part 5: filters

Author: mdeff

notebooks/05_filters.ipynb

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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# 5 Filters\n",
+ "\n",
+ "To filter signals on graphs, we need to define filters. They are represented in the toolbox by the [`pygsp.filters.Filter` class](https://pygsp.readthedocs.io/en/stable/reference/filters.html). Filters are usually defined in the spectral domain. Given the transfer function\n",
+ "\n",
+ "**TODO**\n",
+ "* look at <https://pygsp.readthedocs.io/en/stable/tutorials/intro.html#filters>\n",
+ "* localization as great tool to visualize filters in the vertex domain"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from pygsp import graphs, filters"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## 5.1 Heat diffusion\n",
+ "\n",
+ "**TODO**: show that this is heat diffusion"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "G1 = graphs.Sensor(seed=42)\n",
+ "G1.compute_fourier_basis()\n",
+ "G2 = graphs.Ring(N=100)\n",
+ "G2.compute_fourier_basis()\n",
+ "G2.set_coordinates('line1D')\n",
+ "\n",
+ "TAUS = [0, 5, 100]\n",
+ "DELTA = 10\n",
+ "\n",
+ "fig, axes = plt.subplots(len(TAUS), 3, figsize=(15, 6))\n",
+ "\n",
+ "for i, tau in enumerate(TAUS):\n",
+ " g1 = filters.Heat(G1, tau)\n",
+ " g2 = filters.Heat(G2, tau)\n",
+ " \n",
+ " y = g1.localize(DELTA).squeeze()\n",
+ " G1.plot_signal(y, ax=axes[i, 0])\n",
+ " axes[i, 0].set_axis_off()\n",
+ " axes[i, 0].text(0, -0.2, '$y^T L y = {:.2f}$'.format(y.T @ G1.L @ y))\n",
+ " \n",
+ " G2.plot_signal(g2.localize(G2.N//2), ax=axes[i, 2])\n",
+ " \n",
+ " g1.plot(ax=axes[i, 1])\n",
+ " axes[i, 1].set_xlabel('')\n",
+ " axes[i, 1].set_ylabel('')\n",
+ " text = r'$\\hat{{g}}(\\lambda) = \\exp \\left( \\frac{{-{{{}}} \\lambda}}{{\\lambda_{{max}}}} \\right)$'.format(tau)\n",
+ " axes[i, 1].text(6, 0.5, text, fontsize=15)\n",
+ " \n",
+ "axes[0, 0].set_title('$y = \\hat{{g}}(L) \\delta_{{{}}}$: localized on sensor'.format(DELTA))\n",
+ "axes[0, 1].set_title('$\\hat{g}(\\lambda)$: filter defined in the spectral domain')\n",
+ "axes[0, 2].set_title('$y = \\hat{{g}}(L) \\delta_{{{}}}$: localized on ring graph'.format(G2.N//2))\n",
+ "axes[-1, 1].set_xlabel(\"$\\lambda$: laplacian's eigenvalues / graph frequencies\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## 5.2 Filterbanks\n",
+ "\n",
+ "**TODO**:\n",
+ "* popular filterbanks\n",
+ "* tight vs non-tight"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "G = graphs.Ring(N=20)\n",
+ "G.estimate_lmax()\n",
+ "G.set_coordinates('line1D')\n",
+ "g = filters.HalfCosine(G)\n",
+ "s = g.localize(G.N // 2)\n",
+ "g.plot()\n",
+ "G.plot_signal(s)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## 5.3 Approximations"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**TODO**\n",
+ "* Approximation with Chebyshev polynomials.\n",
+ "* Show computational advantage.\n",
+ "* Show how it smoothes the original filterbank."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## 5.4 Exercise\n",
+ "\n",
+ "Solve the following problem using a graph filter:\n",
+ "$$\\mathbf{x}^* = \\operatorname*{arg\\,min}_{\\mathbf{x} \\in \\mathbb{R}^N} \\|\\mathbf{y} - \\mathbf{x}\\|_2^2 + \\alpha \\mathbf{x}^\\intercal \\mathbf{L} \\mathbf{x},$$\n",
+ "where $y$ is the observed signal, $\\alpha$ is an hyper-parameter which controls the trade-off between the data fidelity term and the smoothness prior."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Your code here."
+ ]
+ }
+ ],
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+ "language": "python",
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