Well-Posed Inhomogeneous Nonlinear Diffusion Scheme for Digital Image Denoising
TL;DR: An inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion is studied.
Abstract: We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. By incorporating edges found at multiple scales via an adaptive edge detector-based indicator function, the proposed scheme removes noise while respecting salient boundaries. We create a smooth transition region around probable edges found and reduce the diffusion rate near it by a gradient-based diffusion coefficient. In contrast to the previous anisotropic diffusion schemes, we prove the well-posedness of our scheme in the space of bounded variation. The proposed scheme is general in the sense that it can be used with any of the existing diffusion equations. Numerical simulations on noisy images show the advantages of our scheme when compared to other related schemes.
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TL;DR: With this extensive review, researchers in image processing will be able to ascertain which of these denoising methods will be best applicable to their research needs and the application domain where such methods are contemplated for implementation.
89 citations
TL;DR: The proposed multiscale Tikhonov-TV (MTTV) and dynamical MTTV methods perform better than many contemporary denoising algorithms in terms of several metrics, including signal-to-noise ratio improvement and structure preservation.
Abstract: Edge preserving regularization using partial differential equation (PDE)-based methods although extensively studied and widely used for image restoration, still have limitations in adapting to local structures We propose a spatially adaptive multiscale variable exponent-based anisotropic variational PDE method that overcomes current shortcomings, such as over smoothing and staircasing artifacts, while still retaining and enhancing edge structures across scale Our innovative model automatically balances between Tikhonov and total variation (TV) regularization effects using scene content information by incorporating a spatially varying edge coherence exponent map constructed using the eigenvalues of the filtered structure tensor The multiscale exponent model we develop leads to a novel restoration method that preserves edges better and provides selective denoising without generating artifacts for both additive and multiplicative noise models Mathematical analysis of our proposed method in variable exponent space establishes the existence of a minimizer and its properties The discretization method we use satisfies the maximum-minimum principle which guarantees that artificial edge regions are not created Extensive experimental results using synthetic, and natural images indicate that the proposed multiscale Tikhonov-TV (MTTV) and dynamical MTTV methods perform better than many contemporary denoising algorithms in terms of several metrics, including signal-to-noise ratio improvement and structure preservation Promising extensions to handle multiplicative noise models and multichannel imagery are also discussed
71 citations
TL;DR: A novel method with local difference value is applied to extract corrupted pixels and the improved method performs well in both edge preservation and noise removing.
63 citations
TL;DR: In this paper, a class of weighted anisotropic diffusion partial differential equations (PDEs) is considered and a well-balanced flow version of the proposed scheme is considered which adds an adaptive fidelity term to the usual diffusion term.
Abstract: Anisotropic diffusion is a key concept in digital image denoising and restoration. To improve the anisotropic diffusion based schemes and to avoid the well-known drawbacks such as edge blurring and ‘staircasing’ artifacts, in this paper, we consider a class of weighted anisotropic diffusion partial differential equations (PDEs). By considering an adaptive parameter within the usual divergence process, we retain the powerful denoising capability of anisotropic diffusion PDE without any oscillating artifacts. A well-balanced flow version of the proposed scheme is considered which adds an adaptive fidelity term to the usual diffusion term. The scheme is general, in the sense that, different diffusion coefficient functions can be utilized according to the need and imaging modality. To illustrate the advantage of the proposed methodology, we provide some examples, which are applied in restoring noisy synthetic and real digital images. A comparison study with other anisotropic diffusion based schemes highlight the superiority of the proposed scheme.
31 citations
Cites background or methods from "Well-Posed Inhomogeneous Nonlinear ..."
...(f) Prasath and Singh [22] — Edge detector based Anisotropic Diffusion (EAD) ∂u ∂t = div α(x)∇u 1 + |∇u|2 /K 2 ...
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...Adaptive schemes comparison results on Kikis128 × 128 synthetic image, (in each sub-figure, the right image shows the surface form of the left image): (a) EED [46]; (b) CED [47]; (c) SAD [48]; (d) ATV [16]; (e) ALD [19]; (f) EAD [22]; (g) original image and its surface form given for comparison; (h) proposed scheme with nonlinear diffusion (WWBF)....
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...There exists various choices [17,19,21,22] for theweight functionα in Eq....
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...5,1](Gσ ⋆∇u(x, t))/(1+|Gσ ⋆ ∇u|2 /K 2) No diffusion at edges Noise remain along edges [19] 1 + Mcχc ,Mc ≫ 0 constant Edge indication Excessive blurring [22] 1 − Gσ ⋆ Canny(u(x, t)) Retains multi-scale edges Cannot handle high noise [37] exp −Θ(Var(2)Nx (u(x, t)), θ)/δ Contextual discontinuities Stippled pattern artifacts [38] 1I + 1Ic exp − (|Gσ ⋆ ∇u(x)| /K)(2) Handles impulse noise Cannot handle textures...
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...Note that the PSNR values are closer together when adaptive fidelity is used (SAD, ATV, ALD, EAD, and our WWBF) in Table 2, but MSSIM values indicate a better performance of the proposed approach....
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TL;DR: A new mathematical anisotropic diffusion function is developed which is able to overcome the drawbacks of the traditional process such as the details loss and the image blur and it converges faster which allows an opportunity to be well implemented in the de-noising process.
31 citations
References
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TL;DR: There is a natural uncertainty principle between detection and localization performance, which are the two main goals, and with this principle a single operator shape is derived which is optimal at any scale.
Abstract: This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.
28,073 citations
TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.
15,225 citations
TL;DR: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.
Abstract: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
12,560 citations
Book•
01 Jan 1992
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Abstract: GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS Definitions And Elementary Properties Approximation Traces Extensions Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Sobolev Functions Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions Isoperimetric Inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS Lp Differentiability ae Approximate Differentiability Differentiability AE for W1,P (P > N) Convex Functions Second Derivatives ae for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES
5,769 citations
Book•
01 Jan 1973
TL;DR: In this article, Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert are described and discussed. But the focus is not on the performance of the operators.
Abstract: Front Cover; Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert; Copyright Page; Table des Matieres; Introduction; CHAPTER I. QUELQUES RESULTATS PRELIMINAIRES; CHAPTER II. OPERATEURS MAXIMAUX MONOTONES; CHAPTER III. EQUATIONS D'EVOLUTION ASSOCIEES AUX OPERATEURS MONOTONES; CHAPTER IV. PROPRIETES DES SEMI-GROUPES DE CONTRACTIONS NON LINEAIRES; APPENDICE : FONCTIONS VECTGRIELLES D ' U N E VARIASLE REELLE -; REFERENCES BIBLIOGRAPHIQUES COMPLEMENTS ET PROBLEMES OUVERTS; BIBLIOGRAPHIE.
3,447 citations