CHAPTER 3 AN OCTREE CONSTRUCTION METHOD WITH THREE TYPES
3.4 The Relation between Object Spatial Resolution and Projection Error Upper
d
| r { max ( ument arg
v l,v l,v
v
* = +
(5) If an octant cannot be defined above, it is called a GGl octant; a GGl octant whose white and black extents exceed the prespecified p value.
The main differences between the octant types of method 1 and method 2 are:
(1) There is no GWl octants in method 1. A octant in the second method is categorized as either as a octant if the white extent + | | is greater than p or a octant if the white extent + | | is smaller than p.
2
GWl 1
GGl rl,v dl,v
1
GBl rl,v dl,v
(2) The definitions of a GBl octant in the two methods are different:
(i) For a GBl2 octant, dl,v >0 and rl,v− dl,v > p for all viewsv∈
[ ]
1,N . (ii) For a GB1l octant, there are two possibilities: dl,v >0 andp d
rl,v− l,v < or dl,v<0 and rl,v−dl,v< p for all viewsv∈
[ ]
1,N . 3.3 The Octant Subdivision Algorithm of the Second Construction MethodWith the new definitions of the octant types the octant subdivision algorithm of the second construction method is given below:
The Octant Subdivision Algorithm:
At each level l = 0, 1, .., L only the GGl octants need to be subdivided into eight child octants. The octant subdivision process is performed at all levels until there are no GGl octants at the final level L.
3.4 The Relation between Object Spatial Resolution and Projection Error Upper Bound
In the first and second construction methods the projection error upper bound parameter p is needed to specify by the user. The proper value of p is determined by
the object spatial resolution. If the object detailed part or cavity has a width less than 2 p then the octant covering the object detailed part will be classified as or octant by the two methods, respectively, so this detail disappears; similarly, the octant covering the object detailed cavity will be classified as octant by the second method, so this detail disappears. Fig. 3.1 shows the depiction of the relation between the object spatial resolution and the projection error upper bound.
1
GBl GBl2
2
GWl
GW octants GB octants
Fig. 3.1. The depiction of the relation between the object spatial resolution and the projection error upper bound.
3.5 Experimental Results
Experiment 1:
In this experiment, we feed the images of a tilted cube used in chapter 2 to method 1 and method 2, respectively. We then list all types of octants generated at each subdivision level, together with the 3D volume of the final constructed models in Tables 3.1 and 3.2 under different error bound and maximum subdivision level of our method and the conventional method. In addition, the graphical display of the constructed object models obtained by the conventional method and the new method is shown in Fig. 3.2.
Table 3.1. Number of the black, grey-black, grey-grey, grey-white and white octants of the constructed cube generated by method 2.
Protrusion=15 Protrusion=7 Protrusion=2 Volume=11755.86 Volume=10333.71 Volume=9345.573
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 1 0 0 0 1 0
1 0 0 8 0 0 0 8 0 0 0 8 0
2 0 0 48 16 0 0 48 16 0 0 48 16
3 12 42 125 205 12 16 151 205 12 2 165 205
4 32 177 264 527 130 115 436 527 234 28 531 527 5 0 891 2 1219 319 775 1139 1255 865 218 1910 1255 6 0 0 0 16 67 3473 300 5272 2098 1330 5527 5325
7 0 151 0 2249 9201 6559 9148 19308
8 18014 0 0 55170
Table 3.2. Number of black, grey-black, grey-grey and white octants of the cube generated by method 1.
Protrusion=15 Protrusion=7 Protrusion=2 Volume=9515.625 Volume=9465.82 Volume=9372.925
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
1 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0
2 0 0 34 14 16 0 0 44 4 16 0 0 46 2 16
3 12 42 52 54 112 12 16 122 27 175 12 2 156 9 189 4 32 145 0 125 114 130 115 288 129 314 234 28 498 33 455
5 319 775 1 789 420 865 218 1528 376 997
6 2 2 0 4 0 3098 1330 2270 2670 2856
7 7593 3315 0 4296 2956
8
Method 1 Protrusion= 15 Protrusion= 7 Protrusion=2
Image of the constructed
model
Method 2 Protrusion= 15 Protrusion= 7 Protrusion=2
Image of the constructed
model
Fig. 3.2. The comparison between construction results of the synthetic cube obtained by method 1 and method 2.
Experiment 2:
In experiment 2,the same real image sets used in chapter 2 are used to compare the performance of our first method and second method.
Table 3.3 shows the size of a projection octant image at different level. We list the numbers of all types of octants generated at each subdivision level, together with the projection error upper bound value and the quality index of the final constructed models in Table 3.4, Table 3.7 and Table 3.10. Table 3.5, Table 3.8 and Table 3.11 are the octant numbers of the octree models constructed by our first method at P= 25, 15 and 7. In Table 3.6, Table 3.9 and Table 3.12, we show the results of applying the GB2 and GW2 classification criteria to the GB1 octant shown in Table 3.5, Table 3.8 and Table 3.11, respectively. As mentioned in chapter two, it is sufficient to use two bits to represent the color of the octant of method 1. However, since we introduce three types of grey ocant in this chapter, it requires three bits to represent the five colors of the octants for method 2. Let num1 and num2 be the total numbers of octants
generated in mehod 1 and method 2, respectively. From Lemma 3-2 introduced later in the next section, we obain that (num1/8) is roughly equal to num2. That is, the total memory space in bits required for method 2 is roughly equal to 3*num2. The toal memory space in bits required for method 1 is 2*num1. So the total number of bits required by mtehod 2 less, compared to method 1. In addition, the graphical display of the constructed object models and the XOR error images obtained by the conventional method and the new method are shown in Figures 3.3 to 3.5.
Besides feeding the image sets used in chapter 2 into method 2, we also apply the second method to construct the octree model of a flower pot and a dinosaur. Fig. 3.6 and Fig. 3.7 are the selected input images and the generated novel views of the constructed results.
Table. 3.3. The radius range of the circle containing the octant projection at different level.
Level
Radius range 0 1 2 3 4 5 6 7 8
Min 327 150 72 35 18 9 4 2 1
Max 331 177 92 47 23 11 6 3 1
Table 3.4. Number of the black, grey-black, grey-grey, grey-white and white octants of the constructed cone generated by the second construction method.
Protrusion=25 Protrusion=15 Protrusion=7 Xor=14213+9889 Xor=9010+5957 Xor=5849+3630
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 4 3 1 0 0 4 3 1 0 0 4 3 1 2 0 0 8 0 24 0 0 8 0 24 0 0 8 0 24 3 0 0 10 22 32 0 0 17 15 32 0 0 27 5 32 4 4 29 0 40 7 4 17 24 51 40 4 6 67 35 104 5 10 87 0 89 6 43 88 105 171 129 6 73 373 0 351 43 7
8
Table 3.5. Number of black, grey-black, grey-grey and white octants of the cone generated by the first construction method.
Protrusion=25 Protrusion=15 Protrusion=7 Xor=56188+24 Xor=30737+125 Xor=10358+1068
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 7 0 1 0 0 7 0 1 0 0 7 0 1 2 0 0 8 0 48 0 0 8 0 48 0 0 8 0 48 3 0 0 32 0 32 0 0 32 0 32 0 0 32 0 32 4 4 44 64 0 144 4 17 91 0 144 4 6 102 0 144 5 0 137 0 0 375 10 191 140 0 387 43 88 298 0 387 6 0 202 0 0 918 73 582 595 0 1134 7 0 934 0 0 3826 8
Table 3.6. Number of grey-black and grey-white leaf octants of the cone generated by applying the second method octant type evaluation criteria on grey-black octants in Table 3.3.
Protrusion=25 Protrusion=15 Protrusion=7
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 0 0 0 0
1 0 0 0 0 0 0
2 0 0 0 0 0 0
3 0 0 0 0 0 0
4 29 15 17 0 6 0 5 9 128 99 92 88 0
6 0 202 418 164
7 170 764
8
Table 3.7. Number of the black, grey-black, grey-grey, grey-white and white octants of the constructed vase generated by the second construction method.
Protrusion=25 Protrusion=15 Protrusion=7 Xor=58496+7317 Xor=33007+4825 Xor=22311+2777
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0 2 0 0 25 16 23 0 0 30 11 23 0 0 34 7 23 3 1 11 62 53 73 1 6 82 43 108 1 0 107 26 138 4 30 165 0 245 56 52 114 132 233 125 88 55 348 125 240
5 47 483 0 483 43 297 524 577 798 588
6 530 1998 0 1828 260 7
8
Table 3.8. Number of black, grey-black, grey-grey and white octants of the vase generated by the first construction method.
Protrusion=25 Protrusion=15 Protrusion=7 Xor=150755+4 Xor=88134+22 Xor=35498+498
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0 2 0 0 41 0 23 0 0 41 0 23 0 0 41 0 23 3 1 11 122 0 194 1 6 127 0 194 1 0 133 0 194 4 30 233 269 0 444 52 114 406 0 444 88 55 477 0 444 5 0 716 0 0 1436 47 1046 658 0 1497 297 524 1498 0 1497 6 0 1043 0 0 4221 530 3153 3002 0 5299 7 0 6117 0 0 17899 8
Table 3.9. Number of grey-black and grey-white leaf octants of the vase generated by applying the second method octant type evaluation criteria on grey-black octants in Table 3.6.
Protrusion=25 Protrusion=15 Protrusion=7
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 0 0 0 0
1 0 0 0 0 0 0
2 0 0 0 0 0 0
3 11 0 6 0 0 0
4 166 67 114 0 55 0 5 78 638 577 469 524 0 6 0 1043 2288 865
7 1157 4960
8
Table 3.10. Number of the black, grey-black, grey-grey, grey-white and white octants of the constructed boy sculpture generated by the second construction method.
Protrusion=25 Protrusion=15 Protrusion=7 Xor=94149+2783 Xor=52185+2410 Xor=30781+2402
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0 2 0 0 38 13 13 0 0 44 7 13 0 0 47 4 13 3 7 32 96 91 78 7 12 148 71 114 7 6 192 34 137 4 49 309 0 347 63 121 263 238 366 196 160 100 630 218 428 5 94 864 0 884 62 747 1032 1046 1373 842 6 867 3677 0 3354 470 7
8
Table 3.11. Number of black, grey-black, grey-grey and white octants of the boy sculpture generated by the first construction method.
Protrusion=25 Protrusion=15 Protrusion=7
Xor=192434+8 Xor=115143+8 Xor=48366+296
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 8 0 0 0 8 0 0 0 8 0 0 2 0 0 51 13 0 0 51 13 0 0 51 0 13 3 7 32 200 169 7 12 220 169 7 6 226 0 169 4 49 414 444 693 121 263 683 693 160 100 855 0 693 5 0 1192 0 2360 94 1803 1132 2435 747 1032 2626 0 2435 6 0 1756 0 7300 867 5771 5210 0 9160 7 0 10985 0 0 30695 8
Table 3.12. Number of grey-black and grey-white leaf octants of the boy sculpture generated by applying the second method octant type evaluation criteria on grey-black octants in Table 3.9.
Protrusion=25 Protrusion=15 Protrusion=7
B GB GG GW W B GB GG GW W B GB GG GW W
0 0 0 0 0 0 0
1 0 0 0 0 0 0
2 0 0 0 0 0 0
3 32 0 12 0 6 0 4 311 103 263 0 100 0 5 106 1086 974 829 1032 0
6 9 1747 4110 1661
7 1861 9124
8
First method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the images
Second method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the images
Fig. 3.3. The comparison between construction results of the cone obtained by the conventional method and the new method.
First method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the image
Second method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the image
Fig. 3.4. The comparison between construction results of the vase obtained by the conventional method and the new method.
First method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the image
Second method Protrusion= 25 Protrusion= 15 Protrusion= 7 Image of the
constructed model Xor error image of the image
Fig. 3.5. The comparison between construction results of the boy sculpture obtained by the conventional method and the new method.
(a) (b)
Fig. 3.6. (a) One of the input image to the second method. (b) The new view generated from the constructed octree model.
(a) (b)
Fig. 3.7. (a) One of the input image to the second method. (b) The new view generated from the constructed octree model.