Chapter 4
Robust methods [Tukey 1977 ; Huber 1981 ; Rousseeuw 1984] are tools for statistics problems in which outliers are an issue. It is well known that the least squares (LS) error estimates can be arbitrarily wrong when outliers are present in the data. A robust procedure is aimed to make solutions insensitive to the influence caused by outliers. That is, its performance should be good with all-inlier data and deteriorates gracefully with increasing number of outliers. The mechanism of robust estimators in dealing with outliers is similar to that of the discontinuity adaptive MRF prior model studied in the previous chapter. This chapter provides a comparative study [Li 1995a] of the two kinds of models based on the results about the DA model and presents an algorithm [Li 1995e] to improve the stability of the robust M-estimator to the initialization.
The conceptual and mathematical comparison comes naturally from the parallelism of the two models: Outliers cause violation of a distributional assumption while discontinuities cause violation of the smoothness assumption. Robustness to outliers is in parallel to adaptation to discontinuities. Detecting outliers is corresponding to inserting discontinuities. The likeness of the two models suggests that results in either model could be used for the other.
Probably for this reason, recent years have seen considerable interests in applying robust techniques to solving computer vision problems. Kashyap and Eom (1988) develop an robust algorithm for estimating parameters in an autoregressive image model where the noise is assumed to be a mixture of a Gaussian and an outlier process. Shulman and Herve (1989) [Shulman and Herve 1989] propose to use Huber's robust M-estimator to compute optical flow involving discontinuities. Stevenson and Delp (1990) use the same estimator for curve fitting. Besl et al. (1988) propose a robust M window operator to prevent smoothing across discontinuities. Haralick et al. (1989) , Kumar and Hanson (1989) and Zhuang et al. (1992) use robust estimators to find pose parameters. Jolion and Meer (1991) identify clusters in feature space based on the robust minimum volume ellipsoid estimator. Boyer et al. (1994) present a procedure for surface parameterization using a robust M estimator. Black et al. (1993,1994) apply a robust operator not only to the smoothness term but also to the data term. Li presents a comparative study on robust models and discontinuity adaptive MRF models [Li 1995a]. He also presents a method for stabilizing robust M estimation with respect to the initialization and convergence [Li 1995e]. Other recent advances in this area can be found in Proceedings of International Conference on Robust Computer Vision [Haralick 1990 ; Forstner and Ruwiedel 1992].
As well known, robust estimation procedures have a serious problem that the estimates are dependent on the initial estimate value; this problem has been successfully overcome by applying the principle of the graduated non-convexity (GNC) method [Blake and Zisserman 1987] for visual reconstruction. The exchange of theoretical results and practical algorithms are useful to the vision community because both MRF and robust models have their applications in vision.