The optimization problem of fitting F to S is described by the objective function shown in Equation (1).

where the residual represents the minimum distance between the i-th point of and its corresponding point on F and indicates the order of the residual. Then is given by:

where is the norm-degree to calculate the distance.

To minimize and find the best surface fit, some variables are tunned. These variables are specific for each situation. Table 1 shows the decision variables for each surface addressed in this paper. On the other hand, norm remains constant in the optimization process and is considered as a parameter of the problem.

Table 1. Decision variables in analytic surfaces fitting.

The number of decision variables and the number of equality constrains in a optimization problem, allow to know the degrees of freedom with the equiation. Table 1 presents the degrees of freedom for each analytic surface addressed in this paper. Notice that corresponds to because the problem is unconstrained.

The Gauss-Newton iterative method for solving non-linear optimization problems uses the Hessian approximation to calculate the next iteration, as is shown in Equation (3).

where is the decision variables vector and is residuals vector.

Notice that in the case in which is not strictly convex, can be singular at some iteration possibly causing the algorithm to diverge. This problem can be overcome by using the Levenberg-Marquardt (LM) Method ([4,5]):

where is the LM parameter and is the identity matrix.

The convexity condition of an objective function and its feasible region determines if a local extrema corresponds to a global extrema or not. In order to evaluate this condition in, it is required to examine the eigenvalues of its Hessian matrix by solving Equation (5)

If all eigenvalues of are positive, then is strictly-convex, but if at least one eigenvalue is equal to zero, is convex [6]. In the case studies on this research, an exact calculation of is not possible. Thus, a numerical calculation is required by approximating partial derivatives numerically.

Some authors have researched the calculation of the distance between a point and an analytic surface. Sappa and Rouhani [7] present a new technique for the estimation a pseudo geometric distance by calculating the height of a small tetrahedron intersecting the surface. This technique is prone to yield accuracy loss in the distance metric when applied to surfaces with high curvatures. Wang and Yu [1] present a comparison of the fitting processes implementing the algebraic, Euclidean, tangent or squared distance for fitting quadric surfaces. Zhou and Salvado [8] compare the geometric and algebraic distances in fitting ellipsoids. The authors estimate the geometric distance as the difference between the length of the ray connecting the point to ellipsoid center and the radius in the intersection of the ray and the surface. This estimation being a fast solution, only works well in cases of quasi-spherical ellipsoids.

Just like Zhou and Salvado [8], Jiang and Cheng [9] classify the surface fitting problem as a non-convex one. The authors do not discuss the convexity analysis neither for the objective function nor for the optimization region. Other references reviewed do not report any classification of the surface fitting problem in terms of the convexity.

Yan, Liu and Wang [10] use Lloyd iterations to reconstruct quadric surfaces from a 3D point cloud. Ying, Yang and Zha [11] fit ellipsoids to data using semidefinite programming obtaining low runtime.

Jiang and Cheng [9] apply a decomposition technique to reduce the dimensions of the optimization space. This implies that the possibilities of dropping into local minima decrease. However, as the approach is out of the geometrical field, coming up with an initial guess of the parameters is not an easy task.

Li and Griffiths [12] fit ellipsoids by a least squares method using quadrics. This method does not require an initial estimation but, as it is not based on real geometrical distances, the results do not provide the best geometric fitting ellipsoid.

References [8,13] report the use of LM method to fit analytic surfaces without mentioning the selection of the LM parameter and its influence on the optimization process.

Numerical optimization strategies are sensitive to the initial guess. The closer to the ideal solution is the initial guess, the less number of iterations in the optimization process. On the other hand, a bad initial guess could make the algorithm to diverge.

Just like Ruiz and Cadavid [14], Kwon et al. [15] use PCA for finding an approximation to axis orientation, center coordinates and radius of point clouds belonging to circular cylindrical surfaces. Because this technique reduces the dimensionality of the data giving the direction of largest dispersion [16], it is limited to cylinders with aspect ratio lower than 5.0 [14] and to cylindrical caps. Similarly, Zhou and Salvado [8] use the eigenvectors of the covariance matrix of the whole data as the axes of the ellipsoid coordinate system. As noted, this technique gives axes that probably will not correspond to the ellipsoid coordinate system in the case of ellipsoidal caps.

Simari and Singh [17] estimate the ellipsoid’s center as the geometric centroid of the data set. This proposal works in the cases of ellipsoids completely sampled but it loses validity when the cloud is only a subsample of the whole ellipsoid.

References [13,18] report the implementation of an algebraic method based on a least squares solution of the general quadric equation to find an initial guess of the parameters of analytic surfaces.

As was shown in the taxonomy conducted in the literature review, there are several issues that remain open in optimized analytic surfaces fitting which are studied in this work: 1) Estimation of the real geometric distance between a point and an ellipsoid, 2) Identification of the effect of the parameters such as the norm k in the distance measurement, 3) Analysis of the optimality conditions effect on the convergence of the algorithm.

A circular cylinder is defined by a radius, an axis vector and its pivot point and respectively. For purposes of this research, no assumption is made on the orientation or position in space of the cylinder from which the data set belongs.

The initial guess of the cylinder’s parameters is obtained with a statistical and geometrical procedure as explained below and shown in Figure 1:

1) Random seed points are selected from and a local neighborhood found for each seed. See Figure 1(a).

2) Crossing each other the line segments defined by a seed point and the normal vector of the best plane fit to, a set of points passing near are found. See Figure 1(b).

3) A PCA is executed over for finding an initial approximation of the cylinder axis and its pivot point, and respectively. See Figure 1(c).

4) An approximation to the cylinder radius is calculated as the average of the minimum distances between and.

This method allows processing both complete cylindrical surfaces and cylindrical caps.

The minimum distance between the point and a circular cylindrical surface is calculated as:

where is the radius of the cylinder and is the orthogonal projection of onto as is shown in Figure 2.

A circular right cone can be defined by an axis vector, an apex and a semi-opening angle.

As is shown in Table 1, an ellipsoid in general position is defined by the center coordinates, the semi axes and the Euler angles.

In order to prove the algorithm for fitting ellipsoids, a subset of the frog shown in Figure 7(a) was taken. In Figure 7(b) the highlighted points to which the ellipsoids were fitted can be seen. The result of the fitting process is presented in Figures 7(c) and (d).

A good initial guess found by an algebraic approach, let to a fast convergence of the algorithm. In Figure 8 it may be seen that the longest fitting process required of 12 iterations for finding the optimum according to the termination criteria. In Figure 9 the initial estimation ellipsoid and the best geometrical ellipsoid fit are shown. Notice that the initial surface wraps most of the points, giving a good starting point for the LM algorithm. Table 2 presents a comparison between the the initial ellipsoids and the optimized ones.

Table 2. Fitting results in frog’s study case.

To test the cylinder and cone fitting algorithms some parts of the fan shown in Figure 10(a) were processed with the algorithms. Figure 10 displays the results of the optimization process of two conical surfaces and one cylinder. As in the case of ellipsoids, the algorithm found the optimal surface after a few iterations. The history of the optimized function for the Fan Data Set is shown in Figure 11. The cones (Figure 12) required 4 iterations while the cylinders (Figure 13) required 6 iterations. The good initial estimation of the surfaces allows the convergence of the algorithm and to reduce the number of iterations, therefore saving computing resources. Geometrical statistics for the Fan Data Set appear in Table 3.

Table 3. Fitting results in fan’s study case.