7. MARCO METODOLÓGICO
7.3 Técnicas e Instrumentos Utilizados
4.4.1.1 Methods
The gold standard extrinsic and intrinsic parameters calculated by the Tsai calibration will have errors associated with each parameter. Four tests were performed to assess the accuracy of the gold standard.
The image in gure 4.7(a) was taken, and the locations of the ducial markers were extracted in the 3D image, using a centre of gravity operator, and in the 2D image by manually clicking on the ducial location. Using corresponding 3D and 2D points, Tsai's algorithm was used to produce a set of gold standard intrinsic and extrinsic parameters. Using these parameters, the projection matrix (matrix
M
in equation 2.1) was formed and the 3D points projected to 2D pixel locations to form perfectly matching 3D and 2D points.Random, zero mean, Gaussian noise with standard deviation
n = 0:
1;
0:
2:::
5:
0 wasadded to each of the 2D points, and a modied Levenberg-Marquardt [Press et al., 1992] non-linear optimisation, was used to optimise the extrinsic parameters only3. This was
repeated 1000 times, and the mean and standard deviation of the recovered extrinsic parameters calculated. In addition, the mean and standard deviation projection and 3D error was calculated for each noise level. This experiment was repeated, adding noise to just the 3D points. The purpose of this simulation was to determine what the eects of noise are on the accuracy of the recovered extrinsic parameters and error measures for a typical setup used in this chapter.
Subsequently, the image in gure 4.7(a) was taken, and the corresponding 2D and 3D point locations extracted as before. A leave-one-out test was used to determine the accuracy of the gold standard parameters. For a set of points, Tsai's algorithm used all but one of the corresponding 2D and 3D point pairs to calculate the gold standard parameters. The remaining point was used to calculate a projection error in millimetres. This was repeated for every combination of points.
4.4.1.2 Results
The graph in gure 4.9(a) shows how the standard deviation of each parameter
t
x;t
y;t
zr
x;r
y;r
zincreases as noise is added to the 2D image points. Similarly, the graph in gure3This is also part of freely available software
4.4 Experiments
114
0 1 2 3 4 5 0 1 2 3 4 5Standard deviation of parameters
Standard deviation of noise
"tx" "ty" "tz" "rx" "ry" "rz" (a) 0 2 4 6 8 10 12 14 0 1 2 3 4 5
Standard deviation of parameters
Standard deviation of noise
"tx" "ty" "tz" "rx" "ry" "rz" (b)
Figure 4.9: (a) Variation in parameters
t
x:::r
z with noise added to the 2D points. (b)4.4 Experiments
115
4.9(b) shows how the parameters vary when noise is added to the 3D points. It can be seen that the noise has greater eect when added to the 3D points. When the noise has a standard deviation 1
:
5 and the noise is added to the 3D points, Tsai's algorithmfails. For both 2D and 3D noise, the
t
z parameter is the most eected. This means thatfor a given gold standard registration,
t
z will be the least accurate parameter.From these graphs in gure 4.9 (a) and (b) we can deduce that in order for all the parameters to have a standard deviation
<
1, the standard deviation of the noise on the 2D pixels must be<
0:
7 pixels, and the standard deviation of the noise in the 3D points must be<
0:
2mm.The graphs in gure 4.10(a)(b) show the mean and standard deviation (on errorbars) of (a) projection error and (b) the 3D error as the noise level is increased. The dierence in projection error and 3D error is immediately apparent. Recall that the parameter most aected by noise is
t
z. An error in the parametert
z will cause a large 3D error but amuch smaller projection error. This explains why graph (b) has larger errors.
13 of the ducials visible in the image in gure 4.7(a) were used for this experiment. Tsai's calibration was performed for every combination of 12 points from the 13 and the mean projection error was calculated as 0.25 mm. Referring to the graph in gure 4.10(a) and assuming that errors are caused entirely by noise on the 3D points suggests that the standard deviation of the noise is likely to be approximately 0
:
2 mm. This would suggest that, using the graph in gure 4.10(b) that the corresponding 3D error is approximately 0:
75 mm. In addition, from the leave one out test, the standard deviation of the parameterst
x;t
y;t
z;r
x;r
y;r
z was 0.07, 0.07, 0.19 mm and 0.06, 0.06, 0.05 degreesrespectively.
4.4.1.3 Conclusions
It was concluded that the gold standard used throughout this chapter is of sucient accuracy for the experiments that follow. The leave one out test revealed that the mean projection error was 0.25 mm and the corresponding mean 3D projection error was likely to be approximately 0
:
75 mm. For most clinical applications, an accuracy of approximately 1mm would be acceptable. With this gold standard, 3D errors of the order of 0:
75 mm would be the best that can be reliably calculated with respect to this quality of gold standard.4.4 Experiments
116
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4Mean (standard deviation on error bars) of projection error
Standard deviation of noise
"2D" "3D" (a) 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mean (standard deviation on error bars) of 3D error
Standard deviation of noise
"2D" "3D"
(b)
Figure 4.10: (a) Mean and standard deviation (on error bars) of projection error as noise is added to 2D or 3D points. (b) Likewise for 3D error.