Thierry Blu

Interpolation revisited [medical images application]

Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. The authors show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, the authors call their use generalized interpolation; they involve a prefiltering step when correctly applied. The authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts. The decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a B spline of the same order with another function that has none. This motivates the use of splines and spline-based functions as a tunable way to keep artifacts in check without any significant cost penalty. The authors discuss implementation and performance issues, and they provide experimental evidence to support their claims.

A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding

This paper introduces a new approach to orthonormal wavelet image denoising. Instead of postulating a statistical model for the wavelet coefficients, we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights. We then minimize an estimate of the mean square error between the clean image and the denoised one. The key point is that we have at our disposal a very accurate, statistically unbiased, MSE estimate-Steins unbiased risk estimate-that depends on the noisy image alone, not on the clean one. Like the MSE, this estimate is quadratic in the unknown weights, and its minimization amounts to solving a linear system of equations. The existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients. Instead, and contrary to the custom in the literature, these coefficients are not considered random any more. We describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near-optimal performance-both regarding quality and CPU requirement-by comparing it with the results of three state-of-the-art nonredundant denoising algorithms on a large set of test images. An interesting fallout of this study is the development of a new, group-delay-based, parent-child prediction in a wavelet dyadic tree

Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix

Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely . These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an RC circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling

Sparse Sampling of Signal Innovations

Sparse sampling of continuous-time sparse signals is addressed. In particular, it is shown that sampling at the rate of innovation is possible, in some sense applying Occams razor to the sampling of sparse signals. The noisy case is analyzed and solved, proposing methods reaching the optimal performance given by the Cramer-Rao bounds. Finally, a number of applications have been discussed where sparsity can be taken advantage of. The comprehensive coverage given in this article should lead to further research in sparse sampling, as well as new applications. One main application to use the theory presented in this article is ultra-wide band (UWB) communications.

The SURE-LET Approach to Image Denoising

We propose a new approach to image denoising, based on the image-domain minimization of an estimate of the mean squared error-Steins unbiased risk estimate (SURE). Unlike most existing denoising algorithms, using the SURE makes it needless to hypothesize a statistical model for the noiseless image. A key point of our approach is that, although the (nonlinear) processing is performed in a transformed domain-typically, an undecimated discrete wavelet transform, but we also address nonorthonormal transforms-this minimization is performed in the image domain. Indeed, we demonstrate that, when the transform is a ldquotightrdquo frame (an undecimated wavelet transform using orthonormal filters), separate subband minimization yields substantially worse results. In order for our approach to be viable, we add another principle, that the denoising process can be expressed as a linear combination of elementary denoising processes-linear expansion of thresholds (LET). Armed with the SURE and LET principles, we show that a denoising algorithm merely amounts to solving a linear system of equations which is obviously fast and efficient. Quite remarkably, the very competitive results obtained by performing a simple threshold (image-domain SURE optimized) on the undecimated Haar wavelet coefficients show that the SURE-LET principle has a huge potential.

Fractional Splines and Wavelets

We extend Schoenbergs family of polynomial splines with uniform knots to all fractional degrees

Image Denoising in Mixed Poisson–Gaussian Noise

\alpha>-1

Cardinal exponential splines: part I - theory and filtering algorithms

. These splines, which involve linear combinations of the one-sided power functions

Monte-Carlo Sure: A Black-Box Optimization of Regularization Parameters for General Denoising Algorithms

x_{+}^{\alpha}=\max(0,x)^{\alpha}

Quantitative Fourier analysis of approximation techniques. I. Interpolators and projectors

, are