248 research outputs found
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page
Continuous Curvelet Transform: I. Resolution of the Wavefront Set
We discuss a Continuous Curvelet Transform (CCT), a transform f â Îf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b â R^2, and orientation θ â [0, 2Ď). The transform is defined by Îf (a, b, θ) = {f, Îłabθ} where
the inner products project f onto analyzing elements called curvelets Îł_(abθ) which are smooth and of rapid decay away from an a by âa rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to âtrackâ the
behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002).
We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Îf (a, x0, θ0) decays rapidly as a â 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of
Îf (a, x0, θ0) for fixed (x0, θ0), as a â 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Îf (a, x, θ) is not of rapid decay as a â 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the âdirectional parabolic square functionâ
S^m(x, θ) = ( Ę|Îf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1/2)
is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study
of Fourier Integral Operators. Smithâs transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their
similarities and differences in resolving the wavefront set
Continuous Curvelet Transform: II. Discretization and Frames
We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length â width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f â Î_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b â R^2, and orientation θ. The transform projects f onto a curvelet Îł_(abθ), yielding coefficient Î_f (a, b, θ) = f, _(Îłabθ); the corresponding curvelet Îł_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smithâs transform is based on true affine parabolic
scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smithâs transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smithâs transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales.
We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^âj, at equispaced intervals in direction, θ_(jâ), = 2Ď2^(âj/2)â, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the âCurvelets 2002â frame constructed by Emmanuel Candès et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but
at fine scales is based on sampling Smithâs transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smithâs continuous transform was intended for use in forming molecular decompositions
of Fourier Integral Operators (FIOâs). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smithâs results, proving that the discrete curvelet transform gives sparse representations of FIOâs of order zero. This yields an alternate proof of a recent result of Candès and Demanet about the sparsity of FIO representations in discrete curvelet frames
Does median filtering truly preserve edges better than linear filtering?
Image processing researchers commonly assert that "median filtering is better
than linear filtering for removing noise in the presence of edges." Using a
straightforward large- decision-theory framework, this folk-theorem is seen
to be false in general. We show that median filtering and linear filtering have
similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise
ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel
noise level in a digital signal. To see dramatic benefits of median smoothing
in an asymptotic setting, the per-pixel noise level should tend to zero (i.e.,
SNR should grow very large). We show that a two-stage median filtering using
two very different window widths can dramatically outperform traditional linear
and median filtering in settings where the underlying object has edges. In this
two-stage procedure, the first pass, at a fine scale, aims at increasing the
SNR. The second pass, at a coarser scale, correctly exploits the nonlinearity
of the median. Image processing methods based on nonlinear partial differential
equations (PDEs) are often said to improve on linear filtering in the presence
of edges. Such methods seem difficult to analyze rigorously in a
decision-theoretic framework. A popular example is mean curvature motion (MCM),
which is formally a kind of iterated median filtering. Our results on iterated
median filtering suggest that some PDE-based methods are candidates to
rigorously outperform linear filtering in an asymptotic framework.Comment: Published in at http://dx.doi.org/10.1214/08-AOS604 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
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