Caputo derivative based nonlinear fractional order variational model for motion estimation in various application oriented spectrum.

Motion information from image sequences (videos) plays a key role in solving a number of real-world problems such as surveillance, traffic management, fire identification, weather prediction, and COVID-19 detection, etc. Generally, this motion is estimated in terms of optical flow between two consecutive image frames. Optical flow is a 2D vector field that illustrates object motion behavior. This paper presents a nonlinear fractional order variational model for estimating optical flow. The objective of this work is to provide dense and discontinuity preserving optical flow for different spectra and make the model robust against outliers. In particular, the presented model generalizes the integer order variational models for fractional order (0, 1). For this purpose, the proposed model is formulated with the help of Charbonnier norm and Caputo fractional derivative. Charbonnier norm is nonlinear in nature, which makes the model robust against noise and outliers, whereas Caputo fractional derivatives are well-capable to deal with discontinuous functions such as images, and therefore, preserve motion discontinuities. The Caputo derivative also manifests long-term memory effect and allows to choose the optimal value of the fractional order that corresponds to a stable solution. The numerical implementation of the formulated variational functional is performed by discretizing the fractional derivative using the Grünwald–Letnikov derivative. The resulting system of equations is solved using multi-variable fixed point iteration scheme. The entire framework is embedded in coarse-to-fine warping strategy, which helps in finding the global extremal. The experimental results are carried out on 20 different application oriented datasets such as fire and smoke, fluid, CXR, etc. The performance of the model is tested using different error measures and demonstrated against several outliers. Error concentration is shown through 3D histograms. The edge preserving nature is also realized through intensity distribution in RGB color channels. A detailed convergence analysis is also provided for the presented model. The validity of the proposed model is also verified through comparisons with other existing models. [ABSTRACT FROM AUTHOR]