Over the past few centuries cameras have greatly evolved to better capture our visual world. However, the fundamental principle has remained the same---the camera obscura. Consequently, though cameras today can capture incredible photographs, they still have certain limitations. For instance, they can capture only 2D scene information. Recent years have seen several efforts to overcome these limitations and extend the capabilities of cameras through the paradigm of computational imaging---capture the scene in a coded fashion, which is then decoded computationally in software. This thesis subscribes to this philosophy. In particular, we present several imaging systems that enable us to overcome limitations of conventional cameras and provide us with flexibility in how we capture scenes.;First, we present a family of imaging systems called radial imaging systems that capture the scene from a large number of viewpoints, instantly, in a single image. These systems consist of a conventional camera looking through a hollow conical mirror whose reflective side is the inside. By varying the parameters of the cone we get a continuous family of imaging systems. We demonstrate the flexibility of this family---different members of this family can be used for different applications. One member is well suited for reconstructing objects with fine geometry such as 3D textures, while another is apt for reconstructing larger objects such as faces. Other members of this family can be used to capture texture maps and estimate the BRDFs of isotropic materials.;We then present an imaging system with a flexible field of view---the size and shape of the field of view can be varied to achieve a desired scene composition in a single image. The proposed system consists of a conventional camera that images the scene reflected in a flexible mirror sheet. By deforming the mirror we can generate a wide and continuous range of smoothly curved mirror shapes, each of which results in a new field of view. This system enables us to realize a wide range of scene-to-image mappings, in contrast to conventional imaging systems that yield a fixed or a fixed set of scene-to-image mappings.;All imaging systems that use curved mirrors (including the ones above) suffer from the problem of defocus due to mirror curvature; due to local curvature effects the entire image is usually not in focus. We use the known mirror shape and camera and lens parameters to numerically compute the spatially varying defocus blur kernel and then explore how we can use spatially varying deconvolution techniques to computationally 'stop-up' the lens---capture all scene elements with sharpness while using larger apertures than what is usually required in curved mirror imaging systems.;Finally, we present an imaging system with flexible depth of field. We propose to translate the image detector along the optical axis during the integration of a single image. We show that by controlling the motion of the detector---its starting position, speed, and acceleration---we can manipulate the depth of field in new and interesting ways. We demonstrate capturing scenes with large depths of field, while using large apertures to maintain high signal-to-noise ratio. We also show how we can capture scenes with discontinuous, tilted or non-planar depths of field.;All the imaging systems presented here subscribe to the philosophy of computational imaging. This approach is particularly attractive as with Moore's law computations become increasingly cheaper, enabling us to push the limits of how cameras can capture scenes.
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