Towards PDE-Based Image Compression

Abstract

While methods based on partial differential equations (PDEs) and variational techniques are powerful tools for denoising and inpainting digital images, their use for image compression was mainly focussing on pre-or postprocessing so far. In our paper we investigate their potential within the decoding step. We start with the observation that edge-enhancing diffusion (EED), an anisotropic nonlinear diffusion filter with a diffusion tensor, is well-suited for scattered data interpolation: Even when the interpolation data are very sparse, good results are obtained that respect discontinuities and satisfy a maximumminimum principle. This property is exploited in our studies on PDE-based image compression. We use an adaptive triangulation method based on B-tree coding for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the EED process. They can be coded in a compact and elegant way that reflects the B-tree structure. Our experiments illustrate that for high compression rates and non-textured images, this PDE-based approach gives visually better results than the widely-used JPEG coding.

Image compression with B-tree coding algorithm enhanced by data modelling with Burrows-Wheeler transformation

“…The interpolation methods are described in section 2 and as a conclusion edgeenhancing diffusion is selected due to its lowest AAE score. Edge Enhancing Diffusion is useful for scattered data interpolation and this method can be used for image reconstruction [6,7]. For compression purposes interpolation quality is not enough if the image data is too expensive to encode.…”

“…In B-tree triangular coding, an image is decomposed into a number of triangular regions that can be recovered in good quality by interpolation from the vertices [1,7]. The decomposition into triangles is stored in a binary tree structure.…”

“…Coding the grey values in all vertices starts by zig-zag traversal of sparse image created with the binary tree structure and storing it in a sequence of grey values. The main difference between EEDC in [7] is in the grey value coder. EEDC uses Huffman coding for coding of the grey values.…”

“…Both perform on various compression ratios so in order to achieve better compression, a novel approach to image compression must be used. In this research partial differential equations (PDEs) are discussed and explained because the PDEs are nowadays successfully applied to problems in image denoising, enhancement, inpainting, segmentation and also image compression [6,7,10,14].…”

Compression of Depth Maps with Segment-Based Homogeneous Diffusion

“…While in the encoding step, only a small subset of all pixels is selected and stored, the missing information is reconstructed by means of PDE-based interpolation when decoding. This idea was introduced by Galić et al in 2005 [2] and extended in 2008 [3]. A further developed version of Schmaltz et al [4] was able to beat JPEG2000.…”