Wavelet-Based Analysis and Estimation of Colored Noise

Goossens, Bart; Aelterman, Jan; Luong, Hiêp; Pižurica, Aleksandra; Philips, Wilfried

doi:10.5772/22839

Cited by 5 publications

(6 citation statements)

References 51 publications

(37 reference statements)

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“…Instead, OAMP demonstrates the generality of its state evolution using three arbitrary chosen values c k = 1, 2, 3. For the algorithm presented in this work, two c k updates are suggested, stated in (24) and (26), which are motivated by VAMP [22] and computed by Stein's Unbiased Risk Estimate (SURE) [24], [25] respectively.…”

Section: A Approximate Message Passingmentioning

confidence: 99%

“…The intrinsically colored aliasing of variable density sampling from a non-uniform spectral density implies that the white state evolution of AMP, OAMP, and VAMP, (4), cannot be relied upon. The primary development of this work is based on the use of the Discrete Wavelet Transform (DWT) to compute a multiresolution decomposition of the power spectrum of the aliasing, as used for colored noise analysis in [26]- [28]. In the wavelet domain, colored aliasing has a structure that resembles a state evolution.…”

Section: B Colored Aliasingmentioning

confidence: 99%

See 1 more Smart Citation

Approximate Message Passing With a Colored Aliasing Model for Variable Density Fourier Sampled Images

Millard¹,

Hess²,

Mailhé³

et al. 2020

IEEE Open J. Signal Process.

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The Approximate Message Passing (AMP) algorithm efficiently reconstructs signals which have been sampled with large i.i.d. sub-Gaussian sensing matrices. Central to AMP is its "state evolution", which guarantees that the difference between the current estimate and ground truth (the "aliasing") at every iteration obeys a Gaussian distribution that can be fully characterized by a scalar. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the aliasing is intrinsically colored, AMP's scalar state evolution is no longer accurate and the algorithm encounters convergence problems. In response, we propose the Variable Density Approximate Message Passing (VDAMP) algorithm, which uses the wavelet domain to model the colored aliasing. We present empirical evidence that VDAMP obeys a "colored state evolution", where the aliasing obeys a Gaussian distribution that can be fully characterized with one scalar per wavelet subband. A benefit of state evolution is that Stein's Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with subbanddependent thresholding that has no free parameters. We empirically evaluate the effectiveness of VDAMP on three variations of Fast Iterative Shrinkage-Thresholding (FISTA) and find that it converges in around 10 times fewer iterations on average than the next-fastest method, and typically a lower mean-squared-error.

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Section: A Approximate Message Passingmentioning

confidence: 99%

Section: B Colored Aliasingmentioning

confidence: 99%

Approximate Message Passing With a Colored Aliasing Model for Variable Density Fourier Sampled Images

Millard¹,

Hess²,

Mailhé³

et al. 2020

IEEE Open J. Signal Process.

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“…In practical applications, colored noise appears in phase alternating line television, color interpolation (demosaicing), image postprocessing techniques such as JPEG compression, thermal cameras, and medical imaging; see, for instance, [12]. The white noise is assumed to be zero mean and the covariance matrix is equal to the identity matrix, i.e., there is no correlation in the noise vector.…”

Section: Image Restoration With Gaussian Colored Noisementioning

confidence: 99%

“…The colored noise is usually assumed to be zero mean, but the noise vector has a nontrivial covariance matrix. The power spectral density function of the colored noise is equal to the power spectral density function times the squares of the filter magnitude response function [12]. Therefore, the colored noise can be viewed as white noise subjected to linear filtering.…”

Section: Image Restoration With Gaussian Colored Noisementioning

confidence: 99%

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Ng¹,

Pan²

2014

SIAM J. Sci. Comput.

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The main aim of this paper is to develop a fast algorithm for solving weighted Toeplitz regularized least squares problems arising from image restoration. Based on augmented system formulation, we develop new Hermitian and skew-Hermitian splitting (HSS) preconditioners for solving such linear systems. The advantage of the proposed preconditioner is that the blurring matrix, weighting matrix, and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use. We show that for a preconditioned system that is derived from a saddle point structure of size (m + n) × (m + n), the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 − λ with |λ| < 1. We also study how to choose the HSS parameter to minimize the magnitude of λ, and therefore the Krylov subspace method applied to solving the preconditioned system converges very quickly. Experimental results for image restoration problems are reported to demonstrate that the performance of the proposed preconditioner is better than the other testing preconditioners.

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“…This model is nevertheless not signal dependent. Another wavelet based method is [9]. The recent method for estimating frequency dependent noise on patches in [10] is probably the closest to our endeavor.…”

mentioning

confidence: 97%

Nonparametric Multiscale Blind Estimation of Intensity-Frequency-Dependent Noise

Colom¹,

Lebrun²,

Buades

et al. 2015

IEEE Trans. on Image Process.

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The camera calibration parameters and the image processing chain which generated a given image are generally not available to the receiver. This happens for example with scanned photographs and for most JPEG images. These images have undergone various nonlinear contrast changes and also linear and nonlinear filters. To deal with remnant noise in such images, we introduce a general nonparametric intensity and frequency-dependent noise model. We demonstrate by simulated and experiments with real images that this model, which requires the estimation of more than 1000 parameters, performs an efficient noise estimation. The proposed noise model is a patch model. Its estimation can therefore be used as a preliminary step to any patch-based denoising method. Our noise estimation method introduces several new tools for performing this complex estimation. One of them is a new sparse patch distance function permitting to find noisy patches with similar underlying geometry. A validation of the noise model and of its estimation method is obtained by comparing its results to ground-truth noise curves for both raw and JPEG-encoded images, and by visual inspection of the denoising results of real images. A fair comparison with the state of the art is also performed.

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Wavelet-Based Analysis and Estimation of Colored Noise

Cited by 5 publications

References 51 publications

Approximate Message Passing With a Colored Aliasing Model for Variable Density Fourier Sampled Images

Approximate Message Passing With a Colored Aliasing Model for Variable Density Fourier Sampled Images

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Nonparametric Multiscale Blind Estimation of Intensity-Frequency-Dependent Noise

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