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Vision problems are inherently ambiguous: Do abrupt brightness changes correspond to object boundaries? Are smooth intensity changes due to shading or material properties? For stereo: Which point in the left image corresponds to which point in the right one? What is the role of color in visual information processing? To answer these (seemingly different) questions we develop an analogy between the role of orientation in organizing visual cortex and tangents in differential geometry. Machine learning experiments suggest using geometry as a surrogate for high-order statistical interactions. The cortical columnar architecture becomes a bundle structure in geometry. Connection forms within these bundles suggest answers to the above questions, and curvatures emerge in key roles. More generally, our path through these questions suggests an overall strategy for solving the inverse problems of vision: decompose the global problems into networks of smaller ones and then seek constraints from these coupled problems to reduce ambiguity. Neural computations thus amount to satisfying constraints rather than seeking uniform approximations. Even when no global formulation exists one may be able to find localized structures on which ambiguity is minimal; these can then anchor an overall approximation. [ABSTRACT FROM PUBLISHER]

Standard approaches to stereo correspondence have difficulty when scene structure does not lie in or near the frontal parallel plane, in part because an orientation disparity as well as a positional disparity is introduced. We propose a correspondence algorithm based on differential geometry, that takes explicit advantage of both disparities. The algorithm relates the 2D differential structure (position, tangent, and curvature) of curves in the left and right images to the Frenet approximation of the (3D) space curve. A compatibility function is defined via transport of the Frenet frames, and they are matched by relaxing this compatibility function on overlapping neighborhoods along the curve. The remaining false matches are concurrently eliminated by a model of “near” and “far” neurons derived from neurobiology. Examples on scenes with complex 3D structures are provided. [ABSTRACT FROM AUTHOR]

Texture segregation has long been attributed to changes in the distribution of elementary features across the visual field [Nature 290 (12) (1981) 91; Biol. Cybernet. 54 (1986) 245]. The study of orientation, a conspicuous feature, has led to models of orientation-based texture segmentation (OBTS) that depend on the magnitude of one or two orientation gradients [Vis. Res. 31 (4) (1991) 679; Vis. Res. 31 (6) (1991) 1073] and influenced further by the relative configuration between the orientation textons and the global orientation edge [Percept. Psychophys. 52 (4) (1992) 255; Vis. Res. 35 (20) (1995) 2863]. Here we show that these models are at best partial and that the notion of orientation gradient has been incompletely used in the study of OBTS. To do so, we first study the behavior of orientation in orientation-defined texture patches. Geometrical analysis identifies two texture curvatures and reveals the incompleteness of previous stimuli. Psychophysical experimentation then demonstrates that segmentation is strongly affected by discontinuities in these curvatures. Importantly, we show that this sensitivity to curvature is independent of the orientation gradients and inconsistent with the simple configural considerations proposed in the past. [Copyright &y& Elsevier]

Locally parallel dense patterns—sometimes called texture flows—define a perceptually coherent structure of particular significance to perceptual organization. We argue that with applications ranging from image segmentation and edge classification to shading analysis and shape interpretation, texture flows deserve attention equal to edge segment grouping and curve completion. This paper develops the notion of texture flow from a geometrical point of view to argue that local measurements of such structures must incorporate two curvatures. We show how basic theoretical considerations lead to a unique model for the local behavior of the flow and to a notion of texture flow "good continuation." This, in turn, translates to a specification of consistency constraints between nearby flow measurements which we use for the computation of globally (piecewise) coherent structure through the contextual framework of relaxation labeling. We demonstrate the results on synthetic and natural images. [ABSTRACT FROM AUTHOR]

While complaints about typical edge operators are common, proposals articulating a notion of the "perfect" edge map are comparatively rare, hindering the improvement of contour enhancement techniques. To address this situation, we suggest that one objective of visual contour computation is the estimation of a clean sketch from a corrupted rendition, the latter modeling noisy and low contrast edge or line operator responses to an image. Our formal model of this clean sketch is the curve indicator random field (CIRF), whose role is to provide a basis for defining edge likelihood models by eliminating the parameter along each curve to create an image of curves. For curves modeled with stationary Markov processes, this ideal edge prior is non-Gaussian and its moment generating functional has a form closely related to the Feynman-Kac formula. This sketch model leads to a nonlinear, minimum mean squared error contour enhancement filter that requires the solution of two elliptic partial differential equations. The framework is also independent of the order of the contour model, allowing us to introduce a Markov process model for contour curvature. We analyze the distribution of such curves and show that its mode is the Euler spiral, a curve minimizing changes in curvature. Example computations using the contour enhancement filter with the curvature-based contour model are provided, highlighting how the filter is curvature-selective even when curvature is absent in the input. [ABSTRACT FROM AUTHOR]

Abstract: Studies of object-based attention (OBA) have suggested that attentional selection is intimately associated with discrete objects. However, the relationship of this association to the basic visual features (‘textons’) which guide the segregation of visual scenes into ‘objects’ remains largely unexplored. Here we study this hypothesized relationship for one of the most conspicuous features of early vision: orientation. To do so we examine how attention spreads through uniform (one ‘object’) orientation-defined textures (ODTs), and across texture-defined boundaries in discontinuous (two ‘objects’) ODTs. Using the divided-attention paradigm we find that visual events that are known to trigger orientation-based texture segregation, namely perceptual boundaries defined by high orientation and/or curvature gradients, also induce a significant cost on attentional selection. At the same time we show that no effect is incurred by the absolute value of the textons, i.e., by the general direction (or, the ‘grain’) of the texture—in conflict with previous findings in the OBA literature. Collectively these experiments begin to reveal the link between object-based attention and texton-based segregation, a link which also offers important cross-disciplinary methodological advantages. [Copyright &y& Elsevier]

Abstract: The association of borders with “figure” rather than “background” provides a topological organizing principle for early vision. Such global influences have recently been shown to have local effects, with neuronal activity modulated by stimulus properties from well outside the classical receptive field. We extend the theoretical analysis of such phenomena by developing the geometry of interaction between shading, boundaries, and boundary ownership for smooth surfaces. The purely exterior edges of smooth objects enjoy a fold-type relationship between shading and boundary, due to foreshortening, while the background is cut off transversely. However, at cusp points in the image mapping the exterior boundary ends abruptly. Since such singular points are notoriously unstable, we conjecture that this process is regularized by a natural quantization of suggestive contours due to physiological boundary-detection mechanisms. The result extends a theorem about how contours must end to one that characterizes surface (Gaussian) curvature in the neighborhood of where they appear to end. Apparent contours and their interaction with local shading thus provide important monocular shape cues. [Copyright &y& Elsevier]