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<!DOCTYPE html
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<title>Optimal Transport in 1-D</title>
<NOSCRIPT>
<DIV STYLE="color:#CC0000; text-align:center"><B>Warning: <A HREF="http://www.math.union.edu/locate/jsMath">jsMath</A>
requires JavaScript to process the mathematics on this page.<BR>
If your browser supports JavaScript, be sure it is enabled.</B></DIV>
<HR>
</NOSCRIPT>
<meta name="generator" content="MATLAB 8.2">
<meta name="date" content="2014-10-20">
<meta name="m-file" content="index">
<LINK REL="stylesheet" HREF="../style.css" TYPE="text/css">
</head>
<body>
<div class="content">
<h1>Optimal Transport in 1-D</h1>
<introduction>
<p>This tour details the computation of discrete 1-D optimal transport with application to grayscale image histogram manipulations.</p>
</introduction>
<h2>Contents</h2>
<div>
<ul>
<li><a href="#1">Installing toolboxes and setting up the path.</a></li>
<li><a href="#8">Optimal Transport and Assignement</a></li>
<li><a href="#13">Grayscale Image Distribution</a></li>
<li><a href="#24">1-D Optimal Assignement</a></li>
<li><a href="#36">Histogram Interpolation</a></li>
</ul>
</div>
<h2>Installing toolboxes and setting up the path.<a name="1"></a></h2>
<p>You need to download the following files: <a href="../toolbox_signal.zip">signal toolbox</a> and <a href="../toolbox_general.zip">general toolbox</a>.
</p>
<p>You need to unzip these toolboxes in your working directory, so that you have <tt>toolbox_signal</tt> and <tt>toolbox_general</tt> in your directory.
</p>
<p><b>For Scilab user:</b> you must replace the Matlab comment '%' by its Scilab counterpart '//'.
</p>
<p><b>Recommandation:</b> You should create a text file named for instance <tt>numericaltour.sce</tt> (in Scilab) or <tt>numericaltour.m</tt> (in Matlab) to write all the Scilab/Matlab command you want to execute. Then, simply run <tt>exec('numericaltour.sce');</tt> (in Scilab) or <tt>numericaltour;</tt> (in Matlab) to run the commands.
</p>
<p>Execute this line only if you are using Matlab.</p><pre class="codeinput">getd = @(p)path(p,path); <span class="comment">% scilab users must *not* execute this</span>
</pre><p>Then you can add the toolboxes to the path.</p><pre class="codeinput">getd(<span class="string">'toolbox_signal/'</span>);
getd(<span class="string">'toolbox_general/'</span>);
</pre><h2>Optimal Transport and Assignement<a name="8"></a></h2>
<p>We consider data \(f \in \RR^{N \times d}\), that can corresponds for instance to an image of \(N\) pixels, with \(d=1\) for
grayscale image and \(d=3\) for color image. We denote \(f = (f_i)_{i=1}^N\) with \(f_i \in \RR^d\) the elements of the data.
</p>
<p>The discrete (empirical) distribution in \(\RR^d\) associated to this data \(f\) is the sum of Diracs \[ \mu_f = \frac{1}{N}
\sum_{i=1}^N \de_{f_i}. \]
</p>
<p>An optimal assignement between two such vectors \(f,g \in \RR^{N \times d}\) is a permutation \(\si \in \Si_N\) that minimizes
\[ \si^\star \in \uargmin{\si \in \Si_N} \sum_{i=1}^N C(f_i,g_{\si(i)}) \] where \(C(u,v) \in \RR\) is some cost function.
</p>
<p>In the following, we consider \(L^p\) costs \[ \forall (u,v) \in \RR^d \times \RR^d, \quad C(u,v) = \norm{u-v}^p \] where
\(\norm{\cdot}\) is the Euclidean norm and \(p\geq 1\).
</p>
<p>This optimal assignement defines the \(L^p\) Wasserstein distance between the associated point clouds distributions \[ W_p(\mu_f,\mu_g)^p
= \sum_{i=1}^N \norm{f_i - g_{\si(i)}}^p = \norm{f - g \circ \si}_p^p \] where \( g \circ \si = (g_{\si(i)})_i \) is the re-ordered
points cloud.
</p>
<h2>Grayscale Image Distribution<a name="13"></a></h2>
<p>We consider here the case \(d=1\), in which case one can compute easily the optimal assignement \(\si^\star\).</p>
<p>Load an image \(f \in \RR^N\) of \(N=n \times n\) pixels.</p><pre class="codeinput">n = 256;
f = rescale( load_image(<span class="string">'lena'</span>, n) );
</pre><p>Display it.</p><pre class="codeinput">clf;
imageplot(f);
</pre><img vspace="5" hspace="5" src="index_01.png"> <p>A convenient way to visualize the distribution \(\mu_f\) is by computing an histogram \( h \in \RR^Q \) composed using \(Q\)
bins \( [u_k,u_{k+1}) \). The histogram is computed as \[ \forall k=1,\ldots,Q, \quad h(p) = \abs{\enscond{i}{ f_i \in [u_k,u_{k+1})
}}. \]
</p>
<p>Number of bins.</p><pre class="codeinput">Q = 50;
</pre><p>Compute the histogram.</p><pre class="codeinput">[h,t] = hist(f(:), Q);
</pre><p>Display this normalized histogram. To make this curve an approximation of a continuous distribution, we normalize \(h\) by
\(Q/N\).
</p><pre class="codeinput">clf;
bar(t,h*Q/n^2); axis(<span class="string">'tight'</span>);
</pre><img vspace="5" hspace="5" src="index_02.png"> <p><i>Exercice 1:</i> (<a href="../missing-exo/">check the solution</a>) Compute and display the histogram of \(f\) for an increasing number of bins.
</p><pre class="codeinput">exo1;
</pre><img vspace="5" hspace="5" src="index_03.png"> <p>Load another image \(g \in \RR^N\).</p><pre class="codeinput">g = rescale( mean(load_image(<span class="string">'fingerprint'</span>, n),3) );
</pre><p>Display it.</p><pre class="codeinput">clf;
imageplot(g);
</pre><img vspace="5" hspace="5" src="index_04.png"> <p><i>Exercice 2:</i> (<a href="../missing-exo/">check the solution</a>) Compare the two histograms.
</p><pre class="codeinput">exo2;
</pre><img vspace="5" hspace="5" src="index_05.png"> <h2>1-D Optimal Assignement<a name="24"></a></h2>
<p>For 1-D data, \(d=1\), one can compute explicitely an optimal assignement \(\si^\star \in \Si_N\) for any cost \(C(u,v) =
\phi(\abs{u-v})\) where \(\phi : \RR \rightarrow \RR\) is a convex function. This is thus the case for the \(L^p\) optimal
transport.
</p>
<p>This is obtained by computing two permutations \( \si_f, \si_g \in \Si_N \) that order the values of the data \[ f_{\si_f(1)}
\leq f_{\si_f(2)} \leq \ldots f_{\si_f(N)} \] \[ g_{\si_g(1)} \leq g_{\si_g(2)} \leq \ldots g_{\si_g(N)}. \]
</p>
<p>An optimal assignement is then optained by assigning, for each \(k\), the index \( i = \si_f(k) \) to the index \( \si_g(k)
\), i.e. \[ \si^\star = \si_g \circ \si_f^{-1}\] where \( \si_f^{-1} \) is the inverse permutation, that satisfies \[ \si_f^{-1}
\circ \si_f = \text{Id} \].
</p>
<p>Note that this optimal assignement \(\si^\star\) is not unique when there are two pixels in \(f\) or \(g\) having the same
value.
</p>
<p>Compute \(\si_f, \si_g\) in \(O(N \log(N))\) operations using a fast sorting algorithm (e.g. QuickSort).</p><pre class="codeinput">[~,sigmaf] = sort(f(:));
[~,sigmag] = sort(g(:));
</pre><p>Compute the inverse permutation \(\sigma_f^{-1}\).</p><pre class="codeinput">sigmafi = [];
sigmafi(sigmaf) = 1:n^2;
</pre><p>Compute the optimal permutation \(\sigma^\star\).</p><pre class="codeinput">sigma = sigmag(sigmafi);
</pre><p>The optimal assignement is used to compute the projection on the set of image having the pixel distribution \(\mu_g\) \[ \Hh_g
= \enscond{m \in \RR^N}{ \mu_m = \mu_g }. \] Indeed, for any \( p > 1 \), the \( L^p \) projector on this set \[ \pi_g( f
) = \uargmin{m \in \Hh_g} \norm{ f - m }_p \] is simply obtained by re-ordering the pixels of \(g\) using an optimal assignement
\(\si^\star \in \Si_N\) between \(f\) and \(g\), i.e. \[ \pi_g( f ) = g \circ \si^\star. \]
</p>
<p>This projection \(\pi_g( f )\) is called the histogram equalization of \(f\) using the histogram of \(g\)</p>
<p>Compute the projection.</p><pre class="codeinput">f1 = reshape(g(sigma), [n n]);
</pre><p>Check the new histogram.</p><pre class="codeinput">clf;
[h,t] = hist(f1(:), p);
bar(t,h*p/n^2);
</pre><img vspace="5" hspace="5" src="index_06.png"> <p>Compare before/after equalization.</p><pre class="codeinput">clf;
imageplot(f, <span class="string">'f'</span>, 1,2,1);
imageplot(f1, <span class="string">'\pi_g(f)'</span>, 1,2,2);
</pre><img vspace="5" hspace="5" src="index_07.png"> <h2>Histogram Interpolation<a name="36"></a></h2>
<p>We now introduce the linearly interpolated image \[ \forall t \in [0,1], \quad f_t = (1-t) f + t g \circ \sigma^{\star} .\]</p>
<p>One can show that the distribution \( \mu_{f_t} \) is the geodesic interpolation in the \(L^2\)-Wasserstein space between
the two distribution \(\mu_f\) (obtained for \(t=0\)) and \(\mu_g\) (obtained for \(t=1\)).
</p>
<p>One can also show that it is the barycenter between the two distributions since it has the following variational characterization
\[ \mu_{f_t} = \uargmin{\mu} (1-t)W_2(\mu_f,\mu)^2 + t W_2(\mu_g,\mu)^2 . \]
</p>
<p>Define the interpolation operator.</p><pre class="codeinput">ft = @(t)reshape( t*f1 + (1-t)*f, [n n]);
</pre><p>The midway equalization is obtained for \(t=1/2\).</p><pre class="codeinput">clf;
imageplot(ft(1/2));
</pre><img vspace="5" hspace="5" src="index_08.png"> <p><i>Exercice 3:</i> (<a href="../missing-exo/">check the solution</a>) Display the progression of the interpolation of the histograms.
</p><pre class="codeinput">exo3;
</pre><img vspace="5" hspace="5" src="index_09.png"> <p class="footer"><br>
Copyright (c) 2010 Gabriel Peyre<br></p>
</div>
<!--
##### SOURCE BEGIN #####
%% Optimal Transport in 1-D
% This tour details the computation of discrete 1-D optimal transport with
% application to grayscale image histogram manipulations.
%% Installing toolboxes and setting up the path.
%%
% You need to download the following files:
% <../toolbox_signal.zip signal toolbox> and
% <../toolbox_general.zip general toolbox>.
%%
% You need to unzip these toolboxes in your working directory, so
% that you have
% |toolbox_signal| and
% |toolbox_general|
% in your directory.
%%
% *For Scilab user:* you must replace the Matlab comment '%' by its Scilab
% counterpart '//'.
%%
% *Recommandation:* You should create a text file named for instance |numericaltour.sce| (in Scilab) or |numericaltour.m| (in Matlab) to write all the
% Scilab/Matlab command you want to execute. Then, simply run |exec('numericaltour.sce');| (in Scilab) or |numericaltour;| (in Matlab) to run the commands.
%%
% Execute this line only if you are using Matlab.
getd = @(p)path(p,path); % scilab users must *not* execute this
%%
% Then you can add the toolboxes to the path.
getd('toolbox_signal/');
getd('toolbox_general/');
%% Optimal Transport and Assignement
% We consider data \(f \in \RR^{N \times d}\), that can corresponds for
% instance to an image of \(N\) pixels, with \(d=1\) for grayscale image
% and \(d=3\) for color image. We denote \(f = (f_i)_{i=1}^N\) with \(f_i
% \in \RR^d\) the elements of the data.
%%
% The discrete (empirical) distribution in \(\RR^d\) associated to this
% data \(f\) is the sum of Diracs
% \[ \mu_f = \frac{1}{N} \sum_{i=1}^N \de_{f_i}. \]
%%
% An optimal assignement between two such vectors \(f,g \in \RR^{N \times d}\)
% is a permutation \(\si \in \Si_N\) that minimizes
% \[ \si^\star \in \uargmin{\si \in \Si_N} \sum_{i=1}^N C(f_i,g_{\si(i)}) \]
% where \(C(u,v) \in \RR\) is some cost function.
%%
% In the following, we consider \(L^p\) costs
% \[ \forall (u,v) \in \RR^d \times \RR^d, \quad C(u,v) = \norm{u-v}^p \]
% where \(\norm{\cdot}\) is the Euclidean norm and \(p\geq 1\).
%%
% This optimal assignement defines the \(L^p\) Wasserstein distance between
% the associated point clouds distributions
% \[ W_p(\mu_f,\mu_g)^p = \sum_{i=1}^N \norm{f_i - g_{\si(i)}}^p = \norm{f - g \circ \si}_p^p \]
% where \( g \circ \si = (g_{\si(i)})_i \) is the re-ordered points cloud.
%% Grayscale Image Distribution
% We consider here the case \(d=1\), in which case one can compute easily
% the optimal assignement \(\si^\star\).
%%
% Load an image \(f \in \RR^N\) of \(N=n \times n\) pixels.
n = 256;
f = rescale( load_image('lena', n) );
%%
% Display it.
clf;
imageplot(f);
%%
% A convenient way to visualize the distribution \(\mu_f\) is by computing
% an histogram \( h \in \RR^Q \) composed using \(Q\) bins \( [u_k,u_{k+1})
% \). The histogram is computed as
% \[ \forall k=1,\ldots,Q, \quad h(p) = \abs{\enscond{i}{ f_i \in [u_k,u_{k+1}) }}. \]
%%
% Number of bins.
Q = 50;
%%
% Compute the histogram.
[h,t] = hist(f(:), Q);
%%
% Display this normalized histogram.
% To make this curve an approximation of a continuous distribution, we
% normalize \(h\) by \(Q/N\).
clf;
bar(t,h*Q/n^2); axis('tight');
%%
% _Exercice 1:_ (<../missing-exo/ check the solution>)
% Compute and display the histogram of \(f\) for an increasing number of bins.
exo1;
%%
% Load another image \(g \in \RR^N\).
g = rescale( mean(load_image('fingerprint', n),3) );
%%
% Display it.
clf;
imageplot(g);
%%
% _Exercice 2:_ (<../missing-exo/ check the solution>)
% Compare the two histograms.
exo2;
%% 1-D Optimal Assignement
% For 1-D data, \(d=1\), one can compute explicitely an optimal assignement
% \(\si^\star \in \Si_N\) for any cost \(C(u,v) = \phi(\abs{u-v})\) where
% \(\phi : \RR \rightarrow \RR\) is a convex function. This is thus the case
% for the \(L^p\) optimal transport.
%%
% This is obtained by computing two permutations \( \si_f, \si_g \in \Si_N \)
% that order the values of the data
% \[ f_{\si_f(1)} \leq f_{\si_f(2)} \leq \ldots f_{\si_f(N)} \]
% \[ g_{\si_g(1)} \leq g_{\si_g(2)} \leq \ldots g_{\si_g(N)}. \]
%%
% An optimal assignement is then optained by assigning, for each \(k\),
% the index \( i = \si_f(k) \) to the index \( \si_g(k) \), i.e.
% \[ \si^\star = \si_g \circ \si_f^{-1}\]
% where \( \si_f^{-1} \) is the inverse permutation, that satisfies
% \[ \si_f^{-1} \circ \si_f = \text{Id} \].
%%
% Note that this optimal assignement \(\si^\star\) is not unique when there
% are two pixels in \(f\) or \(g\) having the same value.
%%
% Compute \(\si_f, \si_g\) in \(O(N \log(N))\) operations using a fast sorting
% algorithm (e.g. QuickSort).
[~,sigmaf] = sort(f(:));
[~,sigmag] = sort(g(:));
%%
% Compute the inverse permutation \(\sigma_f^{-1}\).
sigmafi = [];
sigmafi(sigmaf) = 1:n^2;
%%
% Compute the optimal permutation \(\sigma^\star\).
sigma = sigmag(sigmafi);
%%
% The optimal assignement is used to compute the projection on the set of
% image having the pixel distribution \(\mu_g\)
% \[ \Hh_g = \enscond{m \in \RR^N}{ \mu_m = \mu_g }. \]
% Indeed, for any \( p > 1 \), the \( L^p \) projector on this set
% \[ \pi_g( f ) = \uargmin{m \in \Hh_g} \norm{ f - m }_p \]
% is simply obtained by re-ordering the pixels of \(g\) using an optimal
% assignement \(\si^\star \in \Si_N\) between \(f\) and \(g\), i.e.
% \[ \pi_g( f ) = g \circ \si^\star. \]
%%
% This projection \(\pi_g( f )\) is called the histogram equalization of
% \(f\) using the histogram of \(g\)
%%
% Compute the projection.
f1 = reshape(g(sigma), [n n]);
%%
% Check the new histogram.
clf;
[h,t] = hist(f1(:), p);
bar(t,h*p/n^2);
%%
% Compare before/after equalization.
clf;
imageplot(f, 'f', 1,2,1);
imageplot(f1, '\pi_g(f)', 1,2,2);
%% Histogram Interpolation
% We now introduce the linearly interpolated image
% \[ \forall t \in [0,1], \quad f_t = (1-t) f + t g \circ \sigma^{\star} .\]
%%
% One can show that the distribution \( \mu_{f_t} \) is the geodesic
% interpolation in the \(L^2\)-Wasserstein space between the two distribution
% \(\mu_f\) (obtained for \(t=0\)) and \(\mu_g\) (obtained for \(t=1\)).
%%
% One can also show that it is the barycenter between the two distributions
% since it has the following variational characterization
% \[ \mu_{f_t} = \uargmin{\mu} (1-t)W_2(\mu_f,\mu)^2 + t W_2(\mu_g,\mu)^2 . \]
%%
% Define the interpolation operator.
ft = @(t)reshape( t*f1 + (1-t)*f, [n n]);
%%
% The midway equalization is obtained for \(t=1/2\).
clf;
imageplot(ft(1/2));
%%
% _Exercice 3:_ (<../missing-exo/ check the solution>)
% Display the progression of the interpolation of the histograms.
exo3;
##### SOURCE END #####
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