Variances of Cylinder Parameters Fitted to Range Data

Abstract

Industrial pipelines are frequently scanned with 3D imaging systems (e.g., LADAR) and cylinders are fitted to the collected data. Then, the fitted as-built model is compared with the as-designed model. Meaningful comparison between the two models requires estimates of uncertainties of fitted model parameters. In this paper, the formulas for variances of cylinder parameters fitted with Nonlinear Least Squares to a point cloud acquired from one scanning position are derived. Two different error functions used in minimization are discussed: the orthogonal and the directional function. Derived formulas explain how some uncertainty components are propagated from measured ranges to fitted cylinder parameters.

1. Introduction

Digital representation of an object’s surface is important in many engineering applications. Examples include: manufacturing, modular construction, and preservation of historical artifacts. Frequently, the data needed for such representation are collected with 3D imaging systems. The systems are non-contact, line-of-sight instruments which provide range images r(ϑ, φ) of the object’s surface, where r denotes the distance (measured as a time of flight) from an instrument to a point on a surface, and ϑ and φ are the elevation and the azimuth angles to that point. The scanning process is very fast and current systems can collect datasets containing hundreds of thousands of points within a few seconds [1]. Points collected from a surface may be used to build a 3D geometrical model of a scanned object. For example, a cylinder is frequently modeled in projects where industrial pipelines are scanned [2–5]. Frequently, Nonlinear Least Square (NLS) methods [6–10] are used to fit a model to its corresponding point cloud. An important issue is how to propagate an instrument uncertainty (for example: uncertainty of measured ranges r) to the uncertainties of fitted cylinder parameters. These uncertainties are essential in a quality control process when the as-built model (derived from the point cloud) is checked against the as-designed (theoretical) model [11]. The tolerances used in the design process and the uncertainties of the fitted model parameters are both needed to accept the as-built model. For example, the uncertainties of the fitted coefficients of the cylinder centerline are needed to check, with a given confidence, if two pipes are parallel or to check the plumbness of a cylinder column.

It is commonly accepted by manufacturers and users of 3D imaging systems that the uncertainty of a j-th measured point P_j(X,Y,Z) is caused mainly by an uncertainty of the measured range r_j; an impact of angular uncertainty can be neglected and both the elevation and the azimuth angles of P_j are treated as noise-free parameters. Under this assumption, the closed formulas for variances (i.e., squared standard deviations) of cylinder parameters fitted to data acquired from one scanning location are provided. These variances are an important component in evaluating measurement uncertainty.

The choice of an error function for NLS minimization is a delicate matter. It was shown that the orthogonal error function, which is commonly used in commercial software, yields results that are in disagreement with experiment when a plane is fitted to a point cloud acquired in some experimental conditions [12]. Another error function, the directional error function, gives results that are, on average, in a much better agreement with experiment. Therefore, in this paper, closed formulas for variances of fitted cylinder parameters are derived for both orthogonal and directional error functions. Both error functions depend on four fitting variables: three angles {ϑ, φ, ρ} which determine a cylinder centerline orientation in space, and the distance L of a centerline to the origin of a coordinate system.

2. Variances of Fitted Cylinder Parameters

Points P_j(X, Y, Z) on a surface of an infinitely long regular cylinder of radius R satisfy the following equation

$$\Vert ({\mathit{P}}_j-\mathit{L})\times \mathit{w}\Vert =R,$$ (1)

where the cylinder centerline M is

$$\mathit{M}(q)=\mathit{L}(x,y,z)+\mathit{w}(\vartheta ,\phi )q,$$ (2)

w(ϑ, φ) is a unit vector in the direction of the axis, parameterized by two angles: the elevation ϑ and the azimuth φ, and × stands for a cross product of two vectors. Therefore, the Cartesian coordinates of w(ϑ, φ) may be written as

$$\mathit{w}(\vartheta ,\phi )=[\cos \vartheta \cos \phi ,\cos \vartheta \sin \phi ,\sin \vartheta ].$$ (3)

L is any fixed point on the axis and q is a number defining a location of arbitrary point on the line M. It is convenient to choose L in such a way that its length L = ‖L‖ is the shortest [5]. This happens when L and w are perpendicular and their dot product is zero, L·w = 0. In this setting, L is the distance of the cylinder centerline M from the origin of the coordinate system. The origin and the orientation of the coordinate system (in which experimental points P_j are acquired) are defined by the location and the orientation of a scanner. Vector L may be parameterized as

$$\mathit{L}=Ll,$$ (4a)

where a unit vector l lies on a plane perpendicular to w and

$$l(\vartheta ,\phi ,\rho )=\mathit{u}(\vartheta ,\phi )\cos \rho +\mathit{v}(\vartheta ,\phi )\sin \rho .$$ (4b)

The unit vectors u and v, together with the vector w, form the right-handed orthonormal basis {u, v, w}

$$\mathit{u}=\mathit{v}\times \mathit{w},\mathit{v}=\mathit{w}\times \mathit{u},\mathit{w}=\mathit{u}\times \mathit{v},$$ (5a)

$$\mathit{v}\cdot \mathit{w}=\mathit{w}\cdot \mathit{u}=\mathit{u}\cdot \mathit{v}=0.$$ (5b)

Vectors u and v can be parameterized as follows

$$\mathit{v}(\vartheta ,\phi )=[\sin \vartheta \cos \phi ,\sin \vartheta \sin \phi ,-\cos \vartheta ],$$ (6a)

$$\mathit{u}(\phi )=[\sin \phi ,-\cos \phi ,0].$$ (6b)

The angle ρ in Eq. (4b) defines two coordinates of the unit vector l on the 2D plane perpendicular to w, as shown in Fig. 1. It is easy to check that in this parameterization there are two equivalent sets of parameters defining the same centerline M: {ϑ, φ, ρ, L} and {−ϑ, φ + π, π − ρ, L} which correspond to vectors w and −w.

Fig. 1.

Parameterization of a cylinder for fitting to range data. Unit vectors u,v, and l lie on a plane perpendicular to the unit vector w. Angle ρ is measured between vectors u and l.

Point P_j from Eq. (1) may be written as

$${\mathit{P}}_j=r_j{\mathit{p}}_j({\vartheta }_j,{\phi }_j),$$ (7)

where r_j is a range and a unit vector p_j is equal to

$${\mathit{p}}_j({\vartheta }_j,{\phi }_j)=[\cos {\vartheta }_j\cos {\phi }_j,\cos {\vartheta }_j\sin {\phi }_j,\sin {\vartheta }_j].$$ (8)

It is also convenient to introduce a vector d_j

$${\mathit{d}}_j(\vartheta ,\phi ,{\vartheta }_j,{\phi }_j)={\mathit{p}}_j({\vartheta }_j,{\phi }_j)\times \mathit{w}(\vartheta ,\phi ).$$ (9)

Now, using the above introduced vector d_j together with Eqs. (4) and (5), Eq. (1) can be expressed as

$$\Vert r_j{\mathit{d}}_j-L(\mathit{u}\sin \rho -\mathit{v}\cos \rho )\Vert =R.$$ (10)

Thus, in this parameterization, a regular cylinder of radius R is defined by four parameters: three angles ϑ, φ, ρ and the distance L (in this and the next two sections, a radius R is treated as a known constant; later in Section V this restriction is removed for the orthogonal fitting). Due to imperfections in range measurement, experimental points P_j do not lie exactly on a cylinder surface. Therefore, a model of a cylinder has to be fitted to the experimental dataset P_{N} = {P_j, j = 1,…, N}, where N denotes the number of points. Within the framework of the Least Squares method, the estimated parameters of a cylinder {ϑ^*, φ^*, ρ^*, L^*} are obtained by minimizing the error function

$$E(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})=\frac{1}{N}{\displaystyle \sum _{j=1}^N{E}_j}(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j),$$ (11)

where E_j is the squared distance between the experimental point P_j and its corresponding theoretical point on a cylinder surface. Different definitions of the theoretical point yield different error functions. In this paper we study two error functions: the orthogonal error function E_O and the directional error function E_D, as explained in Figs. 2 and 3 and in the next two sections. The elements of a gradient of the error function ∇E can be calculated as

$$\frac{\partial E}{\partial \alpha }(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})=\frac{1}{N}{\displaystyle \sum _{j=1}^N\frac{\partial E_j}{\partial \alpha }}(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j),\text{for}\alpha =(\vartheta ,\phi ,\rho ,L).$$ (12)

Fig. 2.

Schematic drawing (top view, along the cylinder centerline M) explaining the calculation of the orthogonal error function.

Fig. 3.

Schematic drawing (top view, along the cylinder centerline M) explaining the calculations of the directional error function. Point P_j illustrates the configuration when a ray intersects with the cylinder surface (h_j < R) while point P_k illustrates when there is no intersection (h_k > R).

The knowledge of closed formulas for a gradient ∇E allows application of efficient minimization algorithms to find the fitted parameters {ϑ^*, φ^*, ρ^*, L^*}. Their values depend solely on the experimental dataset P_{N}, i.e.,

$${\vartheta }^{*}={\vartheta }^{*}({\mathit{P}}_{\left\{N\right\}}),{\phi }^{*}={\phi }^{*}({\mathit{P}}_{\left\{N\right\}}),{\rho }^{*}={\rho }^{*}({\mathit{P}}_{\left\{N\right\}}),L^{*}=L^{*}({\mathit{P}}_{\left\{N\right\}}).$$ (13)

The variances of the fitted cylinder parameters var(ϑ^*), var(φ^*), var(ρ^*), and var(L^*) may be calculated following the same general approach developed originally for fitting a sphere to range data [13]. Here, similarly as in the previous study, only range uncertainty is considered and the bearings (ϑ_j, φ_j) of acquired j-th point P_j are treated as noise-free parameters. For other types of instruments, for example coordinate measuring machines (CMM), the above assumption may not be valid and the formulas for variances of fitted cylinder parameters developed in this paper may not be applicable. In addition, it is assumed that measured range r_j is not correlated with r_k for any j ≠ k. When both assumptions are valid, the variances of the fitted cylinder parameters may be estimated by applying to Eq. (13) the uncertainty propagation formula [14]

$$\mathrm{var}({\alpha }^{*})={{\displaystyle \sum _{j=1}^N\left[\frac{\partial {\alpha }^{*}({\mathit{P}}_{\left\{N\right\}})}{\partial r_j}\right]}}^2\mathrm{var}(r_j),\text{for}{\alpha }^{*}=\{{\vartheta }^{*},{\phi }^{*},{\rho }^{*},L^{*}\}.$$ (14)

For every j, the sensitivity vector S_j(P_{N})

$${\mathit{S}}_j({\mathit{P}}_{\left\{N\right\}})=\left[\begin{array}{c}\frac{\partial {\vartheta }^{*}}{\partial r_j}({\mathit{P}}_{\left\{N\right\}})\\ \frac{\partial {\phi }^{*}}{\partial r_j}({\mathit{P}}_{\left\{N\right\}})\\ \frac{\partial {\rho }^{*}}{\partial r_j}({\mathit{P}}_{\left\{N\right\}})\\ \frac{\partial L^{*}}{\partial r_j}({\mathit{P}}_{\left\{N\right\}})\end{array}\right]$$ (15)

can be evaluated by solving the following 4×4 system of linear equations

$$\mathit{H}({\vartheta }^{*},{\phi }^{*},{\rho }^{*},L^{*},{\mathit{P}}_{\left\{N\right\}}){\mathit{S}}_j({\mathit{P}}_{\left\{N\right\}})=-{\mathit{V}}_j({\vartheta }^{*},{\phi }^{*},{\rho }^{*},L^{*},{\mathit{P}}_j)$$ (16)

where the vector V_j is defined by

$${\mathit{V}}_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=\frac{\partial }{\partial r_j}\nabla E(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})$$ (17)

while H is 4×4 Hessian matrix which elements are given by

$${\mathit{H}}_{\alpha ,\beta }(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})=\frac{{\partial }^{2}E}{\partial \alpha \partial \beta }(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})\text{for}\alpha ,\beta =\{\vartheta ,\phi ,\rho ,L\}.$$ (18)

3. Orthogonal Fitting

When fitting a cylinder using the orthogonal error function (see Fig. 2) the theoretical point corresponding to the experimental point P_j is defined as the orthogonal projection of P_j on a cylinder surface. Thus, Eq. (11) takes the form

$$E_O(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})=\frac{1}{N}{\displaystyle \sum _{j=1}^N{(Q_j-R)}^2}$$ (19)

where Q_j is a distance of point P_j to the cylinder axis M, which can be evaluated from Eq. (10)

$$Q_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=\sqrt{r_j{\mathit{d}}_j-L(\mathit{u}\sin \rho -\mathit{v}\cos \rho )]\cdot [r_j{\mathit{d}}_j-L(\mathit{u}\sin \rho -\mathit{v}\cos \rho )]}.$$ (20)

Defining the scalar functions a_j and b_j by means of Eqs. (3), (5), (8), (9) and using the properties of the dot and cross products,

$$a_j(\vartheta ,\phi ,{\mathit{p}}_j)={\mathit{d}}_j\cdot {\mathit{d}}_j=1-{(\mathit{w}\cdot {\mathit{p}}_j)}^2,$$ (21a)

$$b_j(\vartheta ,\phi ,\rho ,{\mathit{p}}_j)={\mathit{p}}_j\cdot l={\mathit{p}}_j\cdot (\mathit{v}\sin \rho +\mathit{u}\cos \rho ),$$ (21b)

the distance Q_j from Eq. (20) may be written as

$$Q_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=\sqrt{a_j(\vartheta ,\phi ,{\mathit{p}}_j)r_j^2-2L{r}_j{b}_j(\vartheta ,\phi ,\rho ,{\mathit{p}}_j)+L^2}.$$ (22)

Explicit expressions for the gradient ∇E_O may be obtained by inserting Eq. (22) into Eq. (19) and using Eq. (12). The minimum $E_O({\vartheta }_O^{*},{\phi }_O^{*},{\rho }_O^{*},L_O^{*})$ can be found numerically using gradient ∇E_O. Then, Eqs. (17) and (18) can be applied to Eq. (19) to get the vector V_j and the matrix H_O. Once the sensitivity vector S_j is evaluated from Eq. (16), the variances of fitted cylinder parameters $\mathrm{var}({\alpha }_O^{*})$ for ${\alpha }_O^{*}=\{{\vartheta }_O^{*},{\phi }_O^{*},{\rho }_O^{*},L_O^{*}\}$ can be calculated from Eq. (14).

4. Directional Fitting

When fitting a cylinder using the directional error function, see Fig. 3, the theoretical point T_j corresponding to the experimental point P_j is defined as an intersection of a ray originating from the instrument and passing through P_j with the cylinder surface

$${\mathit{T}}_j=t_j{\mathit{P}}_j,$$ (23)

where the function t_j(ϑ, φ, ρ, L, P_j) has its value close to one when optimization variables are close to their best-fit values and the theoretical points T_j satisfy Eq. (1) of the cylinder

$$\Vert ({\mathit{T}}_j-\mathit{L})\times \mathit{w}\Vert =R.$$ (24)

A ray need not always intersect a cylinder surface – a similar situation occurs in the directional fitting of a sphere [15]. Whether there is an actual intersection or not can be checked by calculating the distance h_j between the cylinder centerline M given by Eq. (2) and a ray passing through P_j; if h_j < R, then there is an intersection and the theoretical point on a cylinder surface is defined by Eq. (23). Otherwise, if h_j ≥ R, there is no intersection and the theoretical point T_j is defined as a point on a cylinder surface which has the smallest distance to a ray passing through P_j. Assuming that a ray is not parallel to the cylinder centerline M (in which case h_j = L), the distance function h_j(ϑ, φ, ρ, L, p_j) can be calculated as

$$h_j(\vartheta ,\phi ,\rho ,L,{\mathit{p}}_j)=\left|\mathit{L}\cdot ({\mathit{p}}_j\times \mathit{w})\right|/\Vert {\mathit{p}}_j\times \mathit{w}\Vert .$$ (25)

Using a definition of the vectors L, l, w, v and u given in Eqs. (3)–(6), function a_j in Eq. (21a) and the property of the triple vector product, the distance function h_j can be evaluated as

$$h_j(\vartheta ,\phi ,\rho ,L,{\mathit{p}}_j)=L\left|{\mathit{p}}_j\cdot (\mathit{v}\cos \rho -\mathit{u}\sin \rho )\right|/\sqrt{a_j}.$$ (26)

The j-th term in Eq. (11) corresponding to the directional error function E_D is a squared distance between the experimental point P_j and the theoretical point T_j, see Fig. 3, giving

$$E_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=({\mathit{T}}_j-{\mathit{P}}_j)\cdot ({\mathit{T}}_j-{\mathit{P}}_j).$$ (27)

For the case when h_j < R, E_j can be evaluated using the function t_j (ϑ, φ, ρ, L, P_j) which is derived in the Appendix, Eqs. (A1)–(A5). When h_j ≥ R, E_j can be evaluated using h_j(ϑ, φ, ρ, L, p_j) from Eq. (26) and s_j(ϑ, φ, ρ, L, P_j) which is derived in the Appendix, Eqs. (A6)–(A10), giving

$$E_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=\{\begin{array}{cc}{(t_j-1)}^2{r}_j^2& if{h}_j<R\\ {(h_j-R)}^2+{(s_j-1)}^2{r}_j^2& if{h}_j\ge R\end{array}.$$ (28)

From Eq. (28), the j-th term in Eq. (12) for the components of the gradient of the directional error function ∇E_D can be evaluated

$$\frac{\partial E_j}{\partial \alpha }=\{\begin{array}{cc}2r_j^2(t_j-1)\frac{\partial t_j}{\partial \alpha }& if{h}_j<R\\ 2r_j^2(s_j-1)\frac{\partial s_j}{\partial \alpha }+2(h_j-R)\frac{\partial h_j}{\partial \alpha }& if{h}_j\ge R\end{array}\text{for}\alpha =\{\vartheta ,\phi ,\rho ,L\}.$$ (29)

The minimum $E_D({\vartheta }_D^{*},{\phi }_D^{*},{\rho }_D^{*},L_D^{*})$ can be found numerically using gradient ∇E_D. Then, Eqs. (17) and (18) can be applied to E_D and Eq. (29) to get the equations for the vector V_j and the Hessian matrix H_D. Once the sensitivity vector S_j is evaluated from Eq. (16), the variances of fitted cylinder parameters $\mathrm{var}({\alpha }_D^{*})$ for ${\alpha }_D^{*}=\{{\vartheta }_D^{*},{\phi }_D^{*},{\rho }_D^{*},L_D^{*}\}$ can be calculated from Eq. (14).

The derivatives of t_j, s_j, and h_j can be calculated from their definitions. Although the derivative of h_j is not defined when h_j = 0, h_j enters into a j-th term of E_D only when h_j > R, as follows from Eq. (28). Thus, h_j is differentiable in the whole domain where experimental conditions are valid.

5. Discussion

So far, only four fitting variables {ϑ, φ, ρ, L} were used and the radius of a cylinder R was treated as a known constant. When the radius is unknown and needs to be fitted, the orthogonal error function E_O in Eq. (19) may be still effectively minimized in four dimensional space {ϑ, φ, ρ, L} [5]. This is possible because the radius R enters Eq. (19) in a quadratic term. Since a gradient of the error function must be zero when the minimum of E_O is reached, the radius R can be evaluated from the condition

$$\frac{\partial E_O}{\partial R}(\vartheta ,\phi ,\rho ,L,R,{\mathit{P}}_{\left\{N\right\}})=0.$$ (30)

Substituting for E_O Eq. (19), the radius R may be calculated as

$$R\equiv R(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_{\left\{N\right\}})=\frac{1}{N}{\displaystyle \sum _{j=1}^N{Q}_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)},$$ (31)

where Q_j is calculated in Eq. (22). Equation (31) has a simple geometric interpretation: the fitted radius R^* is the algebraic mean of distances of all experimental points P_j from the fitted cylinder centerline defined by $\{{\vartheta }^{*},{\phi }^{*},{\rho }^{*},L^{*}\}$. The variance of the fitted radius $\mathrm{var}(R^{*})$ can be calculated by applying to Eq. (31) the uncertainty propagation formula [14]

$$\mathrm{var}(R^{*})={\displaystyle \sum _{\alpha }{(\frac{\partial R}{\partial \alpha })}^2\mathrm{var}({\alpha }^{*})+{\displaystyle \sum _{j=1}^N{(\frac{\partial R}{\partial r_j})}^2\mathrm{var}(r_j)\text{for}\alpha =\{\vartheta ,\phi ,\rho ,L\}}.}$$ (32)

Individual derivatives $\frac{\partial R}{\partial \alpha }$ are calculated at fitted values ${\alpha }^{*}=\{{\vartheta }^{*},{\phi }^{*},{\rho }^{*},L^{*}\}$ by inserting Eq. (31) into (32) and using derivatives of Q_j. Individual variances var (α^*) of the fitted parameters are calculated in Eq. (14) using the formulas derived in Section III for the orthogonal fitting.

Unfortunately, the above strategy for fitting the unknown cylinder radius R cannot be applied to the directional fitting because a j-th term in the directional error function E_D depends on R in a more complex way and a convenient separation of fitting variables as in Eq. (31) is not possible.

Closed formulas for variances derived in this paper are applicable to a cylinder fitted to a point cloud acquired from only one scanning location. When two or more datasets acquired from different locations are registered together, the formulas presented here cannot be used. This limitation has its root in the way range uncertainties are propagated into uncertainties of fitted parameters: the registration process makes relatively straightforward formulas much more complicated, with variances explicitly depending on the registration parameters. It remains an open question whether fitting a model to one large point cloud consisting of many registered datasets has an advantage over fitting a model to many unregistered clouds.

The difference between the orthogonal and the directional fitting becomes more important for point clouds containing many points acquired at large angle of incidence (i.e., AOI - the angle between a ray coming through the data point P_j and a normal to the cylinder surface at the theoretical point T_j defined in Eq. (23)). Scanning industrial pipelines (where a length of a scanned pipe may be larger than the fitting variable L) may yield a dataset in which AOIs are varying in a large range. Then, the orthogonal fitting may generate a cylinder centerline which is biased and incorrect. In order to quantify the expected differences between both error functions, further research involving computer simulations and field measurements is needed.

Biography

About the author: Marek Franaszek is a physicist in the Intelligent Systems Division of the NIST Engineering Laboratory. His current research interests include mathematical modeling and algorithm development for processing data acquired with 3D imaging systems. The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce.

6. Appendix

This Appendix contains expressions needed for evaluating the directional error function E_D, defined in Eq. (28). The individual j-th term of E_D depends on the distance h_j between the cylinder centerline M and a ray passing through data point P_j. Let us first consider the case when h_j, given by Eq. (26), is less than the cylinder radius R. Eqs. (27) and (24) lead to the following equation for t_j

$$\Vert (t_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}\Vert =\sqrt{[(t_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}]\cdot [(t_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}]}=R,$$ (A1)

where t_j depends on P_j and parameters determining M, i.e., ϑ, φ, ρ, and L. By squaring both sides of Eq. (A1) and using the definitions of a_j and b_j from Eqs. (21a) and (21b), the following quadratic equation for t_j is obtained

$$(a_j{r}_j^2)t_j^2-(2r_{j}L{b}_j)t_j+L^2-R^2=0.$$ (A2)

This quadratic equation has two solutions. From a definition of the theoretical point T_j in Eq. (23), it follows that only t_j > 0 has a physical meaning. Two different experimental settings for collecting data points P_{N} need to be distinguished: a) when a cylinder (for example, a pipe) is scanned from the outside; and b) when scanning is done from inside (for example, in a tunnel or inside a containment vessel for a nuclear reactor).

In the first case, L > R and b_j defined in Eq. (21b) is always positive (a_j from Eq. (21a) is always positive, regardless of a scanner location). Then, from the two real solutions of Eq. (A2), a physical meaning exists only for the smaller one which corresponds to the intersection of a ray with a cylinder surface closer to a scanner location (the origin of coordinates), as shown in Fig. 3. It is useful to define an auxiliary function Δ_j(ϑ, φ, ρ, L, p_j) as

$${\Delta }_j(\vartheta ,\phi ,\rho ,L,{\mathit{p}}_j)=L^2{b}_j^2-a_j(L^2-R^2).$$ (A3)

Substituting Δ_j in the solution of quadratic Eq. (A2) yields the expression for t_j as a function of experimental point P_j and fitting variables

$$t_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=\frac{1}{a_j{r}_j}(L{b}_j-\sqrt{{\mathrm{\Delta }}_j}).$$ (A4)

In the second case, when scanning is done from inside a cylinder and L < R, Δ_j in Eq. (A3) is always positive and $\sqrt{{\Delta }_j}>L{b}_j$. However, the sign of b_j may now vary. Since the requirement that t_j is positive must be uphold, the solution for t_j takes now the following form

$$t_j(\vartheta ,\phi ,L,{\mathit{P}}_j)=\frac{1}{a_j{r}_j}(L{b}_j+\sqrt{{\Delta }_j}).$$ (A5)

Once t_j is known, a j-th term in the directional error function E_D can be calculated for the case when h_j < R according to Eq. (28).

When h_j ≥ R, scanning can be done only from outside and a ray passing through P_j does not intersect with a cylinder surface; then quadratic Eq. (A2) does not have a real solution and t_j cannot be determined. In such a case, a point G_j on a ray is sought which has the smallest distance to the cylinder centerline M, that is

$${\mathit{G}}_j=s_j{\mathit{P}}_j.$$ (A6)

The actual value of s_j can be evaluated by finding a minimum of a squared distance of G_j to M,

$$U_j(s_j)={\Vert (s_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}\Vert }^2=\left[(s_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}\right]\cdot [(s_j{\mathit{P}}_j-\mathit{L})\times \mathit{w}],$$ (A7)

where w is given by Eq. (3) and L by Eqs. (4a) and (4b). Using the properties of multiple dot and cross products, Eqs. (5), (6), and (21), U_j can be expressed as

$$U_j(s_j)=a_j{s}_j^2{r}_j^2+L^2-2s_j{r}_{j}L{b}_j.$$ (A8)

For fixed cylinder parameters ϑ, φ, ρ, L and a given data point P_j, a minimum of U_j can be found from the condition

$$\frac{\partial U_j}{\partial s_j}=0.$$ (A9)

Substituting Eq. (A8) into (A9), the explicit dependence of s_j on fitting variables can be determined

$$s_j(\vartheta ,\phi ,\rho ,L,{\mathit{P}}_j)=L\frac{b_j}{r_j{a}_j}.$$ (A10)

The distance h_j defined in Eq. (26) together with s_j provided in Eq. (A10) are used to calculate a j-th term in the directional error function E_D when h_j ≥ R, according to Eq. (28).