Reconstruction of Irregularly Sampled Image Signals Using Sparse Representations

采样过程是数字信号处理的基本要素。将连续信号从模拟域转换到数字域是必要的，这是利用数字信号处理的方法和算法进行进一步处理的第一步。通常，使用规则采样，其中采样位置规则地排列在相应的网格上。这类采样得到了全面的调查和记录，数字信号处理中的大多数方法都是在规则的网格上使用采样位置。由于一些图像采集系统或为了避免走样而显式地选择采样位置，采样位置可能会在网格上不规则分布。为了进一步处理，这些采样位置必须在规则网格上重建。在以往对不规则采样图像数据的重建工作中，已经证明了这种重建可以通过识别主导基函数和估计其权重来完成。类似于压缩感知，利用了大多数自然信号可以在某些域中稀疏表示的特性。作为第一个目标，从重建不规则采样的图像数据的结果出发，应开发一种重建不规则采样的多维信号的通用方法。在这样做的时候，有必要进行详细的调查，以了解不规则采样对多维信号的影响，并使重建过程适应这个问题。通过使用这些新的见解，将开发一种方法来捕获和重建由高质量的不规则采样帧组成的视频。现有的传感器要么能够捕获高空间分辨率但低时间分辨率的视频，要么反之亦然。通过这种新的方法，可以获得同时具有高空间和高时间分辨率的视频。此外，本文还将从理论上研究不规则采样和基于稀疏性的重建算法与压缩感知之间的关系。此外，还应结合压缩感知中使用的不同重建算法和新的基于稀疏性的重建方法的优点。

The sampling process is a fundamental element of digital signal processing. It is necessary to convert a continuous signal from the analog domain into the digital domain and it is the first step for further processing by methods and algorithms from digital signal processing. Commonly, a regular sampling is used where the sampling positions are regularly arranged on a corresponding grid. This kind of sampling is comprehensively investigated and documented and most methods from digital signal processing work with sampling positions on a regular grid. Due to some image acquisition systems or due to explicitly selecting the sampling positions in order to avoid artifacts from aliasing, the sampling positions may be distributed irregularly on the grid. For further processing, these sampling positions have to be reconstructed on a regular grid. During previous work on the reconstruction of irregularly sampled image data, it has been shown that this reconstruction can be done by identifying the dominant basis function and estimating its weight. Similar to Compressed Sensing, the property that most natural signals can be represented sparsely in certain domains is exploited. As a first objective, starting from the results for the reconstruction of irregularly sampled image data, a general method shall be developed to reconstruct irregularly sampled multidimensional signals. In doing so, detailed investigations are necessary to understand the effect of irregular sampling on multidimensional signals and to adapt the reconstruction process to this problem. By using these new insights, a method shall be developed to capture and reconstruct a video consisting of irregularly sampled frames with high quality. Existing sensors are either able to capture a video with high spatial but low temporal resolution or vice versa. By means of this new method a video with both high spatial and high temporal resolution can be acquired. Moreover, the relationship between irregular sampling followed by a sparsity-based reconstruction algorithm that has to be developed in this work and Compressed Sensing shall be theoretically investigated. Also the benefits of different reconstruction algorithms used in Compressed Sensing and the new sparsity-based reconstruction method shall be combined.