Quotients of index two and general quotients in a space of orderings

In our work we investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field Q(x) is particularly interesting, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index 2 generalizes in an elegant way to quotients of index 2^n for arbitrary finite n. Numerous examples are provided

A fast method for computing the output of rank order filters within arbitrarily shaped windows

Rank order filters are used in a multitude of image processing tasks. Their application can range from simple preprocessing tasks which aim to reduce/remove noise, to more complex problems where such filters can be used to detect and segment image features. There is, therefore, a need to develop fast algorithms to compute the output of this class of filter. A number of methods for efficiently computing the output of specific rank order filters have been proposed [1]. For example, numerous fast algorithms exist that can be used for calculating the output of the median filter. Fast algorithms for calculating morphological erosions and dilations - which are also a special case of the more general rank order filter - have also been proposed. In this paper we present an extension of a recently introduced method for computing fast morphological operators to the more general case of rank order filters. Using our method, we are able to efficiently compute any rank, using any arbitrarily shaped window, such that it is possible to quickly compute the output of any rank order filter. We demonstrate the usefulness and efficiency of our technique by implementing a fast method for computing a recent generalisation of the morphological Hit-or-Miss Transform which makes it more robust in the presence of noise. We also compare the speed and efficiency of this routine with similar techniques that have been proposed in the literature

Lower Bounds for a Polynomial on a basic closed semialgebraic set using geometric programming

$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$. The paper shows that there is an algorithm for computing such a lower bound, based on geometric programming, which applies in a large number of cases. The algorithm extends and generalizes earlier algorithms of Ghasemi and Marshall, dealing with the case $m=0$, and of Ghasemi, Lasserre and Marshall, dealing with the case $m=1$ and $g_1= M-(x_1^d+\cdots+x_n^d)$. Here, $d$ is required to be an even integer $d \ge \max\{2,\deg(f)\}$. The algorithm is implemented in a SAGE program developed by the first author. The bound obtained is typically not as good as the bound obtained using semidefinite programming, but it has the advantage that it is computable rapidly, even in cases where the bound obtained by semidefinite programming is not computable

A new design tool for feature extraction in noisy images based on grayscale hit-or-miss transforms

The Hit-or-Miss transform (HMT) is a well known morphological transform capable of identifying features in digital images. When image features contain noise, texture or some other distortion, the HMT may fail. Various researchers have extended the HMT in different ways to make it more robust to noise. The most successful, and most recent extensions of the HMT for noise robustness, use rank order operators in place of standard morphological erosions and dilations. A major issue with the proposed methods is that no technique is provided for calculating the parameters that are introduced to generalize the HMT, and, in most cases, these parameters are determined empirically. We present here, a new conceptual interpretation of the HMT which uses a percentage occupancy (PO) function to implement the erosion and dilation operators in a single pass of the image. Further, we present a novel design tool, derived from this PO function that can be used to determine the only parameter for our routine and for other generalizations of the HMT proposed in the literature. We demonstrate the power of our technique using a set of very noisy images and draw a comparison between our method and the most recent extensions of the HMT