Abstract—In this paper, we propose K-LLD: a patch-based, locally adaptive denoising method based on clustering the given noisy image into regions of similar geometric structure. In order to effectively perform such clustering, we employ as features the local weight functions derived from our earlier work on steering kernel regression [1]. These weights are exceedingly informative and robust in conveying reliable local structural information about the image even in the presence of significant amounts of noise. Next, we model each region (or cluster)—which may not be spatially contiguous—by “learning ” a best basis describing the patches within that cluster using principal components analysis. This learned basis (or “dictionary”) is then employed to optimally estimate the underlying pixel values using a kernel regression framework. An iterated version of the proposed algorithm is also presented which leads to further performance enhancements. We also introduce a novel mechanism for optimally choosing the local patch size for each cluster using Stein’s unbiased risk estimator (SURE). We illustrate the overall algorithm’s capabilities with several examples. These indicate that the proposed method appears to be competitive with some of the most recently published state of the art denoising methods. Index Terms—Clustering, dictionary learning, image denoising, kernel regression, principal component analysis, Stein’s unbiased risk estimator (SURE). I.

Across the field of inverse problems in image and video processing, nearly all algorithms have various parameters which need to be set in order to yield good results. In practice, usually the choice of such parameters is made empirically with trial and error if no ”ground-truth ” reference is available. Some analytical methods such as cross-validation and Stein’s unbiased risk estimate (SURE) have been successfully used to set such parameters. However, these methods tend to be strongly reliant on restrictive assumptions on the noise, and also computationally heavy. In this paper, we propose a metric Q which is based on singular value decomposition of local image gradients, and provides a quantitative measure of true image content (e.g. visually salient geometric structures such as edges etc.), in the presence of noise and other disturbances. This measure (1) is easy to compute (2) does not require the use of a reference image, (3) reacts reasonably to both blur and random noise, (4) works well even when the noise is not Gaussian. To illustrate its use in selection of algorithmic parameters, the proposed measure is used to automatically and effectively set the parameters of two leading image denoising algorithms. While the experimental focus of this paper is on optimizing denoising filters, the proposed metric can also be used for a large variety of other image and video restoration algorithms such as deblurring, superresolution, and more. In this paper, ample simulated and real data experiments illustrate the effectiveness of the proposed approach for denoising applications. For the sake of completeness, the statistical properties of the proposed metric Q in some special cases are also provided.

We introduce a novel algorithm to reconstruct dynamic MRI data from under-sampled k-t space data. In contrast to classical model based cine MRI schemes that rely on the sparsity or banded structure in Fourier space, we use the compact representation of the data in the Karhunen Louve transform (KLT) domain to exploit the correlations in the dataset. The use of the data-dependent KL transform makes our approach ideally suited to a range of dynamic imaging problems, even when the motion is not periodic. In comparison to current KLT-based methods that rely on a two-step approach to first estimate the basis functions and then use it for reconstruction, we pose the problem as a spectrally regularized matrix recovery problem. By simultaneously determining the temporal basis functions and its spatial weights from the entire measured data, the proposed scheme is capable of providing high quality reconstructions at a range of accelerations. In addition to using the compact representation in the KLT domain, we also exploit the sparsity of the data to further improve the recovery rate. Validations using numerical phantoms and in-vivo cardiac perfusion MRI data demonstrate the significant improvement in performance offered by the proposed scheme over existing methods. I.

Abstract—Non-local means (NLM) provides a powerful framework for denoising. However, there are a few parameters of the algorithm—most notably, the width of the smoothing kernel—that are data-dependent and difficult to tune. Here, we propose to use Stein’s unbiased risk estimate (SURE) to monitor the mean square error (MSE) of the NLM algorithm for restoration of an image corrupted by additive white Gaussian noise. The SURE principle allows to assess the MSE without knowledge of the noise-free signal. We derive an explicit analytical expression for SURE in the setting of NLM that can be incorporated in the implementation at low computational cost. Finally, we present experimental results that confirm the optimality of the proposed parameter selection. Index Terms—Denoising, non-local means, Stein’s unbiased risk estimate. I.

Abstract—Interpolation is the means by which a continuously defined model is fit to discrete data samples. When the data samples are exempt of noise, it seems desirable to build the model by fitting them exactly. In medical imaging, where quality is of paramount importance, this ideal situation unfortunately does not occur. In this paper, we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise. We resort to variational principles, which allow us to impose smoothness constraints on the model for tackling noisy data. Based on shift-, rotation-, and scale-invariant requirements on the model, we show that the-norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose. In addition to Tikhonov-like quadratic regularization, this includes edge-preserving total-variation-like (TV) regularization. We give algorithms to recover the continuously defined model from noisy samples and also provide a data-driven scheme to determine the optimal amount of regularization. We validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of TV-like nonquadratic regularization over Tikhonov-like quadratic regularization. Index Terms—Interpolation, regularization, regularization parameter, splines, Tikhonov functional, total-variation functional.

Foi Abstract—The removal of Poisson noise is often performed through the following three-step procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest. The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied. We introduce optimal inverses for the Anscombe transformation, in particular the exact unbiased inverse, a maximum likelihood (ML) inverse, and a more sophisticated minimum mean square error (MMSE) inverse. We then present an experimental analysis using a few state-of-theart denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the low-count regime. This results in a very ef cient ltering solution that is competitive with some of the best existing methods for Poisson image denoising. Index Terms—denoising, photon-limited imaging, Poisson noise, variance stabilization.

Abstract—Deblurring noisy Poisson images has recently been subject of an increasingly amount of works in many areas such as astronomy or biological imaging. In this paper, we focus on confocal microscopy which is a very popular technique for 3D imaging of biological living specimens which gives images with a very good resolution (several hundreds of nanometers), even though degraded by both blur and Poisson noise. Deconvolution methods have been proposed to reduce these degradations and we focus in this paper on techniques which promote the introduction of explicit prior on the solution. One difficulty of these techniques is to set the value of the parameter which weights the trade-off between the data term and the regularizing term. Actually, only few works have been devoted to the research of an automatic selection of this regularizing parameter when considering Poisson noise so it is often set manually such that it gives the best visual results. We present here two recent methods to estimate this regularizing parameter and we first propose an improvement of these estimators which takes advantage of confocal images. Following these estimators, we secondly propose to express the problem of Poisson noisy images deconvolution as the minimization of a new constrained problem. The proposed constrained formulation is well suited to this application domain since it is directly expressed using the anti log-likelihood of the Poisson distribution and therefore does not require any approximation. We show how to solve the unconstrained and constrained problem using the recent Alternating Direction technique and we present results on synthetic and real data using well-known priors such as Total Variation and wavelet transforms. Among these wavelet transforms, we specially focus on the Dual-Tree Complex Wavelet transform and on the dictionary composed of Curvelets and undecimated wavelet transform. Index Terms—3D confocal microscopy deconvolution, regularizing parameter selection, discrepancy principle, alternating direction method, constrained minimization, Poisson noise. I.

A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a one-to-one representation of signals, which is somewhat constrained but most efficient computationally. The latter are over-complete, but they offer advantages in terms of flexibility (shape of the basis functions) and shift-invariance. In this paper, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: 1) redundant wavelet analysis, 2) wavelet-domain processing, 3) projection of the results onto the wavelet basis, and 4) reconstruction of the signal from its nonredundant wavelet expansion. The wavelet analysis is pyramid-like and is obtained by simple modification of Mallat’s filterbank algorithm (e.g., suppression of the down-sampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of soft-threshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, a new quincunx Mexican-hat-like wavelet transform that is fully reversible and essentially behaves the th Laplacian of a Gaussian.

apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7366--FR+ENGinria-00509447, version 2- 4 Oct 2010Complex wavelet regularization for 3D confocal microscopy deconvolution

regularization for image recovery