Generalized inverses in signal processing and remote sensing

Sibul, Leon H.; Dixon, T.L.; Hillsley, K.L.

doi:10.1109/acssc.1993.342565

Cited by 6 publications

(7 citation statements)

References 5 publications

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“…Transforms and ambiguity functions that were enumerated in the Introduction are coefficients of a continuous unitary group representation given by L/(x) = <f,U(x)g>, (18) where U(x) is the unitary representation of the appropriate group and < , > denotes the inner To derive the reproducing properties we consider a convolution operator on G and apply F-S-G theorem: fL/xLg'1 (x) (19) = fG' U(x)g > <g,U(y 'x)g>d p (x) (20) = IG 1' U(x)g> <U(y)g, U(x)g>d& (x) (21) = <I, U(y)g> I Qs12…”

Section: Reproducing Kernelsmentioning

confidence: 99%

<title>Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms</title>

Sibul

1995

SPIE Proceedings

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We use group representation theory to establish common theoretical foundation for wavelet, Fourier-Wigner, Gabor and short-time Fourier transforms as well as for the narrow and wideband ambiguity functions. These transforms are coefficients of unitary representations of either affine or Heisenberg groups. From this fact, many important properties of these transforms and ambiguity functions, including volume conservation and admissibility conditions, follow. In this paper we use a generalization of the Frobenius-Shur-Godement theorem (generalized resolution of identity) to derive the reproducing kernels associated with these transforms and ambiguity functions. This result has several new applications to the well-established reproducing kernel Hubert space (RKHS) theory. First of all, it establishes the conditions for the general resolution of identity and identifies spaces on which transforms are invertible. These results can be used to solve inverse problems that arise in remote sensing and characterization of stochastic propagation and scattering channels.Since reproducing kernels are positive definite functions, they can be used as approximating functions, analogously to the radial basis functions, for neural network expansions, interpolation and optimization.Because auto-wavelet and auto-Fourier-Wigner transforms are reproducing kernels on a well defined space of functions, we have a powerful method for generating a rich set of 2n dimensional positive definite functions for multi-dimensional interpolation, approximation, and sampling.

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Section: Reproducing Kernelsmentioning

confidence: 99%

<title>Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms</title>

Sibul

1995

SPIE Proceedings

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“…where p represents the continuum of delay and Doppler parameters with domain −, S s (p) is the random spreading function, E x is the transmitted energy [10,11], and R s (p,p) = EfS s (p)S ¤ s (p)g is the signal SF that combines the effects of the backscatter and propagation processes. 3 The signal generated by a collection of M discrete scatterers is…”

Section: The Linear Spreading Modelmentioning

confidence: 99%

“…This is equivalent to (11) and the · i are determined by partial fraction expansion. This situation would also be a rare occurrence as it implies that a subset of the signal elements (reflector responses) have exactly the same energy.…”

Section: Detection and False Alarm Probabilitiesmentioning

confidence: 99%

“…The detection statistic is formed by the unweighted sum of all of the M = ¿ l W bins only L · M of which contain signal energy under H 1 . A CF similar to (11) is then appropriate if the interference is uniform. The repeated factor of the CF expansion corresponds to the M ¡ L filters containing only interference.…”

Section: } D and } F For The Naive Receivermentioning

confidence: 99%

“…The pdfs are derived under the assumption that L · M filters contain signal energy with ® i = 1 + ¹ i . The remaining M ¡ L filters contain only uniform interference with ® 0 = 1 and the pdf model corresponding to the CFs (10), (11) with M, L substituted for L, K is appropriate.…”

Section: (H 0 )mentioning

confidence: 99%

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Detection by incoherent recombination with partial information

Ricker

Cutezo²

2001

IEEE Trans. Aerosp. Electron. Syst.

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Matched filter (MF) detection in spread environmentsis often seriously degraded by the mismatch between the waveform replica and the composite signal formed by the spreading environment. Typically the spreading is caused by multiple delayed reflections due to scatter extent or multipath especially in shallow water sonar applications. It is possible to recover some detector performance by incoherent summation of weighted MF realizations in a process called incoherent recombining (IR). Several IR strategies for Gaussian data that assume varying amounts of prior scattering function (SF) information are examined, their receiver operating characteristics (ROCs) computed, and compared with those of the unrealizable "prescient" receiver (PR). They include optimally weighted and unweighted versions of the maximum likelihood estimator-correlator (EC), and variations of the "at-least-one" (ALO) detector that examines sequences of MF realizations declaring a detection if at least one threshold is crossed. As might be expected, performance improves with the accuracy of the prior information incorporated in the detector formulation. He has conducted research on active sonar echolocation for 31 years with emphasis in detection and estimation, high resolution sonar imaging, scattering functions, waveform design, and wideband (wavelet) processing of sonar signals. His most recent work has been on techniques for echo detection and classification of sonar echos in spread channels based upon their space-time correlative properties. Dr. Ricker has issued over 275 reports and published extensively in IEEE Transactions, JASA, and JUA on various aspects of signal processing and echo location. He holds two patents in the signal processing area. Anthony J. Cutezo received the B.S.E.E. degree from the Pennsylvania State University (PSU), University Park, in 1975 and the M.S.E.E. degree from PSU in 1990.He has been with the Applied Research Laboratory at PSU since 1985 and is currently an Associate Research Engineer engaged in signal processing research. His interests include statistical detection and parameter estimation.

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A model-based estimator-correlator (EC) structure

Ricker

Cutezo

2000

IEEE Trans. Signal Process.

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Generalized inverses in signal processing and remote sensing

Cited by 6 publications

References 5 publications

<title>Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms</title>

<title>Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms</title>

Detection by incoherent recombination with partial information

A model-based estimator-correlator (EC) structure

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