Applications of fractional calculus in computer vision: A survey

Abstract

Fractional calculus is an abstract idea exploring interpretations of differentiation having non-integer order. For a very long time, it was considered as a topic of mere theoretical interest. However, the introduction of several useful definitions of fractional derivatives has extended its domain to applications. Supported by computational power and algorithmic representations, fractional calculus has emerged as a multifarious domain. It has been found that the fractional derivatives are capable of incorporating memory into the system and thus suitable to improve the performance of locality-aware tasks such as image processing and computer vision in general. This article presents an extensive survey of fractional-order derivative-based techniques that are used in computer vision. It briefly introduces the basics and presents applications of the fractional calculus in six different domains viz. edge detection, optical flow, image segmentation, image de-noising, image recognition, and object detection. The fractional derivatives ensure noise resilience and can preserve both high and low-frequency components of an image. The relative similarity of neighboring pixels can get affected by an error, noise, or non–homogeneous illumination in an image. In that case, the fractional differentiation can model special similarities and help compensate for the issue suitably. The fractional derivatives can be evaluated for discontinuous functions, which help estimate discontinuous optical flow. The order of the differentiation also provides an additional degree of freedom in the optimization process. This study shows the successful implementations of fractional calculus in computer vision and contributes to bringing out challenges and future scopes.