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Optic flow is a commonly used term to describe the deformation of an image by a vector field, that is a pixel-to-pixel correspondence between the original and the deformed image. In medical imaging we distinguish two main applications for optic flow, namely apparent motion detection in an image sequence or single-modality image registration. As optic flow is a mathematically ill-posed problem, the calculated vector field will not be the true vector field that brings about the deformation from the original to the deformed image. Our goal is thus to calculate the vector field that approximates the true vector field best.
From all optic flow methods, the class of variational methods shows the best performance. The general idea is to find a vector field that minimizes some energy functional. Using the calculus of variations, this leads to a coupled set of partial differential equations in the vector field components u and v, the so-called Euler equations. To solve for u and v on the image domain, we make a distinction between explicit methods that use a Gauss-Seidel type iterator to converge to the solution and implicit methods that solve the partial differential equation directly. The explicit methods are computationally less complex, but require a large number of iterations before convergence is achieved (this can be decreased substantially by using multigrid schemes), whereas the implicit method is computationally more complex, but might be solved efficiently if the system of equations is sparse.
Our approach is based on standard variational methods from literature and the work of Bart Janssen. We will elucidate both variational methods briefly. The energy functional for the standard methods contains a data term and a smoothness term. The data term is a physical constraint for the conservation of brightness, the smoothness term constraints the spatial derivatives of u and v. This actually solves part of the optic flow ambiguity by encouraging a smoothed vector field. Bart Janssen proposed a variational method that reconstructs a vector field from the motion of so-called multiscale interest points. He also imposes the smoothness constraint on the resulting vector field. A disadvantage of his method is the fact that the linear system to solve is not sparse due to the large support of the Gaussian kernels in Gaussian scale space. We improve on both methods by introducing motion of interest points on ground scale in the standard formalism. Instead of Gaussians, we can now use kernels with a much smaller support, for example B-splines.
We show how we use Mathematica to solve our variational problem on some benchmark sequences, thereby focusing on different data and smoothness terms, different sets of interest points, and the possibilities to improve our code with multigrid schemes.
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