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. High-order 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.
Highlights
Achieving a thorough understanding of the statistics of our visual environment is important from both a biological point of view and an engineering point of view
We analyze image statistics via the distribution of values that result from filtering them with two-dimensional Hermite (TDH) functions
To visualize the results for rank two, we note that the full set of rank two filters can be regarded as points on the surface of an ordinary sphere (Figure 4). This follows from the general observation that the r-th rank of TDH functions is spanned by r + 1 orthonormal filters, so the full set of unit-magnitude filters of rank r may be regarded as the surface of a sphere in (r + 1)-space. In this spherical representation of rank two TDH functions shown in Figure 4, the polar filters correspond to one set of orthogonal directions, the Cartesian filters to a second orthogonal set of directions, and intermediate directions correspond to mixtures of polar or Cartesian filters
Summary
Achieving a thorough understanding of the statistics of our visual environment is important from both a biological point of view and an engineering point of view. The biological relevance is that the statistics of the natural environment are a strong constraint under which visual systems evolve, develop and function [1]. The engineering relevance is that a knowledge of image statistics is important for many problems in computer vision [2], including image de-noising, image classification [3,4,5,6], image compression and texture synthesis [7]. Second-order statistics are concisely captured by the power spectrum, because it is the Fourier transform of the autocorrelation function. While the power spectrum captures important spatial regularities of natural images, such as distance-independent scaling [12], it is far from a complete statistical description of natural images.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
