Quadric surface fitting for sparse range data

Abstract

Surface fitting of 3-D data is one of the basic methods of surface description in computer vision. Most techniques and algorithms presented h the literature are least-squaresbased. In this paper we present a systematic comparison of three commonly used least-squares based methods. The three algorithms were tested on complete sets of quadric surfaces for different combination of noise levels and patch sizes. The technique of M-estimate which is useful in case of gross outliers is also tested. Results are reported for synthetic and real range data. I. INTRODUCTlONClassification of surfaces fitting experimental range data is important in the area of 3-D object recognition [1,2]. A classical approach to obtaining surface representation for two and three dimensional image points is to use linear least squares methods to obtain the best possible solution to the general equation of second degree with the given data constraints [3].The methods are known to be unreliable in presence of noise and/or small samples. In this paper, a systematic study is presented that describes the relationship between noise levels, patch sizes and reliability of surface classification. The set of n sparse range data points are substituted in the general equation of second degree which gives rise to a set of n linear equations. The solution to this system of equations Ax = B can be obtained in three different ways. 1) By solving the system of normal equationsATAx = ATb [7], 2) By singular value decomposition method [4] or 3) By directly computing the eigenvectors of AYA, after substituting A, = [AI b] .A comparison of these three methods is made for different noise levels and for different patch sizes. The results are not satisfactory in case the data contains even a few outlying points. The results can be improved by using the M-estimate method [4,5,61.The different methods' were tested on several sets of synthetic and real data. Complet sets of quadric surfaces were tested. In each case the standard deviation c s of the Gaussian 1. This work was supported by NSF grant IRI-8803665 and State of Michigan Research Excellence Funds. The second author would like to acknowledge Central Michigan University Research Professor Award. 123noise ranged from 0 to 0.05 (The noise level depends on the scale of the data). Four different patch sizes were tested in each case representing data from the entire surface, half surface, quarter surface and from small patch of surface. Similar tests were made for two different types of real range data, a sphere and a cylinder. Both synthetic and real data showed improvement in the results obtained by using the M-estimate method in case of gross outlying points.

A survey of methods for recovering quadrics in triangle meshes

“…as above works well if the data points satisfy the condition that the independent variables (say x, y) are measured without error and the dependent variable (z) has Gaussian noise [Cao and Shrikhande 1991]. Unfortunately, this condition may not be satisfied for real-world data.…”

“…A way of solving this problem is to treat c as a variable [Cao and Shrikhande 1991]. Let A 1 be the matrix A augmented by the vector b = (c, .…”

<title>Image segmentation using Gaussian curvature</title>

“…Consider (4) and xi+yi+ z s= 2 (5) It is sufficient to compute the angle deficit as in equation (1) to get a parameter that approximates the Gaussian curvature ofa surface. For each point in the range image, we apply a 3 x 3 mask.…”

Application of Optimal Ellipsoid Fitting for Curvature Change Tracking in Main Reflector of Complex Antenna Structures

“…Primitive 3D fitting models involve fitting a quadratic surface to a set of points in space, which has applications in 3D reconstruction, pose estimation, the restricted stereo correspondence problem, and object recognition. (4)(5)(6)(7)(8)(9)(10) One of the most used fitting models is the ellipsoid, which is of paramount importance in several fields.…”