A fluid-dynamic based approach to reconnect the retinal vessels in fundus photography

This paper introduces the use of fluid-dynamic modeling to determine the connectivity of overlapping venous and arterial vessels in fundus images. Analysis of the retinal vascular network may provide information related to systemic and local disorders. However, the automated identification of the vascular trees in retinal images is a challenging task due to the low signal-to-noise ratio, nonuniform illumination and the fact that fundus photography is a projection on to the imaging plane of three-dimensional retinal tissue. A zero-dimensional model was created to estimate the hemodynamic status of candidate tree configurations. Simulated annealing was used to search for an optimal configuration. Experimental results indicate that simulated annealing was very efficient on test cases that range from small to medium size networks, while ineffective on large networks. Although for large networks the nonconvexity of the cost function and the large solution space made searching for the optimal solution difficult, the accuracy (average success rate = 98.35%), and simplicity of our novel approach demonstrate its potential effectiveness in segmenting retinal vascular trees

Cortically based optimal transport

We introduce a model for image morphing in the primary visual cortex V1 to perform completion of missing images in time. We model the output of simple cells through a family of Gabor filters and the propagation of the neural signal accordingly to the functional geometry induced by horizontal connectivity. Then we model the deformation between two images as a path relying two different outputs. This path is obtained by optimal transport considering the Wasserstein distance geodesics associated to some probability measures naturally induced by the outputs on V1. The frame of Gabor filters allows to project back the output path, therefore obtaining an associated image stimulus deformation. We perform a numerical implementation of our cortical model, assessing its ability in reconstructing rigidi motions of simple shapes.Comment: 25 page

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

Local and global gestalt laws: A neurally based spectral approach

A mathematical model of figure-ground articulation is presented, taking into account both local and global gestalt laws. The model is compatible with the functional architecture of the primary visual cortex (V1). Particularly the local gestalt law of good continuity is described by means of suitable connectivity kernels, that are derived from Lie group theory and are neurally implemented in long range connectivity in V1. Different kernels are compatible with the geometric structure of cortical connectivity and they are derived as the fundamental solutions of the Fokker Planck, the Sub-Riemannian Laplacian and the isotropic Laplacian equations. The kernels are used to construct matrices of connectivity among the features present in a visual stimulus. Global gestalt constraints are then introduced in terms of spectral analysis of the connectivity matrix, showing that this processing can be cortically implemented in V1 by mean field neural equations. This analysis performs grouping of local features and individuates perceptual units with the highest saliency. Numerical simulations are performed and results are obtained applying the technique to a number of stimuli.Comment: submitted to Neural Computatio

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph- and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a new approach to exploit any anisotropies in data. Via discrete approximations of Lie groups, we develop a graph neural network made of anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global pooling layers. Group equivariance is achieved via equivariant and invariant operators on graphs with anisotropic left-invariant Riemannian distance-based affinities encoded on the edges. Thanks to its simple form, the Riemannian metric can model any anisotropies, both in the spatial and orientation domains. This control on anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic metric) against invariance (isotropic metric) of the graph convolution layers. Hence we open the doors to a better understanding of anisotropic properties. Furthermore, we empirically prove the existence of (data-dependent) sweet spots for anisotropic parameters on CIFAR10. This crucial result is evidence of the benefice we could get by exploiting anisotropic properties in data. We also evaluate the scalability of this approach on STL10 (image data) and ClimateNet (spherical data), showing its remarkable adaptability to diverse tasks.Comment: submitted to NeurIPS'21, https://openreview.net/forum?id=WsfXFxqZXR

Visual Cortex

The neurosciences have experienced tremendous and wonderful progress in many areas, and the spectrum encompassing the neurosciences is expansive. Suffice it to mention a few classical fields: electrophysiology, genetics, physics, computer sciences, and more recently, social and marketing neurosciences. Of course, this large growth resulted in the production of many books. Perhaps the visual system and the visual cortex were in the vanguard because most animals do not produce their own light and offer thus the invaluable advantage of allowing investigators to conduct experiments in full control of the stimulus. In addition, the fascinating evolution of scientific techniques, the immense productivity of recent research, and the ensuing literature make it virtually impossible to publish in a single volume all worthwhile work accomplished throughout the scientific world. The days when a single individual, as Diderot, could undertake the production of an encyclopedia are gone forever. Indeed most approaches to studying the nervous system are valid and neuroscientists produce an almost astronomical number of interesting data accompanied by extremely worthy hypotheses which in turn generate new ventures in search of brain functions. Yet, it is fully justified to make an encore and to publish a book dedicated to visual cortex and beyond. Many reasons validate a book assembling chapters written by active researchers. Each has the opportunity to bind together data and explore original ideas whose fate will not fall into the hands of uncompromising reviewers of traditional journals. This book focuses on the cerebral cortex with a large emphasis on vision. Yet it offers the reader diverse approaches employed to investigate the brain, for instance, computer simulation, cellular responses, or rivalry between various targets and goal directed actions. This volume thus covers a large spectrum of research even though it is impossible to include all topics in the extremely diverse field of neurosciences

Analysis of vessel connectivities in retinal images by cortically inspired spectral clustering

\u3cp\u3eRetinal images provide early signs of diabetic retinopathy, glaucoma, and hypertension. These signs can be investigated based on microaneurysms or smaller vessels. The diagnostic biomarkers are the change of vessel widths and angles especially at junctions, which are investigated using the vessel segmentation or tracking. Vessel paths may also be interrupted; crossings and bifurcations may be disconnected. This paper addresses a novel contextual method based on the geometry of the primary visual cortex (V1) to study these difficulties. We have analyzed the specific problems at junctions with a connectivity kernel obtained as the fundamental solution of the Fokker–Planck equation, which is usually used to represent the geometrical structure of multi-orientation cortical connectivity. Using the spectral clustering on a large local affinity matrix constructed by both the connectivity kernel and the feature of intensity, the vessels are identified successfully in a hierarchical topology each representing an individual perceptual unit.\u3c/p\u3

Analysis of vessel connectivities in retinal images by cortically inspired spectral clustering

Retinal images provide early signs of diabetic retinopathy, glaucoma, and hypertension. These signs can be investigated based on microaneurysms or smaller vessels. The diagnostic biomarkers are the change of vessel widths and angles especially at junctions, which are investigated using the vessel segmentation or tracking. Vessel paths may also be interrupted; crossings and bifurcations may be disconnected. This paper addresses a novel contextual method based on the geometry of the primary visual cortex (V1) to study these difficulties. We have analyzed the specific problems at junctions with a connectivity kernel obtained as the fundamental solution of the Fokker–Planck equation, which is usually used to represent the geometrical structure of multi-orientation cortical connectivity. Using the spectral clustering on a large local affinity matrix constructed by both the connectivity kernel and the feature of intensity, the vessels are identified successfully in a hierarchical topology each representing an individual perceptual unit