CHAPTER 2 FAST OCTREE CONSTRUCTION ENDOWED WITH AN ERROR
2.7 Conclusions and Future Work
In this chapter we propose a fast and efficient octree construction method to obtain the object model from the multiple silhouettes. The computation of the projection of 3D octants onto the 2D image planes is reduced by using the invariant property of cross ratio. A maximum projection error is specified to decide whether an octant subdivision is needed. The experiments are conducted on three real objects to demonstrate the performance of the new method. The results show the improvement of the new method over the conventional method in terms of memory space, computation time, and quality of the construction result. Theoretical analysis on the new method is presented.
In the future, we plan to convert the reconstructed result to a polygonal representation. Then, we can extract textures from the real object images and map them onto the model to obtain the photo-realistic effect. Moreover, further investigations of other effective use of octant subdivisions are underway. Preliminary results of using both grey-grey and grey-white nodes indicate additional reduction on the memory space as well as the computation time is possible. Also, the progressive octree construction process instead of the recursive construction process is also under consideration.
CHAPTER 3
AN OCTREE CONSTRUCTION METHOD WITH THREE TYPES OF GREY OCTANTS
3.1 Introduction
In the previous chapter a new construction method with a new grey octant type, the so-called “grey-black” octant together with an Exclusive-OR projection error control is presented. This first construction method is to be extended to a method with an additional new grey octant type will be described in this chapter. We shall call this method the second construction method. The computer simulation of this second method is given and the analysis on the experimental results in the form of lemmas is given.
3.2 Types of New Grey Octants
There are five octant types in the second method: Bl, Wl, GBl, GWl, and GGl, l = 0, 1, 2, …., L. As before, the bounding circle of the projected octant image has a radius and , the distance map value at the circle center, where v∈[1, N] is the view index and l∈[0, L] is the level index. Assume at each level l∈[0, L] the values of
are nearly equal for v∈[1, N] and the relation: = 2 roughly holds..
The definitions of these octants at a level l∈[0, L] are given below:
v
rl, dl,v
v
rl, rl,v rl+1,v
(1) For all views v∈[1, N] if dl,v> 0 and rl,v< dl,v, then the octant Ol is called a Bl octant.
(2) For at least one view v∈[1, N] if dl,v< 0 and rl,v< |dl,v|, then the octant Ol
is called a Wl octant.
(3) For all views v∈[1, N] if dl,v> 0 and rl,v- |dl,v| < p, then the octant Ol is called a GBl octant; p is the specified projection upper bound and - | | is called the white extent in the projected octant image. The representative view is one with
v
rl, dl,v
|}
d
| r { max ( ument arg
v l,v l,v
v
* = −
(4) For those views v∈[1, N] with dl,v< 0 if rl,v- |dl,v|< p, then the octant Ol is called a GWl octant; - | | is called the lack extent in the projected octant image. The representative view is one with
v
rl, dl,v
|}
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.
3.6 Analytical Analysis
Lemma 3-1:
For a given projection error bound p assume2rL+1,v <p<2rL,v,v∈
[
1,N]
, then the maximum level of subdivision of method 1, , is equal to or.
projection image whose bounding circle diameter is smaller than p. It implies the white extent is smaller than p, so it is a octant; no further octant subdivision is needed. Sometimes, when is closer to p than , if there is no octant at
The maximum subdivision level of method 2, , is no greater than that of method 1, i.e., . In general, .
Because the octants , in the second method are reclassified as:
(1) if both in the first construction method. Therefore, the octants will be further subdivided in the first construction method. Consequently, the first construction method has generally a greater number of subdivision levels, i.e., . Only when there are no octants which are reclassified as octants at the
level , then .
Lemma 3-3a:
Ifrl >P, then GBl1=GBl2. Ifrl < , then P GBl1⊇GBl2 wherer max{rl,v}.
l = v
Proof:
Consider the case where . It implies the second relation “ and ” does not hold, since the fact that
p
rl > dl,v <0
p d
rl,v − l,v < rl,v −dl,v =rl,v +dl,v < p is contradictory to the assumption that . So only the first relation holds.
Furthermore, the first relation also implies the black extent in the octant is equal to
Thus, . Next, consider the case where
1
However, the second relation “ 0
dl,v > rl,v −dl,v < p GB1l GB2l 0
dl,v < andrl,v −dl,v < p” is satisfied by a octant if it exists, but it is definitely not satisfied by a octant since .
Therefore, .
If , then an octant contains the black extent whose length is . Since is greater than p, so this octant will not be contained in a larger with . If black extent is smaller than p, so the octant may be contained in a larger with
, so this octant will not be produced. Thus, .
p
Lemma 3-3c:
Forl =0,1,2,...,L2max, Wl2⊆Wl1andGGl2 ⊆GG1l.
Proof:
By definition a white octant in produced in the second method is also a white octant in and a grey-grey octant is also a octant. Due to the existence of octants in the second method, a less number of octant subdivisions is performed in the second method, so the numbers of white, and grey-grey octants produced in the second method are less, compared to those produced in the first method.
2
Wl 1
Wl GGl2 GG1l
2
GWl
Lemma 3-4a:
Let the GB difference set be defined as , m = L, L+1. There are more octants reclassified as than those reclassified as by the second method for the set of octants belonging to .
2 m 1
m
m GB GB
DGB = −
2
GWm GB2m
1
GBl DGBm
Proof:
It can be shown that if the octants in , l = 0, 1, 2, …, , are further processed in the first method, will be reclassified as either if the white extent > p or if the white extent < p. In the latter case the octant is viewed a octant in the second method. Then, the final octree model is the same as that obtained by the first method.
2
GWl L2max
2
GWl GGl1
1
GBl GB1l
2
GWl
There are eight child octants in a subdivided parent octant. At level m = L and L+1, the projected octant is sufficiently small so that the object silhouette boundary can be approximated by a straight line. The distance map value at the parent octant circle center indicates the shortest distance between the circle center and the object silhouette boundary, so draw a line perpendicular to the object silhouette boundary, as shown in Fig. 3.8. Suppose the object interior is on the left handed side of the boundary whose position is indicated by xB.
Fig. 3.8. The depiction of centroids of the bounding circles for an octant and its parent octant and child octant. (Assume rL < p < rL-1 = 2rL)
Proof of the lemma for the level L:
Now suppose OL-1 is viewed as aGG1L−1octant in the first method and it will be subdivide into eight child octants. First, assume there exists a child octant OL whose circle center is located at x = rL. Later on, a child octant whose circle center is not located on the horizontal axis will be considered. Now suppose OL is aGB1Loctant in the first method. Both octants OL-1 and OL are offspring of a ancestor at a lower level. Then
2
GWl
(a) The parentGG1L−1octant is with a white extent > p.
(b) The childGB1Loctant is with a white extent whose value range is 0 < white extent < p.
(c) Both of the parent and child octants have a black extent less than p.
The above properties imply the following inequalities:
(1) The white extent = 2rL-1 – xB > p, so xB < 2rL-1 –p ( = 2rL + 2rL –p ).
(2) The white extent is given by 0 < 2rL – xB < p, so 2rL – p < xB < 2rL or rL – (p–
rL) < xB < 2rL.
(3) the black extent is given by 0 < xB < p –Δ.
Based on the above inequalities the allowed range of xB is given by rL – (p– rL) < xB < p –Δ.
The range of xB can be further divided into two sub-ranges:
(a) If rL – (p– rL) < xB < rL, then theGB1Lis also as aGWL2octant.
(b) If rL < xB < p –Δ, then theGB1Loctant is also aGB2Loctant.
The ratio of the probabilities of these two cases is given by
ant
Next, to prove the lemma for the case where the level is L+1:
Consider the child octant OL+1, which is classified as a by the first method, and its parent O
1 (c) Since both and are offspring of a ancestor with , so the
black extent in either octant is less than p.
1 1
GBL+ GG1L GWl2 l≤ L−1
The above properties imply the following inequalities:
(1) 2rL – xB > p or xB < 2rL –p ( = 2rL+1 – (p – 2rL+1) ), (2) 0 < 2rL+1 – xB < p or 2rL+1 –p < 0 < xB < 2rL+1, (3) 0 < xB < p – Δ (≈ p)
Combining these inequalities yields 0 < xB < 2rL –p, since 2rL –p < P or rL < p.
The range of xB, [0, 2rL –p], can be written as [0, rL+1]∪[rL+1, 2rL –p].
(1) If 0 < xB < rL+1, then GB1L+1 is a GWL2+1,
One more thing to consider is to remove the assumption that the child octant OL
whose circle center is located at x = rL+1; that is, the child octant considered has its position shifted to the right, see Fig.3.9, then the object silhouette boundary becomes closer to its circle center, so the probability of the octant being also a becomes even larger than that the octant being also a
1
GBL GWL2
1
GBL GBL2, so t the lemma still holds.
Fig. 3.9. The spatial relation depiction of bounding circles for an octant and its two sub-octants
rL+1x rL
Lemma 3-4b
Consider the level l = L or L+1. If
{ } { }
l,v v v,
v rl p max r
min2 < < 2 , then the
octants may produce child octants. However, the child octants are much less in number compared to the child octants produced.
1
GGl−
Bl Bl
GBl
Proof:
Due to the equipment set-up, there is inevitably a small variation in the projected octant radiusrl,v from view to view at each level l.
A parent octant contains a white extent > p for some view , . In this view , the octant is half or even more occupied by the white (i.e. empty) space. Its eight child octants are more likely to be white or grey-black than black. The chance that the child octant is a black octant is rare. This happens only when p is quite small or the range of { , v
−1
GGl Ol−1 v*
[
,Nv*∈1
]
v* Ol−1v ,
rl ∈[1, N]} is sufficiently large. Then due to the discrete nature of the space subdivision, the child octant of a half- or more-occupied octant is black.
Lemma 3-5:
The octree model constructed by the first method contains more terminal octants than the one constructed by the second method. This is also true for the non-terminal octants.
Proof:
The terminal octants of the two constructed octree models consist of the sets { , , l = 0, 1, 2, …, } and { , , l = 0, 1, 2, …, , respectively. The non-terminal octants are the sets of { , l = 0, 1, 2, …, -1}
and { , l = 0, 1, 2, …, -1}. The lemma is a consequence of Lemmas 3a and 3b.
1
Bl 1
W ,l GB1l L1max B ,2l W ,l2 GB2l GWl2 L2max}
1
GGl L1max
2
GGl L2max
A remark is in order here. All the terminal and non-terminal octants are counted in the computer processing time for the octree model construction. However, only the black octants and grey black octants are required to store for displaying the
constructed object model. In these two respects, the second construction method outperforms the first one.
Lemma 3-6:
The octree model constructed by the first method has no less Exclusive-OR projection error than the one constructed by the second method.
The octree model constructed by the first method has no less Exclusive-OR projection error than the one constructed by the second method.