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 (, assessing the impact of targets' geometry distribution on the rotation parameters) and the translation dilution of precision (, assessing the impact of targets' geometry distribution on the translation parameters). Here, the definitions and derivation of relevant formulas of the and are given, the experience conclusions are theoretically proven by the model of and , and an accurate method for determining the optimal placement location of targets and the scanner is proposed by calculating the minimum value of and . Furthermore, we can refer to the model ( and ) as a unified model of the geometric distribution evaluation model, which includes the model in GPS.

1. 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].

Although target-based registration technology is relatively mature [7], studies of the target-based registration precision are still required, such as those on measurement improvement [8,9], the uncertainty of the target center estimation [10], the error propagation for two scans and multiple scans [3,4,11], the directly geo-referenced TLS data precision [3,12], the relationship between the registration precision and the rotation and translation matrices [13], the relationship between the registration precision and the targets’ geometry distribution (TGD) [2], etc. For the relationship between the registration precision and TGD, some researchers have found that an increase in the number of targets can improve the registration precision, and think that the targets will be distributed evenly and will not lie on the same line or be close to such a configuration [2,14,15,16,17,18,19,20]. Fan et al. [12] and Liu et al. [17] used a simulation method to demonstrate that the registration error is inversely proportional to the number of targets and the sum of distances between targets and the barycenter of all targets. Bornaz et al. [21] proved that the registration precision of two scans depends on the overlap ratio adopted (namely the target distribution range of the overlap area), and found that the minimum overlap ratio of 30% is required for assuring a final precision comparable to the range precision of the used instruments. However, all of the above studies have only evaluated the impact of TGD on the registration precision qualitatively through empirical experiments and simulations, while there has been no research conducted on the theoretical evaluation model for it. As a result, we can never know the best location of the scanner and the best TGD. How can we quantitatively evaluate the impact of TGD and describe the relationship between TGD and registration precision? These issues are the focus of this research.

In this study, we first used the theorem of error propagation to constitute a new theoretical evaluation model of the TGD, that is, the rotation dilution of precision () and the translation dilution of precision (); we then theoretically analyzed the model’s existence conditions, the relationship between the model and the number of targets, and the model’s bounds; and finally, we verified the evaluation model of the TGD by conducting experiments.

2. 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 (which can be used to help determine the optimal TGD) and (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.

2.1. 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 are transformed into using the three translation parameters , , and and the three rotation parameters , , and [3,23]. where and represent the same target in and , respectively, whose observation values of coordinates are and ; is the standard rotation matrix; is the translation vector; and

For uniquely determining the above transformation parameters between and , 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 is greater than 3 () and the coordinate of any point in (on a chosen coordinate system) is known, and we employed scanning to obtain the new point cloud in , which was transformed into .

2.2. Calculation of Rotation Parameters

If and , with Equation (1), we can get

From the Rodrigues matrix [21,23], with Equations (2) and (3), we can get where and .

With Equation (4), we can get where and .

If the estimated values of , , and are , , and , respectively, with Equations (5) and (6), the observation equation of rotation parameters can be expressed as where , , and are matrix, matrix, and matrix, respectively, and

Assuming the weight matrix of is , by using the principle of indirect adjustment [24] and , we can obtain the estimated for rotation parameters as

2.3. Calculation of Translation Parameters

As the position of in is , with Equation (1), we can find that the position of in is equal to the value of , namely, the process of determining translation parameters is equivalent to solving the position of in . If the targets are regarded as GPS satellites, 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 and the in is , the observation equation of translation parameters can be expressed as where and .

If the approximation values of translation parameters and corrections of translation parameters , , and are , , and (calculated by the method of Appendix C in [3]) and , , and , then from the linearization theorem [22,24], the linearization form of Equation (10) can be expressed as where ; , , and are matrix, matrix, matrix, respectively; and

Assuming the weight matrix of is , by using the principle of indirect adjustment [24], we can obtain the estimated for translation parameters as

2.4. Quantitative Evaluation Model of TGD

With Equations (9) and (15), based on the theorem of error propagation [24], the covariance and of the rotation parameters and the translation parameter corrections can be obtained as where and are matrices, and 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 parameters , , and and the translation parameters’ corrections , , and are , , , , , and , respectively. With Equation (16), the registration precision (namely, the variances of parameters and ) 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 () 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 () and the translation dilution of precision (), namely where and .

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

The values of and represent the amplification of the unit weight variance, which means the lower the values of and , the higher the solution precisions of the rotation parameters and the translation parameters;

The values of calculated by Equation (19) are related to the coefficient matrix and the weight matrix of , among which is related to TGD (the coordinates of targets ) and the rotation matrix . Namely, when is fixed, the better the quality of TGD, the lower the values of ;

The values of calculated by Equation (20) are related to the coefficient matrix ; and the weight matrix of , among which is related to TGD (the coordinates of targets ) and the position of in . Namely, the better the quality of TGD and the position of , the lower the values of ;

The calculation formula of is identical to the calculation formula of in GPS, so the can be used to evaluate the quality of the received GPS satellites’ distribution. Namely, the and model is a unified evaluation model of the targets’ and GNSS satellites’ geometric distribution.

2.5. Equationuationual Weight Model of and

In constituting the 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 and model, we assume that and are equal to identity matrix , 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 and the rotation angle of the rotation matrix under different TGDs is different and increases monotonically, and the relationship curves corresponding to different TGDs do not intersect. Therefore, we can compare the values of different TGDs under the rotation angle of the rotation matrix by the values of under the rotation angle of the unit matrix (namely, the conclusions of under arbitrary rotation matrix are equivalent to the conclusions under ). Then, we can evaluate the quality of TGD by only the values of under , and Equation (23) can be written as

2.6. 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 in namely, the best position of ; (2) the position of 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 in namely, the best TGD and the best position of .

2.6.1. The Best Position of

The best position of can be identified as follows:

Selecting the possible place of in ;

Obtaining the coordinates of all targets in ;

Using Equations (11), (12), (20) and (24) to calculate the values of under different possible places of ;

When the value of is the minimum, the corresponding place is the best position of .

2.6.2. The Best TGD

The best TGD can be identified as follows:

Selecting the possible place of targets in ;

Obtaining the coordinates of in ;

Setting the number k of targets;

Choosing k places from , and using Equations (6), (20) and (25) to calculate the values of ;

When the value of is the minimum, the corresponding places are the optimum positions of targets, namely, the best TGD.

2.6.3. The Best TGD and the Best Position of

From Section 2.4, it can be known that the model is mainly related to TGD, and the model is related to TGD and the position of ′s origin, which is relative to the selected TGD. Therefore, we firstly determined the best TGD by , and then determined the best position of by . The implementation procedures are as follows:

Selecting the possible place of targets and the possible place of scanner i + 1 in Scan i;

Setting the number k of targets;

Similar to the above, calculating the values of by k different possible places of targets, and selecting the best TGD where the value of is the minimum;

Similar to the above, calculating the values of under different possible places of , and selecting the best position of where the value of is the minimum.

3. Theoretical Analysis

We first theoretically analyzed the existence conditions of and . We then theoretically analyzed the relationship of “ and ” and the number of targets. Finally, we analyzed the bounds of and .

3.1. The Existence Conditions of

The existence condition of is that the matrix is invertible, which is equal to , namely, the rank of is 3.

If all targets and are on the same plane, and assuming the plane equation is , with Equation (13), we can get

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 is less than 3.

Therefore, the existence condition of is that all targets and are not on the same plane, which theoretically proves the experience that “all targets and lie on the same plane” [2,12].

3.2. The Existence Conditions of

The existence condition of is that the matrix is invertible, which is equal to .

With Equation (23), using the property of matrix inversion, we can get where

From the inequality , we know where equality is achieved if and only if , which is equivalent to the situation that any centralized targets in satisfy , namely, the following three situations are true:

All targets are in the plane of ’s coordinate system, and ;

All targets are in the plane of ’s coordinate system, and ;

All targets are in the plane of ’s coordinate system, and .

From the inequality , we also know

Combined with Equations (31) and (36), we can get where equality is achieved if and only if , which is equivalent to the situation that all targets lay on the same line.

From Equation (37), we know that “the value of is smaller when the TGD is closer to a straight line, the value of is larger, and the precision of the rotation parameters solution is worse, while if all targets lay on the same line, ”. Therefore, the existence condition of 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].

3.3. The Relationship Between and the Number of Targets

If more targets are considered (over k), the can be successively augmented by adding row vectors. For example, if there are targets considered, then in which we assume that is nonsingular and is a nonzero vector, and

Yarlagadda et al. [27] proved that increasing the number of satellites will reduce the in GPS applications. Here, we take the same derivation method described by Yarlagadda et al. [27] to prove its effectiveness in . With Equation (38), we can get

By using the inversion formulas of matrix [2], we can get

It is clear that is a positive definite symmetric matrix, which can be denoted as , and is the upper triangular matrix. Let and , where and are real-valued vectors, , and . Through using the property of the matrix trace, we can write

Then, we can get , which means that increasing the number of targets will reduce the value of 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].

3.4. The Relationship Between and the Number of Targets

Similar to , if more targets (over ) are considered, matrix can also be successively augmented by adding row vectors. For example, if there are targets considered, then where

Assume that is nonsingular and , , and 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 and improve the registration precision, which also theoretically proves the experience in Section 3.3.

3.5. Bounds

To find the optimum position of , we need to analyze the bounds of , namely, the minimum of .

Denoting the three eigenvalues of are , , and , and using the property of matrix eigenvalues, we know that , , and are the three eigenvalues of . Then, the can be rewritten as

With Equations (13) and (24), we can get

Let , 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 , which is equivalent to ; that is, “the polyhedron is regular” and “the position of is the barycenter of all targets”. This characteristic theoretically proves the experience that “the best setting position of ” [12].

Furthermore, from Equation (48), it can be seen that the minimum value of is , which shows that increasing the number of targets will reduce the minimum value of .

3.6. Bounds

To find the optimum TGD, we need to analyze the bounds of , namely, the minimum of .

Denoting the three eigenvalues of are , , and , and using the property of matrix eigenvalues, we know that , , and are the three eigenvalues of , so with Equations (19) and (23), we can get where equality is achieved if and only if , which is equivalent to .

Denoting the distance between the barycenter of all targets and the in or are or , respectively, so

Based on the inequality equation , we know where equality is achieved if and only if , , and , which is equivalent to the rotation matrix being the identity matrix .

With Equations (50) and (52), we can then get where equality of Equation (53) is achieved if and only if and .

Equation (53) indicates that the minimum value of is inversely proportional to the sum of the distances 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 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].

4. 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.

4.1. Calculation Method of Registration Precision

The precision of target-based registration can be evaluated by the root mean square errors of rotation parameters () and translation parameters (). The specific experimental processes are as follows:

Step 1: Input the coordinate true values of target and the true values of transformation parameters , , , , , and ;

Step 2: Calculate the target coordinates in : where

Step 3: Assume and the unit weight variance ;

Step 4: If , go to Step 10; if not, continue;

Step 5: Add random noise to the coordinates: where returns a array of random numbers chosen from a normal distribution with the mean and standard deviation as 0 and ;

Step 6: Calculate the approximate values of rotation parameters , , and 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 , go to Step 4;

Step 10: Calculate the root mean square errors of transformation and rotation parameters [3]:

4.2. Experiment I

Since the precision of target-based registration is related to the number of targets, we simulated six targets (see Figure 1) and designed four scenarios: Case A: using three targets ; Case B: using four targets ; Case C: using five targets ; and Case D: using six targets ). We then calculated the , , , and of Case A, B, C, and D with different target distributions and locations of scanners. The specific experimental processes are as follows:

Figure 1

The target geometry of experiment I.

Step 1: Assume and the coordinate true values of targets are

Step 2: Let and . Calculate , , , and of Case A, B, C, and D with different locations of target from Equations (19), (20), (24), (25), (64) and (65) (see Figure 3);

Step 3: Let , , and . Calculate , , , and of Case A, B, C, and D with different locations of targets and from Equations (19), (20), (24), (25), (64) and (65) (see Figure 4);

Step 4: Let , , , and . Calculate , , , and of Case A, B, C, and D with different locations of targets , , and from Equations (19), (20), (24), (25), (64) and (65) (see Figure 5);

Step 5: Let and . Calculate , , , and of Case A, B, C, and D with different locations of scanner from Equations (19), (20), (24), (25), (64) and (65) (see Figure 6).

4.3. Experiment II

To further verify the quantitative evaluation model of the targets’ geometry distribution, we designed another experiment with realistic targets drawn from previous studies [3,23] using an RIEGL VZ-400 laser scanner with different target distributions (see Table 1) and different locations of scanners (see Figure 2). The specific experimental processes are as follows:

Table 1

Different target distributions.

The Name of Distribution	Including Targets	The Name of Distribution	Including Targets
Case A1	p_1, p_2, p_3, p_4, p_5	Case C1	p_1, p_2, p_3
Case B1	p_1, p_2, p_3, p_4	Case C2	p_1, p_2, p_4
Case B2	p_1, p_2, p_3, p_5	Case C3	p_1, p_2, p_5
Case B3	p_1, p_2, p_4, p_5	Case C4	p_1, p_3, p_4
Case B4	p_1, p_3, p_4, p_5	Case C5	p_1, p_3, p_5
Case B5	p_2, p_3, p_4, p_5	Case C6	p_1, p_4, p_5
		Case C7	p_2, p_3, p_4
		Case C8	p_2, p_3, p_5
		Case C9	p_2, p_4, p_5
		Case C10	p_3, p_4, p_5

Figure 2

The target and geometry of experiment II.

Step 1: Assume the locations of scanner , ;

Step 2: Input the coordinates of targets:

Step 3: Calculate , , , and of Case A1 and B1–5 with different locations of scanner from Equations (19), (20), (24), (25), (64) and (65) (see Figure 7);

Step 4: Calculate , , , and of Case C1–10 with different locations of scanner from Equations (19), (20), (24), (25), (64) and (65) (see Figure 8).

4.4. Results Analysis

From Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, it may be concluded that

Figure 3

, , , and of Case A, B, C, and D with respect to one different target.

Figure 4

, , , and of Case A, B, C, and D with respect to two different targets.

Figure 5

, , , and of Case A, B, C, and D with respect to three different targets.

Figure 6

, , , and of Case A, B, C, and D with respect to different .

Figure 7

, , , and of Case A and B1–5 with respect to different scanner i + 1.

Figure 8

, , , and of Case C1–10 with respect to different .

The change of is basically the same as the change of ; the size of and is related to the location of (see Figure 6, Figure 7 and Figure 8) and the number of targets (see Figure 3, Figure 4 and Figure 5), not the location of targets (see Figure 3, Figure 4 and Figure 5);

The farther away the location of + 1 (with respect to different ), the greater the and in Figure 6;

When the number and position of targets change, but the location of the scanner is unchanged, the value of is a constant, the is around a constant, and different numbers of targets (with respect to Case A, B, C, and D) have different constant valued of and ; the more targets (with respect to Case A, B, C, and D), the smaller the and (see Figure 3, Figure 4 and Figure 5);

The change of is also basically the same as the change of ; the size of and is related to the number and position of targets (see Figure 3, Figure 4 and Figure 5), not the location of (see Figure 6, Figure 7 and Figure 8);

The more dispersive the targets (with respect to different locations of targets ), the smaller the and in Figure 3, Figure 4 and Figure 5;

When the location of the scanner changes, but the number and position of targets are unchanged, the value of is a constant, the is around a constant, and different numbers of targets (with respect to Case A, B, C, D, A1, B1–5, and C1–10) have different constant values of and ; the more targets there are (with respect to Case A, B, C, D, A1, B1–5, and C1–10), the smaller the and (see Figure 6, Figure 7 and Figure 8);

The differences between the and the minimum values in cases A1, B1–5, and C1–10 with the minimum are −0.5, −1.6, −1.8, 0.9, −2, −1.7, −1.5, −1.8, −2.1, −1.5, −1.5, −1.6, −1.3, −1.5, −2.0, and −0.8 mm, respectively, which are all less than (half the observation variance, 2.5 mm), so we can use the minimum value with the minimum to represent the minimum value;

We can use and to assess the impact of the targets’ geometry distribution on the rotation parameters and translation parameters, respectively, and use and to help determine the optimal placement location of targets (with respect to the minimum ) and the best location of + 1 (with respect to the minimum ).

5. Conclusions

This research proposes a new evaluation model of TGD (namely, and ) 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 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 and )”, “The best setting position of the is the barycenter of all targets (corresponding to the minimum value)”, “The more dispersive the targets, the higher the registration precision (corresponding to the smaller values)”, “The targets will be not too close to a straight line where bigger exists”, and “The targets will be not too close on the same plane where bigger 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 and 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 and ”, “we also do not use the real TLS field data collection and the actual cases to analyze the application effect of the and 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.