P(0,0)
47
n u v P(0,0) P(4,1) graphic (parallel) projection: ng the starting point of the ray 0,0): origin of image plane 4,1): image pixel at i=4, j=1
description
Orthographic (parallel) projection: Finding the starting point of the ray. P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1. v. n. P(4,1). P(0,0). u. Orthographic (parallel) projection: Finding the starting point of the ray. P(0,0): origin of image plane - PowerPoint PPT Presentation
Transcript of P(0,0)
n
u
v
P(0,0)P(4,1)
Orthographic (parallel) projection:Finding the starting point of the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1
n
u
v
P(0,0)
(4,1) (0,0) 1 4P P u v
P(4,1)
Orthographic (parallel) projection:Finding the starting point of the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1
n
u
v
P(0,0)
(4,1) (0,0) 1 4
(4,1)[0] (0,0)[0] [0] [0]
(4,1)[1] (0,0)[1] 1 [1] 4 [1]
(4,1)[2] (0,0)[2] [2] [2]
P P u v
P P u v
P P u v
P P u v
P(4,1)
Orthographic (parallel) projection:Finding the starting point of the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1
P(0,0)n
u
v
(6) (4,1) 6Q P n
P(4,1)
Orthographic (parallel) projection:Casting the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1Q(6): ray sample point at k=6
Q(6)nnn
n n n
P(0,0)n
u
v
(6) (4,1) 6
(6)[0] (4,1)[0] [0]
(6)[1] (4,1)[1] 6 [1]
(6)[2] (4,1)[2] [2]
Q P n
Q P n
Q P n
Q P n
P(4,1)
Orthographic (parallel) projection:Casting the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1Q(6): ray sample point at k=6
Q(6)nnn
n n n
P(0,0)n
u
v
(6) (4,1) 6
(6)[0] (4,1)[0] [0] 4.3
(6)[1] (4,1)[1] 6 [1] 2.6
(6)[2] (4,1)[2] [2] 22.7
Q P n
Q P n
Q P n
Q P n
P(4,1)
Orthographic (parallel) projection:Casting the ray
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1Q(6): ray sample point at k=6
Q(6)nnn
n n n
(assume)
P(0,0)n
u
v
Orthographic (parallel) projection:Interpolation
P(0,0): origin of image planeP(4,1): image pixel at i=4, j=1Q(6): ray sample point at k=6
Q(6)nnn
n n n
(6) (4,1) 6
(6)[0] (4,1)[0] [0] 4.3
(6)[1] (4,1)[1] 6 [1] 2.6
(6)[2] (4,1)[2] [2] 22.7
Q P n
Q P n
Q P n
Q P n
Q(6)
(assume)
Orthographic (parallel) projection:Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
Orthographic (parallel) projection:Nearest Neighbor Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
( (6)) ( ( (6))Val Q Val NearestNeighbor Q
Orthographic (parallel) projection:Nearest Neighbor Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
( (6)) ( ( (6))
(4.3)
( (6)) ( (6)) (2.6)
(22.7)
Val Q Val NearestNeighbor Q
Round
NearestNeighbor Q Round Q Round
Round
Orthographic (parallel) projection:Nearest Neighbor Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
( (6)) ( ( (6))
(4.3) 4
( (6)) ( (6)) (2.6) 3
(22.7) 23
4
( (6)) 3
23
Val Q Val NearestNeighbor Q
Round
NearestNeighbor Q Round Q Round
Round
Val Q Val
Orthographic (parallel) projection:Nearest Neighbor Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
( (6)) ( ( (6))
(4.3) 4
( (6)) ( (6)) (2.6) 3
(22.7) 23
4
( (6)) 3 30
23
Val Q Val NearestNeighbor Q
Round
NearestNeighbor Q Round Q Round
Round
Val Q Val
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
uu
vv
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0])uu Q trunc Q
vv
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4uu Q trunc Q
vv
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3uu Q trunc Q
vv
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1])
uu Q trunc Q
vv Q trunc Q
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2
uu Q trunc Q
vv Q trunc Q
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2 0.6
uu Q trunc Q
vv Q trunc Q
ww
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2 0.6
(6)[2] ( (6)[2])
uu Q trunc Q
vv Q trunc Q
ww Q trunc Q
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2 0.6
(6)[2] ( (6)[2]) 22.7 22
uu Q trunc Q
vv Q trunc Q
ww Q trunc Q
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2 0.6
(6)[2] ( (6)[2]) 22.7 22 0.7
uu Q trunc Q
vv Q trunc Q
ww Q trunc Q
uuvv
ww
Orthographic (parallel) projection:Trilinear Interpolation
Q(6)
x
y
z
4
2
23
5
2
23
5
3
22
4.3
2.6
22.7
20 24
30
10
32
8
4313
(6)[0] ( (6)[0]) 4.3 4 0.3
(6)[1] ( (6)[1]) 2.6 2 0.6
(6)[2] ( (6)[2]) 22.7 22 0.7
uu Q trunc Q
vv Q trunc Q
ww Q trunc Q
uuvv
ww
0 (6)[0] 4
0 (6)[1] 2
0 (6)[2] 22
x Q
y trunc Q
z Q
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
( , 0, 0)P uu y z 4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
( , 0 1, 0)
x x
P uu y z uu Val y uu Val y
z z
P uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
( , 0, 0 1)
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
P uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
0
( , 0, 0 1) (
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
x
P uu y z uu Val
1 0
0 ) (1 ) ( 0 )
0 1 0 1
x
y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
0
( , 0, 0 1) (
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
x
P uu y z uu Val
1 0
0 ) (1 ) ( 0 )
0 1 0 1
0.3 24 0.7 20 21.2
x
y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
0
( , 0, 0 1) (
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
x
P uu y z uu Val
1 0
0 ) (1 ) ( 0 )
0 1 0 1
0.3 24 0.7 20 21.2
( , 0 1, 0 1)
x
y uu Val y
z z
P uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
0
( , 0, 0 1) (
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
x
P uu y z uu Val
1 0
0 ) (1 ) ( 0 )
0 1 0 1
0.3 24 0.7 20 21.2
0 1 0
( , 0 1, 0 1) ( 0 1 ) (1 ) ( 0 1 )
0 1 0 1
x
y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:1st linear interpolation: along x
Q(6)
x
y
z
20 24
30
10
32
8
4313
uu
0 1 0
( , 0, 0) ( 0 ) (1 ) ( 0 )
0 0
0.3 10 0.7 43 33.1
0 1 0
( , 0 1, 0) ( 0 1 ) (1 ) ( 0 1 )
0 0
0.3 8 0.7 32 24.8
0
( , 0, 0 1) (
x x
P uu y z uu Val y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
x
P uu y z uu Val
1 0
0 ) (1 ) ( 0 )
0 1 0 1
0.3 24 0.7 20 21.2
0 1 0
( , 0 1, 0 1) ( 0 1 ) (1 ) ( 0 1 )
0 1 0 1
0.3 13 0.7 30 24.9
x
y uu Val y
z z
x x
P uu y z uu Val y uu Val y
z z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0)P uu vv z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)P uu vv z vv Val P uu y z vv Val uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)
0.6 24.8 0.4 33.1
P uu vv z vv Val P uu y z vv Val uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)
0.6 24.8 0.4 33.1 28.12
P uu vv z vv Val P uu y z vv Val uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)
0.6 24.8 0.4 33.1 28.12
( , , 0 1)
P uu vv z vv Val P uu y z vv Val uu y z
P uu vv z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)
0.6 24.8 0.4 33.1 28.12
( , , 0 1) ( ( , 0 1, 0 1) (1 ) ( , 0, 0 1)
P uu vv z vv Val P uu y z vv Val uu y z
P uu vv z vv Val P uu y z vv Val uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:2nd linear interpolation: along y
Q(6)
x
y
z
20 24
30
10
32
8
4313
vv
( , , 0) ( ( , 0 1, 0) (1 ) ( , 0, 0)
0.6 24.8 0.4 33.1 28.12
( , , 0 1) ( ( , 0 1, 0 1) (1 ) ( , 0, 0 1)
0.6 24.9 0.4 21.2 23.42
P uu vv z vv Val P uu y z vv Val uu y z
P uu vv z vv Val P uu y z vv Val uu y z
4.3
2.6
22.7
Orthographic (parallel) projection:3rd linear interpolation: along z
Q(6)
x
y
z
4.3
2.6
22.7
20 24
30
10
32
8
4313ww
( , , )P uu vv ww
Val(Q(6))
Orthographic (parallel) projection:3rd linear interpolation: along z
Q(6)
x
y
z
4.3
2.6
22.7
20 24
30
10
32
8
4313ww
( , , ) ( ( , , 0 1) (1 ) ( , , 0)P uu vv ww ww Val P uu vv z ww Val uu vv z
Val(Q(6))
Orthographic (parallel) projection:3rd linear interpolation: along z
Q(6)
x
y
z
4.3
2.6
22.7
20 24
30
10
32
8
4313ww
( , , ) ( ( , , 0 1) (1 ) ( , , 0)
0.7 23.42 0.3 28.12
P uu vv ww ww Val P uu vv z ww Val uu vv z
Val(Q(6))
Orthographic (parallel) projection:3rd linear interpolation: along z
Q(6)
x
y
z
4.3
2.6
22.7
20 24
30
10
32
8
4313ww
( , , ) ( ( , , 0 1) (1 ) ( , , 0)
0.7 23.42 0.3 28.12 24.83
P uu vv ww ww Val P uu vv z ww Val uu vv z
Val(Q(6))
