Reconstructing triangulated surfaces from unorganized points through local skeletal stars

Surface reconstruction from unorganized points arises in a variety of practical situations such as range scanning an object from multiple view points, recovery of biological shapes from twodimensional slices, and interactive surface sketching. [...]Reconstrução da superfície de pontos desorganizados surge em uma variedade de situações práticas, tais como rastreamento de um objeto a partir de vários pontos de vista, a recuperação de formas biológicas de fatias bi-dimensionais, e esboçar superfícies interativas. [...

TVL₁ Planarity Regularization for 3D Shape Approximation

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within. This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy

Some remarks on Heegner point computations

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004

ASSESSMENT OF ULCER WOUNDS USING 3D SKIN SURFACE IMAGING

In medical care, ulcer wound refers to open wound or sore in which certain conditions exist that impede healing. Nonhealing wounds can cause economical and psychological distress for patients. Wound size measurement (top area, true surface area, depth, and volume) is an objective indicator for wound healing. Top area measurement is useful for the follow up of shallow wounds, while true surface area if done accurately can work for all types of wounds. Calculating ulcer volume is crucial since studies showed that wounds start healing from the bottom. Overestimation in top area and true surface area measurement can be solved by digitizing the traced part. The objective of this research is to develop computer algorithms to measure ulcer wound size using 3D surface imaging. The wounds of interest are the wounds located at the leg. The algorithms should construct wound models and compute volume without getting affected by irregularities on wound surface and they should model leg curvature. Two algorithms for constructing wound models and volume computation are developed and evaluated; namely midpoint projection and convex hull approximation (Delaunay tetrahedralization). Parameters that describe the wounds are developed based on real ulcer wound surface images for wound modelling. Wound models representing possible ulcer wounds developed using AutoCAD software are used to investigate the performance of solid reconstruction methods. Results and analysis show that, for volume computation midpoint and convex hull methods can compute volume of leg ulcer without getting affected by irregularities in the healthy skin around the wound. The results show that, for convex hull low errors are produced in cases of regular boundary models excluding the elevated base models. Overestimation in volume for convex hull method can either be due to irregular boundary and/or elevation at the base (both global and local). Surface division is performed prior to convex hull approximation so that the high curvature of the leg and irregularity at the boundary can be represented using a number of linear segments. With the increase in surface division, error due to irregular boundary is reduced. In the case of global curvature, the reconstructed model using convex hull preceded by surface division simulates the leg curvature. Midpoint outperforms convex hull for models excluding elevated base models. Midpoint can construct solids for wound surfaces with local curvature while for surfaces with high global curvature the error is high. Midpoint method is not suitable for shallow and very large wounds

TVL₁shape approximation from scattered 3D data

With the emergence in 3D sensors such as laser scanners and 3D reconstruction from cameras, large 3D point clouds can now be sampled from physical objects within a scene. The raw 3D samples delivered by these sensors however, contain only a limited degree of information about the environment the objects exist in, which means that further geometrical high-level modelling is essential. In addition, issues like sparse data measurements, noise, missing samples due to occlusion, and the inherently huge datasets involved in such representations makes this task extremely challenging. This paper addresses these issues by presenting a new 3D shape modelling framework for samples acquired from 3D sensor. Motivated by the success of nonlinear kernel-based approximation techniques in the statistics domain, existing methods using radial basis functions are applied to 3D object shape approximation. The task is framed as an optimization problem and is extended using non-smooth L1 total variation regularization. Appropriate convex energy functionals are constructed and solved by applying the Alternating Direction Method of Multipliers approach, which is then extended using Gauss-Seidel iterations. This significantly lowers the computational complexity involved in generating 3D shape from 3D samples, while both numerical and qualitative analysis confirms the superior shape modelling performance of this new framework compared with existing 3D shape reconstruction techniques

Genus statistics using the Delaunay tessellation field estimation method: (I) tests with the Millennium Simulation and the SDSS DR7

We study the topology of cosmic large-scale structure through the genus statistics, using galaxy catalogues generated from the Millennium Simulation and observational data from the latest Sloan Digital Sky Survey Data Release (SDSS DR7). We introduce a new method for constructing galaxy density fields and for measuring the genus statistics of its isodensity surfaces. It is based on a Delaunay tessellation field estimation (DTFE) technique that allows the definition of a piece-wise continuous density field and the exact computation of the topology of its polygonal isodensity contours, without introducing any free numerical parameter. Besides this new approach, we also employ the traditional approaches of smoothing the galaxy distribution with a Gaussian of fixed width, or by adaptively smoothing with a kernel that encloses a constant number of neighboring galaxies. Our results show that the Delaunay-based method extracts the largest amount of topological information. Unlike the traditional approach for genus statistics, it is able to discriminate between the different theoretical galaxy catalogues analyzed here, both in real space and in redshift space, even though they are based on the same underlying simulation model. In particular, the DTFE approach detects with high confidence a discrepancy of one of the semi-analytic models studied here compared with the SDSS data, while the other models are found to be consistent.Comment: 14 pages, 9 figures, accepted by Ap