Welcome to Riemannian Computing in Computer Vision

The computer vision community has registered a strong progress over the last few years due to: (1) improved sensor technology, (2) increased computation power, and (3) sophisticated statistical tools. Another important innovation, albeit relatively less visible, has been the involvement of differential geometry in developing vision frameworks. Its importance stems from the fact that despite large sizes of vision data (images and videos), the actual interpretable variability lies on much lower-dimensional manifolds of observation spaces. Additionally, natural constraints in mathematical representations of variables and desired invariances in vision-related problems also lead to inferences on relevant nonlinear manifolds. Riemannian computing in computer vision (RCCV) is the scientific area that integrates tools from Riemannian geometry and statistics to develop theoretical and computational solutions in computer vision. Tools from RCCV has led to important developments in low-level feature extraction, mid-level object characterization, and high-level semantic interpretation of data. In this chapter we provide backgroundmaterial from differential geometry, examples of manifolds commonly encountered in vision applications, and a short summary of past and recent developments in RCCV. We also summarize and categorize contributions of the remaining chapters in this volume.