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Non-Linear Diffusion Flows

Non-Linear Diffusion Flows

This tours details non-linear diffusion PDEs. A good reference for diffusion flows in image processing is [Weickert98].


Installing toolboxes and setting up the path.

You need to download the following files: signal toolbox and 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.


Non-linear Second-order Parabolic PDEs

This tour defines PDE flows that are non-linear extension of the heat equation. Non-linearity is crucial to produce edge-aware flows that do not blur the edges.

It is also important to produce contrast invariant and affine invariant flows.

These flows can be written as \[ \pd{f_t}{t}(x) = G(\nabla f_t(x), \nabla^2 f_t(x)). \] with \(f_0\) defined at time \(t=0\) (whith the slight modification that a blurring is introduced in the Perona-Malick flow to stabilize it).

They are discretized in space by considering a discrete image of \(N = n \times n\) pixels.

n = 256;

We use finite difference operators \(\nabla\) and \(\text{div}=-\nabla^*\) with periodic boundary conditions.

Load an image \(f_0 \in \RR^N\), that will be used to initialize the flow at time \(t=0\).

name = 'hibiscus';
f0 = load_image(name,n);
f0 = rescale( sum(f0,3) );

Display it.


The flow is discretized in time using an explicit time-stepping \[ f^{(\ell+1)}(x) = f^{(\ell)}(x) + \tau G(\nabla f^{(\ell)}(x), \nabla^2 f^{(\ell)}(x)). \] Here \(\tau>0\) should be small enough, and \(f^{(\ell)}\) produces an approximation of \(f_t\) at time \(t=\ell\tau\).

Convolutions can be computed in \(O(N\log(N))\) operations using the FFT, since \[ g = f \star h \qarrq \forall \om, \quad \hat g(\om) = \hat f(\om) \hat h(\om). \]

cconv = @(f,h)real(ifft2(fft2(f).*fft2(h)));

Define a Gaussian blurring kernel of width \(\si\): \[ h_\si(x) = \frac{1}{Z} e^{ -\frac{x_1^2+x_2^2}{2\si^2} }\] where \(Z\) ensures that \(\hat h_\si(0)=1\).

t = [0:n/2 -n/2+1:-1];
[X2,X1] = meshgrid(t,t);
normalize = @(h)h/sum(h(:));
h = @(sigma)normalize( exp( -(X1.^2+X2.^2)/(2*sigma^2) ) );

Set the value of \(\sigma>0\).

sigma = .5;

Define blurring operator.

blur = @(f)cconv(f,h(sigma));

Perona-Malik Flow

The Perona-Malik non-linear diffusion is defined as \[ \pd{f_t}{t} = \text{div}\pa{ g_\la( \norm{\nabla f_t^\si} ) \nabla f_t } \] where \(f^\si = f \star h_\si\).

This PDE was introduced in [PerMal90].

Here, \(g_\la : \RR^+ \rightarrow \RR^+\) is a non-increasing function, that wechose in the following as \[ g_\la(s) = \frac{1}{\sqrt{1 + (s/\la)^2}}. \]

g = @(s,lambda)1./sqrt( 1+(s/lambda).^2 );

Note that in the limit \(\la \rightarrow +\infty\), one recovers the linear heat equation \[ \pd{f_t}{t} = \Delta f_t \] where \(\Delta=\text{div} \circ \nabla\) is the Laplacian.

Define \(A(f) = \norm{\nabla f^\si}\).

amplitude = @(u)repmat( sqrt( sum(u.^2,3) ), [1 1 2]);
A = @(f)amplitude(grad(blur(f)));

Initialize the solution at time \(t=0\).

f = f0;

Set the value of \(\lambda\).

lambda = .01;

Set the value of the descent step size \(\tau>0\).

tau = .2;

Perform one time stepping.

f = f + tau * div( g(A(f),lambda) .* grad(f) );

Final time.

T = .5/lambda;

Number of iteration to reach this final time.

niter = ceil(T/tau);

Exercice 1: (check the solution) Implement the Perona-Malick diffusion flow for \(\la = 10^{-2}\).


Exercice 2: (check the solution) Implement the Perona-Malick diffusion flow for \(\la = 10^{-3}\).


Mean Curvature Flow

In the limit that \(\la \rightarrow +\infty\) and \(\si \rightarrow 0\) the Perona-Malick flow becomes \[ \pd{f_t}{t} = \text{curv}(f_t) \] where \[ \text{curv}(f) = \text{div}\pa{ \frac{\nabla f}{\norm{\nabla f}} }. \] One can show that \(\text{curv}(f)(x)\) is the curvature at location \(x\) of the level set \(\enscond{y}{f(y)=f(x)}\).

This flow is the gradient descent flow associated to the total variation \[ J(f) = \int_{\RR^2} \norm{\nabla f(x)} d x, \] which can be extended to non-smooth functions of bounded variations. Indeed, a (sub)gradient of \(J\) is \( -\text{curv}(f) \).

Total variation regularization was introcued in []

A closely related flow is the so-called mean curvature flow \[ \pd{f_t}{t} = \norm{\nabla f_t} \text{curv}(f_t). \] One can show that this flow is contrast-invariant. This means that for any non-decreasing function \(\phi : \RR \rightarrow \RR\), \(\phi \circ f_t\) is also a solution of the PDE (possibly up to a re-parameterization of the time variable).

One can show that any contrast-invariant flow can be written as \[ \pd{f_t}{t} = \norm{\nabla f_t} \psi( \text{curv}(f_t) ) \] for a non-decreasing function \(\psi : \RR \rightarrow \RR\).

Implement the curv operator. We use a small \(\epsilon\) to avoid division by 0.

epsilon = 1e-6;
amplitude = @(u)sqrt(sum(u.^2,3)+epsilon^2);
normalize = @(u)u./repmat( amplitude(u), [1 1 2]);
curv = @(f)div( normalize(grad(f)) );

Exercice 3: (check the solution) Implement the mean curvature flow.


Affine Invariant Flow

A flow is affine invariant if \(f_t \circ A\) is also a solution of the PDE (possibly up to a re-parameterization of the time variable).

The only affine invariant and contrast invariant flow is \[ \pd{f_t}{t} = \norm{\nabla f_t} \text{curv}(f_t)^{1/3}. \] where \(s^{1/3}= \text{sign}(s) \abs{s}^{1/3}\).

This result was discovered independently in [AlvGuiLiMo93] and [SapTann93]

Exercice 4: (check the solution) Implement the affine-invariant curvature flow.