d) El modelo GTAP: características principales

An in-house geometry calibration phantom was constructed. The phantom, as shown in Figure 6.40, consists of two layers acrylic, eight 5 mm diameter stainless steel beads, and four 35 mm long tungsten bars. The two sheets were connected using four tungsten bars of the same length, so the two layers are parallel and the tungsten bars are perpendicular to the acrylic. Four beads were embedded in each layer.

During the geometry calibration procedure, the phantom was placed on the detector. Since both the detector and phantom were flat, the requirement of the phantom to be

Figure 6.40: CAD drawing of the geometry calibration phantom. The phantom has two acrylic sheets parallel to each other. Four tungsten bars were placed perpendicular to the acrylic layer. Each acrylic plate had four metal beads embedded inside.

parallel to the detector was satisfied. X-ray images of the calibration phantom were acquired. An example image is shown in Figure 6.41. The image went through standard gain and offset corrections. A Matlab program using feature recognition was developed to identify the beads and vertical bars in the projection images. The distances between metal beads were measured. The magnification factors M1 and M2 were calculated using the measured distances and actual distances between beads. Based on the measurements, thez coordinate of the focal spot was calculated.

The centerline of the four projected tungsten bars were found, and the intersection point of them gave the xand ycoordinates of the focal spot. However, due to the uncertainties in finding the centerline of the bars and the finite size of the focal spot,6 _{the four centerlines} do not guarantee intersection at one point. The four lines could intersect at up to 6 different points. Therefore, we need to estimate the position of the intersection point based on the 6_{Due to the uncertainties in the edge detection algorithm. The identified centerline can have errors on the}

Figure 6.41: An example projection image of the geometry calibration phantom. Yellow lines are the centerlines of the four vertical tungsten bars. The intersection point of these four lines gives thex, and ycoordinates of the focal spot. The distance between metal beads can be used to calculate the magnification factor M1 and M2 for the top and bottom layer of the phantom. Thus, the z coordinate of the focal spot can be calculated.

multiple intersections. One simple way to estimate the x and y coordinates is using the averagexand ycoordinates of all intersection points. However, this straightforward method does not always provide the best estimation of x and y.

Why is the average of the intersections not the best?

The main source of error causing multiple intersections is the uncertainty in determining the orientation of the centerlines. Figure 6.42 illustrates one situation.7 _{Centerlines l1,l2,l3,} 7_{This situation happens when the x-ray source is above or close to one or more vertical bars. Since the source}

array contains multiple x-ray sources, it is hard to avoid this situation in geometry calibration. One example is shown in Figure 6.41, the focal spot is close to the top left and top right bars, causing the projected bars to have shorter lengths, which increases the uncertainties in determining the orientations of these two centerlines.

. . l1 . . l2 . . l3 . . l4 • Pavg • P

Figure 6.42: Illustration of the source of error using average coordinates of all intersection points to estimate the focal spot’s x and y coordinates.

and l4 intersect at 6 points, and Pavg is the estimated location calculated using the averaged x and y of all six points. Each centerline is determined by the two end points, assuming the same uncertainty in determining each end point, as indicated by the green circle around each end point. It is clear that the shorter the length of the bar, the larger error on its orientation. Therefore, one would expect the best estimate of the intersection point P to be closer to the lines with a smaller error in orientation, thus further away from the shorter centerlines.

If two centerlines have a very small angle in between, using the average coordinates would result in a large error when estimating the x and y coordinates of the focal spot.8 _{As illus-} trated in Figure 6.43, the intersection point ofl1 andl2is far away from the other intersection points due to the small angle between l1 and l2. The estimate of the projected focal spot using averaged coordinates of all the intersection pointsPavg is close to the intersection with larg error, degrading the accuracy of geometry calibration.

8_{This situation typically occurs with focal spots near the two ends of the source array, where the projection}

. . l1 . l2 . . . l3 . . l4 • Pavg • P

Figure 6.43: Illustration of the source of error in an extreme case using average coordinates of all intersection points as focal spot x and y positions. The intersection point of l1 and l2 is far away from other intersection points due to the small angle between l1 and l2. The averaged point,Pavg, produces a large error.