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66
Fast Discrete Curvelet Transforms
, 2005
"... This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrap ..."
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Cited by 170 (9 self)
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This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n 2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at
Ratedistortion optimized tree structured compression algorithms for piecewise smooth images
 IEEE Trans. Image Processing
, 2005
"... IEEE Transactions on Image Processing This paper presents novel coding algorithms based on tree structured segmentation, which achieve the correct asymptotic ratedistortion (RD) behavior for a simple class of signals, known as piecewise polynomials, by using an RD based prune and join scheme. Fo ..."
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Cited by 76 (16 self)
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IEEE Transactions on Image Processing This paper presents novel coding algorithms based on tree structured segmentation, which achieve the correct asymptotic ratedistortion (RD) behavior for a simple class of signals, known as piecewise polynomials, by using an RD based prune and join scheme. For the one dimensional (1D) case, our scheme is based on binary tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an RD optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying RD behavior � D(R) ∼ c02 −c1R � , thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of O (N log N). We then show the extension of this scheme to the two dimensional (2D) case using a quadtree. This quadtree coding scheme also achieves an exponentially decaying RD behavior, for the polygonal image model composed of a white polygon shaped object against a uniform black background, with low computational cost of O (N log N). Again, the key is an RD optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp.
Nearoptimal detection of geometric objects by fast multiscale methods
 IEEE Trans. Inform. Theory
, 2005
"... Abstract—We construct detectors for “geometric ” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detec ..."
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Cited by 43 (10 self)
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Abstract—We construct detectors for “geometric ” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detection threshold—i.e., the signal strength below which no method of detection can be successful for large dataset size. ii) The optimal computational complexity of a nearoptimal detector, i.e., the complexity required to detect signals slightly exceeding the detection threshold. We describe a general approach to such problems which covers several classes of geometrically defined signals; for example, with onedimensional data, signals having elevated mean on an interval, and, indimensional data, signals with elevated mean on a rectangle, a ball, or an ellipsoid. In all these problems, we show that a naive or straightforward approach leads to detector thresholds and algorithms which are asymptotically far away from optimal. At the same time, a multiscale geometric analysis of these classes of objects allows us to derive asymptotically optimal detection thresholds and fast algorithms for nearoptimal detectors. Index Terms—Beamlets, detecting hot spots, detecting line segments, Hough transform, image processing, maxima of Gaussian processes, multiscale geometric analysis, Radon transform. I.
FAST COMPUTATION OF FOURIER INTEGRAL OPERATORS
, 2007
"... We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operat ..."
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Cited by 39 (10 self)
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We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operator (FIO) of the form ∫ e2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) isan amplitude, and Φ(x, ξ) is a phase function, which is typically as large as ξ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size N by N would require O(N 4) operations. This paper develops a new numerical algorithm which requires O(N 2.5 log N) operations and as low as O ( √ N) in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e2πiΦ(x,ξ) a(x, ξ) into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.
Multiscale Wedgelet Image Analysis: Fast Decompositions And Modeling
 in IEEE Int. Conf. on Image Proc. – ICIP ’02
, 2002
"... The most perceptually important features in images are geometrical, the most prevalent being the smooth contours ("edges") that separate different homogeneous regions and delineate distinct objects. Although wavelet based algorithms have enjoyed success in many areas of image processing, t ..."
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Cited by 24 (8 self)
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The most perceptually important features in images are geometrical, the most prevalent being the smooth contours ("edges") that separate different homogeneous regions and delineate distinct objects. Although wavelet based algorithms have enjoyed success in many areas of image processing, they have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and as a result wavelet based processing algorithms often produce images with ringing around the edges.
ADAPTIVE MULTISCALE DETECTION OF FILAMENTARY STRUCTURES IN A BACKGROUND OF UNIFORM RANDOM POINTS 1
, 2003
"... We are given a set of n points that might be uniformly distributed in the unit square [0,1] 2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C αnorm bounded by β. An ..."
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Cited by 23 (6 self)
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We are given a set of n points that might be uniformly distributed in the unit square [0,1] 2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C αnorm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α,β)> 0, if the number of points sampled from the curve is smaller than T−(α,β)n 1/(1+α) , reliable detection is not possible for large n. We describe a multiscale significantruns algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α,β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will
Searching for a Trail of Evidence in a Maze
, 2007
"... Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables ..."
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Cited by 20 (3 self)
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Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0, 1) while under the alternative, there is an unknown path along which random variables have distribution N(µ, 1), µ> 0, and distribution N(0, 1) away from it. For which values of the mean shift µ can one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j +s) where s = ±1. We show that for paths of length m starting at the origin, the hypotheses become distinguishable (in a minimax sense) if µm ≫ √ log m, while they are not if µm ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold is µm ≈ m −1/4. We
Edgepreserving image denoising and estimation of discontinuous surfaces
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... In this paper, we are interested in the problem of estimating a discontinuous surface from noisy data. A novel procedure for this problem is proposed based on local linear kernel smoothing, in which local neighbourhoods are adapted to the local smoothness of the surface measured by the observed data ..."
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Cited by 17 (7 self)
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In this paper, we are interested in the problem of estimating a discontinuous surface from noisy data. A novel procedure for this problem is proposed based on local linear kernel smoothing, in which local neighbourhoods are adapted to the local smoothness of the surface measured by the observed data. The procedure can therefore remove noise correctly in continuity regions of the surface, and preserve discontinuities at the same time. Since an image can be regarded as a surface of the image intensity function and such a surface has discontinuities at the outlines of objects, this procedure can be applied directly to image denoising. Numerical studies show that it works well in applications, compared to some existing procedures. Index Terms Corners, edges, jumppreserving estimation, local linear fit, noise, nonparametric regression,
Coarsetofine manifold learning
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing — ICASSP ’04
, 2004
"... In this paper we consider a sequential, coarsetofine estimation of a piecewise constant function with smooth boundaries. Accurate detection and localization of the boundary (a manifold) is the key aspect of this problem. In general, algorithms capable of achieving optimal performance require exhau ..."
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Cited by 17 (5 self)
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In this paper we consider a sequential, coarsetofine estimation of a piecewise constant function with smooth boundaries. Accurate detection and localization of the boundary (a manifold) is the key aspect of this problem. In general, algorithms capable of achieving optimal performance require exhaustive searches over large dictionaries that grow exponentially with the dimension of the observation domain. The computational burden of the search hinders the use of such techniques in practice, and motivates our work. We consider a sequential, coarsetofine approach that involves first examining the data on a coarse grid, and then refining the analysis and approximation in regions of interest. Our estimators involve an almost lineartime (in two dimensions) sequential search over the dictionary, and converge at the same nearoptimal rate as estimators based on exhaustive searches. Specifically, for two dimensions, our algorithm requires O(n 7/6) operations for an npixel image, much less than the traditional wedgelet approaches, which require O(n 11/6) operations. 1
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