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

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.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn.

For each pair (n, k) with 1

Measuring the predictive performance of computer-controlled infusion pumps.

Current measures of the performance of computer-controlled infusion pumps (CCIPs) are poorly defined, of little use to the clinician using the CCIP, and pharmacostatistically incorrect. We propose four measures be used to quantitate the performance of CCIPs: median absolute performance error (MDAPE), median performance error (MDPE), divergence, and wobble. These measures offer several significant advantages over previous measures. First, their definitions are based on the performance error as a fraction of the predicted (rather than measured) drug concentration, making the measures much more useful to the clinician. Second, the measures are defined in a way that addresses the pharmacostatistical issue of appropriate estimation of population parameters. Finally, the measure of inaccuracy, MDAPE, is defined in a way that is consistent with iteratively reweighted least squares nonlinear regression, a commonly used method of estimating pharmacokinetic parameters. These measures make it possible to quantitate the overall performance of a CCIP or to compare the predictive performance of CCIPs which differ in either general approach (e.g., compartmental model driven vs. plasma efflux approach), pump mechanics, software algorithms, or pharmacokinetic parameter sets.

Does the maximum entropy method improve sensitivity?

Maximum entropy reconstruction has been used in several fields to produce visually striking reconstructions of positive objects (images, densities, spectra) from noisy, indirect measurements. In magnetic resonance spectroscopy, this technique is notable for its apparent noise suppression and its avoidance of the artifacts that affect discrete Fourier transform spectra of short (zero-extended) data records. In the general case where the length of the reconstructed spectrum exceeds that of the data record or where a convolution kernel is incorporated in the reconstruction, no known analytical solution to the reconstruction problem exists. Consequently, knowledge of the properties of maximum entropy reconstruction has been mainly anecdotal, based on a small selection of published reconstructions. However, in the limiting case where the lengths of the reconstructed spectrum and the data record are the same and a convolution kernel is not applied, the problem can be solved analytically. The solution has a simple structure that helps explain several commonly observed features of maximum entropy reconstructions--for example, the biases in the recovered intensities and the fact that noise near the baseline is more successfully suppressed than is noise superimposed on broad features in the spectrum. The solution also shows that the noise suppression offered by maximum entropy reconstruction could (in this special case) be equally well obtained by a "cosmetic" device: simply displaying the conventional Fourier transform reconstruction using a certain nonlinear plotting scale for the vertical (y) coordinate.