DIFFEOMORPHIC SMOOTHING FOR RETINOTOPIC MAPPING

Abstract

Retinotopic mapping, the mapping of visual input on the retina to cortical neurons, is an important topic in vision science. Typically, cortical neurons are related to visual input on the retina using functional magnetic resonance imaging (fMRI) of cortical responses to slowly moving visual stimuli on the retina. Although it is well known from neurophysiology studies that retinotopic mapping is locally diffeomorphic (i.e., smooth, differentiable, and invertible) within each local area, the retinotopic maps from fMRI are often not diffeomorphic, especially near the fovea, because of the low signal-noise ratio of fMRI. The aim of this study is to develop and solve a mathematical model that produces diffeomorphic retinotopic mapping from fMRI data. Specifically, we adopt a geometry concept, the Beltrami coefficient, as the tool to define diffeomorphism, and model the problem in an optimization framework. Efficient numerical methods are proposed to solve the model. Experimental results with both synthetic and real retinotopy data demonstrate that the proposed method is superior to conventional smoothing methods.

Keywords: Beltrami Coefficient; Diffeomorphic Smoothing; Retinotopic Mapping.

Fig. 1.

Illustration of visual system to the visual cortex.

Fig. 2.

Illustration of several spaces and relations between them.

Fig. 3.

(a) Illustration of approximate the mapping in discrete. (b) The divergence approximation on the vertex ring.

Fig. 4.

Function domain, its mapping result and mapping results with noise: (a) mapping domain on the visual field; (b) mapping result without noise; (c) mapping result with weak noise under Peak Signal-Noise Ratio (PSNR) is 20 dB; (d) mapping result with strong noise (PSNR = 10 dB).

Fig. 5.

Smoothing results for map with big noise: (a) average smoothing; (b) median smoothing; (c) Laplacian smoothing; (d) the proposed smoothing.

Fig. 6.

A subject’s left retinotopic mapping: (a) raw retinotopic mapping; (b)-(e) smoothing results of average, median, Laplacian, and proposed smoothing methods; (f) the raw mapping on cortical surface; (g)-(j) smoothing results on inflated surface of each method, in the same order of (b)-(e); (k) The left inflated cortex.