Feature-Oriented Image Enhancement Using Shock Filters

Osher, Stanley; Rudin, Leonid

doi:10.1137/0727053

Cited by 718 publications

(391 citation statements)

References 12 publications

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“…The aim of nding PDEs that will`clean up' images and generate natural segmentations has led several authors (Gabor 1965;see also Lindenbaum et al 1994;Kramer & Bruckner 1975;Osher & Rudin 1990;Pollak et al 1997) to propose similar PDEs, primarily on ad hoc grounds.…”

Section: Discussionmentioning

confidence: 99%

Mean, median and mode filtering of images

Griffin

2000

Proc. R. Soc. Lond. A

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If a two-dimensional image is simpli ed by repeatedly replacing its values with the mean in an in nitesimal neighbourhood, it evolves according to the di¬usion equation L t = L xx + L yy (subscripts denote di¬erentiation). This equation can alternatively be written in gauge coordinates as L t = L vv + L ww , where the v-direction is tangent to the isophote and the w-direction is along the gradient. The alternative evolution scheme of L t = L vv has also attracted attention. Guichard & Morel showed, in 1996, that this equation describes the operation of repeated in nitesimal median ltering. In this paper it is proved that repeated in nitesimal mode ltering is described by L t = L vv ¡ 2L ww at regular points, and L t = 0 at critical points. Other new results are (i) an approximate equation for median ltering at critical points, and (ii) a derivation of the equation for median ltering at regular points, which generalizes to mean and mode ltering. Finally, the results of numerical implementation of all three ltering schemes are brie®y presented.

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Section: Discussionmentioning

confidence: 99%

Mean, median and mode filtering of images

Griffin

2000

Proc. R. Soc. Lond. A

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“…Chan and Wong [9] applied the Total Variation regularizers to both kernels and images. Another group of methods [10][11][12] did not compute the blur kernels, but studied the reversion of a diffusion process.…”

Section: Related Workmentioning

confidence: 99%

“…filtering to pre-smooth the image and then solve the following shock filtering PDE problem [10] to construct significant step edges:…”

Section: Phase One: Kernel Initializationmentioning

confidence: 99%

Two-Phase Kernel Estimation for Robust Motion Deblurring

Liu

Jia

2010

Lecture Notes in Computer Science

816

1,169

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Abstract.We discuss a few new motion deblurring problems that are significant to kernel estimation and non-blind deconvolution. We found that strong edges do not always profit kernel estimation, but instead under certain circumstance degrade it. This finding leads to a new metric to measure the usefulness of image edges in motion deblurring and a gradient selection process to mitigate their possible adverse effect. We also propose an efficient and high-quality kernel estimation method based on using the spatial prior and the iterative support detection (ISD) kernel refinement, which avoids hard threshold of the kernel elements to enforce sparsity. We employ the TV-1 deconvolution model, solved with a new variable substitution scheme to robustly suppress noise.

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“…The Perona-Malik equation is nonlinear and stimulated many interests in applied mathematics [14, 43, 51, 59, 61]. Over the past two decades, many related mathematical techniques, such as the level set formalism devised by Osher and Sethian [42, 49], Mumford-Shah variational functional [38], and total variation methods [45], have been widely used for image analysis [9, 12, 41, 45, 48]. …”

Section: Introductionmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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Feature-Oriented Image Enhancement Using Shock Filters

Cited by 718 publications

References 12 publications

Mean, median and mode filtering of images

Mean, median and mode filtering of images

Two-Phase Kernel Estimation for Robust Motion Deblurring

High-order fractional partial differential equation transform for molecular surface construction

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