Fast Marching in 2D¶

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

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 $f$.

Display the image.

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

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.

Display the metric $W$.

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

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 $1/W(x)$.