Rotation-invariant connected size-shape pattern spectra

Transcription

1 Rotation-invariant connected size-shape pattern spectra Erik R. Urbach Institute for Mathematics and Computing Science University of Groningen The Netherlands April 28, 2004 BCN Retraite

2 Research topic Connected Morphological operators for Scale and Shape spaces Goal: investigate multi-scale and multi-shape techniques for image processing and texture analysis (pattern recognition). Future: These methods should then be used for improving 3D medical images and texture feature extraction. Now: Using multi-scale/multi-shape techniques for 2D grayscale images consisting of square pixels. BCN Retraite 1

3 Size pattern spectrum BCN Retraite 2

4 Shape pattern spectrum BCN Retraite 3

5 Application Automatic identification of diatoms BCN Retraite 4

6 Results Automatic identification of diatoms Performance on a set of 781 diatom images (based on the ornamentation) consisting of 37 taxa using the C4.5 decision tree classifier. S.e. u.v. S.e. u.v. S.e. b.v. Max-tree Max-tree Moments Moments Unfiltered 78.9% 86.5% 91.9% 71.4% 79.9% Blur 3x3 79.9% 87.3% 91.1% 70.1% 82.9% Median 3x3 81.4% 87.0% 91.2% 73.1% 79.8% Blur 5x5 82.9% 87.1% 90.1% 68.6% 83.5% Median 5x5 82.1% 87.9% 90.7% 71.3% 83.0% BCN Retraite 5

7 Results Performance on rotated images Performance (%) s.e. univariate s.e. bivariate Max tree Performance (%) Rotation angle (degrees) Training set: original 20 s.e. univariate s.e. bivariate Max tree Rotation angle (degrees) Training set: multi-angle Multi-angle training set: images rotated at 0, 1, 5, 15, 30, 45, and 90 degrees. BCN Retraite 6

8 Noise Original σ = 0.64 BCN Retraite 7

9 Results Performance on noisy images s.e. univariate s.e. bivariate Max tree s.e. univariate s.e. bivariate Max tree Performance (%) Performance (%) Std. deviation σ of noise Std. deviation σ of noise Training set: original Training set: noise BCN Retraite 8

10 Conclusions Our method is efficient way for image processing, analysis, and classification based on the size and/or the shape of the components in the image. Future: New attributes, e.g. the orientation of the components Extending for handling images with gaps and bridges Biologically motivated texture classification BCN Retraite 9

11 Max-tree Computation of pattern spectrum using Max-Tree (Subtractive): P 0 9 P 0 6 P 1 6 P 0 3 P Peak components Elongation Area BCN Retraite 10

12 Comparison of pattern spectra S.e. univariate S.e. bivariate Max-tree S.e/attributes hor. linear hor. linear area ver. linear ver. linear elongation I/A 2 45-deg. linear -45-deg. linear square cone Num.openings 6 51 = = = 16 (*) λ , 15, 51, 199 area: pixels I/A 2 : 1/2π..52 Size pattern spectrum 6 3 = = = 16 Normalized yes yes no Derivate yes yes yes Note: Construction of a pattern spectrum using Max-tree is done in a single pass, i.e. it is independent of the number of openings. BCN Retraite 11

13 Timings Timings (seconds) on a 621x501 diatom image Original: = pixels Shape: pixels S.e. u.v. S.e. u.v. S.e. b.v. Max-tree Moments Down-size Down-size Down-size Original Original (shape) Noise σ = Noise σ = 0.64 (shape) BCN Retraite 12

14 Conclusions Advantages: Our method using the Max-tree is rotation-invariant. Our Max-tree method is independent of the number of size and shape classes used. In practice, our Max-tree method is the fastest. Disadvantages: S.e. b.v. method performs the best on the original image set. When only a few openings/closings are needed, the s.e. methods are faster. Computation time of the Max-tree method is not independent of image content. Conclusion: when rotation-invariance is required or when larger pattern spectra are needed, our Max-tree method is the best choice, otherwise s.e. methods are preferable. BCN Retraite 13

15 Max-tree The Max-tree (Salembier) - a (rooted) tree representation: P 0 2 P 1 2 P 0 1 P 1 1 P 2 1 P 0 0 C 0 2 C 0 1 C 1 1 C2 1 C1 2 BCN Retraite 14 C 0 0

16 Attribute opening and thinning 1 Attribute opening and thinning (Breen and Jones): Definition 1 The connected opening Γ x (X) of a set X M at a point x M is the connected component of X that contains x if x X and the empty set otherwise. Definition 2 Let C M be a connected set and T an increasing criterion, then the trivial opening Γ T of C is defined by: Γ T ( ) = { C if C satisfies criterion T, Γ T (C) = otherwise. (1) (2) Definition 3 Let X M be a binary image and T an increasing criterion, then the attribute opening Γ T of X is defined by: Γ T (X) = x X Γ T (Γ x (X)). (3) BCN Retraite 15

17 Attribute opening and thinning 2 Definition 4 Let f be a grayscale image and T an increasing criterion, the grayscale attribute opening γ T is defined as: γ T (f)( x) = max(h : x Γ T (T h (f))). (4) Definition 5 The grayscale attribute thinning υ T of image f and a not necessarily increasing criterion T is defined as: υ T (f)( x) = max(h : x Υ T (T h (f))). (5) BCN Retraite 16

18 Moments Moments: m pq = x p y q f(x, y) dx dy (6) R 2 Central moments: µ pq = (x x) p (y ȳ) q f(x, y) dx dy (7) R 2 Normalized moments: Normalized central moments: n pq = m pq m γ 00 η pq = µ pq µ γ 00 (8) (9) where x = m 10, ȳ = m 01 and γ = p + q m 00 m BCN Retraite 17

19 Moment of Inertia I = n (x x c ) 2 + (y y c ) 2 (10) I = n x 2 + n 0.5 y 2 ( n x)2 n ( n y)2 n (11) x 2 + y 2 dy dx = 1 6 (12) I = n x 2 + n y 2 ( n x)2 n ( n y)2 n + n 6 (13) BCN Retraite 18