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In Vision Geometry '99 we introduced the Gestalt approach to perceptual approximation of surfaces in natural scenes; that is, as a geometric theory retaining certain mathematical properties of surfaces while adhering to the human perceptual organization of vision. The theory of curves follows the same philosophy, relying on optical properties of physical objects whose features in the scale and resolution -- imposed by the observer -- afford 'a one-dimensional Gestalt.' The Gestalt theory of curves and surfaces is part of the Perceptual Geometry of the natural world that hypothetically evolves within intelligent systems capable of retaining partial information from stimuli in 'memory' and visual 'learning' through 'brain plasticity.' Perceptual geometry aims at explaining geometry from the perspective of visual perception, and in turn, how to apply such geometric findings to the ecological study of vision. Perceptual geometry attempts to answer fundamental questions in perception of form and representation of space through synthesis of cognitive and biological theories of visual perception with geometric theories of the physical world. Algorithms in this theory are typically presented based on a combination of a mathematical formulation of eye-movements and multi-scale multi-resolution filtering. In this paper, methods from statistical pattern recognition are applied to explain the learning-theoretic and perceptual analogs of geometric theory of space and its objects optically defined by curves and surfaces. The human visual system recovers depth from the visual stimuli registered on the two-dimensional surface of the retina by means of a variety of mechanisms, from bottom-up (such as stereopsis and motion parallax) to top-down influences. Perception and representation of space and objects within the visual environment rely on combination of a multitude of such mechanisms. The problem of modeling cortical representation of visual information is, therefore, very complex and challenging.
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