Numerical Solution of Partial Differential Equations

Computer Analysis of Images and Patterns

2009

Abstract. 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.

Section: Decodingmentioning

confidence: 99%

Edge-Based Image Compression with Homogeneous Diffusion

Mainberger

Computer Analysis of Images and Patterns

2009

“…Thereby, n is the unit normal vector to the respective segment boundary, and ∂ n u denotes the partial derivative of u in normal direction. The discretisation of this partial differential equation can be done in a straightforward way by using finite differences [17]. Then, as long as we have at least one mask pixel in each segment, there exists an unique solution of the discrete problem (cf.…”

Section: Segment-based Homogeneous Diffusionmentioning

confidence: 99%

Compression of Depth Maps with Segment-Based Homogeneous Diffusion

Hoffmann

Mainberger

Lecture Notes in Computer Science

et al. 2013

Abstract. 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 betweenpixel 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).

“…The discretisation of the occurring derivatives can be done in different ways. We use the popular concept of finite differences, as for example presented in [12]. As notation for the approximation of partial derivatives we use…”

Section: A Closer Look Into Discretisation Issuesmentioning

confidence: 99%

“…Using the method of Brox et al [4] that was the basis for the stereo approach of Slesareva et al [6], we compute w as the minimiser of an energy functional similar to the one from (12).…”

Section: Extension To Variational Optic Flowmentioning

confidence: 99%

Hyperbolic Numerics for Variational Approaches to Correspondence Problems

Zimmer

Breuß

Lecture Notes in Computer Science

et al. 2009

Abstract. Variational approaches to correspondence problems such as stereo or optic flow have now been studied for more than 20 years. Nevertheless, only little attention has been paid to a subtle numerical approximation of derivatives. In the area of numerics for hyperbolic partial differential equations (HDEs) it is, however, well-known that such issues can be crucial for obtaining favourable results. In this paper we show that the use of hyperbolic numerics for variational approaches can lead to a significant quality gain in computational results. This improvement can be of the same order as obtained by introducing better models. Applying our novel scheme within existing variational models for stereo reconstruction and optic flow, we show that this approach can be beneficial for all variational approaches to correspondence problems.