A new numerical method for level set motion in normal direction used in optical flow estimation

Abstract

<p style='text-indent:20px;'>We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.