The problem of this research tackles the process of
fitting two concentric spheres to data, which arises in
computational metrology. There are also many fitting criteria that
could be used effectively, and the most widely used one in
metrology, for example, is that of the sum of squared minimal
distance. However, a simple and robust algorithm assigned for
using the orthogonal distance regression will be proposed in this
paper. A common approach to this problem involves an iteration
process which forces orthogonality to hold at every iteration and
steps of Gauss-Newton type

I. A. Al-Subaihi

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Mathematical Communications 13(2008), 233-239 233Fitting two concentric spheres to data byorthogonal distance regressionI. A. Al-Subaihi∗Abstract. The problem of this research tackles the process of ﬁttingtwo concentric spheres to data, which arises in computational metrology.There are also many ﬁtting criteria that could be used eﬀectively, andthe most widely used one in metrology, for example, is that of the sumof squared minimal distance. However, a simple and robust algorithmassigned for using the orthogonal distance regression will be proposed inthis paper. A common approach to this problem involves an iterationprocess which forces orthogonality to hold at every iteration and steps ofGauss-Newton type.Key words: concentric spheres, orthogonal distance regression, Gauss-Newton methodAMS subject classifications: 65D10Received April 12, 2007 Accepted October 23, 20071. IntroductionTo ﬁnd measured points in space two concentric spheres of the minimum diﬀer-ence in radii that contain all points between them is a problem in computationalmetrology [6, 9]. Let data xi = (xi, yi, zi), i = 1 . . . ,m, be available. In order tomake the problem of this paper more evident, it is recommended to assume that allpoints have errors. Moreover, to ﬁnd two concentric spheres, it is suggested thatsuch suitable criteria goes to its minimum level, where simultaneously each pointhas to be associated with one of the spheres [6].The two concentric spheres are given in parametric representation asx(ap, t) = c+ rcos t1 sin t2sin t1 sin t2cos t2 , 0 ≤ t1 < 2π, 0 < t2 ≤ π (1)out of the smaller sphere, where ap = (c, r)T , c = (c1, c2, c3)T is the common centre,and r is the radius. Replacing ap by aq = (c, R)T in (1) gives the larger parametric∗Department of Science, King Abdulaziz Military Academy, P.O. Box 73052, Riyadh 11538,K.S.A., e-mail: alsubaihi@hotmail.com234 I.A.Al-Subaihisphere. The parametric representations are dependent upon the location parametert, which are independent of any coordinate system [11].Fitting two concentric spheres to data is interpreted as a problem of orthogonaldistance regression (ODR) in [6], which seems to have been the ﬁrst suggested.The problem, in brief, is to ﬁnd two subsets V and W of 1, . . . ,m with V ∩W =φ, V ∪ W = {1, . . . ,m}, |V | ≥ 4, |W | ≥ 4 such that the sum of squared minimaldistancesS(V,W, a) =∑i∈Vmintp2i (ap, t) +∑i∈Wmintq2i (aq, t) (2)is minimized, where a = (c, r, R)T , V and W are the sets of given points associatedwith the smaller and larger spheres respectively, and wherep2i (ap, t) = (a+ r cos t1 sin t2 − xi)2 + (b+ r sin t1 sin t2 − yi)2+ (c+ r cos t2 − zi)2,and q2i (aq, t) represent the larger sphere. The necessary condition,∂S∂a = 0, needsto solve a linear system of 5×5 equations for a at each iteration, until the toleranceis reduced to a suﬃcient small number. The algorithm of minimizing (2) for t isconsidered in section 4.The purpose in this paper, is to ﬁt two concentric spheres to data by an equiva-lent problem to [6], which becomes more appropriate when the data have indepen-dent identically distibuted errors [11]. The orthogonal distance regression problem,which is a nonlinear least squares problem, will be deﬁned in next section. Further,the Jacobian matrix for the Gauss-Newton iteration and the starting points are de-ﬁned in section 3. Furthermore, an algorithm and numerical results are consideredin sections 4 and 5 respectively. In all cases in this paper, ‖.‖ denotes the l2 vectoror matrix norm.2. The ODR problemThe orthogonal distances vector represented in the following formpi = xi − x(ap, ti), i ∈ V,refers to the smaller parametric sphere. Also, qi, i ∈ W is used for the large one,which came as a result of replacing ap by aq . Then the problem given in this paperis to ﬁnd two subsets V , and W by minimizingF (V,W, a, t) =∑i∈V‖pi(ap, ti)‖2 +∑i∈W‖qi(aq, ti)‖2, (3)with respect to a = (c, r, R)T ∈ R5 and ti, i = 1, . . . ,m.Let ti(ap), i ∈ V be such that for any ap, ‖xi − x(ap, ti)‖ is minimized withrespect to ti, i ∈ V , and similarly for ti(aq), i ∈ W . Then the equivalent problemto (3) is the minimization with respect to a = (c, r, R)T ∈ R5 alone ofF (V,W, a) =∑i∈V‖pi(ap, ti(ap))‖2 +∑i∈W‖qi(aq , ti(aq))‖2. (4)Fitting two concentric spheres 235If we deﬁne, for simplicity, the vector δ ∈ Rm which has ith componentδi(a) = ‖xi − c− rαi −Rαi‖,and r = 0 if i ∈W , R = 0 if i ∈ V , whereαi =cos(t1i) sin(t2i)sin(t1i) sin(t2i)cos(t2i) .Then the orthogonal distance regression problem is to ﬁnd two subsets V , and Wsuch as minimizing the objective functionF (V,W, a) = ‖δ‖2. (5)This is a nonlinear problem which can be solved through the application of theGauss-Newton method, or on of its variants, see for example [3, 4, 5, 10, 11, 12].Then the subsets V and W can be determined by calculating, for each i, ‖pi‖ and‖qi‖. If ‖pi‖ < ‖qi‖, then i ∈ V , otherwise in W .3. A Gauss-Newton stepThe Gauss-Newton step d for minimizing (5) is given by ﬁndingmind‖δ +∇aδd‖2,where ∇aδ ∈ Rm×5 a Jacobian matrix has ith row, for any δi = 0∇aδi =− pTi‖pi‖[I αi 0]for i ∈ V,− qTi‖qi‖[I 0 αi]for i ∈W,(6)by deﬁnition of ti(a), and I signiﬁes the identity matrix of order 3. Using theexpression in the right-hand side of (6), the correct vector of partial derivatives ofδi is easily calculated.It is necessary to provide primary values for any iterative algorithm. Actually,it is preferable to minimize the procedures of determining the initial values of a(0),which is considered in [6], in order to cut the long story short. The initial centre ccan be determined by the data mean. The primary values assigned for radii can befound in the following equationsr(0) =1fh, R(0) = fH, 0 < f < 1, (7)where h,H are the minimal and maximal distance of the data points to the startingcentre, respectively. The scalar f is used on the basis of the assumption that notall points with errors are estimated to be lying between the inner and outer sphere,and for R(0) > r(0).236 I.A.Al-Subaihi4. An algorithmThe contents of the last two sections give the ingredients of algorithms which canbe used to solve the orthogonal distance regression problem. For example, we couldconsider an algorithm as followsSTEP 0. Input: The data xi,V (0),W (0), and a tolerance (Tol).STEP 1. Determine: the initial values of a(0).STEP 2. Determine:• ti(a), i = 1, . . . ,m.• The nearest points on the two concentric spheres.• δ(a).• The objective function: F .• The Jacobian matrix:∇aδ(a).STEP 3. Solve: ‖δ(a) +∇aδ(a)d‖2 for d.• if ‖d‖∞ < Tol , Then Go to STEP 7.STEP 4. Use line search.STEP 5. Determine: V (k) and W (k).STEP 6. Go to STEP 2.STEP 7. The process of ﬁtting the two concentric spheres using ODR has beencompleted.The least squares solution is used to determine the values of ti(a), i = 1, . . . ,m,see for example [6, 7], the algorithm [1]ti1 = arctan | yi − c2xi − c1 |.• If xi − c1 < 0, then ti1 = ti1 + π.• If xi − c1 = 0, then– ti1 = π2 , if yi − c2 ≥ 0.– ti1 = 32π, if yi − c2 < 0.ti2 = arctan | (xi − c1) cos(ti1) + (yi − c2) sin(ti1)zi − c3 |• If zi − c3 < 0, then ti2 = ti2 + π,.• If zi − c3 = 0, then– ti2 = π2 , if (xi − c1) cos(ti1) + (yi − c2) sin(ti1) > 0.– ti2 = 32π, if (xi − c1) cos(ti1) + (yi − c2) sin(ti1) < 0.Fitting two concentric spheres 2375. Numerical resultsIn order to verify the performance of the proposed algorithm, it is applied to someexamples. Therefore, the plan adopted in this regard, is to compute a Gauss-Newtonstep d as indicated in the section 3, to be used as an approach to reduce the valueof the objective function F , taking full steps, if possible. The performance shown inthe examples is typical, with a second order rate of convergence usual. The factorf = 0.8 has been used to determine the initial radii, see (7). The symbol γ used inthe following tables refers to the Gauss-Newton step d’s length, and k refers to theiteration number.Example 1 Consider the xi, i = 1, . . . , 16 values given in Table 1, which hasbeen considered in [6]. The Gauss-Newton method is applied with the line searchingfor ﬁtting 16 points with two concentric spheres. The initial value of a(0) wasa(0) = (1.6875, 2.1875, 1.3125, 2.9263, 5.5125)T .Table 2 shows the characterization of V and W , where an integer vector f withfi = 1 if i ∈ V and fi = 2 if i ∈ W . This method gives V = {1, 2, . . . , 8} andW = {11, 12, . . . , 16} in 2 iterations, and converge to the global minimum in 6iterations. Results are shown in Table 3. In the same time, the ﬁnal sets of V andW by [6] take 35 iterations to determine, and 135 iterations to converge to the sameminimum of applying the Gauss-Newton method with line search.1 -1 0 -1 2 2 4 5 1 -1 0 -1 3 2 5 6x 2 1 3 4 2 3 1 1 2 1 3 5 2 3 1 1-1 0 -1 6 -1 -1 6 4 -2 -1 -2 7 -2 -2 6 5Table 1: Example 1 data pointsk f0 1 1 1 2 1 1 2 2 1 1 1 2 1 1 2 21 1 1 1 1 1 1 2 1 2 2 2 2 2 2 2 22-6 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2Table 2: f within the iterations processExample 2 The ﬁrst subset, of size |V |, data points are generated, by takinga particular sphere. The second subset |W | also generated in the same idea andsame centre, but R > r, where |V | = |W | = 50. In other word, the data has beengenerated from the spheres a = (−2, 3, 1, 6, 7)T . Then random perturbations areintroduced for these data. The initial subsets V (0) with |V (0)| = 70 and W (0) with|W (0)| = 30 have been determined by taking a random permutation of the integersfrom 1 to m, and the Matlab command ”randperm” is used for this procedure. Theinitial approximation to a(0) givesa(0) = (−2.3853, 2.8976,−0.3254, 5.5841, 6.7438)T.238 I.A.Al-Subaihik a b c r R F ‖d‖∞ γ1 1.4967 2.9712 2.6177 4.1826 4.7537 4.3229 1.3052 12 1.4439 3.0142 2.8949 4.1108 5.0542 0.59699 3.0056×10−1 13 1.4480 2.7394 3.0126 4.1463 5.1282 0.18462 2.7484×10−1 14 1.4452 2.7358 3.0125 4.1550 5.1357 0.18345 8.7263×10−3 15 1.4452 2.7357 3.0125 4.1550 5.1357 0.18345 1.1021×10−5 16 1.4452 2.7357 3.0125 4.1550 5.1357 0.18345 6.5115×10−8Table 3: Example 1.Results for ﬁtting two concentric spheres to 100 points are shown in Table 4.The subsets V and W have been, as expected, with V = {1, 2, . . . , 50} and W ={51, 52, . . . , 100} in the second iteration. Other choices for V (0) and W (0) eithersizes or data indexes give a similar behaviour.k a b c r R F ‖d‖∞ γ1 -1.8617 3.0984 1.0791 6.3409 6.4127 25.498 1.4045 12 -1.8632 3.1113 1.1265 6.0087 6.9787 0.49438 5.6606×10−1 13 -1.8633 3.1114 1.1266 6.0088 6.9789 0.49437 1.5797×10−4 14 -1.8633e 3.1114 1.1266 6.0088 6.9789 0.49437 1.4517×10−7Table 4: Example 2.6. ConclusionFitting two concentric spheres to data by orthogonal distance regression is consid-ered in this paper. One of the main diﬀerences between applying the Gauss-Newtonmethod to the problem and the method which was applied by [6] is the number oflinear equations need to solve at each iteration. Using the Gauss-Newton methodneeds to solve m × 5 linear equations at each iteration. However, 5 × 5 linearequations at each iteration need to solve for applying [6].Nevertheless, all the numerical results show that the use of the Gauss-Newtonmethod with line search is very rigorous for solving ODR of two concentric spheres.It is worth noting that the number of iterations using the Gauss-Newton methodwith line search has proven to be small, when compared to [6] method. Moreover,The number of iterations on the orthogonal distance regression and the Gauss-Newton method with line search have been not generally aﬀected by the value ofm. The use of the results of [6] for starting point has proved to be crucial for theinitial values of the problem.Fitting two concentric spheres 239AcknowledgmentsI am very pleased to dedicate this paper to professor G.A. Watson, who has con-tributed so much to my understanding of the numerical solution of ODR problems.He is not only a distinctive scholar, but he has always been generous in spirit andgenteel in behavior. Further, I am grateful to the referees for their helpful sugges-tions.References[1] I. A.Al-Subaihi, Using the l1 and l∞ norms in the ﬁtting of curves and sur-faces to data, Ph.D. thesis, University of Dundee, 2006.[2] I. A.Al-Subaihi, G.A.Watson, Algebraic ﬁtting of quadric surfaces to data,Commun. Appl. Anal. 9(2005), 539-548.[3] A.Atieg Numerical methods for ﬁtting curves and surfaces to data by orthog-onal distance regression and some generazations, Ph.D. thesis, University ofDundee, 2005.[4] A.Atieg, G.A.Watson, A class of methods for ﬁtting a curve or surface todata by minimizing the sum of squares of orthogonal distances, J. Comp. Appl.Math. 158(2003) 277-296.[5] H.P.Helfrich, D. Zwick, A trust region method for implicit orthogonal dis-tance regression, Num. Alg. 5(1993), 535-545.[6] H. Spa¨th, Data ﬁtting with a set of two concentric spheres, Math. Commun.11(2006), 63-68.[7] H. Spa¨th, Least-Square ﬁtting with spheres, J. Opt. Theory and Applications96(1998), 191-199.[8] H. Spa¨th, Orthogonal least squares ﬁtting by conic sections, in: Re-cent Advances in Total Least Squares and Errors-in-Variables Techniques,(S. VanHuﬀel, Ed.), SIAM, Philadelphia, 1997, 259-264.[9] V. Srinivasan, How tall is the pyramid of Cheops... And other problems incomputational metrology, SIAM News, 29(1996), 8-9.[10] R. Strebel, D. Sourlier, W.Gander, A comparison of orthogonal leastsquares ﬁtting in coordinate metrology, in: Recent Advances in Total LeastSquares and Errores-in-variables techniques, (S. VanHuﬀel, Ed.), SIAM,Philadelphia, 249-258.[11] D.ATurner, The approximation of cartesian co-ordinate data by parametricorthogonal distance regression, Ph.D. thesis, University of Huddersﬁeld, 1999.[12] G.A.Watson, Some problems in orthogonal distance and non-orthogonal dis-tance regression, Proceedings of Algorithms for Approximation IV(2002), 294-302.

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