Showing papers by "Emmanuel J. Candès published in 2001"

The curvelet transform for image denoising

Abstract: Summary form only given, as follows. We present 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. 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.

Very high quality image restoration by combining wavelets and curvelets

Abstract: We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We develop a methodology to combine wavelets together these new systems to perform noise removal by exploiting all these systems simultaneously. The results of the combined reconstruction exhibits clear advantages over any individual system alone. For example, the residual error contains essentially no visually intelligible structure: no structure is lost in the reconstruction.

Ridgelets and the Representation of Mutilated Sobolev Functions

Abstract: We show that ridgelets, a system introduced in [E. J. Candes, Appl. Comput. Harmon. Anal., 6 (1999), pp. 197--218], are optimal to represent smooth multivariate functions that may exhibit linear singularities. For instance, let $\{u\cdot x - b >0\}$ be an arbitrary hyperplane and consider the singular function $f(x) = 1_{\{u\cdot x - b > 0\}} g(x)$, where g is compactly supported with finite Sobolev L2 norm $\|g\|_{H^s}$, $s > 0$. The ridgelet coefficient sequence of such an object is as sparse as if f were without singularity, allowing optimal partial reconstructions. For instance, the n-term approximation obtained by keeping the terms corresponding to the n largest coefficients in the ridgelet series achieves a rate of approximation of order n-s/d ; the presence of the singularity does not spoil the quality of the ridgelet approximation. This is unlike all systems currently in use, especially Fourier or wavelet representations.