Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images

Shukla, Rahul; Dragotti, Pier Luigi; Do, Minh; Vetterli, Martin

doi:10.1109/tip.2004.840710

Cited by 142 publications

(121 citation statements)

References 26 publications

Supporting

Mentioning

114

Contrasting

Unclassified

Order By: Relevance

“…In fact, during the last few years, several approaches for (locally) adaptive image approximation schemes were developed [2,10,16,17,23,26,28,29,30,32,33], where the approximation schemes are adapted to the (local) image geometry, rather than fixing a basis or a function frame beforehand to approximate f .…”

Section: Introductionmentioning

confidence: 99%

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

Demaret

Iske

2014

Math. Comp.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Demaret

Iske

2014

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…Several other approaches also analyze geometrical structures in images, like polynomial modeling with quadtree segmentation (Shukla et al, 2005), footprints Vetterli, 2001, 2003), multiscale transform (Cohen and Matei, 2001), etc. Apart from exploiting geometrical coherence, multi-directional (M-DIR) processing has also been applied to image denoising and classification.…”

Section: Introductionmentioning

confidence: 99%

Sparse Image Representation by Directionlets

Velisavljević¹,

Vetterli²,

Beferull-Lozano³

et al. 2010

Advances in Imaging and Electron Physics

View full text Add to dashboard Cite

In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency and sparsity of its representation is limited by the spatial symmetry and separability of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges or contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a non-sparse representation. To capture efficiently these elongated structures characterized by geometrical regularity along different directions (not only the horizontal and vertical), a more complex multi-directional (M-DIR) and asymmetric transform is required. We present a lattice-based perfect reconstruction and critically sampled asymmetric M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional (2-D) WT, unlike what is the case for some other directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding asymmetric basis functions, called directionlets, have directional vanishing moments (DVM) along any two directions with rational slopes, which allows for a sparser representation of elongated and oriented features. As a consequence of the improved sparsity, directionlets provide an efficient tool for non-linear approximation (NLA) of images, significantly outperforming the standard 2-D WT. Furthermore, directionlets combined with wavelet-based image compression methods lead to a gain in the performance in terms of both the mean-square error and visual quality, especially at the low bit-rate compression, while having the same complexity. Finally, a shift-invariant non-subsampled version of directionlets is successfully implemented in image interpolation, where critical sampling is not a key requirement.

show abstract

“…Recently a lot of research effort was aimed at representing a natural image exploiting its inherent geometrical structure. Concerning image compression, interesting approaches that follow this idea can be found for example in [2], [3] and [4].…”

Section: Introductionmentioning

confidence: 99%

Image compression using an edge adapted redundant dictionary and wavelets

Peotta

Granai

2006

Signal Processing

View full text Add to dashboard Cite

Low bit rate image coding is an important problem regarding applications such as storage on low memory devices or streaming data on the internet. The state of the art in image compression is to use 2-D wavelets. The advantages of wavelet bases lie in their multiscale nature and in their ability to sparsely represent functions that are piecewise smooth. Their main problem on the other hand, is that in 2-D wavelets are not able to deal with the natural geometry of images, i.e they cannot sparsely represent objects that are smooth away from regular submanifolds. In this paper we propose an approach based on building a sparse representation of the edge part of images in a redundant geometrically inspired library of functions, followed by suitable coding techniques. Best N-terms non-linear approximations in general dictionaries is, in most cases, a NP-hard problem and sub-optimal approaches have to be followed. In this work we use a greedy strategy, also known as Matching Pursuit to compute the expansion. The residual, that we suppose to be the smooth and texture part, is then coded using wavelets. A rate distortion optimization procedure chooses the number of functions from the redundant dictionary and the wavelet basis.

show abstract

Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images

Cited by 142 publications

References 26 publications

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

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

Sparse Image Representation by Directionlets

Image compression using an edge adapted redundant dictionary and wavelets

Contact Info

Product

Resources

About