Lossy image compression methods based on partial differential equations have received much attention in recent years. They may yield high-quality results but rely on the computationally expensive task of finding an optimal selection of data. For the possible extension to video compression, this data selection is a crucial issue. In this context, one could either analyse the video sequence as a whole or perform a frame-by-frame optimisation strategy. Both approaches are prohibitive in terms of memory and run time. In this work, we propose to restrict the expensive computation of optimal data to a single frame and to approximate the optimal reconstruction data for the remaining frames by prolongating it by means of an optic flow field. In this way, we achieve a notable decrease in the computational complexity. As a proof-of-concept, we evaluate the proposed approach for multiple sequences with different characteristics. In doing this, we discuss in detail the influence of possible computational setups. We show that the approach preserves a reasonable quality in the reconstruction and is very robust against errors in the flow field.

We present a strategy for the recovery of a sparse solution of a common problem in acoustic engineering, which is the reconstruction of sound source levels and locations applying microphone array measurements. The considered task bears similarities to the basis pursuit formalism but also relies on additional model assumptions that are challenging from a mathematical point of view. Our approach reformulates the original task as a convex optimisation model. The sought solution shall be a matrix with a certain desired structure. We enforce this structure through additional constraints. By combining popular splitting algorithms and matrix differential theory in a novel framework we obtain a numerically efficient strategy. Besides a thorough theoretical consideration we also provide an experimental setup that certifies the usability of our strategy. Finally, we also address practical issues, such as the handling of inaccuracies in the measurement and corruption of the given data. We provide a post processing step that is capable of yielding an almost perfect solution in such circumstances.

Estimating the shape and appearance of a three dimensional object from flat images is a challenging research topic that is still actively pursued. Among the various techniques available, Photometric Stereo is known to provide very accurate local shape recovery, in terms of surface normals. In this work, we propose to minimise non-convex variational models for Photometric Stereo that recover the depth information directly. We suggest an approach based on a novel optimisation scheme for non-convex cost functions. Experiments show that our strategy achieves more accurate results than competing approaches.

This work analyses several approaches for determining optimal sparse data sets for image reconstructions by means of linear homogeneous diffusion. Two optimisation strategies for finding optimal data locations are presented. The first one impresses through its simplicity and is based on results from spline interpolation theory. However, this approach can only be applied to one dimensional strictly convex and differentiable functions. Due to these restrictions we derive an alternative approach which uses findings from optimal control theory. This new algorithm can be applied on arbitrary signals. Both approaches are analysed for their convergence behaviour. Further, we discuss the problem of selecting good data values for fixed data positions. This problem can be analysed as a least squares problem. An important relationship between the optimal data locations and the data values is derived and we present efficient numerical schemes to obtain these values. Finally, we present a image compression approach based on the findings from this work. Experiments show that is possible to outperform popular compression algorithms.