Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the L median, i.e. the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. Howeve...

Physical quantities referring to angles, like vector direction, color hue, etc., exhibit an inherently periodic nature. Due to this periodicity, digital filters and edge operators proposed for data on the line cannot be applied on such data. In this paper, we introduce filters for angular signals (circular mean, circular median, circular a-trimmed mean, circular modified trimmed mean). Particul...

We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the “continuous k-median (Fermat-Weber) problem” where the goal is to select one or more center points that minimize the average distance to a set of points in a demand region. In such problems, the average is computed a...

We derive a branch-and-bound algorithm to find an optimal inversion median of three signed permutations. The algorithm prunes to manageable size an extremely large search tree using simple geometric properties of the problem and a newly available linear-time routine for inversion distance. Our experiments on simulated data sets indicate that the algorithm finds optimal medians in reasonable tim...

This work shows that the following problems are equivalent, both in theory and in practice: • median filtering : given an n-element vector, compute the sliding window median with window size k, • piecewise sorting : given an n-element vector, divide it in n/k blocks of length k and sort each block. By prior work, median filtering is known to be at least as hard as piecewise sorting: with a sing...

There are two natural orderings in signals: temporal order and rank order. There is no compelling reason to explore only one of these orderings, either in the discrete-time or in the continuous-time case. Nevertheless, the concept of rank order for continuous-time signals remains virtually unstudied, which is in striking contrast to the discrete-time case: ranked order discrete-time filters, of...

When selecting a filter for an application, it is often essential to know the behaviour of the filter in presence of contamination. This robustness of a filter is traditionally explored by means of influence function (IF) and change-of-variance function (CVF). However, as these are asymptotic measures there is uncertainty of the applicability of the obtained results to the finite length filters...

We propose algorithms for maintaining two variants of kd-trees of a set of moving points in the plane. A pseudo kd-tree allows the number of points stored in the two children to differ. An overlapping kd-tree allows the bounding boxes of two children to overlap. We show that both of them support range search operations in O(n) time, where 2 only depends on the approximation precision. When the ...

Natural image matting continues to play a large role in a wide variety of applications. As an ill-posed problem, matting is a very difficult to solve due to its underconstrained nature. Current approaches can require a lot of user input, restrict themselves to a sparse subset of the image, and often make assumptions that are unlikely to hold. In this paper, we pose a way to better satisfy smoot...

This paper presents a new motion estimation algorithm to improve the performance of the existing searching algorithms at a relative low computational cost. We try to amend the incorrect and/or inaccurate estimate of motion with higher precision by using adaptive weighted median filtering and its modifications. The median filter is well-known. A more general filter, called the Adaptively Weighte...