890 research outputs found

    On denoising modulo 1 samples of a function

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    Consider an unknown smooth function f:[0,1]Rf: [0,1] \rightarrow \mathbb{R}, and say we are given nn noisymod1\mod 1 samples of ff, i.e., yi=(f(xi)+ηi)mod1y_i = (f(x_i) + \eta_i)\mod 1 for xi[0,1]x_i \in [0,1], where ηi\eta_i denotes noise. Given the samples (xi,yi)i=1n(x_i,y_i)_{i=1}^{n} our goal is to recover smooth, robust estimates of the clean samples f(xi)mod1f(x_i) \bmod 1. 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

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    Many modern applications involve the acquisition of noisy modulo samples of a function ff, with the goal being to recover estimates of the original samples of ff. For a Lipschitz function f:[0,1]dRf:[0,1]^d \to \mathbb{R}, suppose we are given the samples yi=(f(xi)+ηi)mod1;i=1,,ny_i = (f(x_i) + \eta_i)\bmod 1; \quad i=1,\dots,n where ηi\eta_i denotes noise. Assuming ηi\eta_i are zero-mean i.i.d Gaussian's, and xix_i's form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples f(xi)f(x_i) with a uniform error rate O((lognn)1d+2)O((\frac{\log n}{n})^{\frac{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(xi)mod1f(x_i)\bmod 1 via a kkNN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod 11 estimates from the first stage. Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo 11 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 GG involving the xix_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.Comment: 34 pages, 6 figure

    Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping

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    Consider an unknown smooth function f:[0,1]dRf: [0,1]^d \rightarrow \mathbb{R}, and say we are given nn noisy mod 1 samples of ff, i.e., yi=(f(xi)+ηi)mod1y_i = (f(x_i) + \eta_i)\mod 1, for xi[0,1]dx_i \in [0,1]^d, where ηi\eta_i denotes the noise. Given the samples (xi,yi)i=1n(x_i,y_i)_{i=1}^{n}, our goal is to recover smooth, robust estimates of the clean samples f(xi)mod1f(x_i) \bmod 1. 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 ff (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo 11 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

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    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

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    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

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    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

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    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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