New variational approaches in vision

Anand Rangarajan
University of Florida

Image analysis has a tradition of importing and adapting a host of classical physics based approaches including Lagrangian based variational principles and their associated Euler-Lagrange equations, Hamiltonian dynamics and Hamilton-Jacobi based methods. Approaches in image analysis do not strictly adhere to the classical mechanics sequence of i) first specifying a Lagrangian action principle, ii) deriving the corresponding Euler-Lagrange equation, iii) employing a Legendre transformation to convert the Lagrangian dynamics to a first-order Hamiltonian dynamics, and finally, iv) employing a canonical transformation to derive the Hamilton-Jacobi equation whose solution also yields a solution to the original variational problem. Instead, most research in image analysis uses a combination of one or more of these four approaches depending on the problem at hand. Despite the well known fact that most of classical mechanics is a limiting case of quantum mechanics as Planck's constant tends to zero, there is very little application of quantum mechanical principles in image analysis problems. We exploit a concrete relationship between the classical, non-linear Hamilton-Jacobi equation and the quantum, linear Schrödinger equation in order to derive computationally efficient algorithms which are applicable in image analysis problems where Hamilton-Jacobi theory is used. Examples from distance transform computation, segmentation, path planning and density estimation are showcased.

Presentation (PDF File)

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