Can recent innovations in harmonic analysis `explain' key findings in natural image statistics?

Abstract

Recently, applied mathematicians have been pursuing the goal of sparse coding of certain mathematical models of images with edges. They have found by mathematical analysis that, instead of wavelets and Fourier methods, sparse coding leads towards new systems: ridgelets and curvelets. These new systems have elements distributed across a range of scales and locations, but also orientations. In fact they have highly direction-specific elements and exhibit increasing numbers of distinct directions as we go to successively finer scales. Meanwhile, researchers in natural scene statistics (NSS) have been attempting to find sparse codes for natural images. The new systems they have found by computational optimization have elements distributed across a range of scales and locations, but also orientations. The new systems are certainly unlike wavelet and Gabor systems, on the one hand because of the multi-orientation and on the other hand because of the multi-scale nature. There is a certain degree of visual resemblance between the findings in the two fields, which suggests the hypothesis that certain important findings in the NSS literature might possibly be explained by the slogan: edges are the dominant features in images, and curvelets are the right tool for representing edges. We consider here certain empirical consequences of this hypothesis, looking at key findings of the NSS literature and conducting studies of curvelet and ridgelet transforms on synthetic and real images, to see if the results are consistent with predictions from this slogan. Our first experiment measures the nonGaussianity of Fourier, wavelet, ridgelet and curvelet coefficients over a database of synthetic and photographic images. Empirically the curvelet coefficients exhibit noticeably higher kurtosis than wavelet, ridgelet, or Fourier coefficients. This is consistent with the hypothesis. Our second experiment studies the inter-scale correlation of wavelet coefficient energies at the same location. We describe a simple experiment showing that presence of edges explains these correlations. We also develop a crude nonlinear `partial correlation' by considering the correlation between wavelet parents and children after a few curvelet coefficients are removed. When we kill the few biggest coefficients of the curvelet transform, much of the correlation between wavelet subbands disappears - consistent with the hypothesis. We suggest implications for future discussions about NSS.