Patrick Erik Bradley

Degenerating Families of Dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representations of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are in fact p-adic parameter spaces of dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.

Mumford Dendrograms

An effective p-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat–Tits tree for a p-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field of p-adic numbers. It is shown that fixing p-adic representatives of the residue field allows a natural way of encoding strings by identifying a given alphabet with such representatives. A simple p-adic hierarchic classification algorithm is derived for p-adic numbers, and is applied to strings over finite alphabets. Examples of DNA coding are presented and discussed. Finally, new geometric and combinatorial invariants of time series of p-adic dendrograms are developed.

Using the Relational Model to Capture Topological Information of Spaces

Motivated by research on how topology may be a helpful foundation for building information modelling, a relational database version of the notions of chain complex and chain complex morphism is defined and used for storing cw-complexes and their morphisms, hence instances of building projects and different views upon them, into relational databases. In many cases, this can be done without loss of topological information. The equivalence of categories between sets with binary relations and Alexandrov spaces is proven and used to incorporate the relational complexes into the more general setting of topological databases. For the latter, a topological version of a relational query language is defined by transferring the usual relational algebra operators into topological constructions. In the end, it is proven that such a topological version of relational algebra in general must be able to compute the transitive closure of a relation.

On p-adic classification

A p-adic modification of the split-LBG classification method is presented in which first clusterings and then cluster centers are computed which locally minimize an energy function. The outcome for a fixed dataset is independent of the prime number p with finitely many exceptions. The methods are applied to the construction of p-adic classifiers in the context of learning.

Comparing G-maps with other topological data structures

This article compares two approaches to storing spatial information: On the one hand there are topological datatypes where primitives and their connectivity are explicitly stored, on the other hand there is the G-maps-approach storing abstract “darts” and groups acting on these darts such that their orbits implicitly give the elements and topology of the stored space. First these concepts are mutually related from a categorial viewpoint and, second, their storage complexity is compared.

Cyclic coverings of the p-adic projective line by Mumford curves

Exact bounds for the positions of the branch points for cyclic coverings of the p-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato’s *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory.

Improved Feature Detection in Fused Intensity-Range Images with Complex SIFT (ℂSIFT)

The real and imaginary parts are proposed as an alternative to the usual Polar representation of complex-valued images. It is proven that the transformation from Polar to Cartesian representation contributes to decreased mutual information, and hence to greater distinctiveness. The Complex Scale-Invariant Feature Transform (CSIFT) detects distinctive features in complex-valued images. An evaluation method for estimating the uniformity of feature distributions in complex-valued images derived from intensity-range images is proposed. In order to experimentally evaluate the proposed methodology on intensity-range images, three different kinds of active sensing systems were used: Range Imaging, Laser Scanning, and Structured Light Projection devices (PMD CamCube 2.0, Z+F IMAGER 5003, Microsoft Kinect).

An ultrametric interpretation of building related event data

The long‐term behaviour of the built environment is relevant to practising architects and engineers as well as to investors and policy makers. In contrast to this, the size, structure and dynamics of that important capital of society are not well established. As a first step towards assessing the dynamics of new constructions, refurbishments, demolitions and other building related event variables in urban building stocks in Southwest Germany, a first random sample of event data is examined using the more efficient ultrametric hierarchical classification in order to compare their dynamics. To this end, different ways of binary encodings of the multivariate data are carried out, and their ultrametric classification results compared. It turns out that municipalities of comparable sizes show similar behaviour in contrast to those of differing sizes, which corresponds to previous findings. Consequently, ultrametric methods can be applied to the study of building stock dynamics by revealing inherent hierarchical structure in data.

Deriving spatio-temporal query results in sensor networks

Tracking moving objects in relation to regions of interest, e.g., for pollution control or habitat monitoring, is an important application of Sensor Networks (SN). Research on Moving Object Databases has resulted in sophisticated mechanisms for querying moving objects and regions declaratively. Applying these results to SN in a straightforward way is not possible: First, sensor nodes typically can only determine that an object is in their vicinity, but not the exact position. Second, nodes may fail, or areas may be unobservable. All this is problematic because the evaluation of spatio-temporal queries requires precise knowledge about object positions. In this paper we specify meaningful results of spatio-temporal queries, given those SN-specific phenomena, and say how to derive them from object detections by sensor nodes. We distinguish between objects which definitely fulfill the query and those that could possibly do so, but where those inaccuracies are in the way of a definite answer. We study both spatio-temporal predicates as well as spatio-temporal developments, i.e., sequences of predicates describing complex movement patterns of objects.

A p-Adic RANSAC Algorithm for Stereo Vision Using Hensel Lifting ∗

A p-adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving the relative pose problem in stereo vision is developed. From two 2-adically encoded images a random sample of five pairs of corresponding points is taken, and the equations for the essential matrix are solved by lifting solutions modulo 2 to the 2-adic integers. A recently devised p-adic hierarchical classification algorithm imitiating the known LBG quantization method classifies the solutions for all the samples after having determined the number of clusters using the known intra-inter validity of clusterings. In the successful case, a cluster ranking will determine the cluster containing a 2-adic approximation to the “true” solution of the problem.