Fast Marching in 2D

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This tour explores the use of Fast Marching methods in 2-D.

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warning off
warning on

Shortest Path for Isotropic Metrics

Shortest paths are 2D curves that minimize a weighted length according to a given metric $W(x)$ for $x \in [0,1]^2$. The metric is usually computed from an input image $f(x)$.

The length of a curve $ t \in [0,1] \mapsto \gamma(t) \in [0,1]^2 $ is $$ L(\gamma) = \int_0^1 W(\gamma(t)) \norm{\gamma'(t)} \text{d} t. $$

Note that $L(\gamma)$ is invariant under re-parameterization of the curve $\gamma$.

A geodesic curve $\gamma$ between two points $x_0$ and $x_1$ has minimum length among curves joining $x_0$ and $x_1$, $$ \umin{\ga(0)=x_0, \ga(1)=x_1} L(\ga). $$ A shortest curve thus tends to pass in areas where $W$ is small.

The geodesic distance between the two points is then $d(x_0,x_1)=L(\gamma)$ is the geodesic distance according to the metric $W$.

Pixel values-based Geodesic Metric

The geodesic distance map $D(x)=d(x_0,x)$ to a fixed starting point $x_0$ is the unique viscosity solution of the Eikonal equation $$ \norm{ \nabla D(x)} = W(x) \qandq D(x_0)=0. $$

This equation can be solved numerically in $O(N \log(N))$ operation on a discrete grid of $N$ points.

We load the input image $f$.

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clear options;
n = 300;
name = 'road2';
f = rescale( load_image(name, n) );

Display the image.

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Define start and end points $x_0$ and $x_1$ (note that you can use your own points).

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x0 = [14;161];
x1 = [293;148];

The metric is defined according to $f$ in order to be low at pixel whose value is close to $f(x)$. A typical example is $$ W(x) = \epsilon + \abs{f(x_0)-f(x)} $$ where the value of $ \epsilon>0 $ should be increased in order to obtain smoother paths.

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epsilon = 1e-2;
W = epsilon + abs(f-f(x0(1),x0(2)));

Display the metric $W$.

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Set options for the propagation: infinite number of iterations, and stop when the front hits the end point.

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options.nb_iter_max = Inf;
options.end_points = x1;

Perform the propagation, so that $D(a,b)$ is the geodesic distance between the pixel $x_1=(a,b)$ and the starting point $x_0$. Note that the function |perform_fast_marching| takes as input the inverse of the metric $1/W(x)$.

In [9]:
[D,S] = perform_fast_marching(1./W, x0, options);

Display the propagated distance map $D$. We display in color the distance map in areas where the front has propagated, and leave in black and white the area where the front did not propagate.

In [10]:
hold on;
imageplot( convert_distance_color(D,f) );
h = plot(x0(2),x0(1), '.r'); set(h, 'MarkerSize', 25);
h = plot(x1(2),x1(1), '.b'); set(h, 'MarkerSize', 25);

Exercise 1

Using |options.nb_iter_max|, display the progressive propagation. This corresponds to displaying the front $ \enscond{x}{D(x) \leq t} $ for various arrival times $t$.

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In [12]:
%% Insert your code here.

Geodesic Curve Extraction

Once the geodesic distance map $D(x)$ to a starting point $x_0$ is computed, the geodesic curve between any point $x_1$ and $x_0$ extracted through gradient descent $$ \ga'(t) = - \eta_t \nabla D(\ga(t)), $$ where $\eta_t>0$ controls the parameterization speed of the resulting curve. To obtain unit speed parameterization, one can use $\eta_t = \norm{\nabla D(\ga(t))}^{-1}$.

Recompute the geodesic distance map $D$ on the whole grid.

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options.nb_iter_max = Inf;
options.end_points = [];
[D,S] = perform_fast_marching(1./W, x0, options);

Display $D$.

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colormap jet(256);

Compute the gradient $G_0(x) = \nabla D(x) \in \RR^2$ of the distance map. Use centered differences.

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options.order = 2;
G0 = grad(D, options);

Normalize the gradient to obtained $G(x) = G_0(x)/\norm{G_0(x)}$, in order to have unit speed geodesic curve (parameterized by arc length).

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G = G0 ./ repmat( sqrt( sum(G0.^2, 3) ), [1 1 2]);

Display $G$.

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colormap jet(256);

The geodesic is then numerically computed using a discretized gradient descent, which defines a discret curve $ (\ga_k)_k $ using $$ \ga_{k+1} = \ga_k - \tau G(\ga_k) $$ where $\ga_k \in \RR^2$ is an approximation of $\ga(t)$ at time $t=k\tau$, and the step size $\tau>0$ should be small enough.

Step size $\tau$ for the gradient descent.

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tau = .8;

Initialize the path with the ending point.

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gamma = x1;

Define a shortcut to interpolate $G$ at a 2-D points. Warning: the |interp2| switches the role of the axis ...

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Geval = @(G,x)[interp2(1:n,1:n,G(:,:,1),x(2),x(1)); ...
             interp2(1:n,1:n,G(:,:,2),x(2),x(1)) ];

Compute the gradient at the last point in the path, using interpolation.

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g = Geval(G, gamma(:,end));

Perform the descent and add the new point to the path.

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gamma(:,end+1) = gamma(:,end) - tau*g;

Exercise 2

Perform the full geodesic path extraction by iterating the gradient descent. You must be very careful when the path become close to $x_0$, because the distance function is not differentiable at this point. You must stop the iteration when the path is close to $x_0$.

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%% Insert your code here.

Display the curve on the image background.

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clf; hold on;
h = plot(gamma(2,:),gamma(1,:), '.b'); set(h, 'LineWidth', 2);
h = plot(x0(2),x0(1), '.r'); set(h, 'MarkerSize', 25);
h = plot(x1(2),x1(1), '.b'); set(h, 'MarkerSize', 25);
axis ij;