Depth reconstruction tries to obtain 3D scene geometry from incomplete or low-dimensional data — and it is usually a vital first step in many computational photography tasks. Most image-space geometric representations, however, fail to be general-purpose as they focus on a narrow set of metrics, and do not preserve all information of potential relevance. This dissertation shows that multi-view edges encode all relevant information for supporting higher level computational photography tasks that rely on depth reconstruction. We do this by presenting a novel encoding of multi-view scene geometry from structured light fields, and a reconstruction method for inverting this code. Our model is based on edges in the Epipolar Plane Images (EPIs) of a light field. These edges provide a small number of high-gradient key points and depth labels that can be used to accurately identify occlusion boundaries, and also to anchor the reconstruction in the angular domain for view-consistency. We present a differentiable representation of our model which allows the reconstruction to be optimized via gradient descent on a multi-view reconstruction loss. We evaluate our reconstruction for accuracy, view consistency, and occlusion handling to show that it retains all the geometric information required for higher level computational photography tasks.