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In biological and computational vision, the perception and description of geometric attributes of surfaces in natural scenes has received a great deal of attention. The physical and geometric properties of surfaces related to their optical characteristics depend on their texture. In previous work, we introduced the concept of the Gestalt of a surface. The Gestalt being a geometric object that retains the mathematical properties of the surface. The Gestalt is determined using the theory of Riemannian foliations from differential topology and the concept of an observer's subjective function in a probabilistic manner. In this paper, we continue our study of geometry of natural surfaces with textures that exhibit statistical regularity at some resolution. It appears that all earlier algorithms in computer vision for extraction of shape attributes from texture have made some (piecewise) smoothness assumption about the surfaces under study. However, natural surfaces, as well as most synthetic ones are not smooth in the mathematical sense. Hence, the domain of applicability of current algorithms is severely limited. We propose algorithms to estimate geometric invariants of textured surfaces that are not necessarily smooth, but possess a statistically regular structure. An important step is based on a learning theoretic method for parameterization of textured surfaces. This method takes into account the statistical texture information in a 2D image of the surface. From a dictionary of geometry for the parameter space, a supervised artificial neural network selects the optimal choice for parameterization. As a result, the algorithms for shape from texture (slant, tilt, curvature...) have a more efficient implementation and a faster runtime. In this paper, we explain the significance of statistically symmetric patterns on surfaces. We show how such texture regularity can be used to solve the linearized problem, leaving the full details of the linearization of the Gestalt of surfaces to a forthcoming paper. The solution of the linearized problem together with algorithms to linearized surface Gestalts provide the desired estimates for the geometric features of natural surfaces with statistically regular textures at some scale.
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