Numerical Methods Contents - SAM

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Lemma 5.3.4 (Ncut and Rayleigh quotient). (→ [36, Sect. 2])
With z ∈ {−1, 1} N according to (5.3.9) there holds
∑
d i
Ncut(X) = yT Ay
y T Dy , y := (1 + z) − β(1 − z) , β := z i >0
∑ .
d i
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generalized Rayleigh quotient ρ A,D (y)
z i 0d ∑ d i = β
1+β
. Also observe
i
y T Dy = (1 + z) T D(1 + z) + β 2 (1 − z) T D(1 − z) =
4( ∑ d i + β 2∑ d i ) = 4β ∑
z i >0 z i N
1 x ∗ i } . (5.3.15)
i=1
mean value of entries of x ∗
Ôº¿ º¿
Ôº¿ º¿
3 xs = reshape (V ( : , 2 ) ,m, n ) ; Xidx = find ( xs >(sum(sum( xs ) ) / ( n∗m) ) ) ;
Code 5.3.13: 1st stage of segmentation of greyscale image
1 P = imread ( ’ image .pbm ’ ) ; [m, n ] = size (P) ; [ A,D] = imgsegmat (P) ;
2 [ V, E ] = eig ( f u l l (A+D) , f u l l (D) ) ; % grossly inefficient !