Optimization procedures for the estimation of phase portrait parameters of orientation fields

Oriented patterns in an image often convey important information regarding the scene or the objects contained. Given an image presenting oriented texture, the orientation field of the image is a map that depicts the orientation angle of the texture at each pixel. Rao and Jain developed a method to describe oriented patterns in an image based on the association between the orientation field of a textured image and the phase portrait generated by a pair of linear first-order differential equations. The estimation of the model parameters is a nonlinear, nonconvex optimization problem, and practical experience shows that irrelevant local minima can lead to convergence to inappropriate results. We investigated the performance of four optimization algorithms for the estimation of the optimal phase portrait parameters for a given orientation field. The investigated algorithms are: nonlinear least-squares, linear least-squares, iterative linear least-squares, and simulated annealing. The algorithms are evaluated and compared in terms of the error between the estimated parameters and the parameters known by design, in the presence of noise in the orientation field and imprecision in the initialization of the parameters. The computational effort required by each algorithm is also assessed. Individually, the simulated annealing procedure yielded low fixed-point and parameter errors over the entire range of noise tested, whereas the performance of the other methods deteriorated with higher levels of noise. The use of the result of simulated annealing for the initialization of the nonlinear least-squares method led to further improvement upon the simulated annealing results.