
Linear Diffusion Flows

# Linear Diffusion Flows

This tours studies linear diffusion PDEs, a.k.a. the heat equation. 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.

getd('toolbox_signal/');
getd('toolbox_general/');


## Heat Diffusion

The heat equation reads $\forall t>0, \quad \pd{f_t}{t} = \nabla f_t$ for a function $$f_t : \RR^2 \rightarrow \RR$$ and where $$f_0$$ (the solution at initial time $$t=0$$) is given.

The Laplacian operator reads $\Delta f = \pdd{f}{x_1} + \pdd{f}{x_2}.$

The flow is discretized in space by considering a discrete image of $$N = n \times n$$ pixels.

n = 256;


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

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


Display it.

clf;
imageplot(f0);


The flow is discretized in time using an explicit time-stepping $f^{(\ell+1)} = f^{(\ell)} + \tau \Delta f^{(\ell)}.$ We use finite difference Laplacian $(\Delta f)_i = \frac{1}{h^2}\pa{ f_{i_1+1,i_2}+f_{i_1-1,i_2}+f_{i_1,i_2+1}+f_{i_1,i_2-1}-4f_j }$ where we assume periodic boundary conditions, and where $$h = 1/N$$ is the spacial step size.

h = 1/n;


The step size $$\tau$$ should satisfy $\tau < \frac{h^2}{4}$ for the discretized flow to be stable.

The discrete solution $$f^{(\ell)}$$ converges to the continuous solution $$f_t$$ at time $$t = \tau \ell$$ if both $$\tau \rightarrow 0$$ and $$h \rightarrow 0$$ under the condition $$\tau/h^2 < 1/4$$.

Select a small enough step size.

tau = .5 * h^2/4;


Final time.

T = 1e-3;


Number of iterations.

niter = ceil(T/tau);


Initialize the diffusion at time $$t=0$$.

f = f0;


One step of discrete diffusion.

f = f + tau * delta(f);


Exercice 1: (check the solution) Compute the solution to the heat equation.

exo1;


## Explicit Solution using Convolution

The solution to the heat equation can be computed using a convolution $\forall t>0, \quad f_t = f_0 \star h_t$ where $$\star$$ denotes the convolution of continuous functions $f \star h(x) = \int_{\RR^2} f(y) g(x-y) d y$ and $$h_t$$ is a Gaussian kernel of width $$\sqrt{t}$$ $h_t(x) = \frac{1}{4 \pi t} e^{ -\frac{\norm{x}^2}{4t} }$

One can thus approximate the solution using a discrete convolution. 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 discrete Gaussian blurring kernel of width $$\sqrt{t}$$.

t = [0:n/2 -n/2+1:-1];
[X2,X1] = meshgrid(t,t);
normalize = @(h)h/sum(h(:));
h = @(t)normalize( exp( -(X1.^2+X2.^2)/(4*t) ) );


Define blurring operator.

heat = @(f, t)cconv(f,h(t));


Example of blurring.

clf;
imageplot(heat(f0,2));


Exercice 2: (check the solution) Display the heat convolution for increasing values of $$t$$.

exo2;