טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentSpira Alon
SubjectGeometric Image Evolution on Parametric Surfaces
DepartmentDepartment of Computer Science
Supervisor Professor Ron Kimmel


Abstract

The motion of curves and images in the plane has been researched extensively. Extending these motions to manifolds embedded in spaces of higher dimensions can be beneficial for many applications. This thesis presents new numerical schemes for the flow of curves and images on manifolds as well as for the flow of the manifolds themselves. These schemes are then used to create new applications in image processing and computer vision and to improve existing ones.

First, the thesis introduces an extension of the basic planar curve evolution equations to curves and images painted on manifolds. The use of geodesic curvature flow to create a geometrical scale space for images painted on manifolds is demonstrated. The combination of geodesic curvature flow and geodesic advection is used to implement geodesic active contours which take into account the geometry of the manifold and not only the edges of the image painted on it.

In the second part, the thesis presents an efficient solution to the eikonal equation on parametric manifolds. This numerical scheme is based on Kimmel and Sethian's solution of the equation for triangulated manifolds, but uses the metric tensor of the parametric manifold in order to implement the scheme on the parameterization plane. The numerical scheme is used for creating a short time kernel for the Beltrami image enhancing flow and for extending it to the joint smoothing of manifolds and the images painted on them. It is also used for optimal path planning and for simplifying the construction of an invariant face signature from distances calculated on the face manifold.