Abstract. We model the functional architecture of the early stages of three-dimensional vision by extending the neurogeometric sub-Riemannian model for stereovision introduced in [M. V. Bolelli, G. Citti, A. Sarti, and S. W. Zucker, Front. Comput. Sci., 5 (2024), 1142621]. A new framework for correspondence is introduced that integrates a neural-based algorithm to achieve stereo correspondence locally while simultaneously organizing the corresponding points into global perceptual units. The result is an effective scene segmentation. We achieve this using harmonic analysis on the sub-Riemannian structure and show, in a comparison against Riemannian distance, that the sub-Riemannian metric is central to the solution.

Abstract. Receptive profiles of the primary visual cortex (V1) cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. In this paper we aim to show that the distribution of cells in V1, although not complete to reconstruct the original image, is sufficient to reconstruct the perceived image with subjective constancy. We show that a color constancy image can be reconstructed as the solution of the associated inverse problem, which is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short-range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenization techniques.

Tree-like structures, such as retinal images, are widely studied in computer-aided diagnosis systems for large-scale screening programs. Despite several segmentation and tracking methods proposed in the literature, there still exist several limitations specifically when two or more curvilinear structures cross or bifurcate, or in the presence of interrupted lines or highly curved blood vessels. In this paper, we propose a novel approach based on multi-orientation scores augmented with a contextual affinity matrix, which both are inspired by the geometry of the primary visual cortex (V1) and their contextual connections. The connectivity is described with a 5D kernel obtained as the fundamental solution of the Fokker-Planck equation modeling the cortical connectivity in the lifted space of positions, orientations, curvatures, and intensity. It is further used in a self-tuning spectral clustering step to identify the main perceptual units in the stimuli. The proposed method has been validated on several easy as well as challenging structures in a set of artificial images and actual retinal patches. Supported by quantitative and qualitative results, the method is capable of overcoming the limitations of current state-of-the-art techniques.

This letter presents a mathematical model of figure-ground articulation that takes into account both local and global gestalt laws and is compatible with the functional architecture of the primary visual cortex (V1). The local gestalt law of good continuation is described by means of suitable connectivity kernels that are derived from Lie group theory and quantitatively compared with long-range connectivity in V1. Global gestalt constraints are then introduced in terms of spectral analysis of a connectivity matrix derived from these kernels. This analysis performs grouping of local features and individuates perceptual units with the highest salience. Numerical simulations are performed, and results are obtained by applying the technique to a number of stimuli.

In this paper we reconsider the sub-Riemannian cortical model of image completion introduced in [G. Citti and A. Sarti, J. Math. Imaging Vision, 24 (2006), pp. 307--326]. This model combines two mechanisms, the sub-Riemannian diffusion and the concentration, giving rise to a diffusion driven motion by curvature. In this paper we give a formal proof of the existence of viscosity solutions of the sub-Riemannian motion by curvature. Furthermore we illustrate the sub-Riemannian finite difference scheme used to implement the model and we discuss some properties of the algorithm. Finally results of completion and enhancement on a number of natural images are shown and compared with other models.

The visual systems of many mammals, including humans, are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at the single cell level and geometric processing by means of cell connectivity. We present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low-level object segmentation. The main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher-dimensional spaces. Neural plausibility of the proposed arguments will be discussed.

We designed a strategy for extracting the shapes of cell membranes and nuclei from time lapse confocal images taken throughout early zebrafish embryogenesis using a partial-differential-equation-based segmentation. This segmentation step is a prerequisite for an accurate quantitative analysis of cell morphodynamics during embryogenesis and it is the basis for an integrated understanding of biological processes. The segmentation of embryonic cells requires live zebrafish embryos fluorescently labeled to highlight sub-cellular structures and designing specific algorithms by adapting classical methods to image features. Our strategy includes the following steps: the signal-to-noise ratio is first improved by an edge-preserving filtering, then the cell shape is reconstructed applying a fully automated algorithm based on a generalized version of the Subjective Surfaces technique. Finally we present a procedure for the algorithm validation either from the accuracy and the robustness perspective.

Two-dimensional echocardiography (2DE) is routinely used in clinical practice to measure left ventricular (LV) mass, dimensions, and function. The reliability of these measurements is highly dependent on the ability to obtain nonforeshortened long axis (LA) images of the left ventricle from transthoracic apical acoustic windows. Real time three-dimensional echocardiography (RT3DE) is a novel imaging technique that allows the acquisition of dynamic pyramidal data structures encompassing the entire ventricle and could potentially overcome the effects of LA foreshortening. Accordingly, the aim of this paper was to develop a nearly automated method based on optical flow techniques for the measurement of the left ventricular (LV) LA throughout the cardiac cycle from RT3DE data. The LV LA measurements obtained with the automated technique has been compared with LA measurements derived from manual selection of the LA from a volumetric display of RT3DE data. High correlation (r = .99, SEE = 1.8%, y = .94x + 5.3), no significant bias (-0.18 mm), and narrow limits of agreement (SD: 1.91 mm) were found. The comparison between the LA length derived from 2DE and RT3DE data showed significant underestimation of the 2DE based measurements. In conclusion, this study proves that RT3DE data overcome the effects of foreshortening and indicates that the method we propose allows fast and accurate quantification of LA length throughout the cardiac cycle.

The application of level set techniques to echocardiographic data is presented. This method allows semiautomatic segmentation of heart chambers, which regularizes the shapes and improves edge fidelity, especially in the presence of gaps, as is common in ultrasound data. The task of the study was to reconstruct left ventricular shape and to evaluate left ventricular volume. Data were acquired with a real-time three-dimensional (3-D) echocardiographic system. The method was applied directly in the three-dimensional domain and was based on a geometric-driven scheme. The numerical scheme for solving the proposed partial differential equation is borrowed from numerical methods for conservation law. Results refer to in vitro and human in vivo acquired 3-D + time echocardiographic data. Quantitative validation was performed on in vitro balloon phantoms. Clinical application of this segmentation technique is reported for 20 patient cases providing measures of left ventricular volumes and ejection fraction.

We introduce a new model for multiscale analysis of space-time echocardiographic sequences. The proposed nonlinear partial differential equation, representing the multiscale analysis, filters the sequence while keeping the space-time coherent structures. It combines the ideas of regularized Perona-Malik anisotropic diffusion and the Galilean invariant movie multiscale analysis of Alvarez, Guichard, Lions and Morel. A numerical method for solving the proposed partial differential equation is suggested and its stability is shown. Computational results on synthesized and real sequences are provided. A qualitative and quantitative evaluation of the accuracy of the method is presented.