Marques Pereira Dipartimento di Informatica Universita di Trento Via Inama 7, Trento, TN 38100 ITALY Abstract Learning a map from an input set to an output set is similar to the problem ofreconstructing hypersurfaces from sparse data (Poggio and Girosi, 1990). In this framework, we discuss the problem of automatically selecting "minimal"surface data. The objective is to be able to approximately reconstruct the surface from the selected sparse data. We show that this problem is equivalent to the one of compressing information by data removal andthe one oflearning how to teach. Our key step is to introduce a process that statistically selects the data according to the model.
We address the problem of optical flow reconstruction and in particular theproblem of resolving ambiguities near edges. They occur due to (i) the aperture problem and (ii) the occlusion problem, where pixels on both sides of an intensity edge are assigned the same velocity estimates (and confidence). However, these measurements are correct for just one side of the edge (the non occluded one). Our approach is to introduce an uncertamty field with respect to the estimates and confidence measures. We note that the confidence measuresare large at intensity edges and larger at the convex sides of the edges, i.e. inside corners, than at the concave side. We resolve the ambiguities through local interactions via coupled Markov random fields (MRF). The result is the detection of motion for regions of images with large global convexity.
We open our eyes and we "see" the world in all its color, brightness, and movement. Yet, we have great difficulties when trying to endow our machines with similar abilities. In this paper we shall describe recent developments in the theory of early vision which lead from the formulation of the motion problem as an illposed oneto its solution by minimizing certain "cost" functions. These cost or energy functions can be mapped onto simple analog and digital resistive networks. Thus, we shall see how the optical flow can be computed by injecting currents into resistive networks and recording the resulting stationary voltage distribution at each node. These networks can be implemented in cMOS VLSI circuits and represent plausible candidates for biological vision systems. APERTURE PROBLEM AND SMOOTHNESS ASSUMPTION In this study, we use intensity-based schemes for recovering motion.
Networks for reconstructing a sparse or noisy function often use an edge field to segment the function into homogeneous regions, This approach assumes that these regions do not overlap or have disjoint parts, which is often false. For example, images which contain regions split by an occluding objectcan't be properly reconstructed using this type of network. We have developed a network that overcomes these limitations, using support maps to represent the segmentation of a signal. In our approach, the support ofeach region in the signal is explicitly represented. Results from an initial implementation demonstrate that this method can reconstruct images and motion sequences which contain complicated occlusion.
Many of the processing tasks arising in early vision involve the sclution of ill-posed inverse problems. Two techniques that are often used to solve these inverse problems are reguhuization and Bayesian modeling. Reguhuizatien is used to find a solution that both fits the data and is also sufficiently smooth. Bayesian modeling uses a statistical prior model of the field being estimated to determine an optimal solution. One convenient way of specifying the prior model is to associate an energy function with each possible solution, and to use a Boltzmann distribution to relate the solution energy to its probability. This paper shows that regularization is an example of Bayesian modeling, and that using the regularization energy function for the surface interpolation problem results in a prior model that is fractal (self-aftine over a range of scales). We derive an algorithm for generating typical (fractal) estimates from the posterior distribution. We also show how this algorithm can be used to estimate the uncertainty associated with a regularized solution, and how this uncertainty can be used at later stages of processing. Much of the processing that occurs in the early stages of vision deals with the solution of inverse problems [Hmm, 19771.