Hypergraph Modelling for Geometric Model Fitting

In this paper, we propose a novel hypergraph based method (called HF) to fit and segment multi-structural data. The proposed HF formulates the geometric model fitting problem as a hypergraph partition problem based on a novel hypergraph model. In the hypergraph model, vertices represent data points and hyperedges denote model hypotheses. The hypergraph, with large and "data-determined" degrees of hyperedges, can express the complex relationships between model hypotheses and data points. In addition, we develop a robust hypergraph partition algorithm to detect sub-hypergraphs for model fitting. HF can effectively and efficiently estimate the number of, and the parameters of, model instances in multi-structural data heavily corrupted with outliers simultaneously. Experimental results show the advantages of the proposed method over previous methods on both synthetic data and real images.Comment: Pattern Recognition, 201

Purposive sample consensus: A paradigm for model fitting with application to visual odometry

© Springer International Publishing Switzerland 2015. ANSAC (random sample consensus) is a robust algorithm for model fitting and outliers' removal, however, it is neither efficient nor reliable enough to meet the requirement of many applications where time and precision is critical. Various algorithms have been developed to improve its performance for model fitting. A new algorithm named PURSAC (purposive sample consensus) is introduced in this paper, which has three major steps to address the limitations of RANSAC and its variants. Firstly, instead of assuming all the samples have a same probability to be inliers, PURSAC seeks their differences and purposively selects sample sets. Secondly, as sampling noise always exists; the selection is also according to the sensitivity analysis of a model against the noise. The final step is to apply a local optimization for further improving its model fitting performance. Tests show that PURSAC can achieve very high model fitting certainty with a small number of iterations. Two cases are investigated for PURSAC implementation. It is applied to line fitting to explain its principles, and then to feature based visual odometry, which requires efficient, robust and precise model fitting. Experimental results demonstrate that PURSAC improves the accuracy and efficiency of fundamental matrix estimation dramatically, resulting in a precise and fast visual odometry

The Clustering of the SDSS DR7 Main Galaxy Sample I: A 4 per cent Distance Measure at z=0.15

We create a sample of spectroscopically identified galaxies with $z < 0.2$ from the Sloan Digital Sky Survey (SDSS) Data Release 7, covering 6813 deg$^2$. Galaxies are chosen to sample the highest mass haloes, with an effective bias of 1.5, allowing us to construct 1000 mock galaxy catalogs (described in Paper II), which we use to estimate statistical errors and test our methods. We use an estimate of the gravitational potential to "reconstruct" the linear density fluctuations, enhancing the Baryon Acoustic Oscillation (BAO) signal in the measured correlation function and power spectrum. Fitting to these measurements, we determine $D_{V}(z_{\rm eff}=0.15) = (664\pm25)(r_d/r_{d,{\rm fid}})$ Mpc; this is a better than 4 per cent distance measurement. This "fills the gap" in BAO distance ladder between previously measured local and higher redshift measurements, and affords significant improvement in constraining the properties of dark energy. Combining our measurement with other BAO measurements from BOSS and 6dFGS galaxy samples provides a 15 per cent improvement in the determination of the equation of state of dark energy and the value of the Hubble parameter at $z=0$ ($H_0$). Our measurement is fully consistent with the Planck results and the $\Lambda$CDM concordance cosmology, but increases the tension between Planck$+$BAO $H_0$ determinations and direct $H_0$ measurements.Comment: Accepted by MNRAS, distance likelihood is available in source file