Speakers and abstracts

Speakers

Emre Baspinar (Università di Bologna, Italy)

Antonin Chambolle (CMAP, École Polytechnique, France)

Barbara Gris (LJLL, Sorbonne Université, France)

Jesus Malo (Universitat de València, Spain)

Edoardo Provenzi (Université de Bordeaux, France)

Anna Song (DMA, ENS, France)

Elisa Tartaglia (École des Neurosciences de Paris, France)

Abstracts

A sub-Riemannian model of the visual cortex with frequency and phase and its applications

Emre Baspinar (Università di Bologna, Italy)

Our objective is to develop a geometrical model of vision consistent with the neural characteristics of the visual cortex and study geometric ﬂows for image processing in the relevant model geometry. Our departure point is the visual cortex model of the orientation selective neurons in the cortex, which is presented in [1] by Citti and Sarti. We extend this model and provide a novel sub-Riemannian model of the primary visual cortex which models orientation-frequency selective, phase shifted cortex cell behavior and the associated neural connectivity. We develop an image enhancement algorithm using a multi-frequency Laplace-Beltrami ﬂow in the sub-Riemannian framework of this model. We employ the model framework in order to provide a geometric procedure for multi-feature orientation map construction.

[1] G. Citti and A. Sarti, “A cortical based model of perceptual completion in the roto-translation space,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 307–326, 2006.

Incorporating intuitive prior in image reconstruction

Barbara Gris (LJLL, Sorbonne Université, France)

Image reconstruction consists in recovering an image from an indirect observation (for instance its Radon transform). In general this observation does not allow to determine a unique image and some prior (e.g. image regularity) needs to be incorporated in the reconstruction framework. I will present how one can incorporate intuitive priors about the geometric variation of the image from a reference one using the framework of deformation modules. The framework of deformation modules allow to build deformations satisfying some prior and the idea is to reconstruct an image --from some indirect observations-- as the deformation of the reference one while constraining the deformation to satisfy certain constraints. I will first present this notion of deformation modules and then show how it can be used to perform image
reconstruction.

A "Total variation" with curvature penalization

Antonin Chambolle (CMAP, École Polytechnique)

In this joint work with T. Pock (TU Graz) we propose a convex variant of the total variation which penalizes the curvature of the level lines, and is based on a Gauss map (lifting) of curves to represent curvature dependent energies as convex functionals. Applications to "image inpainting" are presented.

The geometry of the space of the visual stimuli: neural models and psychophysics

Jesus Malo (Universitat de València, Spain)

You can become a TV celebrity by predicting the visibility of patterns!: in 2015 the Structural SIMmilarity index (SSIM) received the glamorous Emmy award of the American TV academy [1,2]. Beyond the glamour of TV industry, image quality reduces to a mathematical problem: describing the geometry of the space of visual stimuli. It is not surprising that Euclidean distance is not a good choice because of two reasons: (a) photographic images live in a low-dimensional manifold inside the huge-dimensional space of all possible visual stimuli, and this implies that some dimensions will have bigger relevance than others. (b) the retina-cortex channel is an adaptive feature extractor, and this implies that it relatively expands some regions of the space while contracts others. Of course these two factors (image statistics and retina-cortex organization) are related [3,4].

In this talk I review the basic facts that any metric should reproduce to be perceptually meaningful. We organized these facts in a database that you can download to check your own distance measure [5]. Then, I show how different neural models induce induce specific non-Euclidean metrics [6], and one can decide which model is best through stimuli synthesized according to the specific metrics [7,8]. The conclusion of this geometric analysis of neural models is a metric that works way better than the acclaimed SSIM, and a procedure to keep improving this model [9].

[1] Wang, Bovik, Sheikh & Simoncelli. IEEE TIP 2004. PDF (+19000 citations!)
[2] 67th EMMY Award of the American TV Academy 2015. VIDEO
[3] Horace Barlow. Network: Comp. Neur. Syst. 2001. PDF
[4] Laparra and Malo. Frontiers in Human Neurosci. 2015. PDF
[5] Malo et al. Database: Psychophysical test-bed, 2018. PDF
[6] Laparra, Muñoz and Malo. JOSA A, 2010. PDF
[7] Wang & Simoncelli. Journal of Vision 2008. PDF
[8] Malo & Simoncelli. Proc. SPIE 2015. PDF
[9] Martinez et al. PLoS ONE 2018 (in press). PDF

Resnikoff's model of perceived colors space

Edoardo Provenzi (Université de Bordeaux, France)

In 1974, H. L. Resnkikoff published an inspiring paper [1] about the intrinsic geometrical structure of the space of perceived colors and the Riemannian metrics on it. The mathematical techniques used range from differential geometry to Jordan algebras theory, passing through representations of Lie groups. These methods, although very common in theoretical physics, are far from being popular among colorimetrists, for this reason, Resnikoff’s paper remained unnoticed for decades. In this seminar, I will present the most salient points of Resnikoff’s ideas.

[1] Resnkikoff, H. L., "Differential geometry and color perception", Journal of Mathematical Biology, 1 (2), 1974.

A Neural Field Model for Color Perception in context

Anna Song (DMA, ENS, France)

We propose a neural field model [1],[7] of color perception in context. This model reconciles into a common framework two opposing perceptual phenomena, simultaneous contrast and chromatic assimilation. Previous works such as [5], showed that they act simultaneously, and can produce larger shifts in color matching when acting in synergy with an oscillating spatial pattern. When an observer looks at some point in an image, its color seems to be perceptually more similar to that of the adjacent locations, while being more dissimilar from that of farther neighbors. The influence of neighbors hence reverses its nature above some characteristic scale. Our model fully exploits the balance between attraction and repulsion in color space, combined at small or large scales in physical space. For that purpose we rely on the theory of opponent colors introduced by Hering [2],[3],[6], and a hypercolumnar structure supposed to code for colors. At some neural mass, the pointwise influence of neighbors is spatially integrated to obtain the final effect that we call a color sensation. Alongside this neural field model, we propose to describe the search for a color match in asymmetric color matching experiments as a mathematical projector over a set of reachable color sensations. Thus, our framework predicts when a perceptual match occurs between two images. We validate it by performing a multiparameter regression of our model to data from [5] and [4], and also our own data. All our results show that we are able to explain the nonlinear behavior of the observed shifts reported in the experiments, along one or two dimensions in color space, which cannot be done using a simple linear model.

[1] B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, 61 (2001).

[2] E. Hering, Outlines of a theory of the light sense.

[3] L. Hurvich, Color vision, Sinauer Associates, 1981.

[4] P. Monnier, Standard definitions of chromatic induction fail to describe induction with s-cone patterned backgrounds, 48 (2008).[5] P. Monnier and S. K Shevell, Chromatic induction from s-cone patterns, 44 (2004), pp. 849–56.

[6] G. W. Wyszecki and W. S. Stiles, Color science: Concepts and methods, quantitative data and formulas, 81 (1968).

[7] H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), pp. 55–80.

Eye-movements as a signature of age-related differences in global planning strategies for spatial navigation

Elisa Tartaglia (École des Neurosciences de Paris, France)

The ability to efficiently ﬁnd alternatives routes when faced with unexpected obstacles along our path is among the most compelling evidence of the ﬂexibility of human behaviour. Although a plethora of plausible computations have been put forward to elucidate how the brain accomplishes effcient goal-oriented navigation, the mechanisms that guide an effective re-planning when facing obstructions are still largely undetermined. There is a fair consensus in postulating that possible alternatives routes are internally replayed sampling from past experiences, however, there is currently no account of the criterion according to which those memories are replayed. Here, we posit that paths, which are expected to be more rewarding are replayed more often and that eye movements are the explicit manifestation of this re-planning strategy. In other words, the visual sampling statistics reﬂects the retrieval of available routes on a mental representation of the environment.

To test our hypothesis, we measured the ability of both young and old human subjects to solve a virtual version of the Tolman maze, while we recorded their eye movements. We used reinforcement learning (RL) to corroborate that eye movements statistics was crucially subtending the decision making process involved in re-planning and that the incorporation of this additional information to the algorithm was necessary to reproduce the behavioral performance of both screened populations.

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