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35
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 23 (8 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Symmetry of Matrix Algebras and Symbolic Calculus for Infinite Matrices
"... We investigate the symbolic calculus for a large class of matrix algebras that are defined by the offdiagonal decay of infinite matrices. Applications are given to the symmetry of highly noncommutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames. ..."
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Cited by 18 (3 self)
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We investigate the symbolic calculus for a large class of matrix algebras that are defined by the offdiagonal decay of infinite matrices. Applications are given to the symmetry of highly noncommutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.
Symmetry and inverseclosedness of matrix algebras and symbolic calculus for infinite matrices
 Trans. Amer. Math. Soc
"... Abstract. We investigate the symbolic calculus for a large class of matrix algebras that are defined by the offdiagonal decay of infinite matrices. Applications are given to the symmetry of some highly noncommutative Banach algebras, to the analysis of twisted convolution, and to the theory of loc ..."
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Cited by 15 (6 self)
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Abstract. We investigate the symbolic calculus for a large class of matrix algebras that are defined by the offdiagonal decay of infinite matrices. Applications are given to the symmetry of some highly noncommutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames. 1.
Sparsity in timefrequency representations
, 2007
"... We consider signals and operators in finite dimension which have sparse timefrequency representations. As main result we show that an Ssparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn / log(n) ..."
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Cited by 11 (6 self)
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We consider signals and operators in finite dimension which have sparse timefrequency representations. As main result we show that an Ssparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn / log(n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.
ENTROPY ENCODING, HILBERT SPACE AND KARHUNENLOÈVE TRANSFORMS
, 2007
"... Abstract. By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding. ..."
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Cited by 7 (7 self)
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Abstract. By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding.
A video time encoding machine
 in Proceedings of the 15th IEEE International Conference on Image Processing (ICIP ’08
, 2008
"... Time encoding is a realtime asynchronus mechanism of mapping analog amplitude information into multidimensional time sequences. We investigate the exact representation of analog video streams with a Time Encoding Machine realized with a population of spiking neurons. We also provide an algorithm th ..."
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Cited by 5 (5 self)
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Time encoding is a realtime asynchronus mechanism of mapping analog amplitude information into multidimensional time sequences. We investigate the exact representation of analog video streams with a Time Encoding Machine realized with a population of spiking neurons. We also provide an algorithm that perfectly recovers streaming video from the spike trains of the neural population. Finally, we analyze the quality of recovery of a spacetime separable video stream encoded with a population of integrateandfire neurons and demonstrate that the quality of recovery increases as a function of the population size. Index Terms — time encoding, video coding, integrateandfire neurons, frames, Gabor wavelets 1.
Population Encoding with HodgkinHuxley Neurons
, 2009
"... The recovery of (weak) stimuli encoded with a population of HodgkinHuxley neurons is investigated. In the absence of a stimulus, the HodgkinHuxley neurons are assumed to be tonically spiking. The methodology employed calls for (i) finding an I/O equivalent description of the HodgkinHuxley neuron ..."
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Cited by 4 (2 self)
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The recovery of (weak) stimuli encoded with a population of HodgkinHuxley neurons is investigated. In the absence of a stimulus, the HodgkinHuxley neurons are assumed to be tonically spiking. The methodology employed calls for (i) finding an I/O equivalent description of the HodgkinHuxley neuron and, (ii) devising a recovery algorithm for stimuli encoded with the I/O equivalent neuron(s). A HodgkinHuxley neuron with multiplicative coupling is I/O equivalent with an IntegrateandFire neuron with a variable threshold sequence. For bandlimited stimuli a perfect recovery of the stimulus can be achieved provided that a Nyquist rate condition is satisfied. A HodgkinHuxley neuron with additive coupling and deterministic conductances is I/O equivalent with a ProjectIntegrateandFire neuron that integrates a projection of the stimulus on the phase response curve. The stimulus recovery is formulated as a spline interpolation problem in the space of finite length bounded energy signals. A HodgkinHuxley
BANACH ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS AND THEIR ALMOST DIAGONALIZATION
, 710
"... Abstract. We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra A over a lattice Λ we associate a symbol class M ∞,A. Then every oper ..."
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Cited by 4 (3 self)
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Abstract. We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra A over a lattice Λ we associate a symbol class M ∞,A. Then every operator with a symbol in M ∞,A is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra A. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on L 2 (R d). If a version of Wiener’s lemma holds for A, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sjöstrand’s class and yields a new proof of a theorem of Beals about the Hörmander class S 0 0,0. 1.
NONLINEAR APPROXIMATION BY SUMS OF EXPONENTIALS AND TRANSLATES
"... In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequ ..."
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Cited by 4 (0 self)
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In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands, if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1–periodic window function as defined in Section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates, if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.
Consistent recovery of stimuli encoded with a neural ensemble
 in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’09
, 2009
"... We consider the problem of reconstructing finite energy stimuli from a finite number of contiguous spikes. The reconstructed signal satisfies a consistency condition: when passed through the same neuron, it triggers the same spike train as the original stimulus. The recovered stimulus has to also mi ..."
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Cited by 3 (3 self)
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We consider the problem of reconstructing finite energy stimuli from a finite number of contiguous spikes. The reconstructed signal satisfies a consistency condition: when passed through the same neuron, it triggers the same spike train as the original stimulus. The recovered stimulus has to also minimize a quadratic smoothness criterion. We show that under these conditions, the problem of recovery has a unique solution and provide an explicit reconstruction algorithm for stimuli encoded with a population of integrateandfire neurons. We demonstrate that the quality of reconstruction improves as the size of the population increases. Finally, we demonstrate the efficiency of our recovery method for an encoding circuit based on threshold spiking that arises in neuromorphic engineering. Index Terms — time encoding, spiking neurons, consistent recovery. 1.