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# Lecture-6 - Lecture-6 Szeliski Mann Picard Szeliski...

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1 Lecture-6 Szeliski, Mann & Picard Szeliski Projective

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2 Projective x x ) , ( t X f ) 1 , ( - ¢ t X f ¢ x = a 1 x + a 2 y + b 1 c 1 x + c 2 y + 1 ¢ y = a 3 x + a 4 y + b 2 c 1 x + c 2 y + 1 Szeliski E = [ f ( ¢ x , ¢ y ) - f ( x , y )] 2 = e 2 1 1 2 1 2 4 3 2 1 1 2 1 + + + + = ¢ + + + + = ¢ y c x c b y a x a y y c x c b y a x a x Projective min
3 Szeliski Motion Vector: [ ] T c c b b a a a a 2 1 2 1 4 3 2 1 = m Szeliski (Levenberg-Marquadet) a kl = e n m k n e n m l b k = - e e n m k b I A m 1 ) ( - + = D l Approximation of Hessian ( J T J, Jacobian) gradient

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4 1 1 1 a y y e a x x e a e ¢ ¢ + ¢ ¢ = 1 1 2 1 2 4 3 2 1 1 2 1 + + + + = ¢ + + + + = ¢ y c x c b y a x a y y c x c b y a x a x = - ¢ ¢ = 2 2 )] , ( ) , ( [ e y x f y x f E Approximation of Hessian A = J T J a kl = e n m k e n m l n J = e 1 m 1 e 1 m 2 e 1 m 3 e 1 m 4 e 1 m 5 e 1 m 6 e 1 m 7 e 1 m 8 e 2 m 1 e 2 m 2 e 2 m 3 e 2 m 4 e 2 m 5 e 2 m 6 e 2 m 7 e 2 m 8 M M M M M M M M e n m 1 e n m 2 e n m 3 e n m 4 e n m 5 e n m 6 e n m 7 e n m 8 J T = e 1 m 1 e 2 m 1 K e n m 1 e 1 m 2 e 2 m 2 K e n m 2 e 1 m 3 e 2 m 3 K e n m 3 e 1 m 4 e 2 m 4 K e n m 4 e 1 m 5 e 2 m 5 K e n m 5 e 1 m 6 e 2 m 6 K e n m 6 e 1 m 7 e 2 m 7 K e n m 7 e 1 m 8 e 2 m 8 K e n m 8
5 Gradient Vector b = - e n e n a 1 n - n e n e n a 2 n - e n e n a 3 n - e n e n a 4 n - e e n b 1 n - e n e n b 2 n - e n e n c 1 n - e n e n c 2 n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 4 3 2 2 2 1 2 4 3 1 2 1 2 1 2 1 4 2 1 3 2 1 2 2 1 1 2 1 2 2 2 1 1 2 1 1 2 2 1 1 4 3 2 1 2 2 1 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 1 + + + + - = ¢ + + + + - = ¢ + + = ¢ = ¢ + + = ¢ + + = ¢ = ¢ = ¢ + + + + - = ¢ + + + + - = ¢ = ¢ + + = ¢ = ¢ = ¢ + + = ¢ + + = ¢ y c x c b y a x a y c y y c x c b y a x a x c y y c x c b y b y y c x c y a y y c x c x a y a y a y y c x c b y a x a y c x y c x c b y a x a x c x b x y c x c b x a x a x y c x c y a x y c x c x a x 1 , 1 2 1 2 4 3 2 1 1 2 1 + + + + = ¢ + + + + = ¢ y c x c b y a x a

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Lecture-6 - Lecture-6 Szeliski Mann Picard Szeliski...

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