Computing the Local Continuity Order of Optical Flow Using Fractional Variational Method

We introduce variational optical flow computation involving priors with fractional order differentiations. Fractional order differentiations are typical tools in signal processing and image analysis. The zero-crossing of a fractional order Laplacian yields better performance for edge detection than the zero-crossing of the usual Laplacian. The order of the differentiation of the prior controls the continuity class of the solution. Therefore, using the square norm of the fractional order differentiation of optical flow field as the prior, we develop a method to estimate the local continuity order of the optical flow field at each point. The method detects the optimal continuity order of optical flow and corresponding optical flow vector at each point. Numerical results show that the Horn-Schunck type prior involving the n + *** order differentiation for 0 < *** < 1 and an integer n is suitable for accurate optical flow computation.

[1]  Chien-Cheng Tseng,et al.  Computation of fractional derivatives using Fourier transform and digital FIR differentiator , 2000, Signal Process..

[2]  Igor M. Sokolov,et al.  Fractional diffusion in inhomogeneous media , 2005 .

[3]  Steven S. Beauchemin,et al.  The computation of optical flow , 1995, CSUR.

[4]  Alain Oustaloup,et al.  Fractional differentiation for edge detection , 2003, Signal Process..

[5]  S. Momani,et al.  Numerical comparison of methods for solving linear differential equations of fractional order , 2007 .

[6]  Mark M. Meerschaert,et al.  A second-order accurate numerical method for the two-dimensional fractional diffusion equation , 2007, J. Comput. Phys..

[7]  Latifa Debbi.  On Some Properties of a High Order Fractional Differential Operator which is not in General Selfadjoint , 2007 .

[8]  Wotao Yin,et al.  A comparison of three total variation based texture extraction models , 2007, J. Vis. Commun. Image Represent..

[9]  Rudolf Gorenflo,et al.  Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion , 2007 .

[10]  Diego A. Murio,et al.  Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP , 2009 .

[11]  I. Podlubny.  Fractional differential equations , 1998 .

[12]  Thomas Brox,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Highly Accurate Optic Flow Computation with Theoretically Justified Warping Highly Accurate Optic Flow Computation with Theoretically Justified Warping , 2022 .

[13]  Michael Felsberg,et al.  alpha Scale Spaces on a Bounded Domain , 2003, Scale-Space.

[14]  Hans-Hellmut Nagel,et al.  An Investigation of Smoothness Constraints for the Estimation of Displacement Vector Fields from Image Sequences , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  J Campos,et al.  Fractional derivatives-analysis and experimental implementation. , 2001, Applied optics.

[16]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[17]  Hans-Hellmut Nagel,et al.  On the Estimation of Optical Flow: Relations between Different Approaches and Some New Results , 1987, Artif. Intell..

[18]  Zhihui Wei,et al.  Fractional Variational Model and Algorithm for Image Denoising , 2008, 2008 Fourth International Conference on Natural Computation.

[19]  Lewis D. Griffin,et al.  Scale Space Methods in Computer Vision , 2003, Lecture Notes in Computer Science.

[20]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[21]  Berthold K. P. Horn,et al.  Determining Optical Flow , 1981, Other Conferences.

[22]  Latifa Debbi.  Explicit solutions of some fractional partial differential equations via stable subordinators , 2006 .

[23]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[24]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[25]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .