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Video Orbits of the Projective Group: A Simple Approach to Featureless Estimation of Parameters
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 1997
"... We present direct featureless methods for estimating the eight parameters of an "exact" projective (homographic) coordinate transformation to register pairs of images, together with the application of seamlessly combining a plurality of images of the same scene, resulting in a single image ..."
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Cited by 98 (9 self)
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We present direct featureless methods for estimating the eight parameters of an "exact" projective (homographic) coordinate transformation to register pairs of images, together with the application of seamlessly combining a plurality of images of the same scene, resulting in a single image (or new image sequence) of greater resolution or spatial extent. The approach is "exact" for two cases of static scenes: 1) images taken from the same location of an arbitrary threedimensional (3D) scene, with a camera that is free to pan, tilt, rotate about its optical axis, and zoom, or 2) images of a flat scene taken from arbitrary locations. The featureless projective approach generalizes interframe camera motion estimation methods that have previously used an affine model (which lacks the degrees of freedom to "exactly" characterize such phenomena as camera pan and tilt) and/or which have relied upon finding points of correspondence between the image frames. The featureless projective approach...
Fast matching pursuit with a multiscale dictionary of gaussian chirps
 IEEE Trans. Signal Process
, 2001
"... Abstract—We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximatedimensional signals with Gaussian chirps at a computational cost ( ) instead of the expected 2 log). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and ..."
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Cited by 38 (1 self)
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Abstract—We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximatedimensional signals with Gaussian chirps at a computational cost ( ) instead of the expected 2 log). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and chirp rate are locally optimized so as to get a “good ” chirp atom, i.e., one for which the correlation with the residual is locally maximized. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a subdictionary of local maxima of the Gaussian Gabor dictionary to accelerate the pursuit further. The efficiency and speed of the method is demonstrated on a sound signal. Index Terms—Adaptive signal processing, approximation methods, chirp modulation, complexity theory, frequency estimation, redundant systems, signal representations, time–frequency analysis. I.
Beyond timefrequency analysis: Energy densities in one and many dimensions
, 1998
"... Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and meas ..."
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Cited by 17 (4 self)
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Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the “A ” content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen’s general construction for joint distributions of arbitrary variables.
An historical account of the `WearComp' and `WearCam' inventions developed for applications in `Personal Imaging'
, 1997
"... We are entering a pivotal era in which we will become inextricably intertwined with computational technology that will become part of our everyday lives in a much more immediate and intimate way than in the past. The recent explosion of interest in socalled "wearable computers" is indicat ..."
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Cited by 14 (0 self)
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We are entering a pivotal era in which we will become inextricably intertwined with computational technology that will become part of our everyday lives in a much more immediate and intimate way than in the past. The recent explosion of interest in socalled "wearable computers" is indicative of this general trend. The purpose of this paper is to provide an historical account of my wearable computer effort, from the 1970s (WearComp0) to present (WearComp7), with emphasis on a particular variation whose origins were in imaging applications. This application, known as `personal imaging', originated as a computerized photographer's assistant which I developed for what many regarded as an obscure photographic technique. However, it later evolved into a more diverse apparatus and methodology, combining machine vision and computer graphics, in a wearable tetherless apparatus, useful in daytoday living. Personal imaging, at the intersection of art, science, and technology, has given rise t...
Detecting Highly Oscillatory Signals by Chirplet Path Pursuit
, 2006
"... This paper considers the problem of detecting nonstationary phenomena, and chirps in particular, from very noisy data. Chirps are waveforms of the very general form A(t) exp(iλ ϕ(t)), where λ is a (large) base frequency, the phase ϕ(t) is timevarying and the amplitude A(t) is slowly varying. Given ..."
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Cited by 14 (2 self)
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This paper considers the problem of detecting nonstationary phenomena, and chirps in particular, from very noisy data. Chirps are waveforms of the very general form A(t) exp(iλ ϕ(t)), where λ is a (large) base frequency, the phase ϕ(t) is timevarying and the amplitude A(t) is slowly varying. Given a set of noisy measurements, we would like to test whether there is signal or whether the data is just noise. One particular application of note in conjunction with this problem is the detection of gravitational waves predicted by Einstein’s Theory of General Relativity. We introduce detection strategies which are very sensitive and more flexible than existing feature detectors. The idea is to use structured algorithms which exploit information in the socalled chirplet graph to chain chirplets together adaptively as to form chirps with polygonal instantaneous frequency. We then search for the path in the graph which provides the best tradeoff between complexity and goodness of fit. Underlying our methodology is the idea that while the signal may be extremely weak so that none of the individual empirical coefficients is statistically significant, one can still reliably detect by combining several coefficients into a
Perspective projections in the spacefrequency plane and fractional Fourier transforms
 Journal of the Optical Society of America A
"... Perspective projections in the spacefrequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented. © 2000 Optical S ..."
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Cited by 6 (4 self)
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Perspective projections in the spacefrequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented. © 2000 Optical Society of America [S07403232(00)006128] OCIS codes: 100.0100, 150.0150.
Fast Ridge Pursuit with a multiscale dictionary of Gaussian chirps
"... We introduce a modied Matching Pursuit algorithm, called Fast Ridge Pursuit, to approximate Ndimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN 2 log N). At each iteration of the pursuit, the best Gabor atom is rst selected, then its scale and c ..."
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Cited by 6 (0 self)
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We introduce a modied Matching Pursuit algorithm, called Fast Ridge Pursuit, to approximate Ndimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN 2 log N). At each iteration of the pursuit, the best Gabor atom is rst selected, then its scale and chirp rate are locally optimized so as to get a good chirp atom. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a subdictionary of local maxima of the Gaussian Gabor dictionary, so as to accelerate further the pursuit. The eciency and speed of the method is demonstrated on a sound signal. EDICS : 2TIFR 1 Introduction There has been a considerable interest last decade in developing analysis techniques to decompose nonstationary signals into elementary components, called atoms, that characterize their salient features. As many signals display both oscillatory phenomena, whic...
A shattered survey of the Fractional Fourier Transform
, 2002
"... In this survey paper we introduce the reader to the notion of the fractional Fourier transform, which may be considered as a fractional power of the classical Fourier transform. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid inter ..."
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Cited by 6 (1 self)
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In this survey paper we introduce the reader to the notion of the fractional Fourier transform, which may be considered as a fractional power of the classical Fourier transform. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid interest in timefrequency analysis methods of signal processing, like wavelets. Like the complex exponentials are the basic functions in Fourier analysis, the chirps (signals sweeping through all frequencies in a certain interval) are the building blocks in the fractional Fourier analysis. Part of its roots can be found in optics where the fractional Fourier transform can be physically realized. We give an introduction to the definition, the properties and computational aspects of both the continuous and discrete fractional Fourier transforms. We include some examples of applications and some possible generalizations.
Approximate weak greedy algorithms
 Advances in Computational Mathematics
, 2001
"... We present a generalization of V. Temlyakov’s weak greedy algorithm, and give a sufficient condition for norm convergence of the algorithm for an arbitrary dictionary in a Hilbert space. We provide two counterexamples to show that the condition cannot be relaxed for general dictionaries. For a clas ..."
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Cited by 5 (3 self)
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We present a generalization of V. Temlyakov’s weak greedy algorithm, and give a sufficient condition for norm convergence of the algorithm for an arbitrary dictionary in a Hilbert space. We provide two counterexamples to show that the condition cannot be relaxed for general dictionaries. For a class of dictionaries with more structure, we give a more relaxed necessary and sufficient condition for convergence of the algorithm. We also provide a detailed discussion of how a “realworld ” implementation of the weak greedy algorithm, where one has to take into account floating point arithmetic and other types of finite precision errors, can be modeled by the new algorithm. Key Words: greedy algorithm, weak greedy algorithm, best mterm approximation, nonlinear approximation, numerical algorithm, computational complexity, redundant systems 1.