A geometric approach to quadratic optimization: an improved method for solving strongly underdetermined systems in CT§

Preliminary results presented at the 8th IASTED Conference on Computer Graphics & Imaging, August 2005.

Figures & data

Table 1. Details of the two test cases.

Test cases	Case 1	Case 2
Image size	255 × 255	255 × 255
Variables	65,025	65,025
Equations	64,980	16,290
Projections	180	90
Rays	361	181
Density range	0.113–0.726	0.4–0.5

Table 2. Case 1—minimal distance and relative error, and iteration number(s) at which they were obtained.

Measure	ART	QUAD	NQUAD
Distance	0.0807, it. 12–13	0.0837, it. 15	0.0803, it. 13
Rel. err.	0.0497, it. 8–9	0.0558, it. 13	0.0521, it. 12

Table 3. Case 2—minimal distance and relative error, and iteration number(s) at which they were obtained.

Measure	ART	QUAD	NQUAD
Distance	0.2803, it. 40	0.2197, it. 40	0.1922, it. 11–14
Rel. err.	0.1373, it. 40	0.0918, it. 37–40	0.0771, it. 10

Figure 1. Tomographic image reconstruction in the discretized model.

Figure 2. Distance measure for Case 2.

Figure 3. Relative error measure for Case 2.

Figure 4. Case 1: phantom and reconstructed images after 10 iterations.

Figure 5. Case 2: phantom and reconstructed images at 10 iterations.

Figure 6. Case 2: phantom and reconstructed images (10 iterations) after histogram equalization.

Figure 7. Case 3: phantom and reconstructed images after 10 iterations.

Figure 8. Case 4: phantom and reconstructed images after 10 iterations.

Table 4. Case 3—minimal distance and relative error, and iteration number(s) at which they were obtained.

Measure	ART	QUAD	NQUAD
Distance	0.0484, it. 10	0.0539, it. 16,17	0.0538, it. 14
Rel. err.	0.0176, it. 8	0.0236, it. 15	0.0230, it. 13,14

Table 5. Case 4—minimal distance and relative error, and iteration number(s) at which they were obtained.

Measure	ART	QUAD	NQUAD
Distance	0.3139 it. 20	0.2611 it. 19,20	0.2385, it. 18–20
Rel. err.	0.1712 it. 20	0.1387, it. 18–20	0.1256, it. 14–20