Morse Description and Geometric Encoding of Digital Elevation Maps

Solé, A.; Caselles, V.; Sapiro, Guillermo; Aràndiga, Francesc

doi:10.1109/tip.2004.832864

Cited by 27 publications

(20 citation statements)

References 45 publications

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“…In that sense they are more related to [23] who apply total variation regularisation in order to modify the coefficients in a wavelet decomposition to reduce oscillatory artifacts. Sometimes PDE-based interpolation strategies have been tailored to specific data sets such as surface data in digital elevation maps [30,60,76]. Moreover, some variational L 1 minimisation ideas play an important role in recent compressed sensing concepts [18].…”

Section: Introductionmentioning

confidence: 99%

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

et al. 2014

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Galić et al. [33] have shown that compression based on edge-enhancing anisotropic diffusion (EED) can outperform the quality of JPEG for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation. However, the reasons for the good performance of EED remained unclear, and they could not outperform the more advanced JPEG 2000. The goals of the present paper are threefold: Firstly, we investigate the compression qualities of various partial differential equations. This sheds light on the favourable properties of EED in the context of image compression. Secondly, we demonstrate that it is even possible to beat the quality of JPEG 2000 with EED if one uses specific subdivisions on rectangles and several important optimisations. These amendments include improved entropy coding, brightness and diffusivity optimisation, and interpolation swapping. Thirdly, we demonstrate how to extend our approach to 3-D and shape data. Experiments on classical test images and 3-D medical data illustrate the high potential of our approach.

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Section: Introductionmentioning

confidence: 99%

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

et al. 2014

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“…Chan and Zhou [8] propose a variational approach with total variation regularisation to minimise oscillations in wavelet decompositions. Work by Solé et al [9] evaluates different PDEs for compression of digital elevation maps, and Liu et al [10] integrate inpainting into existing approaches.…”

Section: Introductionmentioning

confidence: 99%

Progressive modes in PDE-based image compression

Schmaltz

Mach

Mainberger

et al. 2013

2013 Picture Coding Symposium (PCS)

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Abstract-Algorithms based on partial differential equations (PDEs) constitute a relatively novel class of lossy image compression methods. In this paper we introduce a practically relevant extension: We demonstrate how to incorporate progressive modes into these codecs. Since the data in PDE-based codecs is only available at irregular locations, this is a challenging task. We propose two progressive modes: The first one changes the order in which the grey values are stored, while the second additionally distributes the stored information more evenly over the file. Our experiments show that the novel codecs can outperform JPEG and even JPEG 2000 for high compression ratios.

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“…Related variational and PDE methods have also been investigated for more classical interpolation problems such as zooming into an image by increasing its resolution [2,5,6,15,42,46,51]. Some other PDE-based interpolation strategies have been tailored to specific data sets such as level set representations for digital elevation maps [27,48,52]. Moreover, some variational L 1 minimization ideas play an important role in recent compressed sensing concepts [13].…”

Section: Introductionmentioning

confidence: 99%

How to Choose Interpolation Data in Images

Belhachmi¹,

Bucur²,

Burgeth³

et al. 2009

SIAM J. Appl. Math.

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We introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of Γ-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the "importance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius), and study by Γ-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

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Morse Description and Geometric Encoding of Digital Elevation Maps

Cited by 27 publications

References 45 publications

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

Progressive modes in PDE-based image compression

How to Choose Interpolation Data in Images

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