Adaptive image approximation by linear splines over locally optimal delaunay triangulations

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

“…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.…”

“…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.…”

Anisotropic Triangulation Methods in Adaptive Image Approximation

“…In our recent work [8,9,10], adaptive thinning algorithms were applied to image data, and more recently, also to video data [11] to obtain an adaptive approximation scheme for images and videos. The resulting compression methods were shown to be competitive with JPEG2000 (for images) and MPEG4-H264 (for videos).…”

A highly-effective incremental/decremental Delaunay mesh-generation strategy for image representation

“…In recent years, there has been a growing interest in image representations that employ adaptive (i.e., nonuniform) sampling [1][2][3][4][5][6][7] as such representations can, in many applications, have numerous advantages over traditional lattice-based sampling, including greater compactness and the ability to facilitate methods that yield higher quality results or have lower overall complexity. Some of the many applications that can benefit from adaptive sampling include: feature detection [8], pattern recognition [9], computer vision [10], restoration [11], tomographic reconstruction [12], filtering [13], and image/video coding [7,[14][15][16][17][18][19].…”

“…Some of the many applications that can benefit from adaptive sampling include: feature detection [8], pattern recognition [9], computer vision [10], restoration [11], tomographic reconstruction [12], filtering [13], and image/video coding [7,[14][15][16][17][18][19]. Although many classes of adaptively-sampled image representations have been proposed to date [20-23, 1, 24-27], those based on Delaunay triangulations have proven to be particularly effective [6,28,2,4,3,29,30], and are the focus of our work herein. In order to employ a triangle-mesh representation of an image, a means is needed to construct such a representation given an arbitrary lattice-sampled image.…”