{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Volumetric wavelet Data Processing\n", "==================================\n", "\n", "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes.\n", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n", "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n", "$\\newcommand{\\choice}[1]{ \\left\\{ \\begin{array}{l} #1 \\end{array} \\right. }$\n", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n", "$\\newcommand{\\CC}{\\mathbb{C}}$\n", "$\\newcommand{\\RR}{\\mathbb{R}}$\n", "$\\newcommand{\\EE}{\\mathbb{E}}$\n", "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n", "$\\newcommand{\\Ww}{\\mathcal{W}}$\n", "$\\newcommand{\\Vv}{\\mathcal{V}}$\n", "$\\newcommand{\\Nn}{\\mathcal{N}}$\n", "$\\newcommand{\\NN}{\\mathcal{N}}$\n", "$\\newcommand{\\Hh}{\\mathcal{H}}$\n", "$\\newcommand{\\Bb}{\\mathcal{B}}$\n", "$\\newcommand{\\Ee}{\\mathcal{E}}$\n", "$\\newcommand{\\Cc}{\\mathcal{C}}$\n", "$\\newcommand{\\Gg}{\\mathcal{G}}$\n", "$\\newcommand{\\Ss}{\\mathcal{S}}$\n", "$\\newcommand{\\Pp}{\\mathcal{P}}$\n", "$\\newcommand{\\Ff}{\\mathcal{F}}$\n", "$\\newcommand{\\Xx}{\\mathcal{X}}$\n", "$\\newcommand{\\Mm}{\\mathcal{M}}$\n", "$\\newcommand{\\Ii}{\\mathcal{I}}$\n", "$\\newcommand{\\Dd}{\\mathcal{D}}$\n", "$\\newcommand{\\Ll}{\\mathcal{L}}$\n", "$\\newcommand{\\Tt}{\\mathcal{T}}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\al}{\\alpha}$\n", "$\\newcommand{\\la}{\\lambda}$\n", "$\\newcommand{\\ga}{\\gamma}$\n", "$\\newcommand{\\Ga}{\\Gamma}$\n", "$\\newcommand{\\La}{\\Lambda}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\Si}{\\Sigma}$\n", "$\\newcommand{\\be}{\\beta}$\n", "$\\newcommand{\\de}{\\delta}$\n", "$\\newcommand{\\De}{\\Delta}$\n", "$\\newcommand{\\phi}{\\varphi}$\n", "$\\newcommand{\\th}{\\theta}$\n", "$\\newcommand{\\om}{\\omega}$\n", "$\\newcommand{\\Om}{\\Omega}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This numerical tour explores volumetric (3D) data processing." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import division\n", "\n", "import numpy as np\n", "import scipy as scp\n", "import pylab as pyl\n", "import matplotlib.pyplot as plt\n", "\n", "from nt_toolbox.general import *\n", "from nt_toolbox.signal import *\n", "\n", "import warnings\n", "warnings.filterwarnings('ignore')\n", "\n", "%matplotlib inline\n", "%load_ext autoreload\n", "%autoreload 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3D Volumetric Datasets\n", "----------------------\n", "\n", "\n", "We load a volumetric data." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from nt_toolbox.read_bin import *\n", "M = read_bin(\"nt_toolbox/data/vessels.bin\", ndims=3)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "M = rescale(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Size of the image (here it is a cube)." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "n = np.shape(M)[1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can display some horizontal slices." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlEAAAJZCAYAAACJL2CzAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvWnM7Wd57ncte7M9QRICeJ5jbGww8xiIAUdKpJC0Qkqi\n", "5lClrZpKVZVIrTqp7akqNeqnfqh6PjSVTtWTHOUcqWlFQkTUhAwgCCQUMAaCjcHB2BiPmDDYxt4e\n", "Vj/Yv73W+q338nLf4nef9z33JW29e631H57p//yf+3qu+74Xy+Uyg8FgMBgMBoP/bzjtVBdgMBgM\n", "BoPB4DBiFlGDwWAwGAwG+8AsogaDwWAwGAz2gVlEDQaDwWAwGOwDs4gaDAaDwWAw2AdmETUYDAaD\n", "wWCwD8wiajAYDAaDwWAfmEXUoGKxWHxgsVj8YI9/Ty8Wi3/8I77Xby8Wiy8tFosnFovFf7fH7/9o\n", "sVjcuVgsHl4sFn+4WCxe+qO8/2AwOFo4yPlr7Z7vfvb6v63vX7FYLP7lYrH47mKx+M5isfj9F+L+\n", "g4PHLKIGFcvl8l8sl8uXrP9L8p8kuS/JP/0R3+5rSf7zJH+SZCMC7GKxeHWS/zXJB5Kcl+TRJP/L\n", 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"markdown", "metadata": {}, "source": [ "We can display an isosurface of the dataset (here we sub-sample to speed\n", "up the computation)." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAjwAAAGUCAYAAAAxqPiEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XecnVd97/vPeupu06umaKRRl6xi2ZbkChiwAzYlYHoz\n", "MTmEUM7NgRBykpCEe3OBE0ourwMHTogpAUNMCCWm2NgYXGVLttXrjEaa0VRN3TO7PmWdP9YIG1/A\n", "lm1p0Pbv/XrpNdbo2c9ez7PHe757rd9aS2mtEUIIIYSoZNZCN0AIIYQQ4myTwCOEEEKIiieBRwgh\n", "hBAVTwKPEEIIISqeBB4hhBBCVDwJPEIIIYSoeBJ4hBBCCFHxJPAIIYQQouJJ4BFCCCFExZPAI4QQ\n", "QoiKJ4FHCCGEEBVPAo8QQgghKp4EHiGEEEJUPAk8QgghhKh4EniEEEIIUfEk8AghhBCi4kngEUII\n", "IUTFk8AjhBBCiIongUcIIYQQFU8CjxBCCCEqngQeIYQQQlQ8CTxCCCGEqHgSeIQQQghR8STwCCGE\n", "EKLiSeARQgghRMWTwCOEEEKIiieBRwghhBAVTwKPEEIIISqeBB4hhBBCVDwJPEIIIYSoeBJ4hBBC\n", 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{}, "output_type": "display_data" } ], "source": [ "from nt_toolbox.isosurface import *\n", "isosurface(M,.5,3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3D Haar Transform\n", "-----------------\n", "An isotropic 3D Haar transform recursively extracts details wavelet\n", "coefficients by performing local averages/differences along the X/Y/Z axis.\n", "\n", "\n", "We apply a step of Haar transform in the X/Y/Z direction\n", "\n", "Initialize the transform" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "MW = np.copy(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Average/difference along X" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ "MW = np.concatenate(((MW[0:n:2,:,:] + MW[1:n:2,:,:])/np.sqrt(2), (MW[0:n:2,:,:] - MW[1:n:2,:,:])/np.sqrt(2)),0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Average/difference along Y" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "MW = np.concatenate(((MW[:,0:n:2,:] + MW[:,1:n:2,:])/np.sqrt(2), (MW[:,0:n:2,:] - MW[:,1:n:2,:])/np.sqrt(2)),1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Average/difference along Z" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "MW = np.concatenate(((MW[:,:,0:n:2] + MW[:,:,1:n:2])/np.sqrt(2), (MW[:,:,0:n:2] - MW[:,:,1:n:2])/np.sqrt(2)),2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display a horizontal and vertical slice to see the structure of the coefficients." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlEAAAEpCAYAAACz5XfvAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"----------------------------------\n", "An approximation is obtained by keeping only the largest coefficients.\n", "\n", "\n", "We threshold the coefficients to perform $m$-term approximation.\n", "\n", "number of kept coefficients" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from nt_toolbox.perform_thresholding import *\n", "\n", "m = round(.01*n**3)\n", "MWT = perform_thresholding(MW, m, type=\"largest\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Exercise 2__\n", "\n", "Implement the backward transform to compute an approximation $M_1$ from\n", "the coefficients MWT." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "run -i nt_solutions/multidim_2_volumetric/exo2" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [], "source": [ "## Insert your code here." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display the approximation as slices." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlEAAAEpCAYAAACz5XfvAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXm0ZWd9pvfuksQMRvNQJVVJKs0gEBosMdrCCCQjQKbj\n", "xDHGdP7xihOcFdzEwXS63ellVjudOL1Wx247OOkGGhYRCYPEMrKaQUJoQmANIDSgqSRRUkmlicGg\n", "qU7+uPepc/Zz73e37rnSuVVXv2ctraN9ztl7f9+39z11vve8v/frRqNRiqIoiqIoiuWxbrUbUBRF\n", "URRFsTtSX6KKoiiKoiimoL5EFUVRFEVRTEF9iSqKoiiKopiC+hJVFEVRFEUxBfUlqiiKoiiKYgrq\n", "S1TxrNF13Ue6rvv4s/3eZ3CsHV3XHfFsHKsoisJ0Xffvuq77pzM615u6rrt5FucqVk5XOVFFi67r\n", "PpDkD5MckeTHSb6Q5COj0eix1WyX6bpuR5LNo9HojtVuS1EU09F13SVJTkxy0Gg0emKVmzMz6vNr\n", "96aUqGJRuq77wyT/KnNfol6R5PQkG5P8p67r9lrk/XvMtoVFUawVuq7blOS0JA8keddzdI5d+TOq\n", "W+0GFNNRX6KKBXRd94okf5Lkvx2NRhePRqOnR6PRliS/mWRTkvd1XfcnXdf9v13XfarruseSfGD+\n", 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}, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [], "source": [ "s = 2 #width\n", "h = np.exp(-(X**2 + Y**2 + Z**2)/(2*s**2))\n", "h = h/np.sum(h)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The filtering is computed over the Fourier domain." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Mh = np.real(pyl.ifft2(pyl.fft2(Mnoisy,axes=(0,1,2)) * pyl.fft2(pyl.fftshift(h,axes= (0,1,2)),axes=(0,1,2)),axes= (0,1,2)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display denoised slices." ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlEAAAEpCAYAAACz5XfvAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XvQrWd53/frERISAmFkgwgmsSHBHAUyICEhgTg6YVJi\n", 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"imageplot(Mnoisy[:,:,i], \"Noisy\", [1,2,1])\n", "imageplot(Mh[:,:,i], \"Denoised\", [1,2,2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display denoised iso-surface." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAjwAAAGUCAYAAAAxqPiEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XecnVd97/vPeupu06umaKRRl6xi2ZbkChiwAzYlYHoz\n", "MTmEUM7NgRBykpCEe3OBE0ourwMHTogpAUNMCCWm2NgYXGVLttXrjEaa0VRN3TO7PmWdP9YIG1/A\n", "lm1p0Pbv/XrpNdbo2c9ez7PHe757rd9aS2mtEUIIIYSoZNZCN0AIIYQQ4myTwCOEEEKIiieBRwgh\n", "hBAVTwKPEEIIISqeBB4hhBBCVDwJPEIIIYSoeBJ4hBBCCFHxJPAIIYQQouJJ4BFCCCFExZPAI4QQ\n", "QoiKJ4FHCCGEEBVPAo8QQgghKp4EHiGEEEJUPAk8QgghhKh4EniEEEIIUfEk8AghhBCi4kngEUII\n", "IUTFk8AjhBBCiIongUcIIYQQFU8CjxBCCCEqngQeIYQQQlQ8CTxCCCGEqHgSeIQQQghR8STwCCGE\n", "EKLiSeARQgghRMWTwCOEEEKIiieBRwghhBAVTwKPEEIIISqeBB4hhBBCVDwJPEIIIYSoeBJ4hBBC\n", 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{}, "output_type": "display_data" } ], "source": [ "isosurface(M,.5,3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Exercise 3__\n", "\n", "Select the optimal blurring width $s$ to reach the smallest possible\n", "SNR. Keep the optimal denoising Mblur." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAboAAAFHCAYAAADX+4lCAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XmcXGWd7/HPV3ZkF02ABAMMIMMEDGAIsjXIGq6EiIJG\n", "VFQUFYHhegeRcZy8XFEvDopcBhEBr8LIsC9hJ80aICxZ2JeBISGEHSSyBfKbP86JtE0v1dV16jnn\n", "1Pf9evWrq6qrq74n3Se/fp7zLIoIzMzM6uo9qQOYmZkVyYXOzMxqzYXOzMxqzYXOzMxqzYXOzMxq\n", "zYXOzMxqrbBCJ2m0pOmS7pV0j6Qj8senSpov6e78Y6+iMpiZmamoeXSSRgIjI2KWpFWAO4H9gAOA\n", "VyLiF4W8sZmZWQ/LFvXCEbEQWJjfXiTpfmC9/Msq6n3NzMx6ass1OkljgHHArflDh0uaLek0SWu0\n", "I4OZmXWmwgtd3m15LnBkRCwCTgY2AD4MPAUcX3QGMzPrXIVdowOQtBxwKXB5RJzQx9fHAJdExNg+\n", "vuZFOM3M7G9ExJAvfRV2jU6SgNOA+3oWOUnrRMRT+d3JwNz+XqOZAyoLSVMjYmrqHMPhYygHH0M5\n", "VP0Yqp4fmm8AFVbogO2Bg4A5ku7OHzsW+IykDwMBPAYcWmAGMzPrcEWOuryJvq8BXl7Ue5qZmfXm\n", "lVGK0506QAt0pw7QAt2pA7RAd+oALdCdOkALdKcOMEzdqQOkUuhglOGQFFW+RmdmZq3VbF1wi87M\n", 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"----------------------------------------\n", "Denoising is obtained by removing small amplitude coefficients that\n", "corresponds to noise." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Exercise 4__\n", "\n", "Perforn Wavelet denoising by thresholding the wavelet coefficients of\n", "Mnoisy. Test both hard thresholding and soft thresholding to determine\n", "the optimal threshold and the corresponding SNR.\n", "Record the optimal result Mwav." ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAboAAAFKCAYAAABrZZqcAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XeYXGX5xvHvDQmRXgTpCCK9t9AhEAi9g8FIF4mgNBWp\n", "CihNfxZABIKAFCWIhBa6YEIvoYQWQJqA9JLQRcrz++M9myxLNtkyM++ZM/fnuvbambNzZp6dnMyz\n", "b30UEZiZmVXVNLkDMDMzqycnOjMzqzQnOjMzqzQnOjMzqzQnOjMzqzQnOjMzq7S6JTpJC0oaJekx\n", "SY9KOqA4/ktJD0kaK+lmSQvWKwYzMzPVax2dpHmAeSJirKSZgPuBbYH/RMR7xWP2B1aIiL3rEoSZ\n", "mbW8PvV64oh4FXi1uP2+pMeB+SLi8XYPmwl4s+O5kryK3czMviQi1N1z6pbo2pO0MLAScE9x/3hg\n", "V+BDYI3JndOTX6bmUqt0V2APoB9wHnAhEc8jXQtsBowBBhExoZev1fZ8jwM3A5sfCnP8Cs4FLgPu\n", "IuLzXr1GE5J0TEQckzuOnPwe+D0AvwfQ80ZQ3SejFN2WlwIHRsT7ABFxZEQsREocv693DN0i9UPa\n", 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denoised iso-surface with optimal soft thresholding." ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAjwAAAGUCAYAAAAxqPiEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd8XdWV6PHfOucW9WpZVnPvuILB9JbQDakkTAgk4SVD\n", "kkmblBnCTBLSZ1JewkzCpLxUEkoyIYQaIEAAAwZjwL3bsmxZtnrXrWe9P/Y1KIoELpKF5fX9fPxB\n", "vvfcc/bZ9+K7tPbae4uqYowxxhgzlnmj3QBjjDHGmJFmAY8xxhhjxjwLeIwxxhgz5lnAY4wxxpgx\n", "zwIeY4wxxox5FvAYY4wxZsyzgMcYY4wxY54FPMYYY4wZ8yzgMcYYY8yYZwGPMcYYY8Y8C3iMMcYY\n", "M+ZZwGOMMcaYMc8CHmOMMcaMeRbwGGOMMWbMs4DHGGOMMWOeBTzGGGOMGfMs4DHGGGPMmGcBjzHG\n", "GGPGPAt4jDHGGDPmWcBjjDHGmDHPAh5jjDHGjHkW8BhjjDFmzLOAx5iDJCJvE5HdItIlIosO8xx/\n", "FZH/M9xtG+Q6vxSRrx7ma4dso4hMFpFARLzM3x8QkWuOpK1mZIjI+0XkqdFuhzFvFBbwmOOKiJwp\n", "Is+ISLuItIjIchFZcpAv/w7wUVXNB9r7f/EPca2bROTWAQ9r5s9IO5LrHPRrVfVSVR14j8NORIpE\n", "5Oci0iAinSKyWUT+td/zgYisERHp99jXROQXmZ8PBGpdmT+1IvKFEWjnTBH5k4g0Zj5ffxaRmf2e\n", 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"Orthogonal wavelet thresholdings suffers from blocking artifacts.\n", "This can be aleviated by performing a cycle spinning denoising, which\n", "averages the denosing result of translated version of the signal.\n", "\n", "\n", "A typical cycle spinning process is like this.\n", "\n", "Maximum translation." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [], "source": [ "w = 4" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "List of translations." ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [], "source": [ "[dZ, dX, dY] = np.meshgrid(np.arange(0,w),np.arange(0,w),np.arange(0,w))\n", "dX = np.ravel(dX)\n", "dY = np.ravel(dY)\n", "dZ = np.ravel(dZ)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Initialize spinning process." ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Mspin = np.zeros([n,n,n])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Spin." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def circshift(x,v):\n", " x = np.roll(x,v[0], axis = 0)\n", " x = np.roll(x,v[1], axis = 1)\n", " x = np.roll(x,v[2], axis = 2)\n", " return x\n", "\n", "for i in range(w**3):\n", " # shift the image\n", " MnoisyC = circshift(Mnoisy, [dX[i],dY[i],dZ[i]])\n", " # denoise the image to get a result M1\n", " M1 = MnoisyC; # replace this line by some denoising\n", " # shift inverse\n", " M1 = circshift(M1, [-dX[i],-dY[i],-dZ[i]])\n", " # average the result\n", " Mspin = Mspin*(i)/(i+1) + M1/(i+1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Exercise 5__\n", "\n", "Implement cycle spinning hard thresholding with $T=3\\sigma$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "run -i nt_solutions/multidim_2_volumetric/exo5" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "## Insert your code here." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display denoised iso-surface." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "isosurface(Mspin,.5,2,\"Cycle spinning, SNR = %.1f dB\" %snr(M, Mspin))" ] } ], "metadata": { "anaconda-cloud": {}, "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": "3.4.5" } }, "nbformat": 4, "nbformat_minor": 0 }