×

From receptive profiles to a metric model of V1. (English) Zbl 1416.92015

J. Comput. Neurosci. 46, No. 3, 257-277 (2019); correction ibid. 47, No. 2, 231 (2019).
Summary: In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define 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. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abbasi-Sureshjani, S., Favali, M., Citti, G., Sarti, A., ter Haar Romeny, B. M. (2016). Curvature integration in a 5D kernel for extracting vessel connections in retinal images. IEEE Transactions on Image Processing, 2018(27), 606-621. · Zbl 1409.94004
[2] Angelucci, A., Levitt, J. B., Walton, E., Hup, J. M., Bullier, J., Lund, J.S. (2002). Circuits for local and global signal integration in primary visual cortex. The Journal of Neuroscience, 22, 8633-8646.
[3] Angelucci, A., & Bullier, J. (2003). Reaching beyond the classical receptive field of V1 neurons: Horizontal or feedback axons?. Journal of Physiology Paris, 97, 141-154.
[4] Anselmi, F., & Poggio, T. (2010). Representation learning in sensory cortex: a theory. CBMM memo n. 26.
[5] Antoine, J. -P., & Murenzi, R. (1996). Two-dimensional directional wavelets and the scale-angle representation. Signal Processing, 52(3), 241-272. · Zbl 0875.94074
[6] Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc., 66, 937-404. · Zbl 0037.20701
[7] August, J., & Zucker, S. W. (2000). The curve indicator random field: Curve organization via edge correlation. In Boyer, K., & Sarkar, S. (Eds.) Perceptual organization for artificial vision systems. Boston: Kluwer Academic.
[8] Barbieri, D., Citti, G., Sanguinetti, G., Sarti, A. (2014). An uncertainty principle underlying the functional architecture of V1. Journal of Physiology Paris, 106(5-6), 183-193.
[9] Barbieri, D., Cocci, G., Citti, G., Sarti, A. (2014). A cortical-inspired geometry for contour perception and motion integration. J. Math. Imaging Vis., 49(3), 511-529. · Zbl 1291.92074
[10] Bekkers, E. J., Lafarge, M. W., Veta, M., Eppenhof, K. A. J., Pluim, J. P. W., Duits, R. (2018). Roto-translation covariant convolutional networks for medical image analysis. In Schnabel, J. A., Davatzikos, C., Alberola-López, C., Fichtinger, G., Frangi, A. F. (Eds.) Medical image computing and computer assisted intervention - MICCAI 2018 - 21st International Conference, 2018, Proceedings (pp. 440-448). (Lecture Notes in Computer Science; Vol. 11070).
[11] Ben-Shahar, O., Huggins, P., Izo, T., Zucker, S. W. (2003). Cortical connections and early visual function: intra- and inter-columnar processing. J. Physiol. Paris., 97(2-3), 191-208.
[12] Ben-Shahar, O., & Zucker, S. (2004). Geometrical computations explain projection patterns of long-range horizontal connections in visual cortex. Neural Computation, 16(3), 445-476. · Zbl 1111.68610
[13] Bonfiglioli, A., Lanconelli, E., Uguzzoni, F. (2007). Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Math. Berlin: Springer. · Zbl 1128.43001
[14] Boscain, U., Chertovskih, R., Gauthier, J. P., Remizov, A. (2014). Hypoelliptic diffusion and human vision: a semi-discrete new twist. SIAM Journal on Imaging Sciences, 7(2), 669-695. · Zbl 1343.94002
[15] Bosking, W., Zhang, Y., Schoenfield, B., Fitzpatrick, D. (1997). Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. Journal of Neuroscience, 17(6), 2112-2127.
[16] Bressloff, P. C., & Cowan, J. D. (2003). The functional geometry of local and long-range connections in a model of V1. J. Physiol. Paris, 97(2-3), 221-236.
[17] Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J., Wiener, M. C. (2002). What Geometric Visual Hallucinations Tell Us about the Visual Cortex. Neural Computation, 14, 473-491. · Zbl 1037.91083
[18] Citti, G., & Sarti, A. (2006). A Cortical Based Model of Perceptual Completion in the Roto-Translation Space. Journal of Mathematical Imaging and Vision archive, 24(3), 307-326. · Zbl 1478.92100
[19] Cohen, T., & Welling, M. (2016). Group equivariant convolutional networks. Int. Conf. on Machine Learning, 2990-2999.
[20] Cocci, G., Barbieri, D., Sarti, A. (2012). Spatiotemporal receptive fields of cells in V1 are optimally shaped for stimulus velocity estimation. Journal of the Optical Society of America. A, vol. 29, no. 1.
[21] Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5-30. · Zbl 1095.68094
[22] Daugman, J. G. (1985). Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America, A2, 1160-1169.
[23] Deng, C. -X., Li, S., Fu, Z. -X. (2010). The reproducing kernel Hilbert space based on wavelet transform, Proceedings of the 2010 international conference on wavelet analysis and pattern recognition. Qingdao, 370-374.
[24] Dobbins, A., Zucker, S., Cynader, M. (1987). Endstopped neurons in the visual cortex as a substrate for calculating curvature. Nature, 329(6138), 438-441.
[25] Duits, R. (2005). Perceptual organization in image analysis: A mathematical approach based on scale orientation and curvature. Phd thesis: Eindhoven University of Technology.
[26] Duits, R., Felsberg, M., Granlund, G., ter Haar Romeny, B. M. (2007). Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group. International Journal of Computer Vision, 72(1), 79-102.
[27] Duits, R., & Franken, E. M. (2010a). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part I: Linear Left-Invariant Diffusion Equations on SE(2). Quarterly of Applied Mathematics, 68, 293-331. · Zbl 1205.35326
[28] Duits, R., & Franken, E. M. (2010b). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part II: Nonlinear left-invariant diffusions on invertible orientation scores. Quarterly of Applied Mathematics, 68, 255-292. · Zbl 1202.35334
[29] Duits, R., Führ, H., Janssen, B., Bruurmijn, M., Florack, L., van Assen, H. (2013). Evolution Equations on Gabor Transforms and their Applications. Applied and Computational Harmonic Analysis, 35(3), 483-526. · Zbl 1296.35204
[30] Duits, R., Boscain, U., Rossi, F., Sachkov, Y. (2014). Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2). Journal of Mathematical Imaging and Vision, 49(2), 384-417. · Zbl 1296.49040
[31] Elder, J. H., & Goldberg, R. M. (2002). Ecological statistics of Gestalt laws for the perceptual organization of contours. Journal of Vision, 2(4), 5,324-353.
[32] Favali, M., Citti, G., Sarti, A. (2017). Local and Global Gestalt Laws: A Neurally Based Spectral Approach. Neural Computation, 29(2), 394-422. · Zbl 1414.91315
[33] Federer, H. (1969). Geometric measure theory. Berlin: Springer-Verlag. · Zbl 0176.00801
[34] Field, D. J., Hayes, A., Hess, R. F. (1993). Contour integration by the human visual system: evidence for a local association field. Vision Res, 33, 173-193.
[35] Geisler, W. S., Perry, J. S., Super, B. J., Gallogly, D. P. (2001). Edge co-occurrence in natural images predicts contour grouping performance. Vision Research, 41, 711-724.
[36] Gilbert, C. D., Das, A., Ito, M., Kapadia, M., Westheimer, G. (1996). Spatial integration and cortical dynamics. Proceedings of the National Academy of Sciences USA, 93, 615-622.
[37] Gilbert, C. D., & Wu, L. (2013). Top-down influences on visual processing. Nature Reviews Neuroscience, 14, 350-363.
[38] Grossberg, S., & Mingolla, E. (1985). Neural dynamics of perceptual grouping: Textures, boundaries, and emergent segmentations. Perception & Psychophysics, 38(2), 141-171.
[39] Hansen, T., & Neumann, H. (2008). A recurrent model of contour integration in primary visual cortex. J. of Vision, 8(8), 1-25.
[40] Hausdorff, F. (1918). Dimension und ausseres Mass̈. Mathematische Annalen, 79(1-2), 157-179. · JFM 46.0292.01
[41] Hubel, D. H., & Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat visual cortex. J. Physiol. (London), 160, 106-154.
[42] Jones, J. P., & Palmer, L. A. (1987). An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. Journal of Neurophysiology, 58, 1233-1258.
[43] Kapadia, M. K., Westheimer, G., Gilbert, C. D. (1999). Dynamics of spatial summation in primary visual cortex of alert monkeys. Proc Natl Acad Sci USA, 96, 12073-12078.
[44] Karas, P., & Svoboda, D. (2013). Algorithms for efficient computation of convolution, in Design and Architectures for Digital Signal Processing, InTech.
[45] Kruger, N. (1998). Collinearity and parallelism are statistically significant second order relations of complex cell responses. Neural Processing Letters, 8, 117-129.
[46] Lawlor, M., & Zucker, S. W. (2013). Third-order edge statistics: contour continuation, curvature, cortical connections. In Burges, C. J. C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K. Q. (Eds.) Advances in neural information processing systems 26, (Vol. 26 pp. 1763-1771). Red Hook: Curran Associates, Inc.
[47] LeCun, Y., Bengio, Y., Hinton, G. (2015). Deep learning. Nature, 521, 436-444.
[48] Lee, H., Battle, A., Raina, R., Ng, A. Y. (2007). Efficient sparse coding algorithms. In Proceedings of the 19th annual conference on neural information processing systems (pp. 801-808). Cambridge: MIT Press.
[49] Lee, T. S. (1996). Image Representation Using 2D Gabor Wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 10.
[50] Liang, M., & Hu, X. (2015). Recurrent convolutional neural network for object recognition, CVPR.
[51] Martinez, L. M., & Alonso, J. -M. (2003). Complex receptive fields in primary visual cortex. The Neuroscientist, 9(5), 317-331.
[52] Mitchison, G., & Crick, F. (1982). Long axons within the striate cortex: their distribution, orientation, and patterns of connection. Proceedings of National Academy of Sciences USA, 79, 3661-3665.
[53] Montobbio, N., Sarti, A., Citti, G. (2017). A metric model for the functional architecture of the visual cortex (submitted). · Zbl 1439.30091
[54] Montgomery, R. (2002). A tour of subriemannian geometries, their geodesics and applications, Mathematical surveys and monographs, Vol. 91 American mathematical society, Providence, RI. · Zbl 1044.53022
[55] Mumford, D. (1993). Elastica and computer vision. In Bajaj, C (Ed.) Algebraic geometry and its applications (pp. 507-518). Berlin: Springer-Verlag.
[56] Neumann, H., & Mingolla, E. (2001). Computational neural models of spatial integration in perceptual grouping. In Shipley, T.F., & Kellman, P. J. (Eds.) Advances in psychology, 130, From fragments to objects: Segmentation and grouping in vision, 353- 400.
[57] Olshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381, 607-609.
[58] Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Lerer, A.A. (2017). Automatic differentiation in pytorch, in NIPS-W.
[59] Petitot, J. (2008). Neurogeométrie de la vision - Modèles mathématiques et physiques des architectures fonctionnelleś, Éditions de l’École Polytechnique.
[60] Petitot, J., & Tondut, Y. (1999). Vers une neuro-geométrie. Fibrations corticales, structures de contact et contours subjectifs modaux́, Mathématiques, Informatique et Sciences Humaines, vol. 145, 5-101. EHESS, Paris.
[61] Sanguinetti, G., Citti, G., Sarti, A. (2010). A model of natural image edge co-occurrence in the rototranslation group. Journal of Vision 10(14).
[62] Sarti, A., & Citti, G. (2015). The constitution of visual perceptual units in the functional architecture of V1. Journal of Computational Neuroscience, 38(2), 285-300. · Zbl 1410.92028
[63] Sarti, A., Citti, G., Petitot, J. (2008). The symplectic structure of the visual cortex. Biological Cybernetics, 98(1), 33-48. · Zbl 1148.92012
[64] Sifre, L., & Mallat, S. (2013). Rotation, scaling and deformation invariant scattering for texture discrimination, CVPR, IEEE 1233-1240.
[65] Sigman, M., Cecchi, G. A., Gilbert, C. D., Magnasco, M. O. (2001). On a common circle: Natural scenes and Gestalt rules. Proceedings of the National Academy of Sciences, 98(4), 1935-1940.
[66] Spoerer, C. J., McClure, P., Kriegeskorte, N. (2017). Recurrent convolutional neural networks: a better model of biological object recognition. Frontiers in psychology, 8, 1551.
[67] Sturm, K. -T. (1995). On the geometry defined by Dirichlet forms. In Bolthausen, E. et al. (Eds.) Seminar on stochastic analysis, random fields and applications (pp. 231-242). Boston: Birkhäuser. · Zbl 0834.58039
[68] Sturm, K. -T. (1998). Diffusion processes and heat kernels on metric spaces. Annals of Probability, 26(1), 1-55. · Zbl 0936.60074
[69] Vedaldi, A., & Lenc, K. (2015). MatConvNet - convolutional neural networks for MATLAB, Proc. of the ACM Int. Conf. on Multimedia.
[70] Worrall, D. E., Garbin, S. J., Turmukhambetov, D., Brostow, G. J. (2017). Harmonic networks: Deep translation and rotation equivariance. CVPR 5028-5037.
[71] Yen, S. C., & Finkel, L. H. (1998). Extraction of perceptually salient contours by striate cortical networks. Vision Res, 38(5), 719-741.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.