Validation of a Method for the Optimization of Scan Parameters for Measuring with Computed Tomogography

When measuring with computed tomography, the measurement uncertainty of measured values is strongly influenced by the geometry and material of the measured object as well as by the chosen scan parameters, such as those of the X-ray source and detector. Optimizing the scan parameters to achieve a sufficiently high measurement precision while keeping the measurement time as short as possible, is crucial for the efficient use of a coordinate measuring machine. Few tools are available to aid the user at the moment. Therefore scan parameters are often set based on purely empirical studies or subjective assessments of Xray images and thus on the basis of personal experience. A new method for the automatic determination of scan parameters for a given measurement time or a given measurement precision of a single measurement point, as well as a given minimum necessary structural resolution has been developed. It is based on a method for simulating the measurement precision of surface points that are relevant for the measurement task. In this work, values for scan parameters to maintain the highest possible single point precision for a given measurement time and minimum necessary structural resolution have been determined for a multisphere standard using the newly developed setting algorithm and by evaluating empirical studies. The results of both methods are presented and compared.


Introduction
In recent years computed tomography has been established in the area of coordinate metrology as a method for the fast, complete and non-destructive metrological assessment of internal and external geometrical characteristics.The choice of scan parameters significantly influences the measurement uncertainty of measurement values [1].Although the influence of systematic uncertainty contributions, such as those caused by temperature changes or beam hardening, can increasingly be corrected [2,3], random uncertainty contributions and the resultant measurement precision, remain strongly dependent on the choice of scan parameters.For this reason, the optimization of scan parameters to achieve a sufficiently high measurement precision of surface points that are relevant for a measurement task (hereinafter "target points") and the shortest possible measurement time is crucial for the efficient use of a coordinate measuring machine.In the following the validation of an algorithm for optimizing scan parameters (hereinafter "setting algorithm") using a multi-sphere standard is presented.Optimum values for the X-ray settings, such as the X-ray tube voltage and power, the detector settings, such as the integration time and the number of images averaged per projection, as well as the number of projections per scan have been determined both based on the evaluation of empirical studies and by the application of the newly developed setting algorithm described in [5].

Methods for the determination of scan parameters
A manual method on the basis of empirical studies and an algorithm based on the simulation of the measurement process have been used to determine scan parameters.

Manual Determination of Scan Parameters
Empirical studies are performed to manually determine optimum values for scan parameters to maximize measurement precision while adhering to the constraints of the measurement task.In every empirical study, the scan parameter to be optimized is varied in a predefined interval between a minimum and maximum permissible value, while all other scan parameters are kept constant.For each value, a measurement series of 20 measurements is performed under repeatability conditions.In each study, the resulting surface points are averaged in a specified area around predefined target points using the so-called measuring spot strategy [4].In the following, the principle of the measuring spot strategy is described on the basis of figure 1.
Figure 1: Averaging of measurement points using the measuring spot strategy.[4] The area around the target point is defined by a cylinder with a user defined diameter and height (the so called selection tolerance).The axis of the cylinder is oriented orthogonal to the surface of a CAD model that is used as reference.In a first step, the deviation vectors between the measurement points within the cylinder and the corresponding perpendicular points on the surface of the CAD model are calculated (orange) and averaged.The resulting measurement point is the sum of the mean vector and the target point.Then, the standard deviation is calculated from the resulting points for every target point.From this, the standard deviation of a single measurement point is obtained by multiplying the resulting standard deviation by the square root of the number of measurement points within the cylinder.Hereof, the mean standard deviation is determined by averaging the values for each measurement series.Finally, the minimum of the mean standard deviation is calculated and the corresponding value for the scan parameter to be optimized is determined.

Automatic Determination of Scan Parameters
The method used for the automatic optimization of scan parameters is based on the simulation of the standard deviation of the single measurement points and is explained in following on the basis of figure 2.

Defining the Measurement Task
First, the geometry and material of the measurement object have to be defined.The geometry of the object is described by a point cloud in STL format.For this purpose, either a point cloud calculated from a CAD model or a measured point cloud can be used.Additionally, the surface points which are relevant for the measurement task (target points) can be predefined by the user.Depending on the measurement task, a minimum necessary structural resolution (spatial resolution) has to be defined.
The range of possible values for scan parameters can be restricted by adjusting the minimum and maximum value for each scan parameter.Table 1 shows typical minimum and maximum values for scan parameters, such as tube voltage and power, the prefilter and the integration time.To achieve a desired beam hardening of the emitted spectrum by the pre-filter, for example the weakly attenuating pre-filters can be removed from the list of pre-filters available for the setting algorithm.Finally, the user can decide between a maximum permissible measurement time and a maximum permissible standard deviation of single measurement points as optimization criteria.Both optimization criteria restrict the total exposure time available.While the optimal values of time invariant scan parameters, such as tube voltage and power, are independent from the available total exposure time, the optimum of time variant scan parameters, such as the integration time, the number of images averaged per projection and the number of projections per scan are not.

Optimization of Time Invariant Parameters
The algorithm allows the user to automatically determine values for the time invariant parameters tube voltage and tube power as well as the pre-filter with regard to a possibly low mean standard deviation of the target points.Starting with the pre-filter with the lowest attenuation, the combination of tube voltage, tube power and integration time is sought that leads to the highest normalized CNR, while ensuring the boundary conditions defined by the user are respected.In a first step, the maximum permissible tube power that results in the user-defined minimum necessary structural resolution is determined for each tube voltage considered having regard to the influence of the blurring described by the PSF.In addition, the shortest integration time that does not result in underexposure or overexposure of the detection elements is determined.The combination with the highest local normalized CNR is chosen.The process is repeated with the prefilter with the next highest attenuation if no valid combination of tube voltage, tube power and integration time is found.Subsequently, the measurement precision normalized to total exposure time (hereinafter "normalized single point precision") is calculated based on the local normalized CNR averaged over all target points.For the example given in figure 3 the maximum local CNR and the minimum normalized single point precision result when measuring with a tube voltage of 140 kV.

Determination of Total Exposure Time
In case of a given minimum necessary single point precision the minimum necessary total exposure time is calculated from the ratio of the minimum necessary single point precision to the simulated normalized single point precision.In case of a given maximum permissible measurement time the maximum permissible total exposure time is calculated from the difference of the maximum permissible measurement time and an estimation for the time needed for additional processes during the measurement, such as the movement times of the axes of the machine.

Optimization of Time Variant Parameters
The number of projections angles , i needed to avoid aliasing artifacts is calculated using the following relation [6].
Here t is the number of detection elements in direction of the normal of the plane defined by the axis of rotation and the normal of the X-ray detector.The number of images averaged per projection is calculated by dividing the maximum permissible or minimum necessary total integration time by the product of the minimum possible integration time and the minimum necessary number of projection angles.Finally, the number of projection angles is increased by the ratio of maximum permissible or minimum necessary total integration time and the total integration time resulting from the optimization to reduce the difference to the target value.

Results
The methods presented have been applied to determine scan parameters for the measurement of a multi-sphere standard with a Werth TomoScope L coordinate measuring machine with a micro-focus X-ray tube and an energy integrating flat panel detector.
The aim was to minimize the standard deviation of the target points adhering a maximum permissible measurement time of 13.3 minutes (time of a single measurement of the empirical studies that were conducted first).Additionally, the voltage range was limited from 80 kV to 200 kV and the power range was limited from 1 W to 75 W.The range of available pre-filters was not restricted.The red points shown in figure 4 have been used as target points for both methods.A point cloud in STL format of the multi-sphere standard measured previously was used as model of the measurement object for the simulation based setting algorithm and is shown in figure 4 (golden surface).The material of the object was assumed to be ruby (Al2O3) with a density of 0.00405 g 3 .For the measurement of the ruby spheres with 2 mm in diameter a minimum necessary structural resolution of 0.2 mm was defined.

Time Invariant Parameter
The dependence between mean standard deviation normalized by the minimum standard deviation measured and the X-ray tube voltage for a tube power of 75 W is shown in figure 5. Consequently, the highest measurement precision can be achieved when measuring with a tube voltage of 140 kV.This is due to the occurring of the maximum local CNR at 140 kV (simulated values shown in figure 3 (c)) and the standard deviation being inversely proportional to the local CNR.The manual as well as the automatic method for the determination of the tube voltage that leads to a minimum mean standard deviation of the target points show the same result (figure 5, blue dotted line).The dependence between mean standard deviation normalized by the minimum standard deviation measured and the X-ray tube power for a tube voltage of 140 kV is shown in figure 6.Consequently, the highest measurement precision can be achieved when measuring with a tube power of 75 W.The result of the manual as well as of the automatic method for the setting of the tube power that leads to a minimum mean standard deviation of measuring points are the same (figure 6, blue dotted line).Furthermore, the resulting structural resolution has been examined and is sufficient for the measurement task.The measured standard deviation values agree with the theoretical relationship that the change of the standard deviation is inversely proportional to the square root of the change of the tube power (red).The smallest standard deviation of the measuring points was determined with both methods when measuring without prefilter.Subsequently, the results for both, the manual and automatic method, are summarized in table 2.

Time Variant Parameters
The dependence between mean standard deviation normalized by the minimum standard deviation measured and the total exposure time is shown in figure 7. Consequently, the highest measurement precision can be achieved when measuring with the longest exposure time permissible (figure 7, blue dotted line).The measured standard deviation values agree with the theoretical relationship that the change of the standard deviation is inversely proportional to the square root of the change of the total integration time (red).
Figure 7: Dependency of the standard deviation normalized by the minimum standard deviation from total exposure time.The integration time determined with both methods does not lead to an underexposure or overexposure of the detection elements.For both methods relationship (1) was used to calculate the minimum number of projections.Subsequently, the results for both, the manual and automatic method, are summarized in table 3.

Discussion and Conclusion
The manual method based on empirical studies and the simulation-based setting algorithm result in the same settings for the scan parameters tube voltage, tube power and the prefilter.Furthermore, the total integration time resulting from both methods is the same.The standard deviation of the determined surface points at the position of the target points is minimized and hence the reproducibility of the measurement maximized.All boundary conditions defined by the user are met.In practice, empirical studies for the determination of scan parameters are rarely done because of the corresponding high costs and the time needed.The setting algorithm first presented in [5] allows the determination of scan parameter for a given maximum permissible measurement time or a maximum permissible mean standard deviation of the target points in under one minute.Thus, using the setting algorithm can substantially reduce the costs for the determination of scan parameters and by reducing the necessary measurement time per object.

Figure 2 :
Figure 2: Simplified Flow Chart of the Setting Algorithm

Figure 3 :
Figure 3: Simulated contrast (a), noise (b) and contrast noise ratio (c) as function of tube voltage for the example of a multi-sphere standard.Values have been calculated for a tube power of 75 W without pre-filtering.In figure3the simulated values for contrast, noise and contrast noise ratio are shown as function of tube voltage for the example of the measurement of a multi-sphere standard at 75 W tube voltage without prefilter.The difference between the linear attenuation coefficient (hereinafter "gray values") used for the determination of surface points (local contrast) depends on the difference between the minimum and maximum gray value occurring at the edge transition (global contrast) and the blurring described by the point spread function (PSF) that depends, for instance, on the pixel aperture and the focal spot size.The noise of two adjacent gray values at the center of the edge transition (local noise) is approximately equal.The standard deviation of a single surface point is inversely proportional to the ratio of local contrast to local noise (local CNR).[5]

Figure 4 :
Figure 4: Model of the multi-sphere standard as point cloud in STL format (gold) and target points (red).

Figure 5 :
Figure 5: Measured dependency of the standard deviation normalized by the minimum standard deviation from the tube voltage.

Figure 6 :
Figure 6: Dependency of the standard deviation normalized by the minimum standard deviation from the tube power.

Table 1 :
Typical minimum and maximum values for scan parameters for a micro-focus X-ray tube with transmission target and an energy integrating flat panel detector.

Table 2 :
Determined values for time invariant scan parameters.

Table 3 :
Determined values for time variant scan parameters.