Compared to transform-based image compression methods such as JPEG2000, approaches based on partial-differential equations (PDEs) are in a proof-of-concept stage. Nevertheless, R-EED, a codec employing edge-enhancing anisotropic diffusion (EED) and rectangular subdivision, can surpass JPEG2000 quality-wise. However, today's requirements for compression algorithms go beyond pure compression performance. Codecs must also fulfil the feature requirements of specific applications such as online media or medical imaging. We propose three such features for the R-EED codec. By reordering grey values and exploiting the subdivision scheme, we incorporate a progressive mode into R-EED that can outperform JPEG and JPEG2000. Additionally, we show that rectangular subdivision is well-suited for region of interest coding and adapt the quality of image parts according to their importance. Finally, we propose a real-time video player that demonstrates how R-EED-based decoding can be performed efficiently. All of these extensions are compatible with each other and can be used simultaneously. (C) 2015 Elsevier Inc. All rights reserved.

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods.

Galić et al. (Journal of Mathematical Imaging and Vision 31:255–269, 2008) have shown that compression based on edge-enhancing anisotropic diffusion (EED) can outperform the quality of JPEG for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation. However, the reasons for the good performance of EED remained unclear, and they could not outperform the more advanced JPEG 2000. The goals of the present paper are threefold: Firstly, we investigate the compression qualities of various partial differential equations. This sheds light on the favourable properties of EED in the context of image compression. Secondly, we demonstrate that it is even possible to beat the quality of JPEG 2000 with EED if one uses specific subdivisions on rectangles and several important optimisations. These amendments include improved entropy coding, brightness and diffusivity optimisation, and interpolation swapping. Thirdly, we demonstrate how to extend our approach to 3-D and shape data. Experiments on classical test images and 3-D medical data illustrate the high potential of our approach.

Algorithms based on partial differential equations (PDEs) constitute a relatively novel class of lossy image compression methods. In this paper we introduce a practically relevant extension: We demonstrate how to incorporate progressive modes into these codecs. Since the data in PDE-based codecs is only available at irregular locations, this is a challenging task. We propose two progressive modes: The first one changes the order in which the grey values are stored, while the second additionally distributes the stored information more evenly over the file. Our experiments show that the novel codecs can outperform JPEG and even JPEG 2000 for high compression ratios.

The efficient compression of depth maps is becoming more and more important. We present a novel codec specifically suited for this task. In the encoding step we segment the image and extract between-pixel contours. Subsequently we optimise the grey values at carefully selected mask points, including both hexagonal grid locations as well as freely chosen points. We use a chain code to store the contours. For the decoding we apply a segment-based homogeneous diffusion inpainting. The segmentation allows parallel processing of the individual segments. Experiments show that our compression algorithm outperforms comparable methods such as JPEG or JPEG2000, while being competitive with HEVC (High Efficiency Video Coding).

We demonstrate that one can adapt recent diffusion-based image compression techniques such that they become ideally suited for steganographic applications. Thus, the goal is to embed secret images within arbitrary cover images. We hide only a small number of characteristic points of the secret in the cover image, while the remainder is reconstructed with edge-enhancing anisotropic diffusion inpainting. Even when using significantly less than 1

Finding optimal inpainting data plays a key role in the field of image compression with partial differential equations (PDEs). In this paper, we optimise the spatial as well as the tonal data such that an image can be reconstructed with minimised error by means of discrete homogeneous diffusion inpainting. To optimise the spatial distribution of the inpainting data, we apply a probabilistic data sparsification followed by a nonlocal pixel exchange. Afterwards we optimise the grey values in these inpainting points in an exact way using a least squares approach. The resulting method allows almost perfect reconstructions with only 5% of all pixels. This demonstrates that a thorough data optimisation can compensate for most deficiencies of a suboptimal PDE interpolant.

Edges provide semantically important image features. In this paper a lossy compression method for cartoon-like images is presented, which is based on edge information. Edges together with some adjacent grey/colour values are extracted and encoded using a classical edge detector, binary compression standards such as JBIG and state-of-the-art encoders such as PAQ. When decoding, information outside these encoded data is recovered by solving the Laplace equation, i.e. we inpaint with the steady state of a homogeneous diffusion process. For the discrete reconstruction problem, we prove existence and uniqueness and establish a maximum–minimum principle. Furthermore, we describe an efficient multigrid algorithm. The result is a simple codec that is able to encode and decode in real time. We show that for cartoon-like images this codec can outperform the JPEG standard and even its more advanced successor JPEG2000.

There are two main strategies for solving correspondence problems in computer vision: sparse local feature based approaches and dense global energy based methods. While sparse feature based methods are often used for estimating the fundamental matrix by matching a small set of sophistically optimised interest points, dense energy based methods mark the state of the art in optical flow computation. The goal of our paper is to show that this separation into different application domains is unnecessary and can be bridged in a natural way. As a first contribution we present a new application of dense optical flow for estimating the fundamental matrix. Comparing our results with those obtained by feature based techniques we identify cases in which dense methods have advantages over sparse approaches. Motivated by these promising results we propose, as a second contribution, a new variational model that recovers the fundamental matrix and the optical flow simultaneously as the minimisers of a single energy functional. In experiments we show that our coupled approach is able to further improve the estimates of both the fundamental matrix and the optical flow. Our results prove that dense variational methods can be a serious alternative even in classical application domains of sparse feature based approaches.

It is well-known that edges contain semantically important image information. In this paper we present a lossy compression method for cartoon-like images that exploits information at image edges. These edges are extracted with the Marr–Hildreth operator followed by hysteresis thresholding. Their locations are stored in a lossless way using JBIG. Moreover, we encode the grey or colour values at both sides of each edge by applying quantisation, subsampling and PAQ coding. In the decoding step, information outside these encoded data is recovered by solving the Laplace equation, i.e. we inpaint with the steady state of a homogeneous diffusion process. Our experiments show that the suggested method outperforms the widely-used JPEG standard and can even beat the advanced JPEG2000 standard for cartoon-like images.

Estimating the fundamental matrix from a pair of stereo images is one of the central problems in stereo vision. Typically, this estimation is based on a sparse set of point correspondences that has been obtained by a matching of characteristic image features. In this paper, however, we propose a completely different strategy: Motivated by the high precision of recent variational methods for computing the optic flow, we investigate the usefulness of their dense flow fields for recovering the fundamental matrix. To this end, we consider the state-of-the-art optic flow method of Brox et al. (ECCV 2004). Using non-robust and robust estimation techniques for determining the fundamental matrix, we compare the results computed from its dense flow fields to the ones estimated from a RANSAC method that is based on a sparse set of SIFT-matches. Scenarios for both converging and ortho-parallel camera settings are considered. In all cases, the computed results are significantly better than the ones obtained by the RANSAC method – even without the explicit removal of outliers.