Over the last decade, deep neural networks also known as deep learning, combined with large databases and specialized hardware for computation, have made major strides in important areas such as computer vision, computational imaging and natural language processing. However, such frameworks currently suffer from some drawbacks. For example, it is generally not clear how the architectures are to be designed for different applications, or how the neural networks behave under different input perturbations and it is not easy to make the internal representations and parameters more interpretable. In this dissertation, I propose building constraints into feature maps, parameters and and design of algorithms involving neural networks for applications in low-level vision problems such as compressive imaging and multi-spectral image fusion, and high-level inference problems including activity and face recognition. Depending on the application, such constraints can be used to design architectures which are invariant/robust to certain nuisance factors, more efficient and, in some cases, more interpretable. Through extensive experiments on real-world datasets, I demonstrate these advantages of the proposed methods over conventional frameworks.

As a promising solution to the problem of acquiring and storing large amounts of image and video data, spatial-multiplexing camera architectures have received lot of attention in the recent past. Such architectures have the attractive feature of combining a two-step process of acquisition and compression of pixel measurements in a conventional camera, into a single step. A popular variant is the single-pixel camera that obtains measurements of the scene using a pseudo-random measurement matrix. Advances in compressive sensing (CS) theory in the past decade have supplied the tools that, in theory, allow near-perfect reconstruction of an image from these measurements even for sub-Nyquist sampling rates. However, current state-of-the-art reconstruction algorithms suffer from two drawbacks -- They are (1) computationally very expensive and (2) incapable of yielding high fidelity reconstructions for high compression ratios. In computer vision, the final goal is usually to perform an inference task using the images acquired and not signal recovery. With this motivation, this thesis considers the possibility of inference directly from compressed measurements, thereby obviating the need to use expensive reconstruction algorithms. It is often the case that non-linear features are used for inference tasks in computer vision. However, currently, it is unclear how to extract such features from compressed measurements. Instead, using the theoretical basis provided by the Johnson-Lindenstrauss lemma, discriminative features using smashed correlation filters are derived and it is shown that it is indeed possible to perform reconstruction-free inference at high compression ratios with only a marginal loss in accuracy. As a specific inference problem in computer vision, face recognition is considered, mainly beyond the visible spectrum such as in the short wave infra-red region (SWIR), where sensors are expensive.