Quantale Modules and their Operators, with Applications

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology

Model of categories for image processing

Theory of categories has various applications in technology. There are mathematical models that have been built using algebraic structures equipped with different tools of properties from disciplines inside mathematics even inside algebra. We are concentrated on the applications of algebra in image processing. We have seen that basic relations and operations between images, and other used in image compression, already have been described in terms of categories and homomorphism of modules over quantales. They include relations and operations between sets, set inclusion, union, intersection, used in mathematical morphology, and operations, order relation, t-norms in image compression. Focused on the way how these construction of categories of quantale modules are built up we want to show a new construction of an algebraic model through categories as an application of algebraic structures, involving issues from order theory, theory of modules and theory of categories in mathematical morphology and image compression. We start from a model on the theory of categories and theory of modules over a commutative ring. We think to combine the construction of a category from an object with homomorphism of modules over quantales using as a model the same issue over categories of modules and over the category of homomorphism of modules over rings. Our intention is to open the way the study of properties of some important modules and their homomorphism that are immediate objects in image processing, seeing them as categories on their own not as a simple object of a category. Through this we want to import the properties from the treatment existing now to investigate after other properties deriving from our point of view

N-ary Mathematical Morphology

International audienceMathematical morphology on binary images can be fully de-scribed by set theory. However, it is not sucient to formulate mathe-matical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the denition of sup and inf operators. More generally, mathemati-cal morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the no-tion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose an extension of mathematical morphology to an intermediary situation: images composed of a nite number of independent unordered categories

Sabanci-Okan system at ImageClef 2011: plant identication task

We describe our participation in the plant identication task of ImageClef 2011. Our approach employs a variety of texture, shape as well as color descriptors. Due to the morphometric properties of plants, mathematical morphology has been advocated as the main methodology for texture characterization, supported by a multitude of contour-based shape and color features. We submitted a single run, where the focus has been almost exclusively on scan and scan-like images, due primarily to lack of time. Moreover, special care has been taken to obtain a fully automatic system, operating only on image data. While our photo results are low, we consider our submission successful, since besides being our rst attempt, our accuracy is the highest when considering the average of the scan and scan-like results, upon which we had concentrated our eorts

A Unified Algebraic Framework for Fuzzy Image Compression and Mathematical Morphology

In this paper we show how certain techniques of image processing, having different scopes, can be joined together under a common "algebraic roof"

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page

Disconnected Skeleton: Shape at its Absolute Scale

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV: Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape Recognition. Masters thesis, Department of Computer Engineering, Middle East Technical University, May 200