Asymptotic Analysis of Expected Run-lengths of Recursive Median Filters

In this paper we perform asymptotic analysis of run-length distributions of the Recursive Median (RM) filter, for which we rely on Finite State Machine (FSM) representation. We observe radical reduction of complexity of the FSM by standard state reduction techniques, which facilitates statistical analysis. Finally, we discuss the benefits of using the RM sieve instead of the RM filter. We consider the streaking problem of the RM filter. It is demonstrated that the RM filter is not in itself a reliable estimator of location. As the cascading element in the structure of the sieve, however, it is very useful. It turns out that the use of RM sieve reduces the streaking problem to a manageable level. We also give an asymptotic proof on the expected streak length of the RM filter using results of generalised Fibonacci sequences. I. INTODUCTION Statistical properties of median and stack filters have been studied in [6, 7, 11, 12, 13, 16, 17]. Focus has been on properties like output distribution functions, joint distributions, and other statistical descriptions. In the present paper, we will emphasise scale-dependency by examining run length distributions of these systems. Bovik [6] analyses streaking in median filtered images. Streaking can be identified as an effect that produces runs of equal values in the output, runs that have no visual correlate in the input, when image processing is considered. In 1D signal processing streaking produces effects like breakdown of the cross correlation function. These effects are examined here with the aid of mathematical models. Median type filters, at least the types that we are considering, always produce one of the input samples to the output, a property not shared by linear systems. While this property can often be considered useful, e.g. rounding errors are avoided, also problems like streaking appear. Explicitly we will show that the expected run length of the recursive median filter grows exponentially with the filter window length. To achieve this, we use results from the theory of generalised Fibonacci sequences [10]. We will also briefly discuss the benefits of using the RM sieve [1, 4, 5, 16, 17] instead of the RM filter. If correlation characteristics of the RM filter and the RM sieve are compared, it is revealed that the RM filter is not in itself a reliable estimator of location, and it should not be used in data smoothing. As the cascading element in the structure of the sieve, however, the RM filter is very useful. The problem of the RM filter is that of massive streaking. It turns out that the use of RM sieve reduces this problem to a manageable level. Although RM filters have been extensively used in applications, due to some of their agreeable properties, like robustness, effective noise suppression and idempotency, it is our conclusion that extreme care should be taken when RM filters are applied. Also, recent new analytical results supporting this observation have been published; see Alliney [1] for a treatment of the RM sieve in the framework of regularisation theory, and Yli-Harja et al. [14] for an analysis of the sieve structure. For illustrations of the problem of streaking and the solution provided by the Recursive Median sieve, see [16]. It must be emphasised that actually the Datasieve is a much more general concept [5], and here we only consider a 1D selfdual version of it. II. RUN-LENGTH DISTRIBUTIONS In [18], we developed the representations of RM filters in terms of finite automata models. We are thus ready to proceed with finding the run length distribution of RM filters in the case of a binary random input signal. This allows description of the system's behaviour in the scale space analogously to the way Fourier power spectrum describes the frequency response of linear time invariant systems. Both descriptions lose information, and are thus noninvertible, unless additional information is available. Also both descriptions lose all absolute spatial information. Our aim is to look for further properties of run length distributions and similar scale descriptions to provide a useful characterisation of nonlinear median type systems. Provided that we could, from the finite state models of these filters, derive run length distributions, and thus predict the scale-behaviour of the filter, we would be one step closer to a usable stack filter system theory. Let us now describe the run length distribution that we are pursuing to find. For simplicity, let us assume that the input signal x n ( ) is a sequence of iid. random variables with probability 1/2 for occurrence of 1 or 0. Generalisation to Bernoulli trials with probabilities p and 1-p, respectively, is straightforward. Such a signal can be thought to be composed of runs of consecutive equal values, either 1’s or 0’s. Suppose that such a run, r, is picked at random, r being its length. Define the run length density function of the input signal to be φ( ) Pr( ), l r l l = = ≥ 1 . (1) In our case this applies to runs of 1’s and 0’s separately. The corresponding cumulative distribution function is Φ( ) Pr( ) ( ), l r l i l i l = ≤ = ≥ = ∑φ 1 1 . (2) We also have the usual properties E r i i i ( ) ( ) = = ∞ ∑ φ 1 (3) and φ( ) i i= ∞ ∑ = 1 1. (4) If we want to describe the statistic in the sense that a sample, say s, is picked at random, the probability for s to belong to a run of length l is