Analog "neuronal" networks in early vision.

Abstract

Many problems in early vision can be formulated in terms of minimizing a cost function. Examples are shape from shading, edge detection, motion analysis, structure from motion, and surface interpolation. As shown by Poggio and Koch Proc. R. Soc. London, Ser. B 226, 303-323], quadratic variational problems, an important subset of early vision tasks, can be "solved" by linear, analog electrical, or chemical networks. However, in the presence of discontinuities, the cost function is nonquadratic, raising the question of designing efficient algorithms for computing the optimal solution. Recently, Hopfield and Tank [Hopfield, J. J. & Tank, D. W. (1985) Biol. Cylern. 52,[141][142][143][144][145][146][147][148][149][150][151][152] have shown that networks of nonlinear analog "neurons" canibe effective in computing the solution ofoptimization problems. We show how these networks can be generalized to solve the nonconvex energy functionals of early vision. We illustrate this approach by implementing a specific analog network, solving the problem of reconstructing a smooth surface from sparse data while preserving its discontinuities. These results suggest a novel computational strategy for solving early vision problems in both biological and real-time artificial vision systems.This study addresses the use of simple analog networks to implement and solve problems in early vision, such as computing depth from two stereoscopic images, reconstructing and smoothing images from sparsely sampled data, and computing motion. Within the last years, computational studies have provided promising theories ofthe computations necessary for early vision (for partial reviews, see refs. 1-5). A number of early vision tasks can be described within the framework of standard regularization theory (5). Standard regularization analysis can be used to solve these problems in terms of quadratic energy functionals that must be minimized. Previous work by Poggio and Koch (6) showed how to design linear, analog networks for solving regularization problems with quadratic energy functions. The domain of applicability of standard regularization theory is limited, however, by the convexity of the energy functions, which makes it impossible to deal with problems involving true discontinuities. Such problems can be described by nonconvex energy functions involving binary line processes (7-10). More recently Marroquin (11) has proposed an approach to early vision based on the use of Markov random-field models and Bayes estimation theory (11, 33). We will show how these algorithms map naturally onto very simple resistive networks.There has been considerable interest in the computational properties and capabilities of networks of simple, neuroitallike elements (12-15). Recently, flopfield and Tank (16) have shown that analog neuronal networks can provide fast, next-to-optimal solutions to a well-characterized but difficult optimization problem, the "traveling salesman problem."' In this paper we show that networks of simple, analog, or hybrid proce...

Analysis of Retinotopic Maps in Extrastriate Cortex

“…Elongated discontinuities could be marked by "line processes" where cortical gradients went above a threshold (cf. Koch et al, 1986). The two dimensions of somatotopy (parameterized within each domain) could then be interpolated within each domain, more accurate gradients recalculated, and finally local body surface sign determined by measuring the angle between the gradient directions.…”

Response : Computing with Neural Networks

“…Hopfield's potential function provides the link between this optimization problem and its solution in terms of neural network, since in the network the potential is automatically minimized (1); and by proper choice of the network parameters, notably the neuronal interconnections, one can produce any potential function which is a polynomial of a second order (4). A new result is that, approximately, any minimization problem can be broken down into two minimization problems for second-order functionals (7), and thus can be solved by an appropriately constructed neural network.In particular for linear optimization problems this transformation into a second-order miuniization problem (8) and the corresponding network solution appear to be very natural (6,9). In general it is, however, not at all clear under which conditions this transformation of the original problem into a network computation is reasonably economic (for example, in terms of the size of the network).…”

“…In particular for linear optimization problems this transformation into a second-order miuniization problem (8) and the corresponding network solution appear to be very natural (6,9). In general it is, however, not at all clear under which conditions this transformation of the original problem into a network computation is reasonably economic (for example, in terms of the size of the network).…”

Efficient hybrid search for visual reconstruction problems

“…They can be categorized into the deterministic methods [1,18,19] and the stochastic methods [17,20]. Most of the deterministic methods can only deal with simple constraints on the line process [1,19,21]; in addition, they can not find the global minimum solution.…”

“…This is because, in general, the procedure for deriving the dual energy to eliminate the line processes in the GNC algorithm will be extremely complicated. Similarly, both the mean field annealing algorithm [19] and the analog network approaches [18,21] can only deal with specific types of constraints on the line process, for e.g. the energy term for constraining the line processes must have a simple closed-form expression.…”