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Invoking the manifold assumption in machine learning requires knowledge of the manifold's geometry and dimension, and theory dictates how many samples are required. However, in most applications, the data are limited, sampling may not be uniform, and the manifold's properties are unknown; this implies that neighborhoods must adapt to the local structure. We introduce an algorithm for inferring adaptive neighborhoods for data given by a similarity kernel. Starting with a locally conservative neighborhood (Gabriel) graph, we sparsify it iteratively according to a weighted counterpart. In each step, a linear program yields minimal neighborhoods globally, and a volumetric statistic reveals neighbor outliers likely to violate manifold geometry. We apply our adaptive neighborhoods to nonlinear dimensionality reduction, geodesic computation, and dimension estimation. A comparison against standard algorithms using, for example, k-nearest neighbors, demonstrates the usefulness of our approach., (© 2023 Massachusetts Institute of Technology.)

Invariants underlying shape inference are elusive: A variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: It has both image salience, in terms of isophotes, and surface meaning, in terms of surface normal. We relax the notion of occluding contour and, more accurately, the rim on the object that projects to it, to define closed extremal curves. This new shape descriptor is invariant over different renderings. It exists at the topological level, which guarantees an image-based counterpart. It surrounds bumps and dents, as well as common interior shape components, and formalizes the qualitative nature of bump perception. The invariants are biologically computable, unify shape inferences from shading and specular materials, and predict new phenomena in bump and dent perception. Most important, working at the topological level allows us to capture the elusive aspect of bump boundaries.

Border ownership is an intermediate-level visual task: it must integrate (upward flowing) image information about edges with (downward flowing) shape information. This highlights the familiar local-to-global aspect of border formation (linking of edge elements to form contours) with the much less studied global-to-local aspect (which edge elements form part of the same shape). To address this task we show how to incorporate certain high-level notions of distance and geometric arrangement into a form that can influence image-based edge information. The center of the argument is a reaction-diffusion equation that reveals how (global) aspects of the distance map (that is, shape) can be "read out" locally, suggesting a solution to the border ownership problem. Since the reaction-diffusion equation defines a field, a possible information processing role for the local field potential can be defined. We argue that such fields also underlie the Gestalt notion of closure, especially when it is refined using modern experimental techniques. An important implication of this theoretical argument is that, if true, then network modeling must be extended to include the substrate surrounding spiking neurons, including glia., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Many traditional two-view stereo algorithms explicitly or implicitly use the frontal parallel plane assumption when exploiting contextual information since, e.g., the smoothness prior biases toward constant disparity (depth) over a neighborhood. This introduces systematic errors to the matching process for slanted or curved surfaces. These errors are nonnegligible for detailed geometric modeling of natural objects such as a human face. We show how to use contextual information geometrically to avoid such errors. A differential geometric study of smooth surfaces allows contextual information to be encoded in Cartan's moving frame model over local quadratic approximations, providing a framework of geometric consistency for both depth and surface normals; the accuracy of our reconstructions argues for the sufficiency of the approximation. In effect, Cartan's model provides the additional constraint necessary to move beyond the frontal parallel plane assumption in stereo reconstruction. It also suggests how geometry can extend surfaces to account for unmatched points due to partial occlusion.

We propose a theoretical mechanism that enables the elaboration of veins to supply distant cells during leaf development. In contrast to the more standard view that a signal (e.g., auxin) is produced at isolated sites to stimulate growth, we determine the consequences of the hypothesis that auxin is produced at a constant rate in every cell. High concentration sites for auxin emerge naturally in a reaction-diffusion model, together with global information about leaf shape and existing venation. Because the global information is encoded as auxin concentration and its gradient, those signals provide individual cells with sufficient information to determine their own fate. Unlike other models, a single substance suffices for the reaction-diffusion at early, but not initial, stages of development. Neither complex interactions nor predetermination are necessary. We predict angiosperm areolation patterns in simulation, and our model further implies the Sachs Canalization Hypothesis and resolves a dilemma regarding the role of auxin in cell growth.