Ellipse-based Principal Component Analysis for Self-intersecting Curve Reconstruction from Noisy Point Sets

Surface reconstruction from cross cuts usually requires curve reconstruction from planar noisy point samples -- The output curves must form a possibly disconnected 1manifold for the surface reconstruction to proceed -- This article describes an implemented algorithm for the reconstruction of planar curves (1manifolds) out of noisy point samples of a sel-fintersecting or nearly sel-fintersecting planar curve C -- C:[a,b]⊂R→R is self-intersecting if C(u)=C(v), u≠v, u,v∈(a,b) (C(u) is the self-intersection point) -- We consider only transversal self-intersections, i.e. those for which the tangents of the intersecting branches at the intersection point do not coincide (C′(u)≠C′(v)) -- In the presence of noise, curves which self-intersect cannot be distinguished from curves which nearly sel fintersect -- Existing algorithms for curve reconstruction out of either noisy point samples or pixel data, do not produce a (possibly disconnected) Piecewise Linear 1manifold approaching the whole point sample -- The algorithm implemented in this work uses Principal Component Analysis (PCA) with elliptic support regions near the selfintersections -- The algorithm was successful in recovering contours out of noisy slice samples of a surface, for the Hand, Pelvis and Skull data sets -- As a test for the correctness of the obtained curves in the slice levels, they were input into an algorithm of surface reconstruction, leading to a reconstructed surface which reproduces the topological and geometrical properties of the original object -- The algorithm robustly reacts not only to statistical noncorrelation at the self-intersections(nonmanifold neighborhoods) but also to occasional high noise at the nonselfintersecting (1manifold) neighborhood

Methods of obtaining smooth surface in 2D/3D surface reconstruction

Surface reconstruction is an emergent research area in the field of computer aided design and manufacturing. There are various methods / algorithms which are working considerably well for surface reconstruction problem but we cannot say to the best of our knowledge that we got all the solutions. Missing surface can be repaired either by surface patch or by extending boundary curves. However, in both cases, surface smoothening problem arises in form of flat surface. The present paper has been tried to offer a solution to above problem which makes the curve smoother

Imaging starspot evolution on Kepler target KIC 5110407 using light curve inversion

The Kepler target KIC 5110407, a K-type star, shows strong quasi-periodic light curve fluctuations likely arising from the formation and decay of spots on the stellar surface rotating with a period of 3.4693 days. Using an established light-curve inversion algorithm, we study the evolution of the surface features based on Kepler space telescope light curves over a period of two years (with a gap of .25 years). At virtually all epochs, we detect at least one large spot group on the surface causing a 1-10% flux modulation in the Kepler passband. By identifying and tracking spot groups over a range of inferred latitudes, we measured the surface differential rotation to be much smaller than that found for the Sun. We also searched for a correlation between the seventeen stellar flares that occurred during our observations and the orientation of the dominant surface spot at the time of each flare. No statistically-significant correlation was found except perhaps for the very brightest flares, suggesting most flares are associated with regions devoid of spots or spots too small to be clearly discerned using our reconstruction technique. While we may see hints of long-term changes in the spot characteristics and flare statistics within our current dataset, a longer baseline of observation will be needed to detect the existence of a magnetic cycle in KIC 5110407.Comment: 32 pages, 15 figures, accepted to Ap

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat \SL(2,\C)-connections. In this the paper we parametrize the moduli space of flat \SL(2,\C)-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\geq2$ and prove that it is generated by simple factor dressing.Comment: 39 pages; sections about isospectral deformations and about CMC surfaces have been added; the theorems on the reconstruction of surfaces out of spectral data have been improved; 1 figure adde

3D Face Reconstruction from Light Field Images: A Model-free Approach

Reconstructing 3D facial geometry from a single RGB image has recently instigated wide research interest. However, it is still an ill-posed problem and most methods rely on prior models hence undermining the accuracy of the recovered 3D faces. In this paper, we exploit the Epipolar Plane Images (EPI) obtained from light field cameras and learn CNN models that recover horizontal and vertical 3D facial curves from the respective horizontal and vertical EPIs. Our 3D face reconstruction network (FaceLFnet) comprises a densely connected architecture to learn accurate 3D facial curves from low resolution EPIs. To train the proposed FaceLFnets from scratch, we synthesize photo-realistic light field images from 3D facial scans. The curve by curve 3D face estimation approach allows the networks to learn from only 14K images of 80 identities, which still comprises over 11 Million EPIs/curves. The estimated facial curves are merged into a single pointcloud to which a surface is fitted to get the final 3D face. Our method is model-free, requires only a few training samples to learn FaceLFnet and can reconstruct 3D faces with high accuracy from single light field images under varying poses, expressions and lighting conditions. Comparison on the BU-3DFE and BU-4DFE datasets show that our method reduces reconstruction errors by over 20% compared to recent state of the art

Functional principal component analysis of spatially correlated data

This paper focuses on the analysis of spatially correlated functional data. We propose a parametric model for spatial correlation and the between-curve correlation is modeled by correlating functional principal component scores of the functional data. Additionally, in the sparse observation framework, we propose a novel approach of spatial principal analysis by conditional expectation to explicitly estimate spatial correlations and reconstruct individual curves. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov (Xi(s),Xi(t))(Xi(s),Xi(t)) and cross-covariance surface Cov (Xi(s),Xj(t))(Xi(s),Xj(t)) at locations indexed by i and j. Then a anisotropy Matérn spatial correlation model is fitted to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We demonstrate the consistency of our estimates and propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Using simulation studies, we show that these methods have some clear advantages over existing methods of curve reconstruction and estimation of model parameters

Curvature based sampling of curves and surfaces

Efficient sampling methods enable the reconstruction of a generic surface with a limited amount of points. The reconstructed surface can therefore be used for inspection purpose. In this paper a sampling method that enables the reconstruction of a curve or surface is proposed. The input of the proposed algorithm is the number of required samples. The method takes into account two factors: the regularity of the sampling and the complexity of the object. A higher density of samples is assigned where there are some significant features, described by the curvature. The analysed curves and surfaces are described through the B-splines spaces. The sampling of surfaces generated by two or more curves is also discussed