VIEWS: 50 PAGES: 5 POSTED ON: 3/28/2010 Public Domain
The collinearity equations: r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o ) x = xp −c r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o ) r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o ) y = yp −c r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o ) Let’s use the following abbreviations: N x = r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o ) N y = r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o ) D = r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o ) Partial derivatives with respect to interior orientation parameters; (xp, yp, c): The x-coordinate equation ∂x ∂x ∂x r ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o ) N =1 =0 = − 11 =− x ∂x p ∂y p ∂c r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o ) D The y-coordinate equation ∂y ∂y ∂y r ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o ) Ny =0 =1 = − 12 =− ∂x p ∂y p ∂c r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o ) D Partial derivatives with respect to exterior orientation parameters; ground coordinates (Xo, Yo, Zo): The x-coordinate equation ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] = c 11 13 ∂X o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] = c 21 13 ∂Yo [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] = c 31 13 ∂Z o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 The y-coordinate equation ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] = c 12 13 ∂X o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] = c 22 13 ∂Yo [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] = c 32 13 ∂Z o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 In abbreviated form ∂x r D − r13 N x ∂x r D − r23 N x ∂x r D − r33 N x = c 11 = c 21 = c 31 ∂X o D2 ∂Yo D2 ∂Z o D2 ∂y r12 D − r13 N y ∂y r22 D − r23 N y ∂y r32 D − r33 N y =c =c =c ∂X o D2 ∂Yo D2 ∂Z o D2 Partial derivatives with respect to exterior orientation parameters; attitude of image plane (ω, φ, κ): The rotation matrix R is: r11 = cos φ cos κ r12 = -cos φ sin κ r13 = sin φ r21 = cos ω sin κ + sin ω sin φ cos κ r22 = cos ω cos κ - sin ω sin φ sin κ r23 = -sin ω cos φ r31 = sin ω sin κ - cos ω sin φ cos κ r32 = sin ω cos κ + cos ω sin φ sin κ r33 = cos ω cos φ ∂ (Element ) ∂ (Element ) ∂ (Element ) ∂ω ∂φ ∂κ ∂r11 ∂r11 ∂r11 =0 = − sin φ cos κ = − cos φ sin κ = r12 ∂ω ∂φ ∂κ ∂r21 ∂r21 ∂r21 = − sin ω sin κ + cos ω sin φ cos κ = − r31 = sin ω cos φ cos κ = cos ω cos κ − sin ω sin φ sin κ = r22 ∂ω ∂φ ∂κ ∂r31 ∂r31 ∂r31 = cos ω sin κ + sin ω sin φ cos κ = r21 = − cos ω cos φ cos κ = sin ω cos κ + cos ω sin φ sin κ = r32 ∂ω ∂φ ∂κ ∂r12 ∂r12 ∂r12 =0 = sin φ sin κ = − cos φ cos κ = −r11 ∂ω ∂φ ∂κ ∂r22 ∂r22 ∂r22 = − sin ω cos κ − cos ω sin φ sin κ = −r32 = − sin ω cos φ sin κ = − cos ω sin κ − sin ω sin φ cos κ = − r21 ∂ω ∂φ ∂κ ∂r32 ∂r32 ∂r32 = cos ω cos κ − sin ω sin φ sin κ = r22 = cos ω cos φ sin κ = − sin ω sin κ + cos ω sin φ cos κ = − r31 ∂ω ∂φ ∂κ ∂r13 ∂r13 ∂r13 =0 = cos φ =0 ∂ω ∂φ ∂κ ∂r23 ∂r23 ∂r23 = − cos ω cos φ = −r33 = sin ω sin φ =0 ∂ω ∂φ ∂κ ∂r33 ∂r33 ∂r33 = − sin ω cos φ = r23 = − cos ω sin φ =0 ∂ω ∂φ ∂κ The x-coordinate equation ∂x [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )][0 − r31 ⋅ (Y − Yo ) + r21 ⋅ (Z − Z o )] + [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ (Z − Z o )][0 + r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − = −c 13 ∂ω [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂x D[− r31 ⋅ (Y − Yo ) + r21 ⋅ ( Z − Z o )] + N x [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )] = −c ∂ω D2 ∂x [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][− sin φ cosκ ⋅ ( X − X o ) + sin ω cosφ cosκ ⋅ (Y − Yo ) − cosω cosφ cosκ ⋅ (Z − Z o )] = −c 13 ∂φ [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 −c [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )][cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )] [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂x D[− sin φ cosκ ⋅ ( X − X o ) + sin ω cosφ cosκ ⋅ (Y − Yo ) − cosω cosφ cosκ ⋅ ( Z − Z o )] + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Zo )] = −c ∂φ D2 ∂x − D cosκ [− sin φ ⋅ ( X − X o ) + sin ω cosφ ⋅ (Y − Yo ) − cosω cosφ ⋅ ( Z − Zo )] + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )] = −c ∂φ D2 ∂x − D 2 cosκ + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )] = −c ∂φ D2 ∂x = −c [ − D 2 cosκ + N x − N x ⋅ cosκ + N y sin κ ] ∂φ D2 ∂x [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] = −c 13 ∂κ [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂x Ny = −c ∂κ D In Summary: ∂x = −c D[− r31 ⋅ (Y − Yo ) + r21 ⋅ ( Z − Z o )] + N x [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )] ∂x = −c [ − D 2 cosκ + N x − N x ⋅ cosκ + N y sin κ ] ∂x = −c Ny ∂ω D2 ∂φ D2 ∂κ D The y-coordinate equation ∂y [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][0 − r32 ⋅ (Y − Yo ) + r22 ⋅ (Z − Z o )] + [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ (Z − Z o )][0 + r33 ⋅ (Y − Yo ) − r23 ⋅ = −c 13 ∂ω [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y D[− r32 ⋅ (Y − Yo ) + r22 ⋅ ( Z − Z o )] + N y [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )] = −c ∂ω D2 ∂y [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Zo )][sin φ sin κ ⋅ ( X − X o ) − sin ω cosφ sin κ ⋅ (Y − Yo ) + cosω cosφ sin κ ⋅ (Z − Zo )] = −c 13 ∂φ [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Zo )]2 −c [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )][cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )] [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂y D[sin φ sin κ ⋅ ( X − X o ) − sin ω cosφ sin κ ⋅ (Y − Yo ) + cosω cosφ sin κ ⋅ ( Z − Z o )] + N y [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )] = −c ∂φ D2 ∂y D sin κ [sin φ ⋅ ( X − X o ) − sin ω cos φ ⋅ (Y − Yo ) + cos ω cos φ ⋅ ( Z − Z o )] + N y [cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )] = −c ∂φ D2 ∂y D 2 sin κ + N y [cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )] = −c ∂φ D2 ∂y = −c [ D 2 sin κ + N y N x cos κ + N y sin κ ] ∂φ D2 ∂y [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )][−r11 ⋅ ( X − X o ) − r21 ⋅ (Y − Yo ) − r31 ⋅ ( Z − Z o )] = −c 13 ∂κ [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y N =c x ∂κ D In Summary: ∂y D[− r32 ⋅ (Y − Yo ) + r22 ⋅ ( Z − Z o )] + N y [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )] ∂y D 2 sin κ + N y [N x cos κ + N y sin κ ] ∂y N = −c = −c =c x ∂ω D2 ∂φ D2 ∂κ D Partial derivatives with respect to ground coordinates of tie points(X, Y, Z) The x-coordinate equation ∂x ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] =− = −c 11 13 ∂X ∂X o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂x ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] =− = −c 21 13 ∂Y ∂Yo [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 ∂x ∂x r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )] =− = −c 31 13 ∂Z ∂Z o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2 The y-coordinate equation ∂y ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] =− = −c 12 13 ∂X ∂X o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] =− = −c 22 13 ∂Y ∂Yo [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 ∂y ∂y r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )] =− = −c 32 13 ∂Z ∂Z o [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2 In abbreviated form ∂x r D − r13 N x ∂x r D − r23 N x ∂x r D − r33 N x = −c 11 = −c 21 = −c 31 ∂X D2 ∂Y D2 ∂Z D2 ∂y r12 D − r13 N y ∂y r22 D − r23 N y ∂y r32 D − r33 N y = −c = −c = −c ∂X D2 ∂Y D2 ∂Z D2