Dedicated to Manfred Tasche on the occasion of his 65th birthday We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of two-dimensional data. Key words. wavelet transform along pathways, data compression, adaptive wavelet bases, directed wavelets AMS Subject classifications. 65T60, 42C40, 68U10, 94A08 1

Finding efficient geometric representations of images is a central issue to improving image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet basis or frame that produces a sparse representation. For functions that are uniformly regular outside a set of edge curves that are geometrically regular, the main theorem proves that bandelet approximations satisfy an optimal asymptotic error decay rate. A bandelet image compression scheme is derived. For computational applications, a fast discrete bandelet transform algorithm is introduced, with a fast best basis search which preserves asymptotic approximation and coding error decay rates.

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 maximum– minimum 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.

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in

Figure 1: Vectorization of an amulet with 21 holes, using a single topology-preserving gradient mesh. Gradient mesh vector graphics representation, used in commercial software, is a regular grid with specified position and color, and their gradients, at each grid point. Gradient meshes can compactly represent smoothly changing data, and are typically used for single objects. This paper advances the state of the art for gradient meshes in several significant ways. Firstly, we introduce a topology-preserving gradient mesh representation which allows an arbitrary number of holes. This is important, as objects in images often have holes, either due to occlusion, or their 3D structure. Secondly, our algorithm uses the concept of image manifolds, adapting surface parameterization and fitting techniques to generate the gradient mesh in a fully automatic manner. Existing gradient-mesh algorithms require manual interaction to guide grid construction, and to cut objects with holes into disk-like regions. Our new algorithm is empirically at least 10 times faster than previous approaches. Furthermore, image segmentation can be used with our new algorithm to provide automatic gradient mesh generation for a whole image. Finally, fitting errors can be simply controlled to balance quality with storage.

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.

The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal N-term approximations for piecewise Hölder continuous functions with singularities along curves.

We present a method for compression of video clips using data-dependent triangulations. This technique utilizes the time coherence of a video to transfer information from one frame to the next, reducing the computation time for the compression. The results of this method are compared to those obtained with MJPEG and MPEG-2.

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C2-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ’shear ’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.

In earlier work, Demaret and Iske proposed the scattered data coding (SDC) method for (single-rate) coding of arbitrarily-sampled image data. In this paper, several modifications to the SDC method are proposed in order to remove some limitations of the original scheme, improve coding efficiency, and add a progressive lossy-tolossless coding capability. Through experimental results, the proposed method is shown to yield a significant improvement in coding efficiency (relative to the original SDC method) as well as provide an efficient progressive lossy-to-lossless coding capability. Index Terms — image coding, meshes, progressive coding 1.