Geodesic Active Contours (GAC) is a popular image segmentation technique that uses the theory of curve evolution to find the boundaries of objects in an image. Here's a step-by-step guide to implement GAC:

The GAC algorithm is based on the concept of evolving a curve in the direction of its normal vector, with the goal of minimizing a energy functional that depends on the curve's length and its proximity to the object boundaries.

Mathematical Formulation

The GAC algorithm can be formulated as follows:

  1. Define the energy functional: $E(C) = \int_{0}^{L(C)} g(C(s)) ds$ where $C$ is the curve, $L(C)$ is its length, $s$ is the arc-length parameter, and $g$ is a function that depends on the image data.
  2. Define the evolution equation: $\frac{\partial C}{\partial t} = g(C) \kappa(C) \mathbf{n}(C) - \nabla g(C)$ where $\kappa(C)$ is the curvature of the curve, $\mathbf{n}(C)$ is the normal vector to the curve, and $\nabla g(C)$ is the gradient of the function $g$.

Implementation Steps

  1. Image Preprocessing: Apply a Gaussian filter to the input image to reduce noise and enhance the edges.
  2. Initialize the Curve: Initialize the curve $C$ as a simple shape (e.g., a circle or a rectangle) that encloses the object of interest.
  3. Compute the Energy Functional: Compute the energy functional $E(C)$ using the formula above.
  4. Evolve the Curve: Evolve the curve $C$ using the evolution equation above, with a small time step $\Delta t$.
  5. Update the Curve: Update the curve $C$ using the new positions of the curve points.
  6. Repeat Steps 3-5: Repeat steps 3-5 until convergence or a stopping criterion is reached.

Numerical Implementation

To implement the GAC algorithm numerically, you can use the following steps:

  1. Discretize the Curve: Discretize the curve $C$ into a set of points $C_i$, where $i=1,...,N$.
  2. Compute the Curvature: Compute the curvature $\kappa(C_i)$ at each point $C_i$ using a finite difference scheme.
  3. Compute the Normal Vector: Compute the normal vector $\mathbf{n}(C_i)$ at each point $C_i$ using a finite difference scheme.
  4. Compute the Gradient: Compute the gradient $\nabla g(C_i)$ at each point $C_i$ using a finite difference scheme.
  5. Update the Curve Points: Update the curve points $C_i$ using the evolution equation above, with a small time step $\Delta t$.