Active Contours using Level Sets¶

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This tour explores image segementation using level set methods.

Managing level set functions¶

In the level set formalism, the evolution of some curve $ (\ga(t))_{t=0}^1 $ is computed by evolving the zero level of a function $\phi : \RR^2 \rightarrow \RR $ $$ \enscond{\ga(s)}{ s \in [0,1] } = \enscond{x \in \RR^2}{\phi(x)=0}. $$ This corresponds to replacing the parameteric representation $\ga$ of the curve by an implicit representation. This requires an additional dimension (and hence more storage) but ease the handling of topological change of the curve during the evolution.

Discretazion size $n \times n$ of the domain $[0,1]^2$.

One can create a circular shape by using the signed distance function to a circle $$ \phi_1(x) = \sqrt{ (x_1-c_1)^2 + (x_2-c_2)^2 } - r $$ where $r>0$ is the radius and $c \in \RR^2$ the center.

Radius $r$.

Center $c$.

Distance function $\phi_1$.

Exercise 1

Load a square shape $\phi_2$ at a different position for the center.

Display the curves associated to $\phi_1$ and $\phi_2$.

Exercise 2

Compute the intersection and the union of the two shapes. Store the union in $\phi_0$ (phi0) that we will use in the remaining part of the tour.

Mean Curvature Motion.¶

The mean curvature motion corresponds to the minimizing flow of the length of the curve $$ \int_0^1 \norm{\ga'(s)} d s. $$

It is implemeted in a level set formalism by a familly $\phi_t$ of level set function parameterized by an artificial time $t \geq 0$, that satisfies the following PDE $$ \pd{\phi_t}{t} = -G(\phi_t) \qwhereq G(\phi) = -\norm{\nabla \phi} \text{div} \pa{ \frac{\nabla \phi}{\norm{\nabla \phi}} } $$ and where $\nabla \phi_t(x) \in \RR^2$ is the spacial gradient.

This flow is computed using a gradient descent $\phi^{(0)} = \phi_0$ and $$ \phi^{(\ell+1)} = \phi^{(\ell)} - \tau G(\phi^{(\ell)}), $$ where $\tau>0$ is small enough time step.

Maximum time of the evolution $0 \leq t \leq t_{\max}$.

Time step $\tau>0$ (should be small).

Number of iterations.

Initial shape $\phi^{(0)}$ at $t=0$.

We now compute the right hand side of the evolution equation.

Compute the gradient $\nabla \phi$. We use centered differences for the discretization of the gradient.

Norm $\norm{\nabla \phi}$ of the gradient.

Normalized gradient.

The curvature term.

Perform one step of the gradient descent.

Exercise 3

Implement the mean curvature motion.

Edge-based Segmentation with Geodesic Active Contour¶

Geodesic active contours compute loval minimum of a weighted geodesic distance that attract the curve toward the features of the background image.

Note: these active contours should not be confounded with the geodesic shortest paths, that are globally minimizing geodesics between two points. Here the active contour is a close curve progressively decreasing a weighted geodesic length that is only a local minimum (the global minimum would be a single point).

Size of the image.

First we load an image $f_0 \in \RR^{n \times n}$ to segment.

Given a background image $f_0$ to segment, one needs to compute an edge-stopping function $W$. It should be small in area of high gradient, and high in area of large gradient.

We use here $$ W(x) = \al + \frac{\be}{\epsilon + d(x) } \qwhereq d = \norm{\nabla f_0} \star h_a, $$ and where $h_a$ is a blurring kernel of size $a>0$.

Compute the magnitude of the gradient $d_0(x) = \norm{\nabla f_0(x)}$.

Blur size $a$.

Compute the blurring $d = d_0 \star h_a$.

Parameter $\epsilon>0$.

We set the $\al$ and $\be$ parameters to adjust the overall values of $W$ (equivalently we use the function rescale).

Display it.

Exercise 4

Compute an initial shape $\phi_0$ at time $t=0$, for instance a centered square.

Display it.

The geodesic active contour minimizes a weighted length of curve $$ \umin{\ga} \int_0^1 \norm{\ga'(s)} W(\ga(s)) d s $$

The level set implementation of the gradient descent of this energy reads $$ \pd{\phi_t}{t} = G(\phi_t) \qwhereq G(\phi) = -\norm{\nabla \phi} \text{div}\pa{ W \frac{\nabla \phi}{\norm{\nabla \phi}} } $$

This is implemented using a gradient descent scheme. $$ \phi^{(\ell+1)} = \phi^{(\ell)} - \tau G(\phi^{(\ell)}), $$ where $\tau>0$ is small enough.

Gradient step size $\tau>0$.

Final time and number of iteration of the algorithm.

Initial distance function $\phi^{(0)}=\phi_0$.

Note that we can re-write the gradient of the energy as $$ G(\phi) = -W \norm{\nabla \phi} \text{div} \pa{ \frac{\nabla \phi}{\norm{\nabla \phi}} } - \dotp{\nabla W}{\nabla \phi} $$

Pre-compute once for all $\nabla W$.

Exercise 5

Compute and store in G the gradient $G(\phi)$ (right hand side of the PDE) using the current value of the distance function $\phi$.

Do the descent step.

Exercise 6

Implement the geodesic active contours gradient descent.