Numerical Methods Contents - SAM

1 2 3
k ρ (k)
EV ρ (k)
EW ρ (k)
EV ρ (k)
EW ρ (k)
EV ρ (k)
EW
∥ ∥ ∥∥z (k) ∥∥
22 0.9102 0.9007 0.5000 0.5000 0.9900 0.9781
ρ (k) − s·,n
EV := 23 0.9092 0.9004 0.5000 0.5000 0.9900 0.9791
∥
∥
∥z (k−1) ∥∥
,
24 0.9083 0.9001 0.5000 0.5000 0.9901 0.9800
− s·,n
25 0.9075 0.9000 0.5000 0.5000 0.9901 0.9809
ρ (k)
EW := |ρ A(z (k) ) − λ n |
|ρ A (z (k−1) ) − λ n | . 26 0.9068 0.8998 0.5000 0.5000 0.9901 0.9817
27 0.9061 0.8997 0.5000 0.5000 0.9901 0.9825
28 0.9055 0.8997 0.5000 0.5000 0.9901 0.9832
29 0.9049 0.8996 0.5000 0.5000 0.9901 0.9839
30 0.9045 0.8996 0.5000 0.5000 0.9901 0.9844
Observation: linear convergence (→ Def. 3.1.4)
“relative change” ≤ tol:
5.3.2 Inverse Iteration
⎧
∥
∥z
⎪⎨
(k) − z (k−1)∥ ∥ ≤ (1/L − 1)tol ,
∥
∥Az (k)∥ ∥ ∥
∥Az (k−1)∥ ∣
∥ ∣∣∣∣
⎪⎩
∥
∣ ∥
∥z (k) ∥∥ −
∥
∥
∥z (k−1) ∥∥ ≤ (1/L − 1)tol see (3.1.7) .
Estimated rate of convergence
△
✸
Example 5.3.9 (Image segmentation).
Ôº¾ º¿
Given: grey-scale image: intensity matrix P ∈ {0,...,255} m,n , m, n ∈ N
((P) ij ↔ pixel, 0 ˆ= black, 255 ˆ= white)
Ôº¾ º¿
✬
Theorem 5.3.2 (Convergence of direct power method).
Let λ n > 0 be the largest (in modulus) eigenvalue of A ∈ K n,n and have (algebraic) multiplicity
1. Let v,y be the left and right eigenvectors of A for λ n normalized according to ‖y‖ 2 =
‖v‖ 2 = 1. Then there is convergence
∥
∥Az (k)∥ ∥ ∥2 → λ n , z (k) → ±v linearly with rate |λ n−1|
,
|λ n |
✫
where z (k) are the iterates of the direct power iteration and y H z (0) ≠ 0 is assumed.
Remark 5.3.7 (Initial guess for power iteration).
roundoff errors ➤ y H z (0) ≠ 0 always satisfied in practical computations
Usual (not the best!) choice for x (0) = random vector
Remark 5.3.8 (Termination criterion for direct power iteration). (→ Sect. 3.1.2)
Adaptation of a posteriori termination criterion (3.2.7)
✩
✪
△
Ôº¾ º¿
Code 5.3.10: loading and displaying an image
1 M = imread ( ’ eth .pbm ’ ) ;
Loading and displaying images
in MATLAB ✄ 3 f p r i n t f ( ’%dx%d grey scale p i x e l image \ n ’ ,m, n ) ;
2 [m, n ] = size (M) ;
4 figure ; image (M) ; t i t l e ( ’ETH view ’ ) ;
5 c o l = [ 0 : 1 / 2 1 5 : 1 ] ’ ∗ [ 1 , 1 , 1 ] ; colormap ( c o l ) ;
(Fuzzy) task:
Local segmentation
Find connected patches of image of the same shade/color
More general segmentation problem (non-local): identify parts of the image, not necessarily connected,
with the same texture.
Next: Statement of (rigorously defined) problem, cf. Sect. 2.5.2:
Ôº¾ º¿
✎ notation: p k := (P) ij , if k = index(pixel i,j ) = (i − 1)n + j, k = 1,...,N := mn
Preparation: Numbering of pixels 1 . ..,mn, e.g, lexicographic numbering:
pixel set V := {1....,nm}
indexing: index(pixel i,j ) = (i − 1)n + j