interpolationDemo - Orthographic (parallel) projection:...

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Unformatted text preview: 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 u P(0,0) P(4,1) 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 u P(0,0) P (4,1) = P (0, 0) + 1 + 4 u v P(4,1) 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 u P(0,0) P (4,1) = P (0, 0) + 1+ 4 u v P P u v (4,1)[0] (0, 0)[0] [0] [0] (4,1)[1] (0, 0)[1] 1 u[1] 4 v[1] P = P + + (4,1)[2] (0, 0)[2] [2] P P u v[2] P(4,1) Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n Q(6) P(4,1) n nn n v n u P(0,0) Q(6) = P (4,1) + 6 n Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n Q(6) P(4,1) n nn n v n u P(0,0) Q (6) = P (4,1) + 6 n Q P (6)[0] (4,1)[0] (6)[1] (4,1)[1] 6 Q = P + (6)[2] (4,1)[2] Q P n [0] [1] n [2] n Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n Q(6) P(4,1) n nn n v n u P(0,0) Q(6) = P (4,1) + 6 n Q P n 4.3 (6)[0] (4,1)[0] [0] (6)[1] (4,1)[1] 6 n[1] Q = P + = 2.6 (6)[2] (4,1)[2] [2] Q P n 22.7 (assume) Orthographic (parallel) projection: Interpolation P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n Q(6) n nn n v n u P(0,0) Q(6) = P (4,1) + 6 n Q P n 4.3 (6)[0] (4,1)[0] [0] (6)[1] (4,1)[1] 6 n[1] Q = P + = 2.6 (6)[2] (4,1)[2] [2] Q P n 22.7 Q(6) (assume) Orthographic (parallel) projection: Interpolation y x 5 3 22 z 4.3 2.6 22.7 4 2 23 32 30 43 Q(6) 20 24 13 8 10 5 2 23 Orthographic (parallel) projection: Nearest Neighbor Interpolation y Val (Q(6)) = Val ( NearestNeighbor (Q(6)) x z 4.3 2.6 22.7 4 2 23 5 3 22 32 30 43 Q(6) 20 24 13 8 10 5 2 23 Orthographic (parallel) projection: Nearest Neighbor Interpolation y Val (Q (6)) = Val ( NearestNeighbor (Q (6)) Round (4.3) NearestNeighbor (Q (6)) = Round (Q (6)) = Round (2.6) Round (22.7) x z 32 30 43 Q(6) 20 24 13 5 3 22 4.3 2.6 22.7 4 2 23 8 10 5 2 23 Orthographic (parallel) projection: Nearest Neighbor Interpolation y Val (Q (6)) = Val ( NearestNeighbor (Q(6)) 4 Round (4.3) NearestNeighbor (Q(6)) = Round (Q (6)) = Round (2.6) = 3 Round (22.7) 23 4 Val (Q (6)) = Val 3 23 x z 32 30 43 Q(6) 20 24 13 5 3 22 4.3 2.6 22.7 4 2 23 8 10 5 2 23 Orthographic (parallel) projection: Nearest Neighbor Interpolation y Val (Q (6)) = Val ( NearestNeighbor (Q(6)) 4 Round (4.3) NearestNeighbor (Q(6)) = Round (Q (6)) = Round (2.6) = 3 Round (22.7) 23 4 Val (Q (6)) = Val 30 3 = 23 x z 32 30 43 Q(6) 20 24 13 5 3 22 4.3 2.6 22.7 4 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = vv = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = vv = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = vv = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q (6)[1]) = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q (6)[1]) = 2.6 - 2 = ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q (6)[1]) = 2.6 - 2 = 0.6 ww = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q (6)[1]) = 2.6 - 2 = 0.6 ww = Q(6)[2] - trunc(Q(6)[2]) = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q (6)[1]) = 2.6 - 2 = 0.6 ww = Q(6)[2] - trunc(Q(6)[2]) = 22.7 - 22 = y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q(6)[1]) = 2.6 - 2 = 0.6 ww = Q(6)[2] - trunc(Q(6)[2]) = 22.7 - 22 = 0.7 y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: Trilinear Interpolation uu = Q(6)[0] - trunc(Q(6)[0]) = 4.3 - 4 = 0.3 vv = Q(6)[1] - trunc(Q(6)[1]) = 2.6 - 2 = 0.6 ww = Q(6)[2] - trunc(Q(6)[2]) = 22.7 - 22 = 0.7 x Q 4 0 (6)[0] 0 trunc (6)[1] y = Q = 2 (6)[2] 0 z Q 22 y x z 4.3 2.6 22.7 5 3 22 32 30 13 ww 43 Q(6) vv 4 uu 24 20 2 23 8 10 5 2 23 Orthographic (parallel) projection: 1st linear interpolation: along x y x z P (uu , y 0, z 0) = 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x z x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x z x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 10 43 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x z x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 P (uu , y 0 + 1, z 0) = z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P (uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P (uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P (uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 P (uu , y 0, z 0 + 1) = x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P (uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x 0 + 1 x0 P (uu , y 0, z 0 + 1) = uu Val ( y 0 + (1 - uu ) Val ( y 0 ) ) 0 + 1 0 + 1 z z = x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x y x x 0 + 1 0 P (uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P (uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x 0 + 1 x0 P (uu , y 0, z 0 + 1) = uu Val ( y 0 + (1 - uu ) Val ( y 0 ) ) 0 + 1 0 + 1 z z = 0.3 + 0.7 = 21.2 24 20 x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x x x 0 + 1 0 P(uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P(uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x 0 + 1 x0 P(uu , y 0, z 0 + 1) = uu Val ( y 0 + (1 - uu ) Val ( y 0 ) ) 0 + 1 0 + 1 z z = 0.3 + 0.7 = 21.2 24 20 P (uu, y 0 + 1, z 0 + 1) = y x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x x x 0 + 1 0 y 0 + (1 - uu ) ( 0 P(uu , y 0, z 0) = uu ( Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P(uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x 0 + 1 x0 P(uu , y 0, z 0 + 1) = uu Val ( y 0 + (1 - uu ) Val ( y 0 ) ) 0 + 1 0 + 1 z z = 0.3 + 0.7 = 21.2 24 20 x 0 + 1 x0 P (uu, y 0 + 1, z 0 + 1) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) 0 + 1 0 + 1 z z = y x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 1st linear interpolation: along x x x 0 + 1 0 P(uu , y 0, z 0) = uu ( y 0 + (1 - uu ) ( 0 Val ) Val y ) z0 0 z = 0.3 + 0.7 = 33.1 10 43 x 0 + 1 x0 P(uu , y 0 + 1, z 0) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) z0 z0 = 0.3 + 0.7 = 24.8 8 32 x 0 + 1 x0 P(uu , y 0, z 0 + 1) = uu Val ( y 0 + (1 - uu ) Val ( y 0 ) ) 0 + 1 0 + 1 z z = 0.3 + 0.7 = 21.2 24 20 x 0 + 1 x0 P (uu, y 0 + 1, z 0 + 1) = uu ( 0 + 1+ (1 - uu ) ( 0 + 1 Val y ) Val y ) 0 + 1 0 + 1 z z = 0.3 + 0.7 = 24.9 13 30 y x z 4.3 2.6 22.7 32 30 uu 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y y P (uu , vv, z 0) = x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y y P (uu, vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu , y 0, z 0) Val Val = x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y y P (uu, vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu , y 0, z 0) Val Val = 0.6 + 0.4 = 24.8 33.1 x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y y P (uu, vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu , y 0, z 0) Val Val = 0.6 + 0.4 = 28.12 24.8 33.1 x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y P(uu , vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu , y 0, z 0) Val Val = 0.6 + 0.4 = 28.12 24.8 33.1 P (uu, vv, z 0 + 1) = y x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y P(uu , vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu , y 0, z 0) Val Val = 0.6 + 0.4 = 28.12 24.8 33.1 P (uu, vv, z 0 + 1) = vv ( P(uu , y 0 + 1, z 0 + 1) + (1 - vv) (uu , y 0, z 0 + 1) Val Val = y x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 2nd linear interpolation: along y P(uu , vv, z 0) = vv ( P(uu , y 0 + 1, z 0) + (1 - vv) (uu, y 0, z 0) Val Val = 0.6 + 0.4 = 28.12 24.8 33.1 P (uu, vv, z 0 + 1) = vv ( P(uu , y 0 + 1, z 0 + 1) + (1 - vv) (uu, y 0, z 0 + 1) Val Val = 0.6 + 0.4 = 23.42 24.9 21.2 y x z 4.3 2.6 22.7 32 30 vv 20 43 Q(6) 24 13 8 10 Orthographic (parallel) projection: 3rd linear interpolation: along z y P (uu , vv, ww) = x z Val(Q(6)) 4.3 2.6 22.7 32 30 ww 43 Q(6) 20 24 13 8 10 Orthographic (parallel) projection: 3rd linear interpolation: along z y P (uu, vv, ww) = ww ( P (uu , vv, z 0 + 1) + (1 - ww) (uu , vv, z 0) Val Val = x z Val(Q(6)) 4.3 2.6 22.7 32 30 ww 43 Q(6) 20 24 13 8 10 Orthographic (parallel) projection: 3rd linear interpolation: along z y P (uu, vv, ww) = ww ( P (uu , vv, z 0 + 1) + (1 - ww) (uu , vv, z 0) Val Val = 0.7 23.42 + 0.3 28.12 = x z Val(Q(6)) 4.3 2.6 22.7 32 30 ww 43 Q(6) 20 24 13 8 10 Orthographic (parallel) projection: 3rd linear interpolation: along z y P (uu, vv, ww) = ww ( P (uu , vv, z 0 + 1) + (1 - ww) (uu , vv, z 0) Val Val = 0.7 23.42 + 0.3 28.12 = 24.83 x z Val(Q(6)) 4.3 2.6 22.7 32 30 ww 43 Q(6) 20 24 13 8 10 ...
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This note was uploaded on 04/29/2010 for the course CSE 4190.410 taught by Professor Shinyeonggil during the Spring '09 term at Seoul National.

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