{"nbformat_minor": 0, "worksheets": [{"cells": [{"source": ["1-D Haar 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 numerical tour explores 1-D multiresolution analysis with Haar\n", "transform. It was introduced in 1910 by Haar  [Haar1910](#biblio)\n", "and is arguably the first example of wavelet basis."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 2, "cell_type": "code", "language": "python", "metadata": {}, "input": ["addpath('toolbox_signal')\n", "addpath('toolbox_general')\n", "addpath('solutions/wavelet_1_haar1d')"]}, {"source": ["Forward Haar Transform\n", "----------------------\n", "The Haar transform is the simplest orthogonal wavelet transform. It is\n", "computed by iterating difference and averaging between odd and even\n", "samples of the signal.\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 = 512;"]}, {"source": ["First we load a 1-D signal $f \\in \\RR^N$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 4, "cell_type": "code", "language": "python", "metadata": {}, "input": ["name = 'piece-regular';\n", "f = rescale( load_signal(name, N) );"]}, {"source": ["The Haar transform operates over $J = \\log_2(N)-1$ scales.\n", "It computes a series of coarse scale and fine scale coefficients\n", "$a_j, d_j \\in \\RR^{N_j}$ where $N_j=2^j$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 5, "cell_type": "code", "language": "python", "metadata": {}, "input": ["J = log2(N)-1;"]}, {"source": ["The forward Haar transform computed $ \\Hh(f) = (d_j)_{j=0}^J \\cup \\{a_0\\} $.\n", "Note that the set of coarse scale coefficients $(a_j)_{j>0}$ are not\n", "stored.\n", "\n", "\n", "This transform is orthogonal, meaning $ \\Hh \\circ \\Hh^* = \\text{Id} $,\n", "and that there is the following conservation of energy\n", "$$ \\sum_i \\abs{f_i}^2 = \\norm{f}^2 = \\norm{\\Hh f}^2 = \\sum_j \\norm{d_j}^2 + \\abs{a_0}^2. $$\n", "\n", "\n", "One initilizes the algorithm with $a_{J+1}=f$. The set of coefficients\n", "$d_{j},a_j$ are computed from $a_{j+1}$ as\n", "$$ \\forall k=0,\\ldots,2^j-1, \\quad\n", "      a_j[k] = \\frac{a_{j+1}[2k] + a_{j+1}[2k+1]}{\\sqrt{2}}$\n", "  \\qandq\n", "      d_j[k] = \\frac{a_{j+1}[2k] - a_{j+1}[2k+1]}{\\sqrt{2}}. $$\n", "\n", "\n", "\n", "Shortcut to compute $a_j, d_j$ from $a_{j+1}$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 6, "cell_type": "code", "language": "python", "metadata": {}, "input": ["haar = @(a)[   a(1:2:length(a)) + a(2:2:length(a));\n", "                    a(1:2:length(a)) - a(2:2:length(a)) ]/sqrt(2);"]}, {"source": ["Display the result of the one step of the transform."], "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(f); axis('tight'); title('Signal');\n", "subplot(2,1,2);\n", "plot(haar(f)); axis('tight'); title('Transformed');"]}, {"source": ["The output of the forward transform is stored in the variable |fw|.\n", "At a given iteration indexed by a scale |j|, it will store in |fw(1:2^(j+1))|\n", "the variable $a_{j+1}$, and in the remaining entries the variable\n", "$d_{j+1},d_{j+2},\\ldots,d_J$.\n", "\n", "\n", "Initializes the algorithm at scale $j=J$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 8, "cell_type": "code", "language": "python", "metadata": {}, "input": ["j = J;"]}, {"source": ["Initialize |fw| to the full signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 9, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fw = f;"]}, {"source": ["At iteration indexed by $j$,\n", "select the sub-part of the signal containing $a_{j+1}$,\n", "and apply it the Haar operator."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 10, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fw(1:2^(j+1)) = haar(fw(1:2^(j+1)));"]}, {"source": ["Display the signal and its coarse coefficients."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 11, "cell_type": "code", "language": "python", "metadata": {}, "input": ["s = 400; t = 40; \n", "clf;\n", "subplot(2,1,1);\n", "plot(f,'.-'); axis([s-t s+t 0 1]); title('Signal (zoom)');\n", "subplot(2,1,2);\n", "plot(fw(1:2^j),'.-'); axis([(s-t)/2 (s+t)/2 min(fw(1:2^j)) max(fw(1:2^j))]); title('Averages (zoom)');"]}, {"source": ["__Exercise 1__\n", "\n", "Implement a full wavelet Haar transform that extract iteratively wavelet\n", "coefficients, by repeating these steps. Take care of choosing the\n", "correct number of steps."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 12, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo1()"]}, {"collapsed": false, "outputs": [], "prompt_number": 13, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Check that the transform is\n", "orthogonal, which means that the energy of the coefficient is the same\n", "as the energy of the signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "text": ["Should be 0: 0.000.\n"], "output_type": "display_data"}], "prompt_number": 14, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fprintf('Should be 0: %.3f.\\n', (norm(f)-norm(fw))/norm(f));"]}, {"source": ["We display the whole set of coefficients |fw|, with red vertical\n", "separator between the scales.\n", "Can you recognize where are the low frequencies and the high frequencies?"], "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", "plot_wavelet(fw);\n", "axis([1 N -2 2]);"]}, {"source": ["Backward Haar Transform\n", "-----------------------\n", "The backward transform computes a signal $f_1 = \\Hh^*(h)$ from a set of\n", "coefficients $  h = (d_j)_{j=0}^J \\cup \\{a_0\\} $\n", "\n", "\n", "If $h = \\Hh(f)$ are the coefficients of a signal $f$, one recovers\n", "$ f_1 = f $.\n", "\n", "\n", "The inverse transform starts from $j=0$, and computes $a_{j+1}$\n", "from the knowledge of $a_j,d_j$ as\n", "$$\n", "  \\forall k = 0,\\ldots,2^j-1, \\quad\n", "      a_{j+1}[2k] = \\frac{ a_j[k] + d_j[k] }{\\sqrt{2}}$\n", "      a_{j+1}[2k+1] = \\frac{ a_j[k] - d_j[k] }{\\sqrt{2}}$\n", "$$\n", "\n", "\n", "One step of inverse transform"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 16, "cell_type": "code", "language": "python", "metadata": {}, "input": ["ihaar = @(a,d)assign( zeros(2*length(a),1), ...\n", "        [1:2:2*length(a), 2:2:2*length(a)], [a+d; a-d]/sqrt(2) );"]}, {"source": ["Initialize the signal to recover $f_1$ as the transformed coefficient, and\n", "select the smallest possible scale $j=0$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 17, "cell_type": "code", "language": "python", "metadata": {}, "input": ["f1 = fw;\n", "j = 0;"]}, {"source": ["Apply one step of inverse transform."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 18, "cell_type": "code", "language": "python", "metadata": {}, "input": ["f1(1:2^(j+1)) = ihaar(f1(1:2^j), f1(2^j+1:2^(j+1)));"]}, {"source": ["__Exercise 2__\n", "\n", "Write the inverse wavelet transform that computes |f1| from the\n", "coefficients |fw|."], "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": ["exo2()"]}, {"collapsed": false, "outputs": [], "prompt_number": 20, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Check that we have correctly recovered the signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "text": ["Should be 0: 0.000.\n"], "output_type": "display_data"}], "prompt_number": 21, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fprintf('Should be 0: %.3f.\\n', (norm(f-f1))/norm(f));"]}, {"source": ["Haar Linear Approximation\n", "-------------------------\n", "Linear approximation is obtained by setting to zeros the\n", "Haar coefficient coefficients above a certain scale $j$.\n", "\n", "\n", "Cut-off scale."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 22, "cell_type": "code", "language": "python", "metadata": {}, "input": ["j = J-1;"]}, {"source": ["Set to zero fine scale coefficients."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 23, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fw1 = fw; fw1(2^j+1:end) = 0;"]}, {"source": ["Computing the signal $f_1$ corresponding to these coefficients |fw1|\n", "computes the orthogonal projection on the space of signal that are\n", "constant on disjoint intervals of length $N/2^j$."], "metadata": {}, "cell_type": "markdown"}, {"source": ["__Exercise 3__\n", "\n", "Display the reconstructed signal obtained from |fw1|, for a decreasing cut-off scale $j$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 24, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo3()"]}, {"collapsed": false, "outputs": [], "prompt_number": 25, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Haar Non-linear Approximation\n", "-----------------------------\n", "Non-linear approximation is obtained by thresholding low amplitude\n", "wavelet coefficients.\n", "It is defined as\n", "$$ f_T = \\Hh^* \\circ S_T \\circ \\Hh(f) $$\n", "where $S_T$ is the hard tresholding operator\n", "$$ S_T(x)_i = \\choice{\n", "      x_i \\qifq \\abs{x_i}>T, \\\\\n", "      0 \\quad \\text{otherwise}.\n", "  }. $$"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 26, "cell_type": "code", "language": "python", "metadata": {}, "input": ["S = @(x,T) x .* (abs(x)>T);"]}, {"source": ["Set the threshold value."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 27, "cell_type": "code", "language": "python", "metadata": {}, "input": ["T = .5;"]}, {"source": ["Threshold the coefficients."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 28, "cell_type": "code", "language": "python", "metadata": {}, "input": ["fwT = S(fw,T);"]}, {"source": ["Display the coefficients before and after thresholding."], "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_wavelet(fw); axis('tight'); title('Original coefficients');\n", "subplot(2,1,2);\n", "plot_wavelet(fwT); axis('tight'); title('Thresholded coefficients');"]}, {"source": ["__Exercise 4__\n", "\n", "Find the threshold $T$ so that the number of remaining coefficients in\n", "|fwT| is a fixed number $m$. Use this threshold to compute |fwT| and then display\n", "the corresponding approximation $f_1$ of $f$. Try for an increasing number $m$ of coeffiients.\n", "ompute the threshold T"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 30, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo4()"]}, {"collapsed": false, "outputs": [], "prompt_number": 31, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["The Shape of a Wavelet\n", "----------------------\n", "A wavelet coefficient corresponds to an inner product of $f$ with a wavelet Haar atom\n", "$\\psi_{j,k}$\n", "$$ d_j[k] = \\dotp{f}{\\psi_{j,k}} $$\n", "\n", "\n", "The wavelet $\\psi_{j,k}$ can be computed by applying the inverse wavelet\n", "transform to |fw| where |fw[m]=1| and |fw[s]=0| for $s \\neq m$\n", "where $m = 2^j+k$."], "metadata": {}, "cell_type": "markdown"}, {"source": ["__Exercise 5__\n", "\n", "Compute wavelets at several positions and scales."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 32, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo5()"]}, {"collapsed": false, "outputs": [], "prompt_number": 33, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Bibliography\n", "------------\n", "<html><a name=\"biblio\"></a></html>\n", "\n", "\n", "* [Haar1910] Haar A. [Zur Theorie der orthogonalen Funktionensysteme][1], Mathematische Annalen, 69, pp 331-371, 1910.\n", "\n", "[1]:http://dx.doi.org/10.1007/BF01456927"], "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"}]}}}