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

Recently, applied mathematicians have been pursuing the goal of sparse codi ng 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 syste ms have elements distributed across a range of scales and locations, but al so orientations. In fact they have highly direction-specific elements and e xhibit increasing numbers of distinct directions as we go to successively f iner scales. Meanwhile, researchers in natural scene statistics (NSS) have been attempti ng 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-orie ntation 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 repre senting edges. We consider here certain empirical consequences of this hypothesis, looking at key findings of the NSS literature and conducting studies of curvelet a nd 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, ridge let and curvelet coefficients over a database of synthetic and photographic images. Empirically the curvelet coefficients exhibit noticeably higher ku rtosis than wavelet, ridgelet, or Fourier coefficients. This is consistent with the hypothesis. Our second experiment studies the inter-scale correlation of wavelet coeffi cient energies at the same location. We describe a simple experiment showin g that presence of edges explains these correlations. We also develop a cru de nonlinear 'partial correlation' by considering the correlation between w avelet parents and children after a few curvelet coefficients are removed. When we kill the few biggest coefficients of the curvelet transform, much o f the correlation between wavelet subbands disappears-consistent with the h ypothesis. We suggest implications for future discussions about NSS.