Lecture-2-h - Lecture-2 Imaging Geometry Transformations...

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Unformatted text preview: Lecture -2 Imaging Geometry Transformations • • • • • Translation Scaling Rotation Perspective Homogenous 1 Pose Estimation/Image Synthesis Motion Estimation 2 Motion Estimation Object Recognition 3 • Robotics • Image Registration IRS-1C - Washington, DC 4 SPOT - Washington, DC SPOT/IRS-1C Uncorrected SPOT IRS-1C Uncorrected 5 SPOT/IRS-1C Uncorrected 966 m SPOT IRS-1C Uncorrected SPOT/IRS-1C Uncorrected 966 m SPOT IRS-1C Uncorrected 6 IRS-1C/SPOT Registered IRS-1C SPOT 5m Registered Registered IRS-1C to SPOT IRS-1C SPOT Registered 7 Translation È X 2 ˘ È X 1 ˘ Èd x ˘ Í Y ˙ = Í Y ˙ + Íd ˙ Í 2 ˙ Í 1 ˙ Í y˙ Í Z 2 ˙ Í Z1 ˙ Í d z ˙ Î ˚Î˚Î˚ È X 2 ˘ È1 Í Y ˙ Í0 Í 2˙=Í Í Z 2 ˙ Í0 Í˙Í Î 1 ˚ Î0 ÈX2˘ È X1 ˘ ÍY ˙ ÍY ˙ 2˙ Í =TÍ 1 ˙ Í Z2 ˙ Í Z1 ˙ Í˙ Í˙ 1˚ Î Î1˚ È1 0 0 Í0 1 0 T =Í Í0 0 1 Í Î0 0 0 0 1 0 0 È1 Í0 T -1 = Í Í0 Í Î0 0 - dx ˘ 0 - dy ˙ ˙ 1 - dz ˙ ˙ 0 1˚ TT -1 = T -1T = I 0 d x ˘È X1 ˘ 0 d y ˙ Í Y1 ˙ ˙Í ˙ 1 d z ˙ Í Z1 ˙ ˙Í ˙ 0 1 ˚Î 1 ˚ dx ˘ dy ˙ ˙, Translation Matrix dz ˙ ˙ 1˚ È1 Í0 Í Í0 Í Î0 0 0 d x ˘ È1 1 0 d y ˙ Í0 ˙Í 0 1 d z ˙ Í0 ˙Í 0 0 1 ˚ Î0 0 1 0 0 0 0 - d x ˘ È1 1 0 - d y ˙ Í0 ˙=Í 0 1 - d z ˙ Í0 ˙Í 00 1 ˚ Î0 0 0 0˘ 1 0 0˙ ˙ 0 1 0˙ ˙ 0 0 1˚ Scaling 0 0 È1 / S x Í0 1/ S y 0 S-1 = Í Í0 0 1/ Sz Í 0 0 Î0 È X 2 ˘ È X1 ¥ Sx ˘ ÍY ˙ = ÍY ¥S ˙ Í 2˙ Í 1 y˙ Í Z 2 ˙ Í Z1 ¥ S z ˙ Î˚Î ˚ È X 2 ˘ ÈS x ÍY ˙ Í 0 Í 2˙=Í Í Z2 ˙ Í 0 Í˙Í Î 1 ˚ Î0 ÈX 2 ˘ È X1˘ Í˙ Í˙ Í Y2 ˙ = SÍ Y1 ˙ Í Z 2 ˙ Í Z1 ˙ Í˙ Í˙ Î 1 ˚ Î1˚ ÈSx 0 Í 0 Sy S =Í Í0 0 Í Î0 0 0 Sy 0 0 0 0 Sz 0 0˘ È X 1 ˘ 0˙ Í Y1 ˙ ˙Í ˙ 0 ˙ Í Z1 ˙ ˙Í ˙ 1˚ Î 1 ˚ SS -1 = S -1S = I ÈS x Í0 Í Í0 Í Î0 0 0 Sz 0 0˘ 0˙ ˙ 0˙ ˙ 1˚ 0 Sy 0 0 0 0 Sz 0 0˘ È È1 / S x 0 0 Í 0˙ Í Í 0 1/ S y 0 ˙Í 0˙ Í Í 0 0 1/ Sz ˙ ÍÍ 1˚ Í Î 0 0 0 Î 0˘ ˘ È1 ˙ 0 ˙ ˙ Í0 ˙ =Í 0 ˙ ˙ Í0 ˙˙ Í 1 ˚ ˙ Î0 ˚ 0 1 0 0 0 0 1 0 0˘ 0˙ ˙ 0˙ ˙ 1˚ 0˘ ˙ 0˙ , Scaling Matrix 0˙ ˙ 1˚ † 8 Rotation X = R cos f Y = R sin f Y ( X ¢, Y ¢, Z ¢) R X ¢ = R cos(Q + f ) = R cos Q cos f - R sin Q sin f Y ¢ = R sin(Q + f ) = R sin Q cos f + R cos Q sin f X ¢ = X cos Q - Y sin Q Y ¢ = X sin Q + Y cos Q Y’ R q ( X ,Y , Z ) Y f X X’ X Z È X ¢˘ Ècos Q - sin Q 0˘ È X ˘ ÍY ¢ ˙ = Í sin Q cos Q 0˙ ÍY ˙ Í˙Í ˙Í ˙ ÍZ ¢ ˙ Í 0 0 1˙ Í Z ˙ Î˚Î ˚Î ˚ Rotation (continued) Y È1 0 0˘ Í ˙ R = Í0 1 0˙ Í0 0 1˙ Î ˚ v W X u Z Ècos Q - sin Q 0˘ R = Í sin Q cos Q 0˙ Í ˙ Í0 0 1˙ Î ˚ Y Z q v’ Z Y R- b Ècos b 0 - sin b ˘ =Í 0 1 0˙ Í ˙ Í sin b 0 cos b ˙ Î ˚ u’ Q W W’ X Q Y v b X b Z u’ 9 È cos Q sin Q 0˘ ( RqZ ) -1 = Í- sin Q cos Q 0˙ Í ˙ Í0 0 1˙ Î ˚ È cos Q sin Q 0˘ Ècos Q - sin Q 0˘ È1 0 0˘ Í- sin Q cos Q 0˙ Í sin Q cos Q 0˙ = Í0 1 0˙ Í ˙Í ˙Í ˙ Í0 0 1˙ Í 0 0 1 ˙ Í0 0 1 ˙ Î ˚Î ˚Î ˚ Z -1 ZT ( Rq ) = ( Rq ) ( RqZ )( RqZ )T = I Rotation matrices are orthonormal matrices Ï1 if i = j ri .rj = Ì Ó0 otherwise Euler Angles Ècos a cos b cos a sin b sin g - sin a cos g cos a sin b cos g + sin a sin g ˘ a g R = RZ RYb RX = Í sin a cos b sin a sin b sin g + cos a cos g sin a sin b cos g - cos a sin g ˙ Í ˙ Í - sin b ˙ cos b sin g cos b cos g Î ˚ if angles are small È1 R=Í a Í Í- b Î -a 1 g cos Q ª 1 sin Q ª Q b˘ -g ˙ ˙ 1˙ ˚ 10 Perspective Projection (origin at the lens center) (X,Y,Z) World point Image Plane f y Lens 0 Z image -y f = Y Z fY y=Z x=- fX Z Perspective Projection (origin at image center) Image Plane f 0 y image Lens (X,Y,Z) World point Z -y f = Y Z- f fY fX y=x=Z- f Z- f fY fX y= x= f -Z f -Z 11 Perspective Image coordinates Èx˘ È Íy ˙ Í Í ˙=Í Í˙Í Í˙Í Î˚Î ˘È X ˙Í Y ˙Í ˙Í Z ˙Í ˚Î 1 ? ( X , Y , Z ) Æ ÆÆ ÈCh1 ˘ È1 ÍC ˙ Í0 Í h2 ˙ = Í ÍCh 3 ˙ Í0 Í ˙ Í0 ÎCh 4 ˚ Í Î ÈCh1 ˘ È1 ÍC ˙ Í0 Í h2 ˙ = Í ÍCh 3 ˙ Í0 Í ˙ Í0 ÎCh 4 ˚ Í Î 0 1 0 0 0 1 0 0 0 0 1 -1 f World coordinates (kX , kY , kZ , k ), Homogenous transformation CCC ( h1 , h 2 , h 3 ), Inverse homogenous Ch 4 Ch 4 Ch 4 (C h1 , Ch 2 , Ch 3 , Ch 4 ) Æ È1 Í0 Í P = Í0 Í Í0 Î ˘ ˙ ˙ ˙ ˙ ˚ 0 1 0 0 0 1 -1 f 0 0˘ 0˙ ˙ 0˙ ˙ 1˙ ˚ Perspective 0 0 1 -1 f 0˘ È kX ˘ 0˙ Í kY ˙ ˙Í ˙ 0˙ ÍkZ ˙ ˙ 1˙ Í k ˙ ˚Î ˚ 0˘ È kX ˘ È kX ˘ Í ˙ 0˙ Í kY ˙ Í kY ˙ ˙Í ˙ 0˙ ÍkZ ˙ = Í kZ ˙ Í ˙ ˙ 1˙ Í k ˙ Ík - kZ ˙ f˙ ˚Î ˚ Í Î ˚ Ch1 kX fX = = Ch 4 k - kZ f -Z f C kY fY y = h2 = = Ch 4 k - kZ f -Z f x= 12 ...
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