13 research outputs found
Multiple Reflection Symmetry Detection via Linear-Directional Kernel Density Estimation
Symmetry is an important composition feature by investigating similar sides inside an image plane. It has a crucial effect to recognize man-made or nature objects within the universe. Recent symmetry detection approaches used a smoothing kernel over different voting maps in the polar coordinate system to detect symmetry peaks, which split the regions of symmetry axis candidates in inefficient way. We propose a reliable voting representation based on weighted linear-directional kernel density estimation, to detect multiple symmetries over challenging real-world and synthetic images. Experimental evaluation on two public datasets demonstrates the superior performance of the proposed algorithm to detect global symmetry axes respect to the major image shapes
Langevin diffusions on the torus: estimation and applications
We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the OrnsteinâUhlenbeck process. Their likelihood function is a product of transition densities with no analytical expression, but that can be calculated by solving the FokkerâPlanck equation numerically through adequate schemes. We propose three approximate likelihoods that are computationally tractable: (i) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and ShojiâOzaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate transition densities to the exact ones, and investigates the empirical performance of the approximate likelihoods. Finally, two diffusions are used to model the evolution of the backbone angles of the protein G (PDB identifier 1GB1) during a molecular dynamics simulation. The software package sdetorus implements the estimation methods and applications presented in the paper
Directional Statistics in Protein Bioinformatics
This chapter shows how toroidal diffusions are convenient methodological
tools for modelling protein evolution in a probabilistic framework. The chapter
addresses the construction of ergodic diffusions with stationary distributions
equal to well-known directional distributions, which can be regarded as
toroidal analogues of the Ornstein-Uhlenbeck process. The important challenges
that arise in the estimation of the diffusion parameters require the
consideration of tractable approximate likelihoods and, among the several
approaches introduced, the one yielding a specific approximation to the
transition density of the wrapped normal process is shown to give the best
empirical performance on average. This provides the methodological building
block for Evolutionary Torus Dynamic Bayesian Network (ETDBN), a hidden Markov
model for protein evolution that emits a wrapped normal process and two
continuous-time Markov chains per hidden state. The chapter describes the main
features of ETDBN, which allows for both "smooth" conformational changes and
"catastrophic" conformational jumps, and several empirical benchmarks. The
insights into the relationship between sequence and structure evolution that
ETDBN provides are illustrated in a case study.Comment: 26 pages, 13 figure