Results 1  10
of
103
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
"... ..."
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
Abstract

Cited by 207 (22 self)
 Add to MetaCart
This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Oversampled Filter Banks
 IEEE Trans. Signal Processing
, 1998
"... Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a neces ..."
Abstract

Cited by 104 (2 self)
 Add to MetaCart
Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given. Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIR and give a parameterization of a class of filter banks satisfying this property. Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuoustime transforms and allow for very flexible design. We investigate relations of these filter banks to continuoustime filtering and illustrate the design flexibility by giving a procedure for designing maximally flat twochannel filter banks that yield highly regular wavelets with a given number of vanishing moments.
Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets StrangFix
 IEEE Trans. on Signal Processing
, 2007
"... Abstract—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. Cl ..."
Abstract

Cited by 92 (27 self)
 Add to MetaCart
Abstract—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 [34]. 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 circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling. Index Terms—Analogtodigital conversion, annihilating filter method, multiresolution approximations, sampling methods, splines, wavelets. I.
ForWaRD: FourierWavelet Regularized Deconvolution for IllConditioned Systems
 IEEE Trans. on Signal Processing
, 2002
"... We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inhere ..."
Abstract

Cited by 90 (2 self)
 Add to MetaCart
We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inherent in deconvolution, while the wavelet shrinkage exploits the wavelet do main's sparse representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approxi mate meansquarederror (MSE) metric and find that signals with sparser wavelet representa tions require less Fourier shrinkage. ForWaRD is applicable to all illconditioned deconvolution problems, unlike the purely waveletbased Wavelet Vaguelette Deconvolution (WVD), and its es timate features minimal ringing, unlike purely Fourierbased Wiener deconvolution. We analyze ForWaRD's MSE decay rate as the number of samples increases and demonstrate its improved performance compared to the optimal WVD over a wide range of practical samplelengths.
Quantitative Fourier Analysis of Approximation Techniques: Part II  Wavelets
 IEEE Trans. Signal Processing
, 1999
"... In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic ..."
Abstract

Cited by 63 (28 self)
 Add to MetaCart
In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is, in particular, true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the leastsquares approximation error. Another contribution is the application of these results to Bsplines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...
Framing Pyramids
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2003
"... In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal li ..."
Abstract

Cited by 50 (6 self)
 Add to MetaCart
In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous domain frames associated with the LP.
Compressed Sensing of Analog Signals in ShiftInvariant Spaces
, 2009
"... A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that on ..."
Abstract

Cited by 50 (33 self)
 Add to MetaCart
A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for lowrate sampling of continuoustime sparse signals in shiftinvariant (SI) spaces, generated by m kernels with period T. We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m/T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finitelength vectors, we consider sampling of analog signals for which no underlying finitedimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.
Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in splinelike spaces: the Lp theory
 Proc. Amer. Math. Soc
"... Abstract. We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense ” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of splinelike spaces in Lp (Rn). Moreover, the reconstruction can be implemented using fast alg ..."
Abstract

Cited by 42 (5 self)
 Add to MetaCart
Abstract. We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense ” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of splinelike spaces in Lp (Rn). Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical ShannonWhittaker sampling theorem on regular sampling and the PaleyWiener theorem on nonuniform sampling. 1.
General framework for consistent sampling in Hilbert spaces
 Int. J. Wavelets, Multiresolution, Inf. Process
, 2005
"... We introduce a general framework for consistent linear reconstruction in infinitedimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinitedimensional frames. As we show, the linear reconstruction sche ..."
Abstract

Cited by 32 (16 self)
 Add to MetaCart
We introduce a general framework for consistent linear reconstruction in infinitedimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinitedimensional frames. As we show, the linear reconstruction scheme coincides with the socalled oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.