Least Squares Fitting of Ellipsoid Using Orthogonal Distances

Abstract

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).

Design of Hyperboloid Structures

“…Since a hyperboloid can be degenerated into other kinds of elliptic quadrics, such as an elliptic paraboloid. Therefore a proper constraint must be added Bektas [15]. …”

“…The geometric (shortest,orthogonal) distance is defined to be the distance between a data point and its closest point on the hyperboloid. For detailed information on this subject refer to Bektas [15]. The following link can be used for the shortest distance and projection …”

Analysis of Repeatability on Videogrammetry for Infants’ Cranial Deformation

“…An adjustment of the ellipsoid is carried out using Least Square Adjustment (Bektas 2015). The norm L2 is used.…”

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“…Здесь рассматривается проблема подгонки эллипсоида для произвольного множества точек, имеющая фундаментальное значение во многих областях прикладной науки: в астрономии, геоде-зии, цифровой обработке изображений, в робототехнике, в метрологии и др. [7].…”

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