{"worksheets": [{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["Texture Synthesis with PDEs", "===========================", "", "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes.", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$", "$\\newcommand{\\norm}[1]{\\|#1\\|}$", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$", "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$", "$\\newcommand{\\CC}{\\mathbb{C}}$", "$\\newcommand{\\RR}{\\mathbb{R}}$", "$\\newcommand{\\EE}{\\mathbb{E}}$", "$\\newcommand{\\Zz}{\\mathcal{Z}}$", "$\\newcommand{\\Ww}{\\mathcal{W}}$", "$\\newcommand{\\Vv}{\\mathcal{V}}$", "$\\newcommand{\\Nn}{\\mathcal{N}}$", "$\\newcommand{\\NN}{\\mathcal{N}}$", "$\\newcommand{\\Hh}{\\mathcal{H}}$", "$\\newcommand{\\Bb}{\\mathcal{B}}$", "$\\newcommand{\\Ee}{\\mathcal{E}}$", "$\\newcommand{\\Cc}{\\mathcal{C}}$", "$\\newcommand{\\Gg}{\\mathcal{G}}$", "$\\newcommand{\\Ss}{\\mathcal{S}}$", "$\\newcommand{\\Pp}{\\mathcal{P}}$", "$\\newcommand{\\Ff}{\\mathcal{F}}$", "$\\newcommand{\\Xx}{\\mathcal{X}}$", "$\\newcommand{\\Mm}{\\mathcal{M}}$", "$\\newcommand{\\Ii}{\\mathcal{I}}$", "$\\newcommand{\\Dd}{\\mathcal{D}}$", "$\\newcommand{\\Ll}{\\mathcal{L}}$", "$\\newcommand{\\Tt}{\\mathcal{T}}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\al}{\\alpha}$", "$\\newcommand{\\la}{\\lambda}$", "$\\newcommand{\\ga}{\\gamma}$", "$\\newcommand{\\Ga}{\\Gamma}$", "$\\newcommand{\\La}{\\Lambda}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\Si}{\\Sigma}$", "$\\newcommand{\\be}{\\beta}$", "$\\newcommand{\\de}{\\delta}$", "$\\newcommand{\\De}{\\Delta}$", "$\\newcommand{\\phi}{\\varphi}$", "$\\newcommand{\\th}{\\theta}$", "$\\newcommand{\\om}{\\omega}$", "$\\newcommand{\\Om}{\\Omega}$"]}, {"cell_type": "markdown", "metadata": {}, "source": ["This numerical tour explores image synthesis using diffusion equation.", "You might first have a look at the numerical tours on Heat diffusion and", "Total variation minimization that introduces these two notions."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["from __future__ import division", "import nt_toolbox as nt", "from nt_solutions import graphics_3_synthesis_diffusion as solutions", "%matplotlib inline", "%load_ext autoreload", "%autoreload 2"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Heat Flow Synthesis", "-------------------", "It is possible to synthesis a cloudy image by a simple smoothing of a", "noise. Histogram equalization helps to maintain the contrast."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["n = 256"]}, {"cell_type": "markdown", "metadata": {}, "source": ["We blur a noisy image to have a cloud texture."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["M1 = perform_blurring(randn(n, n), 15)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["We impose a flat histogram to enhance the contrast."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["x = linspace(0, 1, n*n)", "M2 = perform_hist_eq(M1, x)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["display"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["imageplot(M1, 'Original', 1, 2, 1)", "imageplot(M2, 'Equalized', 1, 2, 2)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 1__", "", "Perform a synthesis by running a heat diffusion, starting with a random", "noise a time |T=0|. At each step of the diffusion, perform an histogram", "equation to keep the contrast of the texture.", "tep size", "umber of iteration"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo1()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Total Variation Constraints", "---------------------------", "The total variation minimization reduce the number of edges and is", "usually used as a denoising method. It can also be used to perform", "synthesis of singular images.", "", "", "The total variation roughly measures the number of edges present in an", "image."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["M = rescale(load_image('lena', n))", "Gr = grad(M)", "tv = sum(sum(sqrt(sum(Gr.^2, 3)), 2), 1)", "disp(strcat(['Total variation of M = ' num2str(tv) '.']))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 2__", "", "Starting from an initial noise image, perform a total variation", "minimization. At each step of the descent, perform an histogram", "equalization so that the texture has a flat histogram.", "tep size", "void instabilities", "umber of iteration"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo2()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 3__", "", "Perfrom a synthesis that mixes both TV minimization (to reduce the TV", "norm)", "and wavelet histogram equalization (to control the distribution of singularities). Stop the iterations when the", "synthesized image has the same TV norm as the original one."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo3()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}]}], "nbformat": 3, "metadata": {"name": ""}, "nbformat_minor": 0}