Convergence theorems for the Non-Local Means filter

Jin, Qiyu; Grama, Ion; Liu, Quansheng

doi:10.3934/ipi.2018036

Cited by 14 publications

(10 citation statements)

References 15 publications

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“…It is worth noticing that similar convergence results have been established for Gaussian noise in [19,21], but with a different similarity function to adapt the Poisson noise, and the proofs are significantly different due to the different nature of noises and similarity functions; see the comments after the statements of Theorems 1, 2 and 3.…”

Section: Introductionmentioning

confidence: 76%

Poisson Shot Noise Removal by an Oracular Non-Local Algorithm

Jin

Grama²,

Liu³

2021

J Math Imaging Vis

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In this paper we address the problem of denoising images obtained under low light conditions for the Poisson shot noise model. Under such conditions the variance stabilization transform (VST) is no longer applicable, so that the state-of-the-art algorithms which are proficient for the additive white Gaussian noise cannot be applied. We first introduce an oracular non-local algorithm and prove its convergence with the optimal rate of convergence under a Hölder regularity assumption for the underlying image, when the search window size is suitably chosen. We also prove that the convergence remains valid when the oracle function is estimated within a prescribed error range. We then define a realisable filter by a statistical estimation of the similarity function which determines the oracle weight. The convergence of the realisable filter is justified by proving that the estimator of the similarity function lies in the prescribed error range with high probability. The experiments show that under low light conditions the proposed filter is competitive compared with the recent state-of-the-art algorithms.

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Section: Introductionmentioning

confidence: 76%

Poisson Shot Noise Removal by an Oracular Non-Local Algorithm

Jin

Grama²,

Liu³

2021

J Math Imaging Vis

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“…Jin et al. [26] establish convergence theorems for the non‐local means (NLM) filter in removing the additive Gaussian noise.Cheng et al. [27] propose to model the gradient distribution of natural images as spatially variant hyper‐Laplacian and the model is free from tedious tuning trade‐off parameters.…”

Section: Introductionmentioning

confidence: 99%

An image denoising algorithm based on adaptive clustering and singular value decomposition

Wang

et al. 2021

IET Image Processing

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Self‐similarity, a prior of natural images, has attracted much attention. The attribute means that low‐rank group matrices can be constructed from similar image patches. For low‐rank approximation denoising methods based on singular value decomposition (SVD) the ability to accurately construct group matrices with noise and handle singular values are keys. Here, combining image priors, a two‐stage clustering method to adaptively construct group matrices is designed. The method is anti‐noise, that is, when noise levels are high, these matrices are more accurate than that constructed by other algorithms. Then, according to the significance of singular values and singular vectors, singular vectors of the low‐rank estimations are corrected so that the residual noise in the low‐rank estimations is further suppressed. For back projection , the authors use the original noise level and the residual image to adaptively determine projection parameters and new noise levels . So, authors' back projection can provide a good foundation for authors' two‐stage denoising methods, better remove noise and preserve image details. Experimental results show that compared with the existing state‐of‐the‐art denoising algorithms, the proposed algorithm achieves competitive denoising performances in terms of quantitative metrics and preserving details. Especially with the increase of noise, the competitiveness of authors' algorithms is gradually enhanced.

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“…Theoretical analyses and simulation results of Jin et al [40, 41] suggest that larger sizes of the window from which the patches are extracted do not necessarily produce better denoising results. More precisely, there is an optimal value of the window size for the best denoising results [40, 41]. The probability of similarity of patches is higher and the similarity estimation is more accurate for the patches extracted from the same area than for the patches extracted from different areas of the same image.…”

Section: Introductionmentioning

confidence: 99%

“…The probability of similarity of patches is higher and the similarity estimation is more accurate for the patches extracted from the same area than for the patches extracted from different areas of the same image. Building on the results of that work [40, 41], we here propose a denoising method that is based on combining the NLPCA results obtained on different areas of the same image. The method first splits the given image into large pieces.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Poisson image denoising by piecewise principal component analysis and its application in single‐particle X‐ray diffraction imaging

Jin

Miyashita

Tama

et al. 2018

IET Image Processing

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This article describes an improved method for Poisson image denoising that is based on a state-of-the-art Poisson denoising approach known as Non-Local Principal Component Analysis (NLPCA). The new method is referred to as PieceWise Principal Component Analysis (PWPCA). In PWPCA, the given image is first split into pieces, then NLPCA is run on each image piece, and finally, the entire image is reconstituted by a weighted combination of the NLPCA-processed image pieces. Using standard test images with Poisson noise, we show that PWPCA restores images more effectively than state-of-the-art Poisson denoising approaches. In addition, and to the best of our knowledge, we show a first application of such approaches to singleparticle X-ray free-electron laser (XFEL) data. We show that the resolution of 3D reconstruction from XFEL diffraction images is improved when the data are preprocessed with PWPCA. XFELs are currently under rapid development to allow high-resolution biomolecular structure determination at near-physiological conditions. Data analysis methods developments follow these technological advances and are expected to have high impact in structural biology and drug design. This article contributes to these developments. As little experimental single-particle XFEL data is available still, the XFEL experiments shown here were performed with simulated data.

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Convergence theorems for the Non-Local Means filter

Cited by 14 publications

References 15 publications

Poisson Shot Noise Removal by an Oracular Non-Local Algorithm

Poisson Shot Noise Removal by an Oracular Non-Local Algorithm

An image denoising algorithm based on adaptive clustering and singular value decomposition

Poisson image denoising by piecewise principal component analysis and its application in single‐particle X‐ray diffraction imaging

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