Elsevier

Signal Processing

Volume 81, Issue 7, July 2001, Pages 1565-1570
Signal Processing

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Simplified implementation of the recursive median sieve

https://doi.org/10.1016/S0165-1684(01)00054-8Get rights and content

Abstract

In this paper we introduce a simplified implementation of the 1D recursive median sieve. The sieve is a multiscale data analysis method based on iterative application of recursive median filters of increasing window length. We show that this system can be implemented by applying only 3-point median operations, due to the introduction of do not care vertices to the positive Boolean function corresponding to the recursive median filter. This realisation leads to simplified implementation of the sieve structure.

Introduction

Median-type filters are well suited to image processing, where their nonlinear effects are useful. The standard median filter, for example, removes impulsive noise while preserving sharp edges [3]. Many median-type filters, e.g., weighted median and order statistic filters, can be thought of as special cases of stack filters [7] and thus expressed as combinations of minimum and maximum operations. The systems that we concentrate on in this paper belong to recursive and nonrecursive stack filters.

Stack filters always reproduce one of the input samples at the output, a property not shared by linear systems. This is sometimes considered useful, e.g., rounding errors are avoided, and nonpre-existing sample values are not introduced. However, problems like streaking can also appear.

The merit of the recursive median filter over the standard one was discussed in [5], and the benefit of using the recursive median sieve was discussed in [2]. In [8] it was shown that the recursive median filter is not in itself a reliable estimator of location and should not be used in data smoothing. As the cascading element in the structure of the sieve, however, the recursive median filter is very useful. Also, analytical results supporting this observation have recently been published; see [1] for a treatment of the recursive median sieve in the framework of regularisation theory. The deficiency problem of the recursive median filter is that of massive streaking, which can be identified as an effect that produces runs of equal values in the output, when these runs have no correlate in the input. It turns out that the use of the recursive median sieve reduces this problem to a level comparable to that of the standard median filter.

In this paper, we discuss the simplified implementation of recursive median filters and the recursive median sieve. We consider the case when recursive median filters are applied in a cascade of increasing filter window lengths, that is, the recursive median sieve. We show that the recursive median sieve can be implemented in constant time per scale by applying only 3-point median operations.

Finally, we wish to emphasise that actually the datasieve is a much more general concept [2], and here we only consider the ID self-dual version.

Section snippets

Basic definitions

First, some basic definitions. Throughout this paper, we denote real valued signals by capitals, whereas corresponding lower case notation refers to binary variables. The output of the recursive median filter with filter window length 2m+1 is defined byY(n)=median{Y(n−m),…,Y(n−1),X(n),…,X(n+m)},where X and Y denote the input and output signals, respectively.

As was noted earlier, it is sometimes useful to apply recursive median filters in a cascade of increasing filter window lengths; we then

Simplified implementation of the recursive median sieve

It has been shown in [4] that if we suppose zero padding, Eq. (1) can be recast into the formY(n)=maxminY(n−1),max{X(n),X(n+1),…,X(n+m)},min{X(n),X(n+1),…,X(n+m)}.Naturally, throughout this paper, instead of zero padding some other constant may be used. The following proposition is an immediate consequence of Eq. (3).

Proposition 1

Supposing zero padding, the output of the 1D recursive median sieve isXs(n)=maxminXs(n−1),max{Xs−1(n),Xs−1(n+1),…,Xs−1(n+s)},min{Xs−1(n),Xs−1(n+1),…,Xs−1(n+s)},where Xs refers to

Complexity of different realisations of the recursive median sieve

Let us consider the sth stage of the recursive median sieve; that is, the input signal is filtered s times in cascade, each time increasing the filter window size. If the standard implementation of the recursive median filter [5] is used, we then have successive window sizes 3,5,7,…,2s+1. Applying the well-known quicksort-based linear time selection algorithm [6], we can then characterise the complexity of the standard implementation as∑i=1sO(2i+1)=O(s2).On the other hand, taking advantage of

Conclusion

We have shown that the 1D recursive median sieve, based on iterative application of recursive median filters of increasing window length, can be implemented by applying only a single three-point median operation at each scale. This leads to a simplified hardware implementation of the sieve structure.

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