890 research outputs found
On denoising modulo 1 samples of a function
Consider an unknown smooth function , and
say we are given noisy samples of , i.e., for , where denotes noise. Given the
samples our goal is to recover smooth, robust estimates
of the clean samples . We formulate a natural approach for
solving this problem which works with representations of mod 1 values over the
unit circle. This amounts to solving a quadratically constrained quadratic
program (QCQP) with non-convex constraints involving points lying on the unit
circle. Our proposed approach is based on solving its relaxation which is a
trust-region sub-problem, and hence solvable efficiently. We demonstrate its
robustness to noise % of our approach via extensive simulations on several
synthetic examples, and provide a detailed theoretical analysis.Comment: 19 pages, 13 figures. To appear in AISTATS 2018. Corrected typos, and
made minor stylistic changes throughout. Main results unchanged. Added
section I (and Figure 13) in appendi
Denoising modulo samples: k-NN regression and tightness of SDP relaxation
Many modern applications involve the acquisition of noisy modulo samples of a
function , with the goal being to recover estimates of the original samples
of . For a Lipschitz function , suppose we are
given the samples where
denotes noise. Assuming are zero-mean i.i.d Gaussian's, and
's form a uniform grid, we derive a two-stage algorithm that recovers
estimates of the samples with a uniform error rate holding with high probability. The first stage
involves embedding the points on the unit complex circle, and obtaining
denoised estimates of via a NN (nearest neighbor) estimator.
The second stage involves a sequential unwrapping procedure which unwraps the
denoised mod estimates from the first stage.
Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo
data which works with their representation on the unit complex circle. They
formulated a smoothness regularized least squares problem on the product
manifold of unit circles, where the smoothness is measured with respect to the
Laplacian of a proximity graph involving the 's. This is a nonconvex
quadratically constrained quadratic program (QCQP) hence they proposed solving
its semidefinite program (SDP) based relaxation. We derive sufficient
conditions under which the SDP is a tight relaxation of the QCQP. Hence under
these conditions, the global solution of QCQP can be obtained in polynomial
time.Comment: 34 pages, 6 figure
Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping
Consider an unknown smooth function , and
say we are given noisy mod 1 samples of , i.e., , for , where denotes the noise. Given
the samples , our goal is to recover smooth, robust
estimates of the clean samples . We formulate a natural
approach for solving this problem, which works with angular embeddings of the
noisy mod 1 samples over the unit circle, inspired by the angular
synchronization framework. This amounts to solving a smoothness regularized
least-squares problem -- a quadratically constrained quadratic program (QCQP)
-- where the variables are constrained to lie on the unit circle. Our approach
is based on solving its relaxation, which is a trust-region sub-problem and
hence solvable efficiently. We provide theoretical guarantees demonstrating its
robustness to noise for adversarial, and random Gaussian and Bernoulli noise
models. To the best of our knowledge, these are the first such theoretical
results for this problem. We demonstrate the robustness and efficiency of our
approach via extensive numerical simulations on synthetic data, along with a
simple least-squares solution for the unwrapping stage, that recovers the
original samples of (up to a global shift). It is shown to perform well at
high levels of noise, when taking as input the denoised modulo samples.
Finally, we also consider two other approaches for denoising the modulo 1
samples that leverage tools from Riemannian optimization on manifolds,
including a Burer-Monteiro approach for a semidefinite programming relaxation
of our formulation. For the two-dimensional version of the problem, which has
applications in radar interferometry, we are able to solve instances of
real-world data with a million sample points in under 10 seconds, on a personal
laptop.Comment: 68 pages, 32 figures. arXiv admin note: text overlap with
arXiv:1710.1021
Wavelets, ridgelets and curvelets on the sphere
We present in this paper new multiscale transforms on the sphere, namely the
isotropic undecimated wavelet transform, the pyramidal wavelet transform, the
ridgelet transform and the curvelet transform. All of these transforms can be
inverted i.e. we can exactly reconstruct the original data from its
coefficients in either representation. Several applications are described. We
show how these transforms can be used in denoising and especially in a Combined
Filtering Method, which uses both the wavelet and the curvelet transforms, thus
benefiting from the advantages of both transforms. An application to component
separation from multichannel data mapped to the sphere is also described in
which we take advantage of moving to a wavelet representation.Comment: Accepted for publication in A&A. Manuscript with all figures can be
downloaded at http://jstarck.free.fr/aa_sphere05.pd
Denoising modulo samples: k-NN regression and tightness of SDP relaxation
International audienceMany modern applications involve the acquisition of noisy modulo samples of a function f , with the goal being to recover estimates of the original samples of f. For a Lipschitz function f : [0, 1]^d → R, suppose we are given the samples y_i = (f (x_i) + η_i) mod 1; i = 1,. .. , n where η_i denotes noise. Assuming η_i are zero-mean i.i.d Gaussian's, and x_i 's form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples f (x_i) with a uniform error rate O((log n / n)^{1/(d+2)}) holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of f (x_i) mod 1 via a kNN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod 1 estimates from the first stage. The estimates of the samples f(x_i) can be subsequently utilized to construct an estimate of the function f, with the aforementioned uniform error rate. Recently, Cucuringu and Tyagi [7] proposed an alternative way of denoising modulo 1 data which works with their representation on the unit complex circle. They formulated a smoothness regularized least squares problem on the product manifold of unit circles, where the smoothness is measured with respect to the Laplacian of a proximity graph G involving the x_i 's. This is a nonconvex quadratically constrained quadratic program (QCQP) hence they proposed solving its semidefinite program (SDP) based relaxation. We derive sufficient conditions under which the SDP is a tight relaxation of the QCQP. Hence under these conditions, the global solution of QCQP can be obtained in polynomial time
Manifold-valued Image Generation with Wasserstein Generative Adversarial Nets
Generative modeling over natural images is one of the most fundamental
machine learning problems. However, few modern generative models, including
Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued
images that are frequently encountered in real-world applications. To fill the
gap, this paper first formulates the problem of generating manifold-valued
images and exploits three typical instances: hue-saturation-value (HSV) color
image generation, chromaticity-brightness (CB) color image generation, and
diffusion-tensor (DT) image generation. For the proposed generative modeling
problem, we then introduce a theorem of optimal transport to derive a new
Wasserstein distance of data distributions on complete manifolds, enabling us
to achieve a tractable objective under the WGAN framework. In addition, we
recommend three benchmark datasets that are CIFAR-10 HSV/CB color images,
ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we
experimentally demonstrate the proposed manifold-aware WGAN model can generate
more plausible manifold-valued images than its competitors.Comment: Accepted by AAAI 201
Spatially resolved stress measurements in materials with polarization-sensitive optical coherence tomography: image acquisition and processing aspects
We demonstrate that polarization-sensitive optical coherence tomography
(PS-OCT) is suitable to map the stress distribution within materials in a
contactless and non-destructive way. In contrast to transmission
photoelasticity measurements the samples do not have to be transparent but can
be of scattering nature. Denoising and analysis of fringe patterns in single
PS-OCT retardation images are demonstrated to deliver the basis for a
quantitative whole-field evaluation of the internal stress state of samples
under investigation.Comment: 10 pages, 6 figures; Copyright: Blackwell Publishing Ltd 2008; The
definitive version is available at: www.blackwell-synergy.co
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