Image compression by linear splines over adaptive triangulations

Demaret, Laurent; Dyn, Nira; Iske, Armin

doi:10.1016/j.sigpro.2005.09.003

Cited by 116 publications

(117 citation statements)

References 15 publications

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“…[61] and [62]. Interesting adaptive triangulation ideas can be found in [28], [25], and [13]. In the 3-D setting, there have been no pure PDE-based compression methods so far.…”

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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“…[61] and [62]. Interesting adaptive triangulation ideas can be found in [28], [25], and [13]. In the 3-D setting, there have been no pure PDE-based compression methods so far.…”

Section: Introductionmentioning

confidence: 99%

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

et al. 2014

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“…In recent years, geometry-based image processing methods [13][14][15] become more and more popular, some geometric tools or computer graphics tools, such as gradient meshes [16], Delauney triangulations [17] and subdivision are introduced to image processing. In our previous work, we proposed a subdivision based image interpolation method [18].…”

Section: Geometry-based Methodsmentioning

confidence: 99%

Image Interpolation via Scanning Line Algorithm and Discontinuous B-Spline

Liu

Wang

Pang

et al. 2017

MCA

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Image interpolation is a basic operation in image processing. Lots of methods have been proposed, including convolution-based methods, edge modeling methods, point spread function (PSF)-based methods or learning-based methods. Most of them, however, present a high computational complexity and are not suitable for real time applications. However, fast methods are not able to provide artifacts-free images. In this paper we describe a new image interpolation method by using scanning line algorithm which can generate C −1 curves or surfaces. The C −1 interpolation can truncate the interpolation curve at big skipping; hence, the image edge can be kept. Numerical experiments illustrate the efficiency of the novel method.

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“…In that case, however, it is much harder to prove optimal rates for asymptotic N -term approximations, where the technical difficulties are mainly due to the Delaunay criterion. But our greedy approximation algorithm, adaptive thinning [11,13], achieves to construct a sequence of anisotropic Delaunay triangulations {D We may be able to show that for the piecewise affine-linear target functions, i.e., the Birman-Solomjak functions Π Q N f in (4.1), adaptive thinning outputs a sequence of Delaunay triangulations {D * N } N which are "close" to those Delaunay triangulations D N in the proof of Corollary 4.3, along with a sequence of corresponding linear spline interpolants f * N ∈ D * N that approximate f at the same rate as the functions Π Q N f . We prefer to defer this rather delicate point to future work.…”

Section: Concluding Remarks Comparison With Wavelets and Optimalitymentioning

confidence: 99%

“…In previous work, we have developed one such image approximation scheme, termed adaptive thinning, which works with linear splines over anisotropic Delaunay triangulations, and which is locally adaptive to the geometric regularity of the image. As demonstrated in [11,13], adaptive thinning leads to an efficient and competitive image compression method at computational complexity O(N log(N )). Related methods for image approximations by anisotropic triangulations are in [5,8,9,25], see the survey [12] for a comparison of these image approximation methods.…”

Section: Introductionmentioning

confidence: 99%

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Demaret

Iske

2014

Math. Comp.

Self Cite

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Abstract. Anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data, in particular from image data. In previous work, we have proposed a locally adaptive method for efficient image approximation, called adaptive thinning, which relies on linear splines over anisotropic Delaunay triangulations. In this paper, we prove asymptotically optimal N -term approximation rates for linear splines over anisotropic Delaunay triangulations, where our analysis applies to relevant classes of target functions: (a) piecewise linear horizon functions across α Hölder smooth boundaries, (b) functions of W α,p regularity, where α > 2/p−1, (c) piecewise regular horizon functions of W α,2 regularity, where α > 1.

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Image compression by linear splines over adaptive triangulations

Cited by 116 publications

References 15 publications

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

Image Interpolation via Scanning Line Algorithm and Discontinuous B-Spline

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

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