Best (orthogonal) fitting ellipsoid with quaternions

ABSTRACTThis paper proposes a novel method for register partially overlapped three-dimensional (3D) point sets when a complete 3D scanned model is constructed from partially scanned 3D point sets. 3D point sets are the most primitive and fundamental surface representations. 3D acquisition devices such as laser scanners and range image finders are popular sources for acquiring point clouds. With the availability of low cost and highly efficient 3D laser scanners, the popularity of such devices is increasing rapidly in various fields. Generally, a standard scanner can obtain only a partial data to build a complete 3D model of the object. These partially scanned 3D patches need to be aligned together in the same coordinate system to construct a complete 3D model, and this process is widely known as registration. A coarse-to-fine alignment technique is proposed to register two partially overlapped point sets. First, the overlapping regions using a novel sampling strategy are identified and the overlapping regions of two point sets are aligned together. Second, fine registration is carried out, in which an auxiliary pair and surface normal constraints are used to find reliable corresponding points in order to achieve higher accuracy.

Given multiple image data from a set of points in 3D, there are two fundamental questions that can be addressed: (1) What is the structure of the set of points in 3D? (2) What are the positions of the cameras relative to the points? In this paper, we show that for projective views and with structure- and position-defined modulo linear transformations, these problems are are dual in the sense that their solution arises from constraint equations where space point and camera positions occur in a reciprocal way. The problem of computing camera positions from m points in n views can be solved with the same algorithm as the problem of directly reconstructing n+4 points in m-4 views. This unifies different approaches for projective reconstruction: methods based on external calibration and direct methods exploiting constraints that exist between space and image invariants.

In this paper, we present techniques for ellipsoid fitting which are based on minimizing the sum of the squares of the geometric distances between the data and the ellipsoid. The literature often uses "orthogonal fitting" in place of "geometric fitting" or "best-fit". For many different purposes, the best-fit ellipsoid fitting to a set of points is required. The problem offitting ellipsoid is encounteredfrequently intheimage processing, face recognition, computer games, geodesy etc. Today, increasing GPS and satellite measurementsprecisionwill allow usto determine amore realistic Earth ellipsoid. Several studies have shown that the Earth, other planets, natural satellites, asteroids and comets can be modeled as triaxial ellipsoids Burša and Šima (1980), Iz et all (2011). Determining the reference ellipsoid for the Earth is an important ellipsoid fitting application, because all geodetic calculations are performed on the reference ellipsoid. Algebraic fitting methods solve the linear least squares (LS) problem, and are relatively straightforward and fast. Fitting orthogonal ellipsoid is a difficult issue. Usually, it is impossible to reach a solution with classic LS algorithms. Because they are often faced with the problem of convergence. Therefore, it is necessary to use special algorithms e.g. nonlinear least square algorithms. We propose to use geometric fitting as opposed to algebraic fitting. This is computationally more intensive, but it provides scope for placing visually apparent constraints on ellipsoid parameter estimation and is free from curvature bias Ray and Srivastava (2008).

Invariance against rotation of 3D objects is one of the essential properties for 3D shape analysis. Recently proposed algorithms have achieved rotationally invariant 3D point set analysis by using inherently rotation-invariant 3D shape features, i.e., distances and angles among 3D points, as input to Deep Neural Networks (DNNs). The DNNs capture spatial hierarchy and context among the geometric features to produce accurate analytical results. In this article, we delve further into the DNN-based approach to rotation-invariant and highly accurate 3D point set analysis. In particular, we focus our attention on segmentation of 3D point sets, which is one of the most challenging among 3D point set analysis tasks. We propose a novel DNN for 3D point set segmentation called Rotation-invariant and Multi-scale feature Graph convolutional neural network, or RMGnet. Our RMGnet is more flexible than the previous methods as it accepts as input any handcrafted 3D shape features having rotation invariance. In addition, to accurately segment 3D point sets composed of parts having various sizes, we randomize scales at which handcrafted features are extracted and perform multi-resolution analysis of the features by using the DNN. Experimental evaluation demonstrates high segmentation accuracy as well as rotation invariance of the proposed RMGnet.

Our goal is the registration of multiple 3D point clouds obtained from LIDAR scans of underground mines. Such a capability is crucial to the surveying and planning operations in mining. Often, the point clouds only partially overlap and initial alignment is unavailable. Here, we propose an interactive user-assisted point cloud registration system. Guided by the system, the user's role is simply to identify and search for overlapping regions across the point clouds. Specifically, given two point sets, the user clicks on a point in one set, then simply hovers the mouse on the other set to find a matching point. Each mouse position gives rise to a translation, and our system instantly optimises the rotation that aligns the point clouds. Assuming that each individual point set is horizontally levelled with the ground by the level compensator on the LIDAR device, given a candidate 3D translation only one angular parameter needs to be estimated to rotationally align two 3D point sets. We propose a fast rotation search algorithm that delivers globally optimal results in real time. Our method conducts branch-and-bound optimisation with a novel bounding function whose evaluation amounts to simple sorted array operations. This provides smooth and accurate feedback to the user's search for overlapping regions. Our system is intuitive and helps to accelerate the registration of multiple scans.

Common techniques in structure from motion do not explicitly handle foreground occlusions and disocclusions, leading to several trajectories of a single 3D point. Hence, different discontinued trajectories induce a set of (more inaccurate) 3D points instead of a single 3D point, so that it is highly desirable to enforce long continuous trajectories which automatically bridge occlusions after a re-identification step. The solution proposed in this paper is to connect features in the current image to trajectories which discontinued earlier during the tracking. This is done using a correspondence analysis which is designed for wide baselines and an outlier elimination strategy using the epipolar geometry. The reference to the 3D object points can be used as a new constraint in the bundle adjustment. The feature localization is done using the SIFT detector extended by a Gaussian approximation of the gradient image signal. This technique provides the robustness of SIFT coupled with increased localization accuracy.Our results show that the reconstruction can be drastically improved and the drift is reduced, especially in sequences with occlusions resulting from foreground objects. In scenarios with large occlusions, the new approach leads to reliable and accurate results while a standard reference method fails.KeywordsObject PointForeground ObjectBundle AdjustmentSift DescriptorReprojection ErrorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The paper describes a robust method to extract 3D lines from stereo point clouds. This method combines 2D image information with 3D point clouds from a stereo camera. 2D lines are first extracted from the image in the stereo pair, followed by 3D line regression from the back-projected 3D point set of the images points in the detected 2D lines. In this paper, random sample consensus (RANSAC) is used to estimate 3D line from the 3D point set, the Mahalanobis distance from each 3D point to the 3D line is derived, and the statistically motivated distance measure is used to compute the support for the detected 3D line. Experimental results on real environment with high level of clutter, occlusion, and noise demonstrate the robustness of the algorithm.

We develop expressions for measuring the distance between 2D point sets, which are invariant to either 2D affine transformations or 2D similarity transformations of the sets, and assuming a known correspondence between the point sets. We discuss the image normalization to be applied to the images before their comparison so that the computed distance is symmetric with respect to the two images. We then give a general (metric) definition of the distance between images, which leads to the same expressions for the similarity and affine cases. This definition avoids ad hoc decisions about normalization. Moreover, it makes it possible to compute the distance between images under different conditions, including cases where the images are treated asymmetrically. We demonstrate these results with real and simulated images.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this work we consider 3D point sets, which in a typical setting represent unorganized point clouds. Segmentation of these point sets requires first to single out structural components of the unknown surface discretely approximated by the point cloud. Structural components, in turn, are surface patches approximating unknown parts of elementary geometric structures, such as planes, ellipsoids, spheres and so on. The approach used is based on level set methods computing the moving front of the surface and tracing the interfaces between different parts of it. Level set methods are widely recognized to be one of the most efficient methods to segment both 2D images and 3D medical images. Level set methods for 3D segmentation have recently received an increasing interest. We contribute by proposing a novel approach for raw point sets. Based on the motion and distance functions of the level set we introduce four energy minimization models, which are used for segmentation, by considering an equal number of distance functions specified by geometric features. Finally we evaluate the proposed algorithm on point sets simulating unorganized point clouds.

Scanning devices acquire geometric information from the surface of an object in the form of a 3D point set. Such point sets, as any data obtained by means of physical measurement, contain some noise. To create an accurate model of the scanned object, this noise should be resolved before or during the process of surface reconstruction. In this paper, we develop a statistical technique to estimate the noise in a scanned point set. The noise is represented as normal distributions with zero mean and their variances determine the amount of the noise. These distributions are estimated with a variational Bayesian method, which is known to provide more robust estimations than point estimate methods, such as maximum likelihood and maximum a posteriori. Validation experiments and further tests with real scan data show that the proposed technique can accurately estimate the noise in a 3D point set.

This paper deals with the problem of precise automatic estimation of the surface geometry of pot sherds uncovered at archaeological excavation sites using dense 3D laser-scan data. Critical to ceramic fragment analysis is the ability to geometrically classify excavated sherds, and, if possible, reconstruct the original pots using the sherd fragments. To do this, archaelogists must estimate the pot geometry in terms of an axis and associated profile curve from the discovered fragments. In this paper, we discuss an automatic method for accurately estimating an axis/profile curve pair for each archeological sherd (even when they are small) based on axially symmetric implicit polynomial surface models. Our method estimates the axis/profile curve for a sherd by finding the axially symmetric algebraic surface which best fits the measured set of dense 3D points and associated normals. We note that this method will work on 3D point data alone and does not require any local surface computations such as differentiation. Axis/profile curve estimates are accompanied by a detailed statistical error analysis. Estimation and error analysis are illustrated with application to a number of sherds. These fragments, excavated from Petra, Jordan, are chosen as exemplars of the families of geometrically diverse sherds commonly found on an archeological excavation site. We then briefly discuss how the estimation results may be integrated into a larger pot reconstruction program.

A camera equipped with a transparent shield can be modeled using the pinhole camera model and residual error vectors defined by the difference between the estimated ray from the pinhole camera model and the actual three-dimensional (3D) point. To calculate the residual error vectors, we employ sparse calibration data consisting of 3D points and their corresponding 2D points on the image. However, the observation noise and sparsity of the 3D calibration points pose challenges in determining the residual error vectors. To address this, we first fit Gaussian Process Regression (GPR) operating robustly against data noise to the observed residual error vectors from the sparse calibration data to obtain dense residual error vectors. Subsequently, to improve performance in unobserved areas due to data sparsity, we use an additional constraint; the 3D points on the estimated ray should be projected to one 2D image point, called the ray constraint. Finally, we optimize the radial basis function (RBF)-based regression model to reduce the residual error vector differences with GPR at the predetermined dense set of 3D points while reflecting the ray constraint. The proposed RBF-based camera model reduces the error of the estimated rays by 6% on average and the reprojection error by 26% on average.

A new mesh reconstruction scheme for approximating a surface from a set of unorganized 3D points is proposed. The proposed method, called a shrink-wrapped boundary face (SWBF) algorithm, produces the final surface by iteratively shrinking the initial mesh generated from the definition of the boundary faces. SWBF surmounts the genus-0 spherical topology restriction of previous shrink-wrapping-based mesh generation techniques and can be applied to any type of surface topology. Furthermore, SWBF is significantly faster than a related algorithm of Jeong and others, as SWBF requires only a local nearest-point-search in the shrinking process. Our experiments show that SWBF is very robust and efficient for surface reconstruction from an unorganized point cloud.

This paper addresses the problem of segmenting an unstructured set of 3D points sampled from a physical model. Given such a set of points the first problem is that of creating a polyhedral approximation of the model. For that we introduce an algorithm which extends Boissonnat's work and generates a triangular surface approximation. The proposed method allows the reconstruction of objects with arbitrary genus and proposes an automatic termination procedure. The next step in the process concerns the segmentation of the data points into regions for which each may be fitted by a single surface. Here we summarize our experiences with a region growing technique based on angle between normals criteria. Using just first order derivative estimations it is shown that the method is able to classify segments into predefined second order surface classes.

Generative modeling is an approach to computer-assisted geometric modeling. The goal of the approach is to allow convenient and high-level specification of shapes, and provide tools for rendering and analysis of the specified shapes. Shapes include curves, surfaces, and solids in 3D space, as well as higher-dimensional entities such as surfaces deforming in time, and solids with a spatially varying mass density. Shape specification in the approach involves combining low-dimensional entities, especially 2D curves, into higher-dimensional shapes. This combination is specified through a powerful shape description language which builds multidimensional parametric functions. The language is based on a set of primitive operators on parametric functions which include arithmetic operators, vector and matrix operators, integration and differentiation, constraint solution and global optimization. Although each primitive operator is fairly simple, high-level shapes and shape building operators can be defined using recursive combination of the primitive operators. The approach encourages the modeler to build parameterized families of shapes rather than single instances. Shapes can be parameterized by scalar parameters (e.g., time or joint angle) or higher-dimensional parameters (e.g., a curve controlling how the scale of a cross section varies as it is translated). Such parameterized shapes allow easy modification of the design, since the modeler can interact with parameters that relate to high-level properties of the shape. In contrast, many geometric modeling systems use a much lower-level specification, such as through sets of many 3D control points. Tools for rendering and analysis of generative models are developed using the concept of interval analysis. Each primitive operator on parametric functions has an inclusion function method, which produces an interval bound on the range of the function, given an interval bound on its domain. With these inclusion functions, robust algorithms exist for computing solutions to nonlinear systems of constraints and global minimization problems, when these problems are expressed in the modeling language. These algorithms, in turn, are developed into robust approximation techniques to compute intersections, CSG operations, and offset operations.