### Table 2: Improvement in the reconstruction of 3D mirror-symmetric con gurations from noisy 2D perspective projections. with symmetrization performed prior, following, or both prior and following the reconstruction, as discussed above. The reconstructions were compared to the real (measured) 3D coordinates of the object. The results are given in Table 3. In the second example we took images of the object at ve di erent positions (Fig. 7a). 18 feature points were manually extracted from each of the three images (visually displayed as black crosses in Fig. 7b). The 3D object was reconstructed using the invariant reconstruction method with symmetrization performed prior, following, or both prior and following the reconstruction, as

1997

"... In PAGE 12: ... The percentage of improvement between the reconstruction with no symmetry assumption and the reconstruction with correction for symmetry was calculated and averaged over the simulations (7500 trials). The results are given in Table2 . Using greater than 0:1 the percentage of improve- ment breaks down, although when using orthographic projections the improvement is signi cant up to = 0:3.... ..."

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### Table 3: Improvement in the reconstruction of a real 3D mirror-symmetric object from three 2D images. The error (average per point) is given in cm, where the object size is approximately 30cm. In the second example we took images of the object at ve di erent positions (Fig. 7a). 18 feature points were manually extracted from each of the three images (visually displayed as black crosses in Fig. 7b). The 3D object was reconstructed using the invariant reconstruction method with symmetrization performed prior, following, or both prior and following the reconstruction, as discussed above. The reconstructions were compared to the real (measured) 3D coordinates of the object. The results are given in Table 4. It can be seen that in this example the symmetrization prior to reconstruction was more e ective than following reconstruction. This is due to the fact that the 3D reconstruction itself produced a relatively mirror-symmetric object. 13

1997

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### Table 3: Improvement in the reconstruction of a real 3D mirror-symmetric object from three 2D images. The error (average per point) is given in cm, where the object size is aproximately 30cm. In the second example we took images of the object at ve di erent positions (Fig. 6a). 18 feature points were manually extracted from each of the three images (visually displayed as black crossed in Fig. 6b). The 3D object was reconstructed using the invariant reconstruction method with symmetrization performed prior, following, or both prior and following the reconstruction, as discussed above. The reconstructions were compared to the real (measured) 3D coordinates of the object. The results are given in Table 4. It can be seen that in this example the symmetrization prior to reconstruction was more e ective than following reconstruction. This is due to the fact that the 3D reconstruction itself produced a relatively mirror-symmetric object.

### Table 2. Improvement in reconstruction of 3D mirror-symmetric con g- urations from noisy 2D perspective projections.

"... In PAGE 7: ... The percentage of improvement between the reconstruction with no symme- try assumption and the reconstruction with correction for symmetry was calcu- lated and averaged over the simulations (7500 trials). The results are given in Table2 . Using greater than 0:1 the percentage of improvement breaks down, although when using orthographic projections the improvement is signi cant up to = 0:3.... ..."

### Table 1: The error and % improvement of the reconstruction of 3D mirror-symmetric con gurations from noisy 2D projections.

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### Table 2: Improvement in the reconstruction of 3D mirror-symmetric con gurations from noisy 2D perspective projections.

### Table 6.3: Improvement in reconstruction of a real 3D mirror-symmetric object from three 2D images. Reconstructions with correction for symmetry are compared with the recon- struction with no symmetry assumption. The di erences between the recon- structions and the original 3D mirror-symmetric object were measured as the least squared-distance between the reconstructed object and the original object. The improvement in the reconstruction is measured as the percent- age of decrease in the di erence between the reconstruction and the original mirror-symmetric object when correction for symmetry is performed com- pared to the case where no symmetry assumption is used.

1993

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### Table 1. The error and % improvement of the reconstructions of 3D mirror-symmetric con gurations from noisy 2D projections. In order to obtain some statistical appraisal of the improvement obtained by

### Table 6.1: Quality of the reconstructions of 3D mirror-symmetric con gu- rations from noisy 2D projections. The di erences between the reconstructions and the original 3D mirror- symmetric con gurations are measured by the mean squared-distance re- quired to move points of the reconstructed con guration in order to obtain the original con guration. As seen in the examples, reconstruction of 3D mirror-symmetric con gurations from noisy 2D projected data, can be greatly improved by correcting for symmetry either prior and/or following reconstruction. Although, correcting for symmetry prior to recon- struction improves the result, correcting for symmetry following reconstruction generally gives a greater improvement. Not surprisingly, the greatest improvement in reconstruc- tion is obtained when correction for symmetry is performed both prior and following reconstruction.

1993

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### Table 4: Improvement in reconstruction of a real 3D mirror-symmetric object from three 2D images. The error (average per point) is given in cm, where the object size is approximately 80cm.

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"... In PAGE 10: ... The reconstructions were compared to the real (measured) 3D coordinates of the object. The results are given in Table4 . It can be seen that in this example the symmetrization prior to reconstruction was more e ective than following reconstruction.... ..."

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