Image compression with anisotropic triangulations

Abstract

We propose a new image compression method based on geodesic Delaunay triangulations. Triangulations are generated by a progressive geodesic meshing algorithm which exploits the anisotropy of images through a farthest point sampling strategy. This seeding is performed according to anisotropic geodesic distances which force the anisotropic Delaunay triangles to follow the geometry of the image. Geodesic computations are performed using a Riemannian Fast Marching, which recursively updates the geodesic distance to the seed points. A linear spline approximation on this triangulation allows to approximate faithfully sharp edges and directional features in images. The compression is achieved by coding both the coefficients of the spline approximation and the deviation of the geodesic triangulation from an Euclidean Delaunay triangulation. Numerical results show that taking into account the anisotropy improves the approximation by isotropic triangulations of complex images. The resulting encoder competes well with waveletbased encoder such as JPEG-2000 on geometric images.

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

“…However, for all but the smallest images, this is infeasible due to the large amount of possible subsets: Selecting e.g. 10 % of the pixels of a 256 × 256 image leaves 65536 6554 ≈ 3.8 · 10 9250 (13) options. Moreover, saving the positions of the points is quite expensive in this case, since there is no regular pattern behind the position of optimal points.…”

“…[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.…”

Anisotropic Triangulation Methods in Adaptive Image Approximation

“…The anisotropic meshing problem, as pointed out in [1], can be interpreted as the search for a criterion based on a locally modified metric, according to which triangulations are then constructed. To connect this viewpoint with the concept of anisotropic triangulations, the Euclidean metric corresponds to a uniform triangulation, whereas metrics whose unit balls are disks of varying sizes lead to isotropic adaptive triangulations.…”

“…A quite natural choice for H seems to be the Hessian. One construction of anisotropic triangulations for image approximation, based on an anisotropic geodesic metric, has recently been proposed in [1]. Instead of taking the Hessian matrix as tensor structure matrix, a regularized version of the gradient tensor is used in [1].…”

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

“…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.…”