{"nbformat_minor": 0, "worksheets": [{"cells": [{"source": ["Advanced Wavelet Thresholdings\n", "==============================\n", "\n*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_matlab/) 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", "$\\newcommand{\\Om}{\\Omega}$\n"], "metadata": {}, "cell_type": "markdown"}, {"source": ["This numerical tour present some advanced method for denoising that makes\n", "use of some exoting 1D thresholding functions, that in some cases give\n", "better results than soft or hard thresholding."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 2, "cell_type": "code", "language": "python", "metadata": {}, "input": ["warning off\n", "addpath('toolbox_signal')\n", "addpath('toolbox_general')\n", "addpath('solutions/denoisingwav_3_advanced')\n", "warning on"]}, {"source": ["Generating a Noisy Image\n", "------------------------\n", "Here we use an additive Gaussian noise.\n", "\n", "\n", "First we load an image."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 3, "cell_type": "code", "language": "python", "metadata": {}, "input": ["n = 256;\n", "name = 'hibiscus';\n", "M0 = load_image(name,n);\n", "M0 = rescale( sum(M0,3) );"]}, {"source": ["Noise level."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 4, "cell_type": "code", "language": "python", "metadata": {}, "input": ["sigma = .08;"]}, {"source": ["Then we add some Gaussian noise to it."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 5, "cell_type": "code", "language": "python", "metadata": {}, "input": ["M = M0 + sigma*randn(size(M0));"]}, {"source": ["Compute a 2D orthogonal wavelet transform."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 6, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Jmin = 3;\n", "MW = perform_wavelet_transf(M,Jmin,+1);"]}, {"source": ["Semi-soft Thresholding\n", "----------------------\n", "Hard and soft thresholding are two specific non-linear diagonal\n", "estimator, but one can optimize the non-linearity to capture the\n", "distribution of wavelet coefficient of a class of images.\n", "\n", "\n", "Semi-soft thresholding is a familly of non-linearities that interpolates between soft and hard thresholding.\n", "It uses both a main threshold |T| and a secondary threshold |T1=mu*T|.\n", "When |mu=1|, the semi-soft thresholding performs a hard thresholding,\n", "whereas when |mu=infty|, it performs a soft thresholding."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 7, "cell_type": "code", "language": "python", "metadata": {}, "input": ["T = 1; % threshold value\n", "v = linspace(-5,5,1024);\n", "clf;\n", "hold('on');\n", "plot(v, perform_thresholding(v,T,'hard'), 'b--');\n", "plot(v, perform_thresholding(v,T,'soft'), 'r--');\n", "plot(v, perform_thresholding(v,[T 2*T],'semisoft'), 'g');\n", "plot(v, perform_thresholding(v,[T 4*T],'semisoft'), 'g:');\n", "legend('hard', 'soft', 'semisoft, \\mu=2', 'semisoft, \\mu=4');\n", "hold('off');"]}, {"source": ["__Exercise 1__\n", "\n", "Compute the denoising SNR for different values of |mu| and different\n", "value of |T|. Important: to get good results, you should not threshold\n", "the low frequency residual.\n", "list/sigma, mulist,"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 8, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo1()"]}, {"collapsed": false, "outputs": [], "prompt_number": 9, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["One can display, for each |mu|, the optimal SNR (obtained by testing many\n", "different |T|). For |mu=1|, one has the hard thresholding, which gives\n", "the worse SNR. The optimal SNR is atained here for |mu| approximately\n", "equal to 6."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 10, "cell_type": "code", "language": "python", "metadata": {}, "input": ["err_mu = compute_min(err, 2);\n", "clf;\n", "plot(mulist, err_mu, '.-');\n", "axis('tight');\n", "set_label('\\mu', 'SNR');"]}, {"source": ["Soft and Stein Thresholding\n", "---------------------------\n", "Another way to achieve a tradeoff between hard and soft thresholding is\n", "to use a soft-squared thresholding non-linearity, also named a Stein estimator.\n", "\n", "\n", "We compute the thresholding curves."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 11, "cell_type": "code", "language": "python", "metadata": {}, "input": ["T = 1; % threshold value\n", "v = linspace(-4,4,1024);"]}, {"source": ["hard thresholding"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 12, "cell_type": "code", "language": "python", "metadata": {}, "input": ["v_hard = v .* (abs(v)>T);"]}, {"source": ["soft thresholding"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 13, "cell_type": "code", "language": "python", "metadata": {}, "input": ["v_soft = v .* max( 1-T./abs(v), 0 );"]}, {"source": ["Stein thresholding"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 14, "cell_type": "code", "language": "python", "metadata": {}, "input": ["v_stein = v .* max( 1-(T^2)./(v.^2), 0 );"]}, {"source": ["We display the classical soft/hard thresholders."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 15, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "hold('on');\n", "plot(v, v_hard, 'b');\n", "plot(v, v_soft, 'r');\n", "plot(v, v_stein, 'k--');\n", "hold('off');\n", "legend('Hard', 'Soft', 'Stein');"]}, {"source": ["__Exercise 2__\n", "\n", "Compare the performance of Soft and Stein thresholders, by determining\n", "the best threshold value."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 16, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo2()"]}, {"collapsed": false, "outputs": [], "prompt_number": 17, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}], "metadata": {}}], "nbformat": 3, "metadata": {"kernelspec": {"name": "matlab_kernel", "language": "matlab", "display_name": "Matlab"}, "language_info": {"mimetype": "text/x-matlab", "name": "matlab", "file_extension": ".m", "help_links": [{"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md", "text": "MetaKernel Magics"}]}}}