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19
Sampling signals with finite rate of innovation
 IEEE Transactions on Signal Processing
, 2002
"... Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials ..."
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Cited by 350 (67 self)
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Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic “bandlimited and sinc kernel ” case. In particular, we show how to sample and reconstruct periodic and finitelength streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinitelength signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and errorcorrection coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems. Index Terms—Analogtodigital conversion, annihilating filters, generalized sampling, nonbandlimited signals, nonuniform splines, piecewise polynomials, poisson processes, sampling. I.
Sampling and reconstruction of signals with finite rate of innovation in the presence of noise
 IEEE Transactions on Signal Processing
, 2005
"... Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely, certain signals of finite rate of innovation [24]. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time an ..."
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Cited by 77 (2 self)
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Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely, certain signals of finite rate of innovation [24]. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, Vetterli et al. considered the case of deterministic, noiseless signals, and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become illconditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze in detail a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finitelength signals and develop schemes based on a Gaussian kernel, which avoid the problem of illconditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results.
Directionlets: Anisotropic Multidirectional Representation with Separable Filtering
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2004
"... In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges a ..."
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Cited by 58 (6 self)
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In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (MDIR) and anisotropic transform is required. We present a new latticebased perfect reconstruction and critically sampled anisotropic MDIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard twodimensional (2D) WT. The corresponding anisotropic basis functions (directionlets) have directional vanishing moments (DVM) along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation (NLA) of images, achieving the approximation power O(N −1.55), which is competitive to the rates achieved by the other oversampled transform constructions.
SuperResolution From Unregistered and Totally Aliased Signals Using Subspace Methods
, 2007
"... In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the analogtodigital (A/D) converter, etc. A lowpass filter is usually applied before the sampling operation to avoid aliasing. However, when multiple copies are ..."
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Cited by 28 (8 self)
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In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the analogtodigital (A/D) converter, etc. A lowpass filter is usually applied before the sampling operation to avoid aliasing. However, when multiple copies are available, it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal. If the different copies have unknown relative offsets, this is a nonlinear problem in the offsets and the signal coefficients. They are not easily separable in the set of equations describing the superresolution problem. Thus, we perform joint registration and reconstruction from multiple unregistered sets of samples. We give a mathematical formulation for the problem when there are sets of samples of a signal that is described by expansion coefficients. We prove that the solution of the registration and reconstruction problem is generically unique
Nonuniform average sampling and reconstruction in multiply generated shiftinvariant spaces
 Constr. Approx
"... Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals ..."
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Cited by 23 (12 self)
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Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals with finite rate of innovation, which includes finitelygenerated shiftinvariant spaces, twisted shiftinvariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with nonuniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, that has welllocalized average sampler, can be found to be welllocalized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case that the original signal comes from a finitelygenerated shiftinvariant space. average sampling, ideal sampling, signals with finite rate of innovation, shiftKey words. invariant spaces
Frames in spaces with finite rate of innovations
 Adv. Comput. Math
"... Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applic ..."
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Cited by 20 (14 self)
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Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space Vq(Φ, Λ) is generated by a family of welllocalized molecules Φ of similar size located on a relativelyseparated set Λ using ℓ q coefficients, and hence is locally finitelygenerated. Moreover that space Vq(Φ, Λ) includes finitelygenerated shiftinvariant spaces, spaces of nonuniform splines, and the twisted shiftinvariant space in Gabor (Wilson) system as its special cases. Use the welllocalization property of the generator Φ, we show that if the generator Φ is a frame for the space V2(Φ, Λ) and has polynomial (subexponential) decay, then its canonical dual (tight) frame has the same polynomial (subexponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator Φ for the space Vq(Φ, Λ) with q = 2, and of the polynomial (subexponential) decay property of the mask associated with a refinable function that has polynomial (subexponential) decay. Advances in Computational Mathematics, to appear 1.
Compression of ECG as Signal with Finite Rate of Innovation
"... Abstract — Compression of ECG (electrocardiogram) as a signal with finite rate of innovation (FRI) is proposed in this paper. By modelling the ECG signal as the sum of bandlimited and nonuniform linear spline which contains finite rate of innovation (FRI), sampling theory is applied to achieve effec ..."
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Cited by 11 (2 self)
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Abstract — Compression of ECG (electrocardiogram) as a signal with finite rate of innovation (FRI) is proposed in this paper. By modelling the ECG signal as the sum of bandlimited and nonuniform linear spline which contains finite rate of innovation (FRI), sampling theory is applied to achieve effective compression and reconstruction of ECG signal. The simulation results show that the performance of the compression of ECG as a signal with FRI is quite satisfactory in preserving the diagnostic information as compared to the classical sampling scheme which uses the sinc interpolation. I.
Reconstruction of Wireless UWB Pulses by Exponential Sampling
 Filter, Wireless Sensor Network
, 2010
"... Measurement and reconstruction of wireless pulses is an important scheme in wireless ultra wide band (UWB) technology. In contrary to the bandlimited analog signals, which can be recovered from evenly spaced samples, the reconstruction of the UWB pulses is a more demanding task. In this work we des ..."
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Cited by 3 (2 self)
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Measurement and reconstruction of wireless pulses is an important scheme in wireless ultra wide band (UWB) technology. In contrary to the bandlimited analog signals, which can be recovered from evenly spaced samples, the reconstruction of the UWB pulses is a more demanding task. In this work we describe an exponential sampling filter (ESF) for measurement and reconstruction of UWB pulses. The ESF is constructed from parallel filters, which has exponentially descending impulse response. A pole cancellation filter was used to extract the amplitudes and time locations of the UWB pulses from sequentially measured samples of the ESF output. We show that the amplitudes and time locations of p sequential UWB pulses can be recovered from the measurement of at least 2p samples from the ESF output. For perfect reconstruction the number of parallel filters in ESP should be 2p. We study the robustness of the method against noise and discuss the applications of the method.