{"nbformat_minor": 0, "worksheets": [{"cells": [{"source": ["Signal Denoising with Wavelets\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 tour uses wavelets to perform signal denoising using\n", "thresholding estimators. Wavelet thresholding properites were\n", "investigated in a series of papers by Donoho and\n", "Johnstone, see for instance [DonJohn94](#biblio) and [DoJoKePi95](#bibli)."], "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_1_wavelet_1d')\n", "warning on"]}, {"source": ["Loading a Signal and Making Noise\n", "---------------------------------\n", "Here we consider a simple additive Gaussian white noise.\n", "\n", "\n", "Size $N$ of the signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 3, "cell_type": "code", "language": "python", "metadata": {}, "input": ["N = 2048*2;"]}, {"source": ["First we load the 1-D signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 4, "cell_type": "code", "language": "python", "metadata": {}, "input": ["name = 'piece-regular';\n", "f0 = load_signal(name, N);\n", "f0 = rescale(f0,.05,.95);"]}, {"source": ["Variance $\\si^2$ of the noise."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 5, "cell_type": "code", "language": "python", "metadata": {}, "input": ["sigma = 0.05;"]}, {"source": ["Generate a noisy signal $f = f_0 + w$ where $w \\sim \\Nn(0,\\si^2 \\text{Id})$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 6, "cell_type": "code", "language": "python", "metadata": {}, "input": ["f = f0 + randn(size(f0))*sigma;"]}, {"source": ["Display."], "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": ["clf;\n", "subplot(2,1,1);\n", "plot(f0); axis([1 N 0 1]);\n", "title('Clean signal');\n", "subplot(2,1,2);\n", "plot(f); axis([1 N 0 1]);\n", "title('Noisy signal');"]}, {"source": ["Hard Thresholding vs. Soft Thresholding\n", "---------------------------------------\n", "A thresholding $\\Theta : \\RR^N \\rightarrow \\RR^N$\n", "is a non-linear function that operates diagonaly, i.e.\n", "$$ \\forall i, \\quad \\Theta(x)_i = \\th(x_i) $$\n", "where $\\th : \\RR \\rightarrow \\RR$ is the 1-D thresholding function.\n", "\n", "\n", "The most important thresholding are the hard thresholding\n", "(related to $\\ell^0$ minimization) and the soft thresholding (related\n", "to $\\ell^1$ minimization).\n", "\n", "\n", "The hard thresholding reads\n", "$$ \\Theta^0_T(x)_i  = \\choice{\n", "      x_i \\qifq \\abs{x_i}>T, \\\\\n", "      0 \\quad \\text{otherwise}.\n", " }$\n", "$$"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 8, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Theta0 = @(x,T)x .* (abs(x)>T);"]}, {"source": ["The soft thresholding reads\n", "$$ \\Theta^1_T(x)_i  = \\max\\pa{ 0, 1-\\frac{T}{\\abs{x}} } x\n", "$$"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 9, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Theta1 = @(x,T)max(0, 1-T./max(abs(x),1e-9)) .* x;"]}, {"source": ["Display the thresholding."], "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": ["t = linspace(-3,3,1024)'; T = 1;\n", "clf;\n", "plot( t, [Theta0(t,T), Theta1(t,T)], 'LineWidth', 2 );\n", "axis('equal'); axis('tight');\n", "legend('\\Theta^0', '\\Theta^1');"]}, {"source": ["Wavelet Thresholding\n", "--------------------\n", "It is possible to perform non linear denoising by thresholding the\n", "wavelet coefficients of $f$.\n", "\n", "\n", "Shortcut for the foward orthogonal wavelet transform $W$ and the inverse\n", "wavelet transform $W^{-1}=W^*$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 11, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.ti = 0; Jmin = 4;\n", "W  = @(f) perform_wavelet_transf(f,Jmin,+1,options);\n", "Wi = @(fw)perform_wavelet_transf(fw,Jmin,-1,options);"]}, {"source": ["Compute the wavelet coefficients $x=W(f)$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 12, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x = W(f);"]}, {"source": ["Threshold the wavelet coefficients $\\tilde x=\\Theta_0(x)$.\n", "Here we use $T=3\\si$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 13, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x1 = Theta0(x, 3*sigma);"]}, {"source": ["Display the wavelet coefficients $W(f)$ and the hard thresholded\n", "coefficients $\\Theta_0(W(f))$. Note how the wavelet\n", "coefficients are contaminated by a small amplitude Gaussian white noise, most of which are removew by thresholding."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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ZgcE1RleqPg8+zyyFmdODcIYAg8ss3aMxOyAb3fSxssLhr2+O7aAdTiHCQzJPGzq7CJkE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\nhCTAAAhJgAEQ0v8BOUAYGId9PKAAAAAASUVORK5CYII=\n", "output_type": "display_data"}], "prompt_number": 14, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "subplot(2,1,1);\n", "plot_wavelet(x,Jmin); axis([1 N -1 1]);\n", "title('W(f)');\n", "subplot(2,1,2);\n", "plot_wavelet(Theta0(W(f),T),Jmin); axis([1 N -1 1]);\n", "title('\\Theta_0(W(f))');"]}, {"source": ["Reconstruct from the thresholded coefficients the final estimator $\\tilde f = W^*(\\tilde x)$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 15, "cell_type": "code", "language": "python", "metadata": {}, "input": ["f1 = Wi(x1);"]}, {"source": ["Display noisy and denoised signal."], "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": ["clf;\n", "subplot(2,1,1);\n", "plot(f); axis([1 N 0 1]);\n", "title('f');\n", "subplot(2,1,2);\n", "plot(f1); axis([1 N 0 1]);\n", "title('f_1');"]}, {"source": ["Given a thresholding $\\Theta$ (for instance $\\Theta^0$ or $\\Theta^1$), one\n", "thus defines a wavelet thresholding estimator as\n", "$$ \\Theta_W(f) = W^* \\circ \\Theta \\circ W. $$\n", "\n", "\n", "Operator to re-inject the coarse scale noisy coefficients.\n", "Improves a little bit the result of soft thresholding denoising (because\n", "of the bias)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 17, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x = W(f); \n", "reinject = @(x1)assign(x1, 1:2^Jmin, x(1:2^Jmin));"]}, {"source": ["Define the soft and hard thresholding estimators."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 18, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Theta0W = @(f,T)Wi(Theta0(W(f),T));\n", "Theta1W = @(f,T)Wi(reinject(Theta1(W(f),T)));"]}, {"source": ["The denoising performance of an estimator $\\tilde f = \\Theta_W(f)$ is\n", "measured using the $L^2$ error to the (unknown) ground trust $f_0$.\n", "One usually expresses it in dB using the SNR\n", "$$ SNR =  -10\\log_{10}\\pa{ \\frac{\\norm{f_0-\\tilde f}^2}{\\norm{f_0}^2} }. $$"], "metadata": {}, "cell_type": "markdown"}, {"source": ["__Exercise 1__\n", "\n", "Display the evolution of the denoising SNR\n", "when $T$ varies.\n", "Store in |fBest0| and |fBest1| the optimal denoising results.\n", "etrieve best"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 19, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo1()"]}, {"collapsed": false, "outputs": [], "prompt_number": 20, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Display the results.\n", "For 1-D signals, hard thresholding seems to outperform soft thresholding.\n", "For natural images, on contrary, soft thresholding seems to be better."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 21, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "subplot(2,1,1);\n", "plot(fBest0); axis([1 N 0 1]); e = snr(fBest0,f0);\n", "title(['Hard, SNR=' num2str(e,3) 'dB']);\n", "subplot(2,1,2);\n", "plot(fBest1); axis([1 N 0 1]); e = snr(fBest1,f0);\n", "title(['Soft, SNR=' num2str(e,3) 'dB']);"]}, {"source": ["Translation Invariant Wavelet Transform\n", "---------------------------------------\n", "Orthogonal wavelet transforms are not translation invariant.\n", "It means that the processing of an image and of a translated version of\n", "the image give different results. A translation invariant wavelet\n", "transform is implemented by ommitting the sub-sampling at each stage of\n", "the transform. This correspond to the decomposition of the image in a\n", "redundant familly of $N (J+1)$ atoms where $N$ is the number of\n", "samples and $J$ is the number of scales of the transforms.\n", "\n", "\n", "The foward and backward transform algorithm is the so-called \"a trou\"\n", "algorithm, that was introduced in [Holsch87](#biblio). See also [Fowler05](#biblio) for a\n", "review. This algorithm runs in $O(J N)$ operations.\n", "\n", "\n", "The invariant transform is obtained using the same function, by\n", "activating the switch |options.ti=1|."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 22, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.ti = 1;\n", "W  = @(f) perform_wavelet_transf(f,Jmin,+1,options);\n", "Wi = @(fw)perform_wavelet_transf(fw,Jmin,-1,options);"]}, {"source": ["Compute the invariant transform."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 23, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fw = W(f);"]}, {"source": ["|fw(:,:,1)| corresponds to the low scale residual.\n", "Each |fw(:,1,j)| corresponds to a scale of wavelet coefficient, and has\n", "the same size as the original signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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hPKV2gGUz2jGurgP7/bU6kPWSntfI9Nrb3tjGzpPhC7uG4rF2/x4vvnf/2PnJ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\nRK5CYII=\n", "output_type": "display_data"}], "prompt_number": 24, "cell_type": "code", "language": "python", "metadata": {}, "input": ["nJ = size(fw,3)-4;\n", "clf;\n", "subplot(5,1, 1);\n", "plot(f0); axis('tight');\n", "title('Signal');\n", "i = 0;\n", "for j=1:3\n", "    i = i+1;\n", "    subplot(5,1,i+1);\n", "    plot(fw(:,1,nJ-i+1)); axis('tight');\n", "    title(strcat(['Scale=' num2str(j)]));\n", "end\n", "subplot(5,1, 5);\n", "plot(fw(:,1,1)); axis('tight');\n", "title('Low scale');"]}, {"source": ["Translation Invariant Wavelet Denoising\n", "---------------------------------------\n", "Orthogonal wavelet denoising does not performs very well because of its\n", "lack of translation invariance. A much better result is obtained by not sub-sampling the wavelet\n", "transform, which leads to a redundant tight-frame.\n", "\n", "\n", "Translation invariant denoising using cycle spinning is introduced in\n", "[CoifDon95](#biblio). We uwe here the a trou algorithm which is faster.\n", "\n", "\n", "Operator to re-inject the coarse scales."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 25, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x = W(f); \n", "reinject = @(x1)assign(x1, 1:N, x(1:N));"]}, {"source": ["Define the soft and hard thresholding estimators."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 26, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Theta0W = @(f,T)Wi(Theta0(W(f),T));\n", "Theta1W = @(f,T)Wi(reinject(Theta1(W(f),T)));"]}, {"source": ["__Exercise 2__\n", "\n", "Display the evolution of the denoising SNR\n", "when $T$ varies.\n", "Store in |fBest0| and |fBest1| the optimal denoising results.\n", "etrieve best"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 27, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo2()"]}, {"collapsed": false, "outputs": [], "prompt_number": 28, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Display the results."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 29, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "subplot(2,1,1);\n", "plot(fBest0); axis([1 N 0 1]); e = snr(fBest0,f0);\n", "title(['Hard TI, SNR=' num2str(e,3) 'dB']);\n", "subplot(2,1,2);\n", "plot(fBest1); axis([1 N 0 1]); e = snr(fBest1,f0);\n", "title(['Soft TI, SNR=' num2str(e,3) 'dB']);"]}, {"source": ["Bibliography\n", "------------\n", "<html><a name=\"biblio\"></a></html>\n", "\n", "\n", "* [DonJohn94] D. L. Donoho and I. M. Johnstone, [Ideal spatial adaptation via wavelet shrinkage][1], Biometrika, vol. 81, pp. 425-455, 1994.\n", "* [DoJoKePi95] Donoho, D.L., I.M. Johnstone, G. Kerkyacharian and D. Picard, [Wavelet Shrinkage: Asymptopia][2], J. Roy. Statist. Soc. B 57 2, 301-369,1995.\n", "* [CoifDon95] R.R. Coifman and D.L. Donoho, _Translation-Invariant De-Noising_, in Wavelets and Statistics, A. Antoniadis and G. Oppenheim, Eds. San Diego, CA: Springer-Verlag, Lecture notes 1995.\n", "* [Fowler05] J. E. Fowler, [The redundant discrete wavelet transform and additive noise][3], IEEE Signal Processing Letters, vol. 12, issue 9, pp. 629-632, 2005.\n", "* [Holsch87] M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, _A real-time algorithm for signal analysis with the help of the wavelet transform_, in Wavelets: Time-Frequency Methods and Phase Space, Springer-Verlag, 1989, pp. 286 297, 1987\n", "\n", "[1]:http://dx.doi.org/10.1093/biomet/81.3.425\n", "[2]:http://www.jstor.org/stable/2345967\n", "[3]:http://dx.doi.org/10.1109/LSP.2005.853048"], "metadata": {}, "cell_type": "markdown"}], "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"}]}}}