Multimodal sparse signal representations and their role in Compressed Sensing

压缩感知（CS）在医学成像领域具有巨大的影响，因为它允许将昂贵且耗时的数据采集过程转移到信号重建任务。例如，在磁共振成像（MRI）中，图像采集过程受到k空间中的空间傅立叶系数的顺序采样的影响，使得MRI成为相当慢的模态。这是一个严重的限制，一方面降低了临床环境中患者的吞吐量，另一方面使得难以捕获移动的身体部位，例如在心脏MRI中。在计算机断层扫描（CT）（另一种医学成像模式）中，图像对比度反映X射线的衰减。重复或长期扫描是有风险的，因为患者暴露在高辐射剂量下。CS依赖于两个简单但强大的原则，即稀疏信号表示和感测域与稀疏表示域之间的不相干性。基于这些原则，恢复保证已经得到了完美重建所需的测量数量的下限。一般来说，解析给出的基础，如，例如傅立叶或小波基，被用来稀疏地表示广泛的信号类别。如今，通过利用旨在找到合适的帧或字典的方法来获得现有技术的恢复结果，所述合适的帧或字典由于其对感兴趣的特定信号类别的适应而提供更稀疏的信号表示。学习适当的信号表示在很大程度上取决于正确的选择和训练样本的数量。这些重要的问题是通过调查样本的学习过程的复杂性。在这个项目中，我们的目标是调查CS和稀疏信号表示在多模式的情况下。通常，同一个场景或物体可以通过一种以上的模态来感知，而无需太多额外的努力。在许多场景中，由于特定的硬件设置，甚至固有地提供这种多模态感测。例如，来自不同模态的图像的融合已经成为医学成像中的常见实践，导致提供组合采集的系统，例如PET-CT或PET-MRI。该项目解决了上述多模态稀疏信号表示及其在压缩感知中的作用背景下的四个密切相关的研究问题。(1)如何对模态的统计依赖性进行建模，以及如何学习这样的模型？(2)为了找到模型参数的可靠估计，需要多少训练数据？(3)为了保证从几个多模态测量中重建，信号模型和传感矩阵的必要和充分条件是什么？(4)收集的结果在重建精度方面如何影响双峰医学成像，以及我们如何处理未注册的双峰测量？

Compressed Sensing (CS) has an enormous impact in the field of Medical Imaging, since it allows shifting the expensive and time consuming data acquisition process to a signal reconstruction task. In Magnetic Resonance Imaging (MRI) for example, the image acquisition process suffers from the sequential sampling of spatial Fourier coefficients in k-space, making MRI a rather slow modality. This is a severe limitation that on the one hand reduces the throughput of patients in the clinical environment and on the other hand makes it hard to capture moving body parts like in cardiac MRI for example. In Computed Tomography (CT), another medical imaging modality, the image contrast reflects the attenuation of X-rays. Repeated or long-term scans are risky because the patients are exposed to high radiation doses. CS relies on two simple, yet powerful principles, namely a sparse signal representation and the incoherence between the sensing domain and the sparse representation domain. Based on these principles, recovery guarantees have been derived that give a lower bound on the number of measurements needed for perfect reconstruction. In general, analytically given bases like, e.g. the Fourier or wavelet basis, are employed to sparsely represent a wide range of signal classes. Nowadays, state of the art recovery results are obtained by utilizing methods aiming at finding a suitable frame or dictionary that provides a sparser signal representation due to its adaptation to the particular signal class of interest. Learning an adequate signal representation heavily depends on the right choice and the number of training examples. These important issues are tackled by the investigation of the sample complexity of the learning process. In this project, we aim at investigating CS and sparse signal representations in a multimodal context. Often, one and the same scene or object can be sensed through more than one modality, without much additional effort. In many scenarios such multimodal sensing is even provided inherently due to the particular hardware setting. For example, the fusion of images from different modalities has become common practice in medical imaging, leading to systems that provide a combined acquisition like e.g. PET-CT or PET-MRI. The project addresses four strongly interconnected research questions in the above-mentioned context of multimodal sparse signal representations and their role in compressed sensing. (1) How can the statistical dependency of the modalities be modeled and how can such a model be learned? (2) How much training data is needed in order to find reliable estimates of the model s parameters? (3) What are necessary and sufficient conditions on the signal model and the sensing matrix in order to guarantee reconstruction from few multimodal measurements? (4) How do the gathered results impact bimodal medical imaging in terms of reconstruction accuracy and how can we deal with unregistered bimodal measurements?