Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D

Grossauer, Harald; Scherzer, Otmar

doi:10.1007/3-540-44935-3_16

Cited by 62 publications

(41 citation statements)

References 18 publications

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“…the tensorial analogue to the average Euclidean distance (9). The results confirm in both cases the visual impression that anisotropic nonlinear diffusion is the favourable interpolant for tensor fields.…”

Section: Methodssupporting

confidence: 73%

“…If the data are not available on a regular grid, scattered data interpolation techniques have been proposed [7,15]. More recently, also interpolation methods based on variational formulations and nonlinear partial differential equations (PDEs) have been advocated [4,11], in particular for so-called inpainting methods [12,5,9], where the image data are only corrupted in specific areas. Nonlinear PDEs allow to design discontinuity-preserving interpolants.…”

Section: Introductionmentioning

confidence: 99%

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Tensor Field Interpolation with PDEs

Weickert

Welk

2006

Mathematics and Visualization

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We present a unified framework for interpolation and regularisation of scalar-and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued scattered data interpolation and for tensor field inpainting. By choosing suitable differential operators, interpolation methods using radial basis functions are covered. Our experiments show that a novel interpolation technique based on anisotropic diffusion with a diffusion tensor should be favoured: It outperforms interpolants with radial basis functions, it allows discontinuity-preserving interpolation with no additional oscillations, and it respects positive semidefiniteness of the input tensor data.

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Section: Methodssupporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Tensor Field Interpolation with PDEs

Weickert

Welk

2006

Mathematics and Visualization

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“…-In [12] we proposed to use the Ginzburg-Landau equation for digital inpainting purposes. Originally this equation was developed by Ginzburg & Landau [13] to phenomenologically describe phase transitions in superconductors near their critical temperature.…”

Section: Introductionmentioning

confidence: 99%

“…This property makes the real valued Ginzburg-Landau equation a reasonable method for high quality inpainting of binary images, i.e., level sets. In [12] and in this paper we focus on inpainting of gray-valued or color images. For this purpose we use the complex valued Ginzburg-Landau equation.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Iterative Methods for Solving a Ginzburg-Landau Equation

Borzì

Grossauer

Scherzer

2005

Int J Comput Vision

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Abstract. Very recently we have proposed to use a complex GinzburgLandau equation for high contrast inpainting, to restore higher dimensional (volumetric) data (which has applications in frame interpolation), improving sparsely sampled data and to fill in fragmentary surfaces. In this paper we review digital inpainting algorithms and compare their performance with a Ginzburg-Landau inpainting model. For the solution of the Ginzburg-Landau equation we compare the performance of several numerical algorithms. A stability and convergence analysis is given and the consequences for applications to digital inpainting are discussed.

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“…More recently this filling-in effect has also become the main feature of PDE-based inpainting methods such as [10,11,16,34,50,65]. Here one aims at restoring missing informations in certain corrupted image areas by means of second or higher-order PDEs.…”

mentioning

confidence: 99%

Image Compression with Anisotropic Diffusion

et al. 2008

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Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for remov- ing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widelyused JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec.

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Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D

Cited by 62 publications

References 18 publications

Tensor Field Interpolation with PDEs

Tensor Field Interpolation with PDEs

Analysis of Iterative Methods for Solving a Ginzburg-Landau Equation

Image Compression with Anisotropic Diffusion

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