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**Evaluation of inverse finite element techniques for gradient based motion estimation.**
*(English)*
Zbl 1052.68140

Blackledge, Johnathan M. et al., Image processing III: Mathematical methods, algorithms and applications. Proceedings of the 3rd IMA conference, De Montford Univ., Leicester, GB, September 2000. Chichester: Horwood Publishing for The Institute of Mathematics and its Applications (ISBN 1-898563-72-1/hbk). Horwood Publishing Series: Mathematics and Applications, 200-220 (2001).

Summary: The analysis of digital image sequences for detecting, tracking and estimating the motion of objects is one of the most useful and challenging areas of digital image processing. Reliable, stable and fast computation of the optical flow is important where optical flow is defined as the distribution of apparent velocities of movement of brightness patterns in a sequence of images. The task essentially involves determining an approximation to a velocity field that transforms one image onto the next in the sequence.

An approach is described for the design, implementation and evaluation of an image processing algorithm to compute optical flow using inverse finite element methods. Our approach focuses on the development and comparison of techniques for computing optical flow using standard and non-standard finite element methods which were discussed in previous work. The fundamental finite volume technique for computing optical flow originally designed by B. K. P. Horn and B. G. Schunck [(*) Artif. Intell. 17, 185–203 (1981)] is enhanced. Horn and Schunck use finite volume methods to approximate the brightness derivatives at a point in the centre of a cube formed by eight image brightness measurements. Solution can often lead to slow convergence of iterative methods.

In this paper the alternative approach of finite element methods is investigated and evaluated for the estimation of the partial derivatives. This has the advantage of a rigorous mathematical formulation, along with speed of reconstruction, conceptual simplicity and ease of implementation. These are particularly important for situations in which the implementation using finite difference or finite volume methods (e.g., variable and adaptive grids) becomes complicated.

A new technique (FET) in two dimensions is derived, with results presented and discussed. Future work will focus on the development of finite element-based algorithms to handle variable scale for focusing on objects of interest in a scene, i.e., those with motion. Special Finite Element Methods may be used to replace or combine the smoothing terms of (*) with diffusion constructs. There is also the possibility of using local iterative processes over a multiresolution hierarchy of grids to ease computation and speed up convergence for these types of problems.

For the entire collection see [Zbl 0979.00033].

An approach is described for the design, implementation and evaluation of an image processing algorithm to compute optical flow using inverse finite element methods. Our approach focuses on the development and comparison of techniques for computing optical flow using standard and non-standard finite element methods which were discussed in previous work. The fundamental finite volume technique for computing optical flow originally designed by B. K. P. Horn and B. G. Schunck [(*) Artif. Intell. 17, 185–203 (1981)] is enhanced. Horn and Schunck use finite volume methods to approximate the brightness derivatives at a point in the centre of a cube formed by eight image brightness measurements. Solution can often lead to slow convergence of iterative methods.

In this paper the alternative approach of finite element methods is investigated and evaluated for the estimation of the partial derivatives. This has the advantage of a rigorous mathematical formulation, along with speed of reconstruction, conceptual simplicity and ease of implementation. These are particularly important for situations in which the implementation using finite difference or finite volume methods (e.g., variable and adaptive grids) becomes complicated.

A new technique (FET) in two dimensions is derived, with results presented and discussed. Future work will focus on the development of finite element-based algorithms to handle variable scale for focusing on objects of interest in a scene, i.e., those with motion. Special Finite Element Methods may be used to replace or combine the smoothing terms of (*) with diffusion constructs. There is also the possibility of using local iterative processes over a multiresolution hierarchy of grids to ease computation and speed up convergence for these types of problems.

For the entire collection see [Zbl 0979.00033].

### MSC:

68U10 | Computing methodologies for image processing |

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\textit{J. V. Graham} et al., in: Image processing III: Mathematical methods, algorithms and applications. Proceedings of the 3rd IMA conference, De Montford Univ., Leicester, GB, September 2000. Chichester: Horwood Publishing for The Institute of Mathematics and its Applications. 200--220 (2001; Zbl 1052.68140)