ODIF FOR MORPHOLOGICAL FILTERS Sari Peltonen

In this paper our recently introduced method called output distributional influence function (ODIF) is used for the evaluation of the robustness properties of the morphological filters. Several examples of the ODIFs for the dilation, the closing, and the clos-opening are given and explained carefully. For each of these morphological filters the effect of filter length is examined by using the ODIFs for the expectation and the variance. The robustness properties of the filters are also compared to each other and the effect of the distribution of the contamination is investigated for the closing as an example of realistic filtering conditions. 1. INFLUENCE FUNCTION Influence function (IF) is a useful heuristic tool of robust statistics introduced by Hampel [1, 2] under the name influence curve (IC) for studying the performance of filters under noisy conditions. Definition 1 . The IF of estimatorT at underlying probability distributionF is given by IF(x;T; F ) = lim t! 0+ T ((1 t)F + t x) T (F ) t for thosex where this limit exists. In this definition x is the probability measure which puts mass 1 at the pointx. The IF gives the effect that an infinitesimal contamination at point x has on the estimator T when divided by the mass of the contamination. So the IF gives asymptotic bias caused by the contamination and thus characterizes properties of the estimator as the number of observations approaches infinity. 2. CHANGE-OF-VARIANCE FUNCTION The IF gives only one aspect of robustness of an estimator, namely local robustness of the asymptotic value of the estimator. Another important aspect is the local robustness of the asymptotic variance. Asymptotic variance of estimator T at F denoted by V (T;F ) is defined to be the variance of p N [T (FN ) T (F )] asN ! 1, whereFN is the empirical distribution of sample (X1;X2; : : : ; XN ). Local robustness of the asymptotic variance can be characterized by the change-of-variance function (CVF) defined as follows, [3]. Definition 2. The CVF of estimator T atF is defined as CVF(x;T; F ) = lim t! 0+ V (T; (1 t)F + t x) V (T;F ) t for thosex where this limit exists. 3. OUTPUT DISTRIBUTIONAL INFLUENCE FUNCTION Since the IF is an asymptotic measure, it describes properties of infinite length filters which may differ from those of finite length filters used in the real world filtering applications. It would be more useful and more interesting to examine properties of these finite length filters rather than the asymptotic properties. In the case where the output distribution of a filter can be expressed in a closed form as a function of the distribution functions of the input samples we introduced in [5] output distributional influence function (ODIF) for analysing the robustness of the finite length filters. It is also required for the IF and the CVF that the filters are Fisher consistent, i.e., they measure asymptotically the correct value. This prohibits the analysis of morphological filters by these asymptotic methods but the ODIFs do not have this limitation. We assume here that the input samples are independent and identically distributed (i.i.d.) random variables. First we need a way to denote the output distribution function of a filter when a fraction" of the input samples has different distribution than the rest of the samples. We denote by H(1 ")F+"Gy ( ) the output distributionHF ( ) of the filter where every occurrence of the common distribution functionF of the input samples is replaced by (1 ")F+"Gy andGy can be any distribution function with mean y. As usual, we defineh(1 ")F+"Gy (x) = d dx H(1 ")F+"Gy (x). We gave the following definition for the ODIF for the distribution function in [5]. Definition 3. Let the output distribution function of a filter be HF ( )whereF ( ) is the common distribution function of the input samples and let Gy( ) be a distribution function having mean y. Then the ODIF for the distribution function ( ) is (x; y) = lim "! 0+ H(1 ")F+"Gy (x) HF (x) " for thosex andy where this limit exists. If the output distribution functionHF ( ) does not contain any derivative ofF , the ODIF for the distribution function ( ) can be expressed as, [5] (x; y) = hF (x) f(x) (Gy(x) F (x)) : (1) In [5] we defined the ODIF in the same way as for the distribution function in Definition 3 also for the density function and moments. The ODIFs for the density function, the expectation, and the variance were derived to be ! (x; y) = d dx (x; y); (2)