Establishment of a New Quantitative Evaluation Model of the Targets’ Geometry Distribution for Terrestrial Laser Scanning

The precision of target-based registration is related to the geometry distribution of targets, while the current method of setting the targets mainly depends on experience, and the impact is only evaluated qualitatively by the findings from empirical experiments and through simulations. In this paper, we propose a new quantitative evaluation model, which is comprised of the rotation dilution of precision (rDOP, assessing the impact of targets’ geometry distribution on the rotation parameters) and the translation dilution of precision (tDOP, assessing the impact of targets’ geometry distribution on the translation parameters). Here, the definitions and derivation of relevant formulas of the rDOP and tDOP are given, the experience conclusions are theoretically proven by the model of rDOP and tDOP, and an accurate method for determining the optimal placement location of targets and the scanner is proposed by calculating the minimum value of rDOP and tDOP. Furthermore, we can refer to the model (rDOP and tDOP) as a unified model of the geometric distribution evaluation model, which includes the DOP model in GPS.


Introduction
Terrestrial laser scanning (TLS) can provide a three-dimensional (3D) spatial point cloud dataset of the objects' surfaces. The spatial resolution of the data is much higher than that of conventional surveying methods [1]. Due to occluded surfaces and limitations in the view of a scanner, we usually need to make several scans from different setups of the scanner in order to survey a quite large and complex object [2,3]. These point clouds (scans) must first be registered to a chosen coordinate system before a coherent parametric description of the object can be formed [4]. Target-based registration with two scans is one of the most common registration approaches and is often performed using a 3D rigid body transformation algorithm [5,6].

Methods
There are two kinds of situations in practical applications, which are "we need to determine the optimal setting position of scanner where TGD is known" and "we need to determine the optimal TGD where the position of scanner is known". The unit of the rotation parameters is different from the unit of translation parameters, and the calculation results of translation and rotation will interact with each other when the transformation parameters are dependent on calculation models. For these reasons, we will first introduce the common registration model of two scans. We will then present the calculation of rotation parameters using the Rodrigues matrix [3,6]. Thirdly, we will propose a new calculation method of the translation parameter (similar to spatial distance resection in GPS [22]), which can ensure that the parameters of translation and rotation are computed independently. Fourthly, we will propose a new quantitative evaluation model of TGD, namely rDOP (which can be used to help determine the optimal TGD) and tDOP (which can be used to help determine the optimal setting position of scanner). Finally, we will derive an equal weight model of TGD and propose a set of model application schemes.

Registration Model of Two Scans
In the context of TLS, registration is the transformation of multiple point clouds (scans) into the coordinate system of a chosen scan [2]. The rigid body transformation operation of registration is expressed in Equation (1), in which the point clouds in Scan i + 1 are transformed into Scan i using the three translation parameters t x , t y , and t z and the three rotation parameters r a , r b , and r c [3,23].
where p i j and p i+1 j represent the same target in Scan i and Scan i + 1, respectively, whose observation values of coordinates are (x i j , y i j , z i j ) and (x i+1 j , y i+1 j , z i+1 j ); R is the standard 3 × 3 rotation matrix; T is the 3 × 1 translation vector; and For uniquely determining the above transformation parameters between Scan i and Scan Scan i + 1, we usually need to use three or more targets with known 3D coordinates [2,21], and these targets are placed in the overlap locations between the two point-clouds.
In this study, we assumed that the number of targets k is greater than 3 (k ≥ 3) and the coordinate of any point in Scan i (on a chosen coordinate system) is known, and we employed scanning to obtain the new point cloud in Scan Scan i + 1, which was transformed into Scan i.

Calculation of Rotation Parameters
, with Equation (1), we can get From the Rodrigues matrix [21,23], with Equations (2) and (3), we can get With Equation (4), we can get If the estimated values of r a , r b , and r c arer a ,r b , andr c , respectively, with Equations (5) and (6), the observation equation of rotation parameters can be expressed as where A r(k) , L r(k) , and δ r are 3k × 3 matrix, 3k × 1 matrix, and 3 × 1 matrix, respectively, and Assuming the weight matrix of L r(k) is P r(k) , by using the principle of indirect adjustment [24] and V T PV = min, we can obtain the estimated δ r for rotation parameters as

Calculation of Translation Parameters
As the position of scanner i + 1 in Scan i + 1 is (0, 0, 0), with Equation (1), we can find that the position of scanner i + 1 in Scan i is equal to the value of T, namely, the process of determining translation parameters is equivalent to solving the position of scanner i + 1 in Scan i. If the targets are regarded as GPS satellites, scanner i + 1 is regarded as a GPS receiver, and the calculation method of translation parameters is equivalent to solving the position of the GPS receiver by GPS satellites, namely, spatial distance resection in GPS [22], which will not be affected by the estimated precision of rotation parameters.
If the observation value of distance between scanner i + 1 and the jth target in Scan i + 1 is d i+1 j , the observation equation of translation parameters can be expressed as where d i+1 If the approximation values of translation parameters and corrections of translation parameters t x , t y , and t z are t x,0 , t y,0 , and t z,0 (calculated by the method of Appendix C in [3]) and ε t x , ε t y , and ε t z , then from the linearization theorem [22,24], the linearization form of Equation (10) can be expressed as where T = T 0 + δ t ; A t(k) , L t(k) , and δ t are k × 3 matrix, k × 1 matrix, 3 × 1 matrix, respectively; and Assuming the weight matrix of L t(k) is P t(k) , by using the principle of indirect adjustment [24], we can obtain the estimated δ t for translation parameters as

Quantitative Evaluation Model of TGD
With Equations (9) and (15), based on the theorem of error propagation [24], the covariance D δ r δ r and D δ t δ t of the rotation parameters δ r and the translation parameter corrections δ t can be obtained as where D δ r δ r and D δ t δ t are 3 × 3 matrices, and σ 0 is the unit weight variance, usually determined in the initial processing before registration.
As the trace of a real-symmetric matrix is equal to the trace of its corresponding diagonal matrix and the parameters' variance-covariance matrix is a real-symmetric matrix, we usually use the trace of the parameters' variance-covariance matrix in the precision evaluation of parameters, such as point precision evaluation. For this reason, we assume that the variances of rotation parametersr a ,r b , andr c and the translation parameters' corrections ε t x , ε t y , and ε t z are σ a , σ b , σ c , σ t x , σ t y , and σ t z , respectively. With Equation (16), the registration precision (namely, the variances of parameters δ r and δ t ) can be obtained as where tr(.) is the trace of the matrix. In GPS positioning, the impact of the satellites' geometry distribution on the positioning quality is evaluated by the dilution of precision (DOP) values [25][26][27]. Similarly, we can also build a quantitative evaluation model of the impact of TGD on the registration precision, that is, the rotation dilution of precision (rDOP) and the translation dilution of precision (tDOP), namely where G k = A T r(k) P r(k) A r(k) and H k = A T t(k) P t(k) A t(k) . With Equations (17)- (20), we can find that the registration precision of rotation parameters and translation parameters are From the above evaluation model of TGD, we can find that 1.
The values of rDOP and tDOP represent the amplification of the unit weight variance, which means the lower the values of rDOP and tDOP, the higher the solution precisions of the rotation parameters and the translation parameters; 2.
The values of rDOP calculated by Equation (19) are related to the coefficient matrix A r(k) and the weight matrix P r(k) of L r(k) , among which A r(k) is related to TGD (the coordinates of targets p i jc ) and the rotation matrix R. Namely, when R is fixed, the better the quality of TGD, the lower the values of rDOP; 3.
The values of tDOP calculated by Equation (20) are related to the coefficient matrix A t(k) ; and the weight matrix P t(k) of L t(k) , among which A t(k) is related to TGD (the coordinates of targets p i j ) and the position of scanner i + 1 in Scan i. Namely, the better the quality of TGD and the position of scanner i + 1, the lower the values of tDOP; 4.
The calculation formula of tDOP is identical to the calculation formula of DOP in GPS, so the tDOP can be used to evaluate the quality of the received GPS satellites' distribution. Namely, the rDOP and tDOP model is a unified evaluation model of the targets' and GNSS satellites' geometric distribution.

Equationuationual Weight Model of rDOP and tDOP
In constituting the DOP model for evaluating the impact of the selected GNSS satellite geometry [25][26][27], we usually assume that the weight matrix is an identity matrix. Additionally, in all the empirical experiments and simulations of the TGD impact on the registration precision [2,12,[14][15][16][17][18][19][20], we assume that the weight matrix is an identity matrix. For these reasons and for convenience of the following analysis on the nature of the rDOP and tDOP model, we assume that P r(k) and P t(k) are equal to identity matrix I, and use the equal weight least squares method to compute the registration parameters in Equations (9) and (15). With Equations (6)- (8) and Equations (12)-(14), we then get where Through the simulation method similar to [13], we can find that the relationship curve between rDOP and the rotation angle of the rotation matrix R under different TGDs is different and increases monotonically, and the relationship curves corresponding to different TGDs do not intersect. Therefore, we can compare the rDOP values of different TGDs under the rotation angle of the rotation matrix R by the values of rDOP under the rotation angle of the unit matrix I 3 (namely, the conclusions of rDOP under arbitrary rotation matrix R are equivalent to the conclusions under R = I 3 ). Then, we can evaluate the quality of TGD by only the values of rDOP under R = I 3 , and Equation (23) can be written as

Model Application Scheme
In order to use our proposed evaluation model, here, we give the implementation procedures for three kinds of situations: (1) the position of all targets are known, so we need to determine the optimum setting position of scanner i + 1 in Scan i, namely, the best position of scanner i + 1; (2) the position of scanner i + 1 is known, so we need to determine the optimum setting positions of targets, namely, the best TGD; and (3) we need to determine both the optimum positions of targets and the scanner i + 1 in Scan i, namely, the best TGD and the best position of scanner i + 1.

The Best Position of Scanner i + 1
The best position of Scanner ii + 1 can be identified as follows: (1) Selecting the possible place o i 1 , · · · , o i m of scanner i + 1 in Scan i; (2) Obtaining the coordinates of all targets in Scan i; (3) Using Equations (11), (12), (20) and (24) to calculate the values of tDOP under different possible places of scanner i + 1; (4) When the value of tDOP is the minimum, the corresponding place is the best position of scanner i + 1.

The Best TGD
The best TGD can be identified as follows: (1) Selecting the possible place p i 1 , · · · , p i m of targets in Scan i; (2) Obtaining the coordinates of scanner i + 1 in Scan i; (3) Setting the number k of targets; (4) Choosing k places from p i 1 , · · · , p i m , and using Equations (6), (20) and (25) to calculate the values of rDOP; (5) When the value of rDOP is the minimum, the corresponding places are the optimum positions of targets, namely, the best TGD.

The Best TGD and the Best Position of Scanner i + 1
From Section 2.4, it can be known that the rDOP model is mainly related to TGD, and the tDOP model is related to TGD and the position of scanner i + 1 s origin, which is relative to the selected TGD. Therefore, we firstly determined the best TGD by rDOP, and then determined the best position of scanner i + 1 by tDOP. The implementation procedures are as follows: (1) Selecting the possible place p i 1 , · · · , p i m of targets and the possible place o i 1 , · · · , o i m of scanner i + 1 in Scan i; (2) Setting the number k of targets; (3) Similar to the above, calculating the values of rDOP by k different possible places of targets, and selecting the best TGD where the value of rDOP is the minimum; (4) Similar to the above, calculating the values of tDOP under different possible places of scanner i + 1, and selecting the best position of scanner i + 1 where the value of tDOP is the minimum.

Theoretical Analysis
We first theoretically analyzed the existence conditions of rDOP and tDOP. We then theoretically analyzed the relationship of "rDOP and tDOP" and the number of targets. Finally, we analyzed the bounds of rDOP and tDOP.

The Existence Conditions of tDOP
The existence condition of tDOP is that the matrix H k is invertible, which is equal to |H k | 0, namely, the rank of A t(k) is 3.
If all targets and scanner i + 1 are on the same plane, and assuming the plane equation is Then, Equation (26) minus Equation (27) is With Equations (12), (13) and (28), we can get where the equation has a non-zero solution if and only if the rank of A t(k) is less than 3. Therefore, the existence condition of tDOP is that all targets and scanner i + 1 are not on the same plane, which theoretically proves the experience that "all targets and scanner i + 1 should not lie on the same plane" [2,12].

The Existence Conditions of rDOP
The existence condition of rDOP is that the matrix G k is invertible, which is equal to |G k | 0. With Equation (23), using the property of matrix inversion, we can get where From the inequality x i y i ) 2 , we know where equality is achieved if and only if τ x,j = τ y,j = τ z,j = 0, which is equivalent to the situation that any centralized targets p i+1 jc in Scan i + 1 satisfy (R + I 3 )p i+1 jc = 0, namely, the following three situations are true: 1.
All targets are in the plane x i oy i of Scan i's coordinate system, and R = All targets are in the plane x i oz i of Scan i's coordinate system, and R = All targets are in the plane y i oz i of Scan i's coordinate system, and R = Combined with Equations (31) and (36), we can get where equality is achieved if and only if τ x,j = γ xy τ y,j = γ xz τ z,j , which is equivalent to the situation that all targets lay on the same line. From Equation (37), we know that "the value of |G k | is smaller when the TGD is closer to a straight line, the value of rDOP is larger, and the precision of the rotation parameters solution is worse, while if all targets lay on the same line, |G k | = 0". Therefore, the existence condition of rDOP is that all targets are not on the same line, or the above three situations are not satisfied, which theoretically proves the experience that "the TLS targets should be distributed evenly over the overlapped space and should not lie on the same line or be close" [2,17].

The Relationship Between tDOP and the Number of Targets
If more targets are considered (over k), the A t(k) can be successively augmented by adding row vectors. For example, if there are k + 1 targets considered, then in which we assume that A t(k) is nonsingular and β k+1 is a nonzero vector, and β k+1 = l x,k+1 m y,k+1 n z,k+1 = Yarlagadda et al. [27] proved that increasing the number of satellites will reduce the DOP in GPS applications. Here, we take the same derivation method described by Yarlagadda et al. [27] to prove its effectiveness in tDOP. With Equation (38), we can get By using the inversion formulas of matrix [2], we can get It is clear that H −1 k is a positive definite symmetric matrix, which can be denoted as H −1 k = U T U, and U is the upper triangular matrix. Let η = β k+1 H −1 k and µ = β k+1 U T , where η and µ are 1 × 3 real-valued vectors, ηη T ≥ 0, and µµ T ≥ 0. Through using the property of the matrix trace, we can write Then, we can get tr(H −1 k+1 ) < tr(H −1 k ), which means that increasing the number of targets will reduce the value of tDOP and improve the registration precision, which theoretically proves the experience that "the more targets, the higher the registration precision" [2,[17][18][19][20]27].

The Relationship Between rDOP and the Number of Targets
Similar to tDOP, if more targets (over k) are considered, matrix A r(k) can also be successively augmented by adding row vectors. For example, if there are k + 1 targets considered, then (1) , where Assume that A r(k) is nonsingular and α k+1(1) , α k+1(2) , and α k+1(3) are nonzero vectors. By taking the same derivation method of Equations (38)-(42), we can get Therefore, increasing the number of targets will reduce the value of rDOP and improve the registration precision, which also theoretically proves the experience in Section 3.3.

tDOP Bounds
To find the optimum position of scanner i + 1, we need to analyze the bounds of tDOP, namely, the minimum of tDOP.
Denoting the three eigenvalues of H k are λ t,1 , λ t,2 , and λ t,3 , and using the property of matrix eigenvalues, we know that 1 λ t,1 , 1 λ t,2 , and 1 λ t,3 are the three eigenvalues of H −1 k . Then, the tDOP can be rewritten as With Equations (13) and (24), we can get Let f = 1 λ t,1 + 1 λ t,2 + 1 λ t,3 + µ(λ t,1 + λ t,2 + λ t,3 − k), and using the method of Lagrange multipliers as described in [15], we can get The equality of Equation (48) is achieved if and only if λ t,1 = λ t,2 = λ t,3 = k 3 , which is equivalent to k j=1 l 2 x,j = k j=1 m 2 y,j = k j=1 n 2 z,j = k 3 ; that is, "the polyhedron p i 1 p i 2 · · · p i k is regular" and "the position of scanner i +1 is the barycenter of all targets". This characteristic theoretically proves the experience that "the best setting position of scanner i + 1 is the barycenter of all targets" [12].
Furthermore, from Equation (48), it can be seen that the minimum value of tDOP is 3 √ k , which shows that increasing the number of targets will reduce the minimum value of tDOP.

rDOP Bounds
To find the optimum TGD, we need to analyze the bounds of rDOP, namely, the minimum of rDOP.
Denoting the three eigenvalues of G k are λ r,1 , λ r,2 , and λ r,3 , and using the property of matrix eigenvalues, we know that 1 λ r,1 , 1 λ r,2 , and 1 λ r,3 are the three eigenvalues of G −1 k , so with Equations (19) and (23), we can get where equality is achieved if and only if λ r,1 = λ r,2 = λ r,3 = 2 τ 2 z,j . Denoting the distance between the barycenter of all targets and the jth target in Scan i or Scan i + 1 are d i jc or d i+1 jc , respectively, so Based on the inequality equation (x + y) 2 ≤ 2(x 2 + y 2 ), we know where equality is achieved if and only if x i jc = x i+1 jc , y i jc = y i+1 jc , and z i jc = z i+1 jc , which is equivalent to the rotation matrix R being the identity matrix I 3 .
With Equations (50) and (52), we can then get where equality of Equation (53) (53) indicates that the minimum value of rDOP is inversely proportional to the sum of the distances d i jc from the targets to the barycenter of all targets. Therefore, without considering the precision of target extraction, the more targets disperse, the smaller the minimum rDOP and the higher the registration precision. This theoretically proves the experience that "the more dispersive the targets, the higher the registration precision" [3,17,18].

Experimental Verification
In practical applications, we might calculate all the registration parameters together (while the above model is deduced by separating translation and rotation parameters), so we need to analyze the applicability of the quantitative evaluation model of the TGD without separating translation and rotation. For these reasons, we first introduce the method of calculating the registration precision without separating translation and rotation, then design two experiments to verify the quantitative evaluation model of the TGD, and finally analyze the experiments' results.

Calculation Method of Registration Precision
The precision of target-based registration can be evaluated by the root mean square errors of rotation parameters (RMSE r ) and translation parameters (RMSE t ). The specific experimental processes are as follows: Step 1: Input the coordinate true values of target p i j ( j = 1, 2, · · · , k) and the true values of transformation parameters r a , r b , r c , t x , t y , and t z ; Step 2: Calculate the target coordinates in Scan i + 1: where Step 3: Assume j 0 = 1 and the unit weight variance σ 0 = 5mm; Step 4: If j 0 > 1000, go to Step 10; if not, continue; Step 5: Add random noise to the coordinates: where normrnd(0, σ 0 , 3, 1) returns a 3 × 1 array of random numbers chosen from a normal distribution with the mean and standard deviation as 0 and σ 0 ; Step 6: Calculate the approximate values of rotation parameters r a,0 , r b,0 , and r c,0 by Equations (8) and (9); Step 7: Calculate the approximate values of translation parameters by Equation (1): where Step 8: Calculate the estimated transformation parameters [3]: where Step 9: If j 0 = j 0 + 1, go to Step 4; Step 10: Calculate the root mean square errors of transformation and rotation parameters [3]:

Results Analysis
From Figures 3-8, it may be concluded that (a) The change of tDOP is basically the same as the change of RMSE t ; the size of tDOP and RMSE t is related to the location of scanner i + 1 (see  and the number of targets (see , not the location of targets (see ; (b) The farther away the location of scanner i + 1 (with respect to different T 0 ), the greater the tDOP and RMSE t in Figure 6; (c) When the number and position of targets change, but the location of the scanner is unchanged, the value of tDOP is a constant, the RMSE t is around a constant, and different numbers of targets (with respect to Case A, B, C, and D) have different constant valued of tDOP and RMSE t ; the more targets (with respect to Case A, B, C, and D), the smaller the tDOP and RMSE t (see ; (d) The change of rDOP is also basically the same as the change of RMSE r ; the size of rDOP and RMSE r is related to the number and position of targets (see , not the location of scanner i + 1 (see Figures 6-8); (e) The more dispersive the targets (with respect to different locations of targets p i 1 , p i 2 , p i 3 ), the smaller the rDOP and RMSE r in , so we can use the RMSE t minimum value with the minimum tDOP to represent the RMSE t minimum value; (h) We can use rDOP and tDOP to assess the impact of the targets' geometry distribution on the rotation parameters and translation parameters, respectively, and use rDOP and tDOP to help determine the optimal placement location of targets (with respect to the minimum rDOP) and the best location of scanner i + 1 (with respect to the minimum tDOP).

Conclusions
This research proposes a new evaluation model of TGD (namely, rDOP and tDOP ) for the first time, and quantitatively verifies that the model can be used to assess the impact of TGD on the registration precision by experiments, which show that "the change of rDOP / tDOP is basically the same as the change of the registration precision". In addition, this research also mathematically proves the existing experiences of TGD by the proposed model, such as "The more targets, the higher the registration precision (corresponding to the smaller rDOP and tDOP )", "The best setting position of the Scanner i + 1 is the barycenter of all targets (corresponding to the minimum tDOP value)", "The more dispersive the targets, the higher the registration precision (corresponding to the smaller rDOP values)", "The targets will be not too close to a straight line where bigger rDOP exists", and "The targets will be not too close on the same plane where bigger tDOP exists".

Conclusions
This research proposes a new evaluation model of TGD (namely, rDOP and tDOP) for the first time, and quantitatively verifies that the model can be used to assess the impact of TGD on the registration precision by experiments, which show that "the change of rDOP/tDOP is basically the same as the change of the registration precision". In addition, this research also mathematically proves the existing experiences of TGD by the proposed model, such as "The more targets, the higher the registration precision (corresponding to the smaller rDOP and tDOP)", "The best setting position of the Scanner i + 1 is the barycenter of all targets (corresponding to the minimum tDOP value)", "The more dispersive the targets, the higher the registration precision (corresponding to the smaller rDOP values)", "The targets will be not too close to a straight line where bigger rDOP exists", and "The targets will be not too close on the same plane where bigger tDOP exists".
If the targets are considered as control points or satellites, we can use the model to help design the optimal control network in engineering surveying and geodetic surveying or the optimal satellite constellation in GNSS. Therefore, we conclude that the proposed rDOP and tDOP model can be considered a unified evaluation model of the TGD, control point distribution, and satellite constellation.
However, it should be noted that "we only theoretically analyze the equal weight model of rDOP and tDOP", "we also do not use the real TLS field data collection and the actual cases to analyze the application effect of the rDOP and tDOP model", "the experiments do not consider targets' positioning precision that are affected by many factors (such as the height of scanner/targets, scanning distance, incident angel, material type of targets, etc. [28])", and "our experiments do not consider other applications such as the engineering surveying, the geodetic surveying, aerial photogrammetry and so on". In the future, we will conduct more experiments and simulations to verify our model's applications.
Author Contributions: R.Y. was the scientific responsible and coordinator of the research group, and responsible for the derivation of relevant theories in the paper; R.Y. and X.M. conceived the original ideas and worked on preparing the original draft; Z.X., Y.L. and Y.Y. assisted in results compiling and writing the manuscript; Z.X., Y.L. and H.Z. supervised the research work. All authors have read and agreed to the published version of the manuscript.