2. Direct imaging for point targets From Born approximation the reflected signal at the j-th transducer is M N f()=邋G(x,x,)G(x,x() k=1i=1 where G(x,x): the Greens function the reflectivity of the k-th scatters k located at x he location of the i-th transducers (過两科学大季 A Direct Numerical Imaging Method 6 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets ❖ From Born approximation, the reflected signal at the j-th transducer is 6 0 G x ( , ) : x 0 0 1 1 ˆ ( ) ( , ) ( , ) ( ) M N j j k k i k i k i r G G e w t w = = = 邋 x x where the Greens function : k t the reflectivity of the k-th scatters located at xk . : i x the location of the i-th transducers
2. Direct imaging for point targets We have M P(w)=a gig k=1 s=(xx)G"(x…x)LG"(x Ey Rank(P(w))=M (過两科学大季 A Direct Numerical Imaging Method 7 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets We have 7 0 0 1 0 0 0 0 1 2 ( ) , ( , ), ( , ), , ( , ) M T k k k k T k k k N k P g g g G G G w t = = = 轾犏臌 å v v v x x x L R ank P M ( ( )) . w =
2. Direct imaging for point targets Define the time reversal matrix R()@Po)p(p)a(点 k,k≠=1 a k, k) The point spread function kke In the well-resolved case Right singular vector Singular value (過学故时大学 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method 2 2 0 0 0 0 0 0 ( ) , ( ) . T T k k k k k k k k P g g g P g g g w t w t = = v v v v v v 2. Direct imaging for point targets 8 Define the time reversal matrix ( ) 0 0 0 0 , 1 , , 1 ( ) ( ) ( )= ( , ) M T T T k k k k k k k k M k k k k k k R P P g g g g w w w t t ⅱ ⅱ ? ¢= ⅱ ¢= × L G å å v v v v @ @ x x In the well-resolved case The point spread function Right singular vector Singular value
2. Direct imaging for point targets : the signal space SVD vs=spanu,u, L, u,i dVN. the noise space The imaging function (x l p. cf al-a kox)? i P k=1 (過两科学大季 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets 9 P( ) w The imaging function V S : the signal space SVD V N: the noise space 2 2 2 2 1 1 1 ( ) ( ) ( ) S ( ) ( ) N V k k I x g P g g g u = = = - - ? å v v v v v x x x x 1 2 { , , , } S N V span = v v v u u u L
2. Direct imaging for point targets An improved imaging function X n?1 x ax Where I(x)=22 g(x u k=1 (過两科学大季 A Direct Numerical Imaging Method 10 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets 10 An improved imaging function max ( ) ( ) , 1. ( ) n n I x I x n I x 轾 = ? 犏犏犏臌 where 2 1 ( ) ( ) . M T k k I x g x u = = å