtRRLJUzyXJOwgaFU8ys33A\nMuAW4G4zuwrYA8wDcPdnzOxuYBtwBPimu/81pdhFJAX1XH340ggvzRxh/e8C340TlIjkRz0aRSRC\nhYKIRKhQEJEI3SWZE/UqlKJSpiAiESoUGlSa40FT0ku7UvWhCXFvZmrkPXH3KdIoZQoiEtHxmUKt\nasBoZ+RqU381u6166b4FSZsyBRGJcvfcH4Dn+ejt7fXe3t6GX8s75rT/dj3a7jFQ1/cx7wKhCIVC\n6dFKhUPSsRTt79MjlUddhYKqDyISlXeWUKRMofQY7YyZ99k07TN63n+fHqk+lCmISOM6/pJkNaVe\ni6Xfi6Bo8Uj7sjB9zzeIAo/RWK0PQtZfzKz7JKgAalub3H16rZVUfRCRCFUfahh+ttSNUNLulCmI\nSIQyhQapbi/tTpmCiESoUBCRCFUfRKoowqXovChTEJGIeqaNuw24HDjo7meEy74HXAG8CzwPfNXd\nXw9fWwJcBfwVuNbd16YUu6SgkwdvGe1ycycdk3oyhZ8Ds4ctWwec4e5nAs8BSwDMbCowHzg9fM+P\nzWxMYtGKSOrqmUvyMTPrHrbs4YqnTwBfCH+fC9zp7u8AL5jZTuA84HeJRCuSgk5uP6gmiYbGrwF3\nhb9PJCgkSvaFy6QJWY7HqD4Rnf23V4pVKJjZUoIp529v4r0LgYVx9i8iyWu6UDCzfyFogJzpQ7da\n7gdOqVhtUrjsKO6+ElgZbquwd0m2szjzULQDZUfVNXVJ0sxmAzcCn3P3typeWg3MN7NjzGwyMAV4\nMn6YIpKVmuMpmNkdwMXAScBLwDKCqw3HAK+Eqz3h7leH6y8laGc4Alzv7n01g1CmUNPwxrBGz25q\nTBPqHE9Bg6y0qEZv4VYBIGiQFRFphjIFkc6hTEFEGqdCQUQiVCiISIQKBRGJUKEgIhEqFEQkoijD\nsR0C3gx/5u0kFEclxRHVynGcWs9KheinAGBmA/VcQ1UcikNxpBuHqg8iEqFCQUQiilQorMw7gJDi\niFIcUW0fR2HaFESkGIqUKYhIARSiUDCz2Wa2w8x2mtniDPd7ipk9ambbzOwZM7suXD7ezNaZ2R/D\nnydmEMsYM/u9mf0qxxhOMLN7zOxZM9tuZp/KKY4bwv/HVjO7w8w+kFUcZnabmR00s60Vy0bct5kt\nCT+3O8zs0pTj+F74v9liZveb2QlpxJF7oRDOC/HvwBxgKvClcP6ILBwBvuXuU4EZwDfDfS8G+t19\nCtAfPk/bdcD2iud5xPAj4CF3Pw04K4wn0zjMbCJwLTA9nHxoDMFcIlnF8XOOnuek6r5TnuekWhzZ\nzLfi7rk+gE8BayueLwGW5BTLA8BngR1AV7isC9iR8n4nEXzYPgP8KlyWdQzHAy8QtjNVLM86jonA\nXmA8Qee6XwH/mGUcQDewtdYxGP5ZBdYCn0orjmGv/RNwexpx5J4pMPQhKMllrohwwptzgA3ABHcf\nDF86AExIefc/JBgI972KZVnHMBl4GfhZWI251cw+mHUc7r4f+D7wIjAIHPZg8qGsj0elkfad52f3\na0Bp/NNE4yhCoZA7MzsOuJdgoNk3Kl/zoOhN7RKNmZXm6dw00jppxxAaC0wDVrj7OQTdziMpehZx\nhPX1uQSF1MnAB83syqzjGEme+y6JM99KPYpQKNQ9V0QazOz9BAXC7e5+X7j4JTPrCl/vAg6mGMI/\nAJ8zs93AncBnzOy/M44BgrPLPnffED6/h6CQyDqOWcAL7v6yu/8FuA+4IIc4Ko2078w/uxXzrfxz\nWEAlHkcRCoWNwBQzm2xm4wgaTFZnsWMzM+CnwHZ3/0HFS6uBBeHvCwjaGlLh7kvcfZK7dxP87b92\n9yuzjCGM4wCw18w+ES6aCWzLOg6CasMMMzs2/P/MJGjwzDqOSiPtO9N5TjKbbyXNRqMGGlR6CFpT\nnweWZrjfCwlSwS3A5vDRA3yIoOHvj8AjwPiM4rmYoYbGzGMAzgYGwuPxS+DEnOL4V+BZYCvwXwRz\njGQSB3AHQVvGXwiyp6tG2zewNPzc7gDmpBzHToK2g9Jn9T/SiEM9GkUkogjVBxEpEBUKIhKhQkFE\nIlQoiEiECgURiVChICIRKhREJEKFgohE/D9WZUwnkR/NLAAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.imshow(data + proj_t[3,14], cmap=plt.cm.gray);"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Where they intersect:"
]
},
{
"cell_type": "code",
"execution_count": 227,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"both = data + proj_t[3,14]\n",
"plt.imshow((both > 1.1).astype(int), cmap=plt.cm.gray);"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The measurement from the CT scan would be a small number here:"
]
},
{
"cell_type": "code",
"execution_count": 228,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"2.1374953737965541"
]
},
"execution_count": 228,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.resize(proj, (l//7,l))[3,14]"
]
},
{
"cell_type": "code",
"execution_count": 229,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"proj += 0.15 * np.random.randn(*proj.shape)"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### About *args"
]
},
{
"cell_type": "code",
"execution_count": 230,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"a = [1,2,3]\n",
"b= [4,5,6]"
]
},
{
"cell_type": "code",
"execution_count": 231,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"c = list(zip(a, b))"
]
},
{
"cell_type": "code",
"execution_count": 232,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 4), (2, 5), (3, 6)]"
]
},
"execution_count": 232,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c"
]
},
{
"cell_type": "code",
"execution_count": 233,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 2, 3), (4, 5, 6)]"
]
},
"execution_count": 233,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"list(zip(*c))"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"### The Projection (CT readings)"
]
},
{
"cell_type": "code",
"execution_count": 234,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(7,7))\n",
"plt.imshow(np.resize(proj, (l//7,l)), cmap='gray')\n",
"plt.axis('off')\n",
"plt.savefig(\"images/proj.png\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Regresssion"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we will try to recover the data just from the projections (the measurements of the CT scan)"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### Linear Regression: $Ax = b$"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Our matrix $A$ is the projection operator. This was our 4d matrix above (angle, location, x, y) of the different x-rays:"
]
},
{
"cell_type": "code",
"execution_count": 203,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 203,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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mibhZpUoV49fv9LTTTjPWt8mVG7744gtjfUf1oKWbUT2E5eaoUaOM9d2lSxdj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\nAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(12,12))\n",
"plt.title(\"A: Projection Operator\")\n",
"plt.imshow(proj_operator.todense().A, cmap='gray')"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"We are solving for $x$, the original data. We (un)ravel the 2D data into a single column."
]
},
{
"cell_type": "code",
"execution_count": 202,
"metadata": {
"hidden": true,
"scrolled": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 202,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(5,5))\n",
"plt.title(\"x: Image\")\n",
"plt.imshow(data, cmap='gray')\n",
"\n",
"plt.figure(figsize=(4,12))\n",
"# I am tiling the column so that it's easier to see\n",
"plt.imshow(np.tile(data.ravel(), (80,1)).T, cmap='gray')"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"Our vector $b$ is the (un)raveled matrix of measurements: "
]
},
{
"cell_type": "code",
"execution_count": 238,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 238,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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4gWyMMca0AN31QL65m87bytgmXWO7dI3t0jW2y7bYJl3TcnbpFg3ZGGOMMTl2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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(8,8))\n",
"plt.imshow(np.resize(proj, (l//7,l)), cmap='gray')\n",
"\n",
"plt.figure(figsize=(10,10))\n",
"plt.imshow(np.tile(proj.ravel(), (20,1)).T, cmap='gray')"
]
},
{
"cell_type": "markdown",
"metadata": {
"heading_collapsed": true
},
"source": [
"#### Scikit Learn Linear Regression"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"from sklearn.linear_model import Lasso\n",
"from sklearn.linear_model import Ridge"
]
},
{
"cell_type": "code",
"execution_count": 126,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 126,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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3DrS8evWqSWOfJIbfgHbQwLVaza6MfCIVNCKaAFi5sbFhSUo46gRBYJoTbe/9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\nk6cQeaHgyZOnEP0/7dQsdMhwS3oAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Reconstruction with L2 (Ridge) penalization\n",
"rgr_ridge = Ridge(alpha=0.2)\n",
"rgr_ridge.fit(proj_operator, proj.ravel())\n",
"rec_l2 = rgr_ridge.coef_.reshape(l, l)\n",
"plt.imshow(rec_l2, cmap='gray')"
]
},
{
"cell_type": "code",
"execution_count": 179,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"2304"
]
},
"execution_count": 179,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"18*128"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"hidden": true
},
"outputs": [],
"source": [
"18 x 128 x 128 x 128"
]
},
{
"cell_type": "code",
"execution_count": 178,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
"(2304, 16384)"
]
},
"execution_count": 178,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"proj_operator.shape"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"hidden": true
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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JYgS/fPPNN7z33nuuz/eiEvywdu1aOnfunPX5ZIUgO2mFet9EJWQj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\n/B8DgYMcgRCkOgAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Reconstruction with L1 (Lasso) penalization\n",
"# the best value of alpha was determined using cross validation\n",
"# with LassoCV\n",
"rgr_lasso = Lasso(alpha=0.001)\n",
"rgr_lasso.fit(proj_operator, proj.ravel())\n",
"rec_l1 = rgr_lasso.coef_.reshape(l, l)\n",
"plt.imshow(rec_l1, cmap='gray')"
]
},
{
"cell_type": "markdown",
"metadata": {
"hidden": true
},
"source": [
"The L1 penalty works significantly better than the L2 penalty here!"
]
}
],
"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": "3.6.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}