Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images

Abstract

Natural image statistics play a crucial role in shaping biological visual systems, understanding their function and design principles, and designing effective computer-vision algorithms. Highorder statistics are critical for conveying local features, but they are challenging to study -largely because their number and variety is large. Here, via the use of two-dimensional Hermite (TDH) functions, we identify a covert symmetry in high-order statistics of natural images that simplifies this task. This emerges from the structure of TDH functions, which are an orthogonal set of functions that are organized into a hierarchy of ranks. Specifically, we find that the shape (skewness and kurtosis) of the distribution of filter coefficients depends only on the projection of the function onto a 1-dimensional subspace specific to each rank. The characterization of natural image statistics provided by TDH filter coefficients reflects both their phase and amplitude structure, and we suggest an intuitive interpretation for the special subspace within each rank.

Bilateral Symmetry Strengthens the Perceptual Salience of Figure against Ground

“…Although symmetry has been discussed in terms of a major grouping principle or law of good Gestalt since Wertheimer and Metzger [1,2], the specific effects symmetry may produce on feature grouping, figure-ground segregation, visual discrimination, or time to respond to visual configurations have become subject to systematic quantitative investigation in perceptual science only recently. In the case of visual perception, symmetry may be conceived as a geometric property that yields configurational simplicity and, therefore, represents an ecological advantage for information processing [30]. Symmetry may also be seen as a perceptual feature that attracts attention, enhances configural salience, and facilitates grouping [5,[31][32][33].…”

A rotation-equivariant convolutional neural network model of primary visual cortex

“…We use the two-dimensional Hermite functions in polar coordinates, which form a steerable, orthonormal basis (Fig. 1C; see also Victor et al, 2006;Hu & Victor, 2016). For filters of size k, we use all 2d Hermite functions up to rank k, which means we have k(k + 1)/2 basis functions.…”