Multimodal sparse signal representations and their role in Compressed Sensing

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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?