Compressive sensing is a method for efficiently acquiring and reconstructing a signal, through solving underdetermined linear systems. In imaging, compressive sensing theory facilitates the stable reconstruction of an image from a number of measurements fewer than the number of pixels of the image, provided the scene in question is compressible by an algorithm. This approach allows for sub-Nyquist image acquisition, achieving efficient imaging with reduced data requirements. In the realm of compressive sensing theory, the principle of sparsity—a concept that has been and continues to be foundational in numerous scientific fields—facilitates efficient estimations, compression, dimensionality reduction, and modeling. The significant role of sparsity could guide the development of more efficient data acquisition protocols. The discovery that a compressible signal can be efficiently captured through a number of incoherent measurements proportional to its level of information carries far-reaching implications, affecting a wide array of potential applications, such as data compression, data acquisition and inverse problem. In this thesis, we introduce a novel imaging technique that utilizes compressed sensing for rapid and efficient acquisition. This involves identifying the optimal pairing of sensing basis and sparse representation through deep learning. Additionally, we explore the application of the sparse representation in image segmentation of calcium imaging data. The sparse representation achieved contributes to background suppression, thereby enhancing subsequent anatomical analysis.

In this thesis, we first present highly innovative end-to-end optimized frameworks for compressive imaging. Conventional compressive imaging heavily relies on computational algorithms for object reconstruction without incorporating optimized designs for the optical hardware in the imaging acquisition system, and the sparse latent representation is determined empirically. This approach often proves insufficiently effective for certain data-driven problems, where a more integrated and adaptive solution might be required. Conversely, our frameworks not only optimize object reconstruction algorithms but also incorporate programmable photonics devices, such as spatial light modulators and diffractive optical elements, to enhance imaging speed, throughput, and overall performance. Within this optimization framework, we introduce a novel imaging modality named Deep Compressive Imaging via Optimized-Pattern Scanning (DeCIOPS). DeCIOPS employs an optimizable illumination pattern scanned across the object, gathering low-resolution measurements through a single-pixel detector, and reconstructing a high-resolution object. Utilizing a fully differentiable simulation model for image formation and object reconstruction, DeCIOPS faithfully maps the ground truth object into a reconstructed one. Leveraging an innovative auto-encoder comprising a deep neural network and compressive sensing algorithm, we optimize the illumination pattern and accurately reconstruct the object from a minimal number of samples, facilitating high imaging speed. Furthermore, to enhance the performance of pattern-scanning based imaging, we introduce the Incoherence-Enhanced Generative Adversarial Deep Compressive Imaging (IE-GADCI) algorithm. This algorithm optimizes both the sensing basis (related to the illumination pattern) and sparse representation basis (related to image reconstruction) by leveraging the adversarial duality between them.

In addition to applications in computational imaging, deep compressive sensing could facilitate background suppression in image processing. This advancement proves particularly beneficial for segmenting neurons within calcium imaging data, offering a significant tool for analyzing neural activity with greater precision. We introduce DeepCaImX, a pioneering end-to-end deep learning approach that seamlessly integrates an iterative shrinkage-thresholding algorithm with a long-short-term-memory neural network, delivering swift and hyper-parameter-free results. DeepCaImX represents a multi-task, multi-class, and multi-label segmentation method, incorporating a compressive-sensing-inspired neural network featuring recurrent and fully connected layers. It notably stands as the first neural network capable of simultaneously generating precise neuronal footprints and extracting clean neuronal activity traces from calcium imaging data. We hold firm confidence in the potential of our optimized compressive imaging approach to significantly bolster the throughput of calcium imaging, thereby assuming a pivotal role in neuroscience research.

While our thesis predominantly focuses on computational imaging modalities and novel image processing methodology, the concept of end-to-end joint optimization of imaging hardware and object reconstruction algorithms holds promise for a series of other imaging applications. These encompass wide-field high-speed super-resolution imaging and hyperspectral imaging, avenues which we eagerly anticipate exploring in future research endeavors.