Subjective geometry and geometric psychology

Abstract“Subjective geometry” is a term coined by Weintraub and Krantz to describe the distortion imposed upon geometric patterns by the visual system itself—so-called optical illusions. The latter are widely regarded as being generated by misplaced “constancy” effects, i.e., they are regarded as stemming from the invariance of an object's appearance under wide variations in viewing conditions, such as obliquity, rotations, etc. The invariances represented by these constancies—shape constancy, size constancy, etc.—are spatiotemporal invariants of certain Lie subgroups of P4(R) ⊕ CO(1, 3) ⊕ GL(4, R) that govern Euclidean and non-Euclidean geometry. Tha Euclidean subgroups describe a Cyclopean visual world; the non-Euclidean, a binocular (bipolar) world of hyperbolic nature, according to the work of Luneburg, Blank, Indow, and others. The visual field of view is itself a geometric object involvingnot only “figure” and “ground” but also visual contours (orbits of the Lie groups involved), linear perspective, interposition, and contact and symplectic structures. The retina and “cortical retina” are both covered by a family of “circular-surround” cellular response fields (of a “Mexican hat” nature) which constitute an atlas for the visual manifold S. Upon this manifold are defined certain equivariant vector bundles that account for constancy phenomena and certain jet bundles, arising out of the vector bundles by prolongation, that generate the differential invariants characterizing higher form perception. The resultant theory of perceptual-cognitive processing has been termed “geometric psychology,” in analogy to MacLane's “geometrical mechanics” and Brockett–Hermann–Mayne's “geometry of systems,” the mathematical structure being very similar in all three instances. Functorial maps from the category GvFB(S) of equivariant fibre bundles to the simplicial category and the category of simplicial objects complete the theory by extending the perceptual system to cognitive phenomena and information-processing psychology

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Hyperbolic planforms in relation to visual edges and textures perception

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g. optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in [1, 2] to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

A metric model of the visual cortex

The purpose of this thesis is the development of a model for the geometry of the connectivity of the primary visual cortex (V1), by means of functional analysis tools on metric measure spaces. The metric structure proposed to describe the internal connections of V1 implements a notion of correlation between neurons, based on their feature selectivity: this is expressed through a connectivity kernel that is directly induced by the local feature analysis performed by the cells. Such kernel carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. Moreover, its construction can be applied to banks of filters not necessarily obtained through a group representation, and possibly only numerically known. This model is then applied to insert biologically inspired connections in deep learning algorithms, to enhance their ability to perform pattern completion in image classification tasks. The main novelty in our approach lies in its ability to recover global geometric properties of the functional architecture of V1 without imposing any parameterization or invariance, but rather by exploiting the local information naturally encoded in the behavior of single V1 neurons in presence of a visual stimulus

Cartography

The terrestrial space is the place of interaction of natural and social systems. The cartography is an essential tool to understand the complexity of these systems, their interaction and evolution. This brings the cartography to an important place in the modern world. The book presents several contributions at different areas and activities showing the importance of the cartography to the perception and organization of the territory. Learning with the past or understanding the present the use of cartography is presented as a way of looking to almost all themes of the knowledge

Learning Equivariant Representations

State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video recognition tasks. In this thesis, we extend equivariance to other kinds of transformations, such as rotation and scaling. We propose equivariant models for different transformations defined by groups of symmetries. The main contributions are (i) polar transformer networks, achieving equivariance to the group of similarities on the plane, (ii) equivariant multi-view networks, achieving equivariance to the group of symmetries of the icosahedron, (iii) spherical CNNs, achieving equivariance to the continuous 3D rotation group, (iv) cross-domain image embeddings, achieving equivariance to 3D rotations for 2D inputs, and (v) spin-weighted spherical CNNs, generalizing the spherical CNNs and achieving equivariance to 3D rotations for spherical vector fields. Applications include image classification, 3D shape classification and retrieval, panoramic image classification and segmentation, shape alignment and pose estimation. What these models have in common is that they leverage symmetries in the data to reduce sample and model complexity and improve generalization performance. The advantages are more significant on (but not limited to) challenging tasks where data is limited or input perturbations such as arbitrary rotations are present

Camera positioning for 3D panoramic image rendering

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London.Virtual camera realisation and the proposition of trapezoidal camera architecture are the two broad contributions of this thesis. Firstly, multiple camera and their arrangement constitute a critical component which affect the integrity of visual content acquisition for multi-view video. Currently, linear, convergence, and divergence arrays are the prominent camera topologies adopted. However, the large number of cameras required and their synchronisation are two of prominent challenges usually encountered. The use of virtual cameras can significantly reduce the number of physical cameras used with respect to any of the known camera structures, hence adequately reducing some of the other implementation issues. This thesis explores to use image-based rendering with and without geometry in the implementations leading to the realisation of virtual cameras. The virtual camera implementation was carried out from the perspective of depth map (geometry) and use of multiple image samples (no geometry). Prior to the virtual camera realisation, the generation of depth map was investigated using region match measures widely known for solving image point correspondence problem. The constructed depth maps have been compare with the ones generated using the dynamic programming approach. In both the geometry and no geometry approaches, the virtual cameras lead to the rendering of views from a textured depth map, construction of 3D panoramic image of a scene by stitching multiple image samples and performing superposition on them, and computation of virtual scene from a stereo pair of panoramic images. The quality of these rendered images were assessed through the use of either objective or subjective analysis in Imatest software. Further more, metric reconstruction of a scene was performed by re-projection of the pixel points from multiple image samples with a single centre of projection. This was done using sparse bundle adjustment algorithm. The statistical summary obtained after the application of this algorithm provides a gauge for the efficiency of the optimisation step. The optimised data was then visualised in Meshlab software environment, hence providing the reconstructed scene. Secondly, with any of the well-established camera arrangements, all cameras are usually constrained to the same horizontal plane. Therefore, occlusion becomes an extremely challenging problem, and a robust camera set-up is required in order to resolve strongly the hidden part of any scene objects. To adequately meet the visibility condition for scene objects and given that occlusion of the same scene objects can occur, a multi-plane camera structure is highly desirable. Therefore, this thesis also explore trapezoidal camera structure for image acquisition. The approach here is to assess the feasibility and potential of several physical cameras of the same model being sparsely arranged on the edge of an efficient trapezoid graph. This is implemented both Matlab and Maya. The quality of the depth maps rendered in Matlab are better in Quality

Automatic Reconstruction of Textured 3D Models

Three dimensional modeling and visualization of environments is an increasingly important problem. This work addresses the problem of automatic 3D reconstruction and we present a system for unsupervised reconstruction of textured 3D models in the context of modeling indoor environments. We present solutions to all aspects of the modeling process and an integrated system for the automatic creation of large scale 3D models