In this chapter we present Gibbs-Markov models for 2-D motion in the context of their application to video coding and processing. We study nonlinear trajectory model that incorporates both velocity and acceleration. Although the maximum a posteriori probability criterion is the preferred choice for most motion estimation algorithms based on Gibbs-Markov models, we discuss the more general Bayesian criterion, including the merits of several loss functions. We describe various models for the

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... ihood and prior probability distributions, but we concentrate on pixel-, block-and region-based motion models. We propose a new motion model that incorporates acceleration into the affine model. This contribution is mainly theoretical, however we present some experimental results to underline essential differences between models discussed. In Video Data Compression for Multimedia Computing, ch. 4, Kluwer Acad. Pub., 1997 the rapid development of digital video compression techniques in which 2-D motion plays an essential role. Efficient encoding of time-varying images is essential for economical use of network or storage resources in the provision of video services. Image sequences can be compressed by independent coding of each frame (intraframe coding) or by straightforward extension of spatial coding techniques to three dimensions (e.g., 3-D transform coding). However, such approaches ignore the fact that the majority of new information (innovations) in a time-varying image is carried by motion. Since the correlation of image intensity or color is very high along the direction of motion, the knowledge of motion helps in removing interimage redundancy, as is the case in predictive or hybrid (predictive/transform or predictive/subband) coding compensated for motion. In fact, algorithms currently used in videoconferencing, digital and high-definition TV are of the hybrid type [40] . Other applications that can greatly benefit from the knowledge of motion are sampling structure conversion and noise reduction. Again, due to the high correlation along motion trajectories, motion-compensated interpolation is the most effective tool in both cases. In the sampling structure conversion missing samples can be reliably recovered, whereas in noise reduction noise can be suppressed without altering image features. A very good discussion of those and related issues can be found in [40] . The remainder of this chapter is organized as follows. In the next two sections, a framework for the description of motion models is established. Then, various statistical estimation criteria are discussed. Finally, pixel-, block-and region-based Gibbs-Markov motion models are presented. The main focus of this contribution is the description of various motion models; some experimental results are included but more can be found in our previous publications.