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%GSP_DEMO_WAVELET Introduction to spectral graph wavelet with the GSPBox
%
% The wavelets are a special type of filterbank. In this demo, we will
% show how you can very easily construct a wavelet frame and apply it to
% a signal. If you want to do find an interactive demo of the wavelet, we
% encourage you to use the sgwt_demo2 of the sgwt toolbox. It can be
% downloaded at:
%
% http://wiki.epfl.ch/sgwt/documents/sgwt_toolbox-1.02.zip
%
% The sgwt toolbox has the same core as the GSPBox and all his functions
% have equivalent in the GSPBox ( Except the demos ;-) ).
%
% In this demo we will show you how to compute the wavelet coefficients
% of a graph and visualize them. First, let's load a graph :
%
% G = gsp_bunny();
%
% This graph is a nearest-neighbor graph of a pointcloud of the Stanford
% bunny. It will allow us to get interesting visual results using
% wavelets.
%
% At this stage we could compute the full Fourier basis using
% GSP_COMPUTE_FOURIER_BASIS, but this would take a lot of time, and can
% be avoided by using Chebychev polynomials approximations. This operation
% is implemented in most function and is thus completely transparent.
%
% Simple filtering
% ----------------
%
% Before tackling wavelets, we can see the effect of one filter localized
% on the graph. So we can first design a few heat kernel filters :
%
% taus = [1, 10, 25, 50];
% Hk = gsp_design_heat(G, taus);
%
% Let's now create a signal as a Kronecker located on one vertex (e.g.
% the vertex 100) :
%
% S = zeros(G.N, 1);
% vertex_delta = 83;
% S(vertex_delta) = 1;
%
% Sf_vec = gsp_filter_analysis(G, Hk, S);
% Sf = gsp_vec2mat(Sf_vec, length(taus));
%
% and plot the filtered signal :
%
% param_plot.cp = [0.1223, -0.3828, 12.3666];
%
% figure;
% subplot(221)
% gsp_plot_signal(G,Sf(:,1), param_plot);
% axis square
% title(sprintf('Heat diffusion tau = %d', taus(1)));
% subplot(222)
% gsp_plot_signal(G,Sf(:,2), param_plot);
% axis square
% title(sprintf('Heat diffusion tau = %d', taus(2)));
% subplot(223)
% gsp_plot_signal(G,Sf(:,3), param_plot);
% axis square
% title(sprintf('Heat diffusion tau = %d', taus(3)));
% subplot(224)
% gsp_plot_signal(G,Sf(:,4), param_plot);
% axis square
% title(sprintf('Heat diffusion tau = %d', taus(4)));
%
% Figure 1: Heat diffusion at different scales
%
%
% Visualizing wavelets atoms
% --------------------------
%
% Let's now replace the filtering by the heat kernel by a filter bank of
% wavelets. We can create a filter bank using one of the design functions
% such as GSP_DESIGN_MEXICAN_HAT :
%
% Nf = 6;
% Wk = gsp_design_mexican_hat(G, Nf);
%
% We can plot the filter bank spectrum :
%
% figure;
% gsp_plot_filter(G,Wk);
%
% Figure 2: Wavelets filterbank (Original)
%
%
% As we can see, the wavelets atoms are stacked on the low frequency part
% of the spectrum. If we want to get a better coverage of the graph
% spectrum we can use the function GSP_DESIGN_WARPED_TRANSLATES :
%
% param_filter.filter = Wk;
% Wkw = gsp_design_warped_translates(G,Nf,param_filter);
%
% Now let's plot the new filter bank :
%
% figure;
% gsp_plot_filter(G,Wkw);
%
% Figure 3: Wavelet filterbank (spectrum adaptated)
%
%
% We can see that the wavelet atoms are much more spread along the graph
% spectrum. We can visualize the filtering by one atom as we did with the
% heat kernel, by placing a Kronecker delta at one specific vertex and
% filter using the filter bank :
%
% S = zeros(G.N*Nf,Nf);
% S(vertex_delta) = 1;
% for ii=1:Nf
% S(vertex_delta+(ii-1)*G.N,ii) = 1;
% end
%
% Sf = gsp_filter_synthesis(G,Wkw,S);
%
% We can plot the resulting signal for the different scales :
%
% figure;
% subplot(221)
% gsp_plot_signal(G,Sf(:,1), param_plot);
% axis square
% mu = mean(Sf(:,1));
% sigma = std(Sf(:,1));
% c_scale = 4;
% caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
% title('Wavelet 1');
%
% subplot(222)
% gsp_plot_signal(G,Sf(:,2), param_plot);
% axis square
% mu = mean(Sf(:,2));
% sigma = std(Sf(:,2));
% caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
% title('Wavelet 2');
%
% subplot(223)
% gsp_plot_signal(G,Sf(:,3), param_plot);
% axis square
% mu = mean(Sf(:,3));
% sigma = std(Sf(:,3));
% caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
% title('Wavelet 3');
%
% subplot(224)
% gsp_plot_signal(G,Sf(:,4), param_plot);
% axis square
% mu = mean(Sf(:,4));
% sigma = std(Sf(:,4));
% caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
% title('Wavelet 4');
%
% Figure 4: A few wavelets atoms
%
%
% Curvature estimation
% --------------------
%
% As a last and more applied example, let us try to estimate the
% curvature of the underlying 3D model by only using only spectral
% filtering on the graph.
%
% A simple way to accomplish that is to use the
% coordinates map [x, y, z] and filter it using the wavelets defined
% above. We obtain a 3-dimensional signal [hat(x), hat(y), hat(z)]*
% which describes variation along the 3 coordinates :
%
% s_map = G.coords;
% s_map_out = gsp_filter_analysis(G, Wk, s_map);
% s_map_out = gsp_vec2mat(s_map_out, Nf);
%
% Finally we can get the curvature estimation by taking the l_1 or
% l_2 norm of the filtered signal :
%
% dd = s_map_out(:,:,1).^2 + s_map_out(:,:,2).^2 + s_map_out(:,:,3).^2;
% dd = sqrt(dd);
%
% Let's now plot the result to observe that we indeed have a measure of
% the curvature :
%
% figure;
% subplot(221)
% gsp_plot_signal(G,dd(:,2), param_plot);
% axis square
% title('Curvature estimation scale 1');
% subplot(222)
% gsp_plot_signal(G,dd(:,3), param_plot);
% axis square
% title('Curvature estimation scale 2');
% subplot(223)
% gsp_plot_signal(G,dd(:,4), param_plot);
% axis square
% title('Curvature estimation scale 3');
% subplot(224)
% gsp_plot_signal(G,dd(:,5), param_plot);
% axis square
% title('Curvature estimation scale 4');
%
% Figure 5: Curvature estimation using wavelet feature
%
%
%
% Url: http://lts2research.epfl.ch/gsp/doc/demos/gsp_demo_wavelet.php
% Copyright (C) 2013-2016 Nathanael Perraudin, Johan Paratte, David I Shuman.
% This file is part of GSPbox version 0.7.0
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% If you use this toolbox please kindly cite
% N. Perraudin, J. Paratte, D. Shuman, V. Kalofolias, P. Vandergheynst,
% and D. K. Hammond. GSPBOX: A toolbox for signal processing on graphs.
% ArXiv e-prints, Aug. 2014.
% http://arxiv.org/abs/1408.5781
% Author: Johan Paratte
% Date : 21 August 2014
%% Initialization
clear;
close all;
%% Load the graph of the bunny
G = gsp_bunny();
G = gsp_estimate_lmax(G);
%% Heat kernel
taus = [10, 15, 20, 30];
Hk = gsp_design_heat(G, taus);
S = zeros(G.N, 1);
vertex_delta = 83;
S(vertex_delta) = 1;
Sf_vec = gsp_filter_analysis(G, Hk, S);
Sf = gsp_vec2mat(Sf_vec, length(taus));
param_plot.cp = [0.1223, -0.3828, 12.3666];
subplot(221)
gsp_plot_signal(G,Sf(:,1), param_plot);
axis square
title(sprintf('Heat diffusion tau = %d', taus(1)));
subplot(222)
gsp_plot_signal(G,Sf(:,2), param_plot);
axis square
title(sprintf('Heat diffusion tau = %d', taus(2)));
subplot(223)
gsp_plot_signal(G,Sf(:,3), param_plot);
axis square
title(sprintf('Heat diffusion tau = %d', taus(3)));
subplot(224)
gsp_plot_signal(G,Sf(:,4), param_plot);
axis square
title(sprintf('Heat diffusion tau = %d', taus(4)));
%% Wavelets
Nf = 6;
Wk = gsp_design_mexican_hat(G, Nf);
figure;
gsp_plot_filter(G,Wk);
param_filter.filter = Wk;
Wkw = gsp_design_warped_translates(G,Nf,param_filter);
figure;
gsp_plot_filter(G,Wkw);
S = zeros(G.N*Nf,Nf);
S(vertex_delta) = 1;
for ii=1:Nf
S(vertex_delta+(ii-1)*G.N,ii) = 1;
end
Sf = gsp_filter_synthesis(G,Wk,S);
figure;
subplot(221)
gsp_plot_signal(G,Sf(:,1), param_plot);
axis square
mu = mean(Sf(:,1));
sigma = std(Sf(:,1));
c_scale = 4;
caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
title('Wavelet 1');
subplot(222)
gsp_plot_signal(G,Sf(:,2), param_plot);
axis square
mu = mean(Sf(:,2));
sigma = std(Sf(:,2));
caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
title('Wavelet 2');
subplot(223)
gsp_plot_signal(G,Sf(:,3), param_plot);
axis square
mu = mean(Sf(:,3));
sigma = std(Sf(:,3));
caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
title('Wavelet 3');
subplot(224)
gsp_plot_signal(G,Sf(:,4), param_plot);
axis square
mu = mean(Sf(:,4));
sigma = std(Sf(:,4));
caxis([mu - c_scale*sigma, mu + c_scale*sigma]);
title('Wavelet 4');
%% Curvature estimation
s_map = G.coords;
s_map_out = gsp_filter_analysis(G, Wk, s_map);
s_map_out = gsp_vec2mat(s_map_out, Nf);
dd = s_map_out(:,:,1).^2 + s_map_out(:,:,2).^2 + s_map_out(:,:,3).^2;
dd = sqrt(dd);
figure;
subplot(221)
gsp_plot_signal(G,dd(:,2), param_plot);
axis square
title('Curvature estimation scale 1');
subplot(222)
gsp_plot_signal(G,dd(:,3), param_plot);
axis square
title('Curvature estimation scale 2');
subplot(223)
gsp_plot_signal(G,dd(:,4), param_plot);
axis square
title('Curvature estimation scale 3');
subplot(224)
gsp_plot_signal(G,dd(:,5), param_plot);
axis square
title('Curvature estimation scale 4');
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