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Consider classes of signals which have a ̄nite number of degrees of freedom per unit of time, and call this number the rate of innovation of a signal. Examples of signals with ̄nite rate of innovation include stream of Diracs (e.g. the Poisson process), non-uniform splines and piecewise polynomials. Eventhough these signals are not bandlimited, we show that(More)
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(More)
We propose a new approach to image denoising, based on the <i>image-domain minimization </i>of an estimate of the mean squared error-Stein's <i>unbiased risk estimate </i>(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,(More)
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. We show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, we call their use generalized(More)
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(More)
We consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white Gaussian noise. To achieve this, we propose to minimize <i>Stein's</i> <i>unbiased</i> <i>risk</i> <i>estimate</i> (SURE) which provides a means of assessing the true mean-squared error (MSE) purely from the measured data(More)
We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function phi(x). We first give the expression for the cases of phi's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support(More)
We propose a general methodology (PURE-LET) to design and optimize a wide class of transform-domain thresholding algorithms for denoising images corrupted by mixed Poisson-Gaussian noise. We express the denoising process as a linear expansion of thresholds (LET) that we optimize by relying on a purely data-adaptive unbiased estimate of the mean-squared(More)
We present a general Fourier-based method that provides an accurate prediction of the approximation error as a function of the sampling step T . Our formalism applies to an extended class of convolution-based signal approximation techniques, which includes interpolation, generalized sampling with prefiltering, and the projectors encountered in wavelet(More)
Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function(More)