Results

This section presents some example results of the hierarchical view evaluation algorithm described in the previous section. The results are compared to those of the straightforward view evaluation method. In a straightforward method, when the feasible view volumes are evaluated with a patch, all the feasible viewcells and the patch are first subdivided to their highest resolutions, and then every smallest viewcell is evaluated with every smallest patch element. Table 4.1 shows some of the parameter values used in the examples. The results

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were obtained on a laptop computer with an Intel Pentium 4-M 1.6GHz CPU, and 768 MB of DDR RAM.

Parameter Value Scanner angular sampling density, 1/α 5 samples/degree

Surface sampling density requirement, D 20 samples/inch2 Scanner maximum range, R_{max 600 inches = 50 feet}

Angle-of-incidence threshold, φ_max 70°

Surface voxel width 2 inches

Smallest viewcell width 4 inches

Smallest patch element length ≤ 4 inches Table 4.1: Some parameter values used in the experiment.

The scene is a synthetic midsize living room, represented as a polygonal model. Figure 4.14(a) shows the triangle mesh created from the first range scan. The image shows the scanner position where the scan is made and shows all the false surfaces. The ceiling of the room is not shown. Figure 4.14(b) shows the feasible view volumes and the true surface voxels in the partial scene model. The feasible view volumes are made up of viewcells of different sizes. There are 8,481 initial viewcells, and the total volume is equivalent to 28,802 smallest viewcells (each is 4 inches wide).

Figure 4.14: (a) The triangle mesh created from the first range scan of a living room. The scanner position is shown at the origin of the axes. Occlusion surfaces are shown in red, hole- boundary surfaces in blue, and image-boundary surfaces in green. The ceiling of the room is not shown. (b) The octree partial scene model is shown with the feasible view volumes (cyan). The feasible view volumes are made up of viewcells of different sizes. There are 8,481 initial viewcells, and the total volume is equivalent to 28,802 smallest viewcells (each is 4 inches wide).

The under-sampled surface voxels in the octree scene model are then grouped into 971 planar patches, each no larger than 10 feet × 10 feet. One of the occlusion surface patches is shown in Figure 4.15(a) in magenta color. The patch has 1020 smallest elements. This patch is then evaluated with the feasible view volumes. Figure 4.15(a) and (b) show the result when the straightforward and the hierarchical view evaluation methods are used, respectively. The best 500 candidate viewcells are shown. The three axes originate from the highest-score viewcell. The results appear to be almost identical, but the computation time of the straightforward method is more than 20 times that of the hierarchical method. In the hierarchical method, the user-specified relative representative error bound of s

(

)

%

s =25

ε .

Figure 4.16 shows the accumulated results when a second patch is evaluated with the feasible view volumes. This patch has 1023 smallest elements. Again, the best 500 candidate viewcells are shown. The results appear identical, but the difference in the computation times is large.

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Figure 4.15: The results of evaluating the patch with the feasible view volumes. The best 500 candidate viewcells are shown. The three axes originate from the highest-score viewcell.

Figure 4.16: The accumulated results of evaluating another patch with the feasible view volumes. The best 500 candidate viewcells are shown.

In the next experiment, the hierarchical view evaluation method is run with different relative representative error bounds, ε_s, on the new scan sampling density evaluations. The tests are on the patch shown in Figure 4.15. The hierarchical method is run using ε_s =50%,

%

s =25

ε ,and ε_s =5%. The resulting viewcells’ scores are compared to those computed by the straightforward method, and the relative errors in their scores are shown in Figure 4.17.

hierarchical method 17.8 seconds straightforward method 262.4 seconds (a) (b) straightforward method 259.6 seconds (a) (b) hierarchical method 11.9 seconds

In the graphs, there appears to be some anomaly in the score relative errors for ε_s =25%, and ε_s =5%, where some relative errors are not bounded by ε_s. Actually, this is not unexpected because εs is only a bound on the relative error of s

(

)

(

)

(

) ( )

(

)

max . The main source of the large score relative errors is the “cut-off” effect of the min function. Despite this anomaly, the majority of the score relative errors are within their respective bounds.

Table 4.2 shows the computation times when the patch shown in Figure 4.15 is evaluated with different values of ε_s. The higher the value of ε_s, the shorter the computation time. This demonstrates the ability of the hierarchical method to trade-off between computational accuracy and speed.

Figure 4.17: The relative errors of the viewcells’ scores when they are evaluated with relative representative error bounds of εs =50%, εs =25% and εs =5% on s

(

)

0 0.5 1 1.5 2 2.5 3 x 104 0 5 10 15 20 25 30 35 40 45 View S co re R e la tiv e E rro r (% ) epsilon s = 50% epsilon s = 25% epsilon s = 5%

100

%

s =5

ε ε_s =25% ε_s =50%

Computation time 14.8 sec 11.9 sec 10.4 sec

Number of sampling

density evaluations 18,260,821 7,562,709 2,765,184

Table 4.2: The effects of using different values of ε_s to bound the relative representative error of s

(

)