A framework for intrinsic image processing on surfaces

After many years of study, the subject of image processing on the plane, or more generally in Euclidean space is well developed. However, more and more practical problems in different areas, such as computer vision, computer graphics, geometric modeling, and medical imaging, inspire us to consider imaging on surfaces beyond imaging on Euclidean domains. Several approaches, such as implicit representation approaches and parameterization approaches, are investigated about image processing on surfaces. Most of these methods require certain preprocessing to convert image problems on surfaces to image problems in Euclidean spaces. In this work, we use differential geometry techniques to directly study image problems on surfaces. By using our approach, all plane image variation models and their algorithms can be naturally adapted to study image problems on surfaces. As examples, we show how to generalize Rudin–Osher–Fatemi (ROF) denoising model [1] and convexified Chan–Vese (CV) [2] segmentation model on surfaces, and then demonstrate how to adapt popular algorithms to solve the total variation related problems on surfaces. This intrinsic approach provides us a robust and efficient method to directly study image processing, in particular, total variation problems on surfaces without requiring any preprocessing.