Sparse Stochastic Processes and Discretization of Linear Inverse Problems

Bostan, Emrah; Kamilov, Ulugbek S.; Nilchian, Masih; Unser, Michaël

doi:10.1109/tip.2013.2255305

Cited by 38 publications

(36 citation statements)

References 39 publications

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“…It therefore implies that a small fraction of all coefficients of a sparse signal carries most of its energy. This definition highlights the limitations of Gaussian priors and suggests using more refined statistical models to tackle practical image-processing tasks [3].…”

Section: Introductionmentioning

confidence: 99%

Statistics of wavelet coefficients for sparse self-similar images

Fageot

Bostan

Unser

2014

2014 IEEE International Conference on Image Processing (ICIP)

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We study the statistics of wavelet coefficients of non-Gaussian images, focusing mainly on the behaviour at coarse scales. We assume that an image can be whitened by a fractional Laplacian operator, which is consistent with an ∥ω∥ −γ spectral decay. In other words, we model images as sparse and self-similar stochastic processes within the framework of generalised innovation models. We show that the wavelet coefficients at coarse scales are asymptotically Gaussian even if the prior model for fine scales is sparse. We further refine our analysis by deriving the theoretical evolution of the cumulants of wavelet coefficients across scales. Especially, the evolution of the kurtosis supplies a theoretical prediction for the Gaussianity level at each scale. Finally, we provide simulations and experiments that support our theoretical predictions.

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Section: Introductionmentioning

confidence: 99%

Statistics of wavelet coefficients for sparse self-similar images

Fageot

Bostan

Unser

2014

2014 IEEE International Conference on Image Processing (ICIP)

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“…Finally, the desired signal is reconstructed from the sub-signals by K-means clustering algorithm. Bostan and Kamilov et al [25] proposed a novel statistically-based discretization paradigm and derive a class of maximum a posterior (MAP) estimators for solving ill-conditioned linear inverse problems. It proposes the theory of sparse stochastic processes, which specifies the continuous -domain signals as solutions of linear stochastic differential equations.…”

Section: Work Related To Sparse Signal Processing-based Multipath MImentioning

confidence: 99%

Review on Sparse-Based Multipath Estimation and Mitigation: Intense Solution to Counteract the Effects in Software GPS Receivers

Elango¹,

Kumar²,

Kumar³

et al. 2018

Multifunctional Operation and Application of GPS

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Multipath is the major concern in GPS receivers that fade the actual GPS signal causes positioning error up to 10 m so special care need to be taken to mitigate the multipath effects. Numerous methods like hardware based antenna arrays technique, receiver based narrow correlator receiver, double -delta discriminator, Adaptive Multipath Estimator, Wavelet Transformation and Particle filter, Kalman filter based post receiver methods etc. used to resolve the problem. But some of the methods can only reduce code multipath error but not effective in eliminating carrier multipath error. Most of these techniques are based on the assumption that the Line-of-Sight (LOS) signal is stronger than the Non-Line of-Sight (NLOS) signals. However, in the scenarios where the LOS signal is weaker than the composite multipath signal, this approach may result in a bias in code tracking. In this chapter, different types of multipath mitigation and its limitation are described. The recent development in sparse signal processing based blind channel estimation is investigated to compensate the multipath error. The Rayleigh and Rician fading model with different multipath parameters are simulated to test the urban scenario. The inverse problem of finding the GPS signal is addressed based on the deconvolution approach. To solve linear inverse problems, the suitable kind of appropriate objective function has been formulated to find the signal of interest. By exploiting this methods, the signal is observed and the carrier and code tracking loop parameters are computed with minimal error.

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“…Assuming periodic boundary conditions, this is directly solved by using the fast Fourier transform [10]. The final step of the ADMM is a trivial update.…”

Section: Phase Reconstructionmentioning

confidence: 99%

Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy

Bostan

Froustey

Rappaz

et al. 2014

2014 IEEE International Conference on Image Processing (ICIP)

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We present a variational reconstruction algorithm for the phaseretrieval problem by using the differential interference contrast microscopy. Principally, we rely on the transport-of-intensity equation that specifies the sought phase as the solution of a partial differential equation. Our approach is based on an iterative reconstruction algorithm involving the total variation regularisation which is efficiently solved via the alternating direction method of multipliers. We illustrate the applicability of the method via real data experiments. To the best of our knowledge, this work demonstrates the performance of such an iterative algorithm on real data for the first time.

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Sparse Stochastic Processes and Discretization of Linear Inverse Problems

Cited by 38 publications

References 39 publications

Statistics of wavelet coefficients for sparse self-similar images

Statistics of wavelet coefficients for sparse self-similar images

Review on Sparse-Based Multipath Estimation and Mitigation: Intense Solution to Counteract the Effects in Software GPS Receivers

Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy

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