Tangent bundle curve completion with locally connected parallel networks

The Concept of Symmetry and the Theory of Perception

Perceptual constancy refers to the fact that the perceived geometrical and physical characteristics of objects remain constant despite transformations of the objects such as rigid motion. Perceptual constancy is essential in everything we do, like recognition of familiar objects and scenes, planning and executing visual navigation, visuomotor coordination, and many more. Perceptual constancy would not exist without the geometrical and physical permanence of objects: their shape, size, and weight. Formally, perceptual constancy and permanence of objects are invariants, also known in mathematics and physics as symmetries. Symmetries of the Laws of Physics received a central status due to mathematical theorems of Emmy Noether formulated and proved over 100 years ago. These theorems connected symmetries of the physical laws to conservation laws through the least-action principle. We show how Noether's theorem is applied to mirror-symmetrical objects and establishes mental shape representation (perceptual conservation) through the application of a simplicity (least-action) principle. This way, the formalism of Noether's theorem provides a computational explanation of the relation between the physical world and its mental representation.

Tangent Bundle Elastica and Computer Vision

Visual curve completion, an early visual process that completes the occluded parts between observed boundary fragments (a.k.a. inducers), is a major problem in perceptual organization and a critical step toward higher level visual tasks in both biological and machine vision. Most computational contributions to solving this problem suggest desired perceptual properties that the completed contour should satisfy in the image plane, and then seek the mathematical curves that provide them. Alternatively, few studies (including by the authors) have suggested to frame the problem not in the image plane but rather in the unit tangent bundleR (2) × S(1), the space that abstracts the primary visual cortex, where curve completion allegedly occurs. Combining both schools, here we propose and develop a biologically plausible theory of elastica in the tangent bundle that provides not only perceptually superior completion results but also a rigorous computational prediction that inducer curvatures greatly affects the shape of the completed curve, as indeed indicated by human perception.

Evaluating contextual processing in diffusion MRI: application to optic radiation reconstruction for epilepsy surgery

Diffusion MRI and tractography allow for investigation of the architectural configuration of white matter in vivo, offering new avenues for applications like presurgical planning. Despite the promising outlook, there are many pitfalls that complicate its use for (clinical) application. Amongst these are inaccuracies in the geometry of the diffusion profiles on which tractography is based, and poor alignment with neighboring profiles. Recently developed contextual processing techniques, including enhancement and well-posed geometric sharpening, have shown to result in sharper and better aligned diffusion profiles. However, the research that has been conducted up to now is mainly of theoretical nature, and so far these techniques have only been evaluated by visual inspection of the diffusion profiles. In this work, the method is evaluated in a clinically relevant application: the reconstruction of the optic radiation for epilepsy surgery. For this evaluation we have developed a framework in which we incorporate a novel scoring procedure for individual pathways. We demonstrate that, using enhancement and sharpening, the extraction of an anatomically plausible reconstruction of the optic radiation from a large amount of probabilistic pathways is greatly improved in three healthy controls, where currently used methods fail to do so. Furthermore, challenging reconstructions of the optic radiation in three epilepsy surgery candidates with extensive brain lesions demonstrate that it is beneficial to integrate these methods in surgical planning.