[...]... linear system as - L ~ L ~ f ~ ( ~ ' , y')h(~, y, ~', y') d~'ay' (1. 3-4 ) Now, a linear system is called space-invariant if the impulse response h(x, y, x', y') depends only on x - x', y - y', that is, A (~. 3-5 ) h ( ~ , u, ~', y') - h ( ~ - ~', y - y') Thus for linear space-invariant systems, the output (1. 3-4 ) can be rewritten as fo(~, y) - L ~ L ~ f ~ ( ~ ' , fo(X, y) y ' ) h ( ~ - ~', y - y') a ~ ' d... and rtl 0 i Yl -~ - n l X l / R , (2. 4-7 a) n20t = v2 + n 2 x 2 / R (2. 4-7 b) Using Eq (2. 4-6 ), (Eq 2. 4-7 ) and the fact that x z = x2, we obtain V2 - R hi n2 + Vl (2. 4-8 ) The matrix-vector equation relating the coordinates of the ray after refraction to those before refraction becomes (~2 )- , 2 _p 1 ~ ( 0)1 (~)~1 , (2. 4-9 a) where the quantity p given as p Tt2 721 \z-, -r R lu][Q.A-O]'~'I is termed... with period NA can be formed as [Antoniou (1979)]" OO fp(nA) - ~ f ( n A + r N A ) (1. 2-1 ) r=-oo The discrete function f(nA) may be formed by the discrete values of a continuous function f ( x ) evaluated at the points x - nL The discrete Fourier transform (DFT) of fp(nA) is defined as N-1 Fp(mK )- E fp(nA)exp(jmnKA) , K- - 2~ NA" (1. 2-2 ) n=0 The inverse DFT is defined as N-1 fp(nA) - ~ Fp(mK)exp( -. .. will find a path that extremizes t(z) with respect to variations in z We thus set d t ( z ) / d z = 0 to get a-~ _ [h ~-' F-(d-z)2] 1/2 - - z [h 2+z2] 1/2 (2. 2-2 ) or sin Oi = sinO~ (2. 2-3 a) 0~ - 0r (2. 2-3 b) so that We can readily check that the second derivative of t(z) is positive so that the result obtained corresponds to the least time principle Equation (2. 2-3 ) states that the angle of incidence... continuous (d~)~ - (dx)~ + (d~) ~, (2. 3-3 ) where we restrict ourselves to two dimensions Also, from Figure 2.6, dz/ds - (2. 3-4 ) s i n O & ~A Z Figure 2.6 The path of a ray in a medium with a continuous inhomogeniety 2.3 Refraction in an Inhomogeneous Medium 17 Combining (2. 3-2 ), (2. 3-3 ) and (2. 3-4 ), we obtain dx ~) 2 = (~) n2 T~21siTt20 1 - - (2. 3-5 a) 1 or alternatively, by differentiating with respect... O . Ting-Chung. Contemporary optical image processing with MATLAB / Ting-Chung Poort, Partha P. Banerjee 1st ed. p. cm. ISBN 0-0 8-0 4378 8-5 Cnardeover) 1. Image proe~sing. 2. MATLAB. I. Banerjee, . 7: Contemporary Topics in Optical Image Processing 7.1 Theory of Optical Heterodyne Scanning 7.1.1 Bipolar incoherent image processing 7.1.2 Optical scanning holography 7.2 Acousto-Optic Image. Lens Image Processing Systems 5.1 Impulse Response and Single Lens Imaging System 5.2 Two-Lens Image Processing System 5.3 Examples of Coherent Image Processing 5.4 Incoherent Image Processing