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Variations on Variational Principles for Vision


By Olivier Faugeras

(Research Director, INRIA, France)

Review by:

Georg Langs

Vienna University of Technology


Dr. Olivier Faugeras is  leading the INRIA ODYSEE laboratory studying biological and computer vision. A focus lies in the understanding of natural perception mechanisms, in order to take advantage of their structure in the design of computer vision algorithms. Topics range from applications like medical imaging where, for example, non-rigid multi-modal registration is a challenging but highly relevant problem, to the theory of variational approaches and partial differential equations in computer vision.


Olivier Faugeras' introduction Nature is thrifty in its resources gave the leading topic to his talk. He explored 3 examples where variational principles which are closely related to physics can contribute to elegant solutions of computer vision problems.


In shape theory, the idea of viewing curves as points on a manifold of all possible shapes enables one to handle deformations as journeys along paths on the manifold which minimize a certain energy criterion. Shapes can be compared by the Hausdorff distance or extensions like rigidified Hausdorff distance. Thereby, the  trajectory on the shape manifold can be controlled in order to deal with correct correspondences of points on the curves. It allows for the computation of empirical mean and covariance of sets of shapes or combined shape and texture information without the explicit use of landmarks. It also opens the possibility to define shape priors for image or volume segmentation. However, manually annotated landmarks remain important in improving the quality of correspondences on curves since semantic context still eludes criteria formulated purely on shapes.


In multi modal image matching, the correlation ratio and the mutual information are popular similarity measures. In a variational framework approaches for mono modal registration can be generalized to more complex multi-modal cases. 


Multi-image stereo can be formulated as an energy minimization problem, too. In a variational framework dense depth recovery and even the integration of time are tackled by back projecting the texture onto a suitable surface.


Faugeras concluded that the Euler-Lagrange equations are the fundamental equations in computer vision.  Question were concerned with the choice of the right distance function or the problem of similarities on curves. Faugeras answered that the appropriate distance function depends on the complexity, although local minima seem to be relatively independent from the choice. Singularities on curves pose a problem that has to be addressed by the careful design of a set of possible shapes.