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This paper is concerned with the multi-resolution issue used in many computer vision applications. Such approaches are very popular to optimize a cost function that, in most of the situations, has been linearized for mathematical facility reasons. In general, a multi-resolution setup consists in a redefinition of the problem at a different resolution level where the mathematical assumptions (usually linearity) hold. Following a coarseto-fine strategy, a usual process consists in 1) optimizing the large scales and 2) use this result as an initial condition for the estimation at finer scales. Such process is repeated until the plain image resolution. One of the main drawbacks of such downscaling approach is its incapacity to correct the eventual errors that have been made at larger scales. These latter are indeed propagated along the scales and disturb the final result. In this paper, we suggest a new formulation of the multi-resolution setup where we exploit some smoothing techniques issued from optimal control theory and in particular variational data assimilation. The time is here artificial and is related to the various scales we are dealing with. Following a consistent mathematical framework, we define an original downscaling/upscaling technique to perform the multi-resolution. We validate this approach by defining a simple optical flow estimation technique based on Lucas-Kanade. Experimental results on synthetic data demonstrate the efficiency of this new methodology.

OPTIMAL CONTROL THEORY FOR MULTI-RESOLUTION PROBLEMS IN COMPUTER VISION - Application to Optical-flow Estimation