Results 1  10
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18
Stochastic Models for Sparse and PiecewiseSmooth Signals
"... Abstract—We introduce an extended family of continuousdomain stochastic models for sparse, piecewisesmooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classica ..."
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Cited by 22 (17 self)
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Abstract—We introduce an extended family of continuousdomain stochastic models for sparse, piecewisesmooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finiterateofinnovation signals within Gelfand’s framework of generalized stochastic processes. We then focus on the class of scaleinvariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, splinetype signals that are piecewisesmooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the sametype spectral signature. We prove that the generalized Poisson processes have a sparse representation in a waveletlike basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TVdenoising algorithm. Index Terms—Fractals, innovation models, Poisson processes, sparsity, splines, stochastic differential equations, stochastic processes,
Invariances, Laplacianlike wavelet bases, and the whitening of fractal processes
, 2009
"... In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fraction ..."
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Cited by 11 (8 self)
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In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBmlike processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.
Leftinverses of fractional Laplacian and sparse stochastic processes
, 2012
"... The fractional Laplacian (−△) γ/2 commutes with the primary coordination transformations in the Euclidean space Rd:dilation,translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 <γ <d, itsinverseistheclassicalRieszpotentialIγ which is dilationinvar ..."
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Cited by 7 (6 self)
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The fractional Laplacian (−△) γ/2 commutes with the primary coordination transformations in the Euclidean space Rd:dilation,translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 <γ <d, itsinverseistheclassicalRieszpotentialIγ which is dilationinvariant and translationinvariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential Iγ to any noninteger number γ larger than d and show that it is the unique leftinverse of the fractional Laplacian (−△) γ/2 which is dilationinvariant and translationinvariant. We observe that, for any 1 ≤ p ≤∞and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that Iγ f is not pintegrable. We then introduce the new unique leftinverse Iγ,p of the fractional Laplacian (−△) γ/2 with the property that Iγ,p is dilationinvariant (but not translationinvariant) and that Iγ,p f is pintegrable for any Schwartz function f. We finally apply that linear operator Iγ,p with p = 1 to solve the stochastic partial differential equation (−△) γ/2 � = w with white Poisson noise as its driving term w.
Bayesian Estimation for ContinuousTime Sparse Stochastic Processes
"... Abstract—We consider continuoustime sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal inbetween (interpolation problem). By relying on tools from the theory of splines, we derive ..."
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Cited by 6 (6 self)
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Abstract—We consider continuoustime sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal inbetween (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum meansquare error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with wellknown techniques for the recovery of sparse signals, such as the norm and Log ( relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.
SelfSimilarity: Part I  Splines and Operators
 IEEE TRANS. SIGNAL PROCESS
, 2007
"... The central theme of this pair of papers (Parts I and II in this issue) is selfsimilarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of Lsplines; these are defined in the following terms: @ A is a c ..."
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Cited by 1 (0 self)
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The central theme of this pair of papers (Parts I and II in this issue) is selfsimilarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of Lsplines; these are defined in the following terms: @ A is a cardinal Lspline iff v @ A a ‘ “ @ A, where L is a suitable pseudodifferential operator. Our starting point for the construction of “selfsimilar” splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a twoparameter family of generalized fractional derivatives,, where is the order of the derivative and is an additional phase factor. We specify the corresponding Lsplines, which yield an extended class of fractional splines. The operator is used to define a scaleinvariant energy measure—the squared Pnorm of the th derivative of the signal—which provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order P, which admits a stable representation in a Bspline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order P. We also establish a formal link between the regularization parameter and the cutoff frequency of the smoothing spline filter: H P. Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions.
OPTIMAL INTERPOLATION OF FRACTIONAL BROWNIAN MOTION GIVEN ITS NOISY SAMPLES
"... We consider the problem of estimating a fractional Brownian motion known only from its noisy samples at the integers. We show that the optimal estimator can be expressed using a digital Wienerlike filter followed by a simple timevariant correction accounting for nonstationarity. Moreover, we pro ..."
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We consider the problem of estimating a fractional Brownian motion known only from its noisy samples at the integers. We show that the optimal estimator can be expressed using a digital Wienerlike filter followed by a simple timevariant correction accounting for nonstationarity. Moreover, we prove that this estimate lives in a symmetric fractional spline space and give a practical implementation for optimal upsampling of noisy fBm samples by integer factors. 1.
Wavelets and Advanced Biomedical Imaging
"... Abstract—Our purpose in this talk is to advocate the use of wavelets for advanced bioimaging. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a concise multiresolution representation of images and that they can be computed most efficiently. We then discuss a s ..."
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Abstract—Our purpose in this talk is to advocate the use of wavelets for advanced bioimaging. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a concise multiresolution representation of images and that they can be computed most efficiently. We then discuss a simple but remarkably effective imagedenoising procedure that essentially amounts to discarding small wavelet coefficients (softthresholding); we show that this type of algorithm is the solution of a variational problem that promotes sparse solutions. We argue that the underlying principle of wavelet regularization is a powerful concept that can be used advantageously in a variety of inverse imagereconstruction problems, including MRI and computed tomography. We illustrate our point by presenting a novel waveletbased deconvolution algorithm for 3D fluorescence microscopy, as well as some preliminary results for dynamic PET reconstruction. We will also discuss wavelet techniques for the analysis of functional MRI data and optical microscopy (extended depth of field). Index Terms—Wavelets, sparsity, denoising, softthresholding, regularization, deconvolution, image reconstruction
MMSE ESTIMATION OF SPARSE LÉVY PROCESSES 1 MMSE Estimation of Sparse Lévy Processes
"... Abstract—We investigate a stochastic signalprocessing framework for signals with sparse derivatives, where the samples of a Lévy process are corrupted by noise. The proposed signal model covers the wellknown Brownian motion and piecewiseconstant Poisson process; moreover, the Lévy family also ..."
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Abstract—We investigate a stochastic signalprocessing framework for signals with sparse derivatives, where the samples of a Lévy process are corrupted by noise. The proposed signal model covers the wellknown Brownian motion and piecewiseconstant Poisson process; moreover, the Lévy family also contains other interesting members exhibiting heavytail statistics that fulfill the requirements of compressibility. We characterize the maximumaposteriori probability (MAP) and minimum meansquare error (MMSE) estimators for such signals. Interestingly, some of the MAP estimators for the Lévy model coincide with popular signaldenoising algorithms (e.g., totalvariation (TV) regularization). We propose a novel noniterative implementation of the MMSE estimator based on the beliefpropagation algorithm performed in the Fourier domain. Our algorithm takes advantage of the fact that the joint statistics of general Lévy processes are much easier to describe by their characteristic function, as the probability densities do not always admit closedform expressions. We then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradientsparse signals. Index Terms—Lévy process, stochastic modeling, sparsesignal estimation, non linear reconstruction, totalvariation estimation, belief propagation (BP), message passing. I.
1Bayesian Estimation for ContinuousTime Sparse Stochastic Processes
"... Abstract—We consider continuoustime sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal inbetween (interpolation problem). By relying on tools from the theory of splines, we derive ..."
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Abstract—We consider continuoustime sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal inbetween (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum meansquare error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with wellknown techniques for the recovery of sparse signals, such as the `1 norm and Log (`1`0 relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.