The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a/spl grave/ trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.

I. Introduction

Over the last decade, there has been abundant interest in wavelet methods for noise removal in signals and images. In many hundreds of papers published in journals throughout the scientific and engineering disciplines, a wide range of wavelet-based tools and ideas have been proposed and studied. Initial efforts included very simple ideas like thresholding of the orthogonal wavelet coefficients of the noisy data, followed by reconstruction. Later efforts found that substantial improvements in perceptual quality could be obtained by translation invariant methods based on thresholding of an undecimated wavelet transform. More recently, “tree-based” wavelet denoising methods were developed in the context of image denoising, which exploit the tree structure of wavelet coefficients and the so-called parent-child correlations which are present in wavelet coefficients of images with edges. Also, many investigators have experimented with variations on the basic schemes—modifications of thresholding functions, level-dependent thresholding, block thresholding, adaptive choice of threshold, Bayesian conditional expectation nonlinearities, and so on. Extensive efforts by a large number of researchers have produced a body of literature which exhibits substantial progress overall, achieved by combining a sequence of incremental improvements.