arXiv subjects

Martin Brooks

Publications and source records attributed to Martin Brooks.

6 recordsLinked to original sources

arXiv2017

A Topological Approach to Scaling in Financial Data

There is a large body of work, built on tools developed in mathematics and physics, demonstrating that financial market prices exhibit self-similarity at different scales. In this paper, we explore the use of analytical topology to characterize financial price series. While wavelet and Fourier transforms decompose a signal into sets of wavelets and power spectrum respectively, the approach presented herein decomposes a time series into components of its total variation. This property is naturally suited for the analysis of scaling characteristics in fractals.

q-fin.TR↗

arXiv2016

Retinal Vessel Segmentation Using A New Topological Method

A novel topological segmentation of retinal images represents blood vessels as connected regions in the continuous image plane, having shape-related analytic and geometric properties. This paper presents topological segmentation results from the DRIVE retinal image database.

cs.CG↗

arXiv2016

Persistence Lenses: Segmentation, Simplification, Vectorization, Scale Space and Fractal Analysis of Images

A persistence lens is a hierarchy of disjoint upper and lower level sets of a continuous luminance image's Reeb graph. The boundary components of a persistence lens's interior components are Jordan curves that serve as a hierarchical segmentation of the image, and may be rendered as vector graphics. A persistence lens determines a varilet basis for the luminance image, in which image simplification is a realized by subspace projection. Image scale space, and image fractal analysis, result from applying a scale measure to each basis function.

cs.CV↗

arXiv2016

Varilets: Additive Decomposition, Topological Total Variation, and Filtering of Scalar Fields

Continuous interpolation of real-valued data is characterized by piecewise monotone functions on a compact metric space. Topological total variation of piecewise monotone function f:X->R is a homeomorphism-invariant generalization of 1D total variation. A varilet basis is a collection of piecewise monotone functions { $g_i$ |i = 1...n}, called varilets, such that every linear combination $\sum a_ig_i$ has topological total variation $\sum |a_i|$. A varilet transform for $f$ is a varilet basis for which $f =\sum α_ig_i$. Filtered versions of $f$ result from altering the coefficients $α_i$.

cs.CG↗

arXiv2007

An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation

Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem using the l_inf norm, and we present an optimal linear time algorithm based on novel formalism. Moreover, given a precomputation in time O(n log n) consisting of a labeling of all extrema, we compute any optimal segmentation in constant time. We compare experimentally its performance to two piecewise linear segmentation heuristics (top-down and bottom-up). We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.

cs.DB↗

arXiv2007

An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation

Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting an array in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem, present an optimal linear time algorithm based on novel formalism, and compare experimentally its performance to a linear time top-down regression algorithm. We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.

cs.DS↗