Vector-attribute filters

Transcription

1 Vector-attribute filters Erik R. Urbach, Niek J. Boersma, and Michael H.F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands April 2005

2 Outline Purpose Binary vector-attribute filters Gray-scale vector-attribute filters Moment invariants as vector-attribute Image analysis using pattern spectra Conclusions

3 Purpose Attribute filters use a criterion to remove or preserve connected components (flat zones) based on their attributes. So far attribute filters have always been based on one or more scalar attributes, e.g. area, perimeter, elongation, number of holes. Binary image X Nuts Bolts PRO : Computationally efficient for image filtering and analysis. CON : How to remove/preserve components based on their similarity with a given shape.

4 Vector-attribute filters Removing objects that are similar enough to a given shape. Example: removing objects that are similar enough (ɛ) to the reference shape (letter A). Original image X ɛ = 0.01 ɛ = 0.10 ɛ = 0.15 A value of ɛ = 0 means only those shapes are removed that are exactly the same as the reference shape.

5 Binary attribute thinning Trivial thinning Φ T of a connected set C with criterion T is C if C satisfies T, and is empty otherwise. Furthermore, Φ T ( ) =. Binary connected opening Γ x (X) of set X at point x M yields the connected component of X containing x if x X, and otherwise. Definition 1. The binary attribute thinning (non-increasing grain filter) Φ T of set X with criterion T is given by Φ T (X) = x X Φ T (Γ x (X)) (1) A multi-variate attribute thinning Φ {Ti} (X) with scalar attributes {τ i } and their corresponding criteria {T i }, with 1 i N, preserves a component C if i : T i, T i = τ i (C) r i : N Φ {Ti} (X) = Φ T i (X). (2) i=1

6 Binary vector-attribute thinning C is preserved if τ(c) Υ satisfies criterion T r,ɛ τ (C) = d( τ(c), r) ɛ. Dissimilarity measure d : Υ Υ R quantifies the difference between τ(c) and r. A binary vector-attribute thinning Φ r,ɛ τ (X), with D-dimensional vectors from a space Υ R D, removes the connected components of a binary image X whose vector-attributes differ less than ɛ from a reference vector r Υ. Definition 2. The vector-attribute thinning Φ r,ɛ τ of X with respect to a reference vector r and using vector-attribute τ and scalar value ɛ is given by Φ τ r,ɛ(x) = {x X T τ r,ɛ(γ x (X))}. (3) Possible choice for d: Euclidean distance d( u, v) = v u. Any dissimilarity measure can be used (such as Mahalanobis distance). Since the triangle inequality d(a, c) d(a, b) + d(b, c) is not required, d need not be a distance.

7 Gray-scale vector-attribute thinning Extension to gray-scale using threshold decomposition: φ τ r,ɛ(f) = sup{h T τ r,ɛ(γ x (X h (f)))}, (4) where threshold set X h (f) is defined as: X h (f) = {x M f(x) h}. Example: removing letters from image f consisting of nested versions of the letters A, B, and C. f φ τ S A,ɛ (f) φ τ S B,ɛ (f) φ τ S C,ɛ (f) Note: efficient algorithms (such as Max-tree) exists to compute gray-scale attribute thinnings directly without using the expensive threshold decomposition.

8 Moment invariants as vector-attribute Hu s set of seven moment invariants are invariant to rotation, scaling and translation. Krawtchouk moment invariant form a set of discrete and orthogonal moment invariants. Flusser and Suk developed a set of complete and independent moment invariants Problem of Krawtchouk moment invariants when the reference shape is not rotationally symmetric: angle used here is the orientation instead of the direction of the shape

9 Moment invariants as vector-attribute Influence of orientation on dissimilarity using moment invariants of Hu and Krawtchouk: rotated versions of letter A are compared with original A, double-sized A, half-sized A, and letter B (dotted line): Error A Larger A Smaller A B Rotation angle (degrees) Error A Larger A Smaller A B Rotation angle (degrees) Error A Larger A Smaller A B Rotation angle (degrees) Hu Krawtchouk Krawtchouk d( τ(c i ), τ(s 1 )) d( τ(c i ), τ(s 1 )) min 2 n=1 d( τ(c i ), τ(s n )) Note that the letter B should always be more dissimilar to A than any rotated or scaled A. Furthermore, rotation- and scale-invariance is only approximated in the digital case.

10 Hu s Moment invariants Using vector-attribute thinning with Hu s set of 7 moment invariants as vector-attribute to remove from image X the letters A, B, and C respectively. X Φ τ S A,0.010 (X) Φ τ S B,0.013 (X) Φ τ S C,0.010 (X) X Φ τ S A,ɛ (X) X Φ τ S B,ɛ (X) X Φ τ S C,ɛ (X)

11 Pattern spectrum Definition 3. A binary shape granulometry is a set of operators (thinnings) {β r } with r from some totally ordered set Λ, with the following three properties for all r, s Λ and λ > 0. anti-extensiveness: β r (X) X (5) scale-invariance: β r (X λ ) =(β r (X)) λ (6) nesting relationship: β r (β s (X)) =β max(r,s) (X), (7) The (shape) pattern spectrum is defined for a granulometry {β r }: (s β (X))(u) = da(β r(x)) dr, (8) r=u where A(X) denotes the Lebesgue measure in R n, which is the area if n = 2.

12 Pattern spectrum Shape granulometry using shape family F r, which is a set containing the reference shapes of the first r letters of the alphabet. A shape is removed from image X if it resembles at least one of the reference shapes. Amount of detail removed 9 x Number of objects removed Number of letters in F Number of letters in F X Pattern spectrum Shape histogram Y 1 = X Φ τ F 1,ɛ Y 2 = Y 1 Φ τ F 2,ɛ Y 3 = Y 2 Φ τ F 3,ɛ Y 4 = Y 3 Φ τ F 4,ɛ Y 5 = Y 4 Φ τ F 5,ɛ

13 Conclusions New class of attribute filters and granulometries whose attributes are vector instead of scalar values. Attribute thinnings were implemented using Salembier s Max-tree algorithm. Other algorithms for connected (gray-scale) thinnings, such as level line trees, can also be used. Alternative moment invariants such as the complex moment invariants of Flusser and Suk should be investigated. More research is also needed to determine better ways for selecting the parameters like ɛ and the order and the choice of shape classes. Other dissimilarity measures than the Euclidean distance should be investigated (adaptive system such as genetic algorithm, Mahalanobis distance only if multiple reference instances of the target class are available). Because only examples of the target class are used, the filtering problem resembles one-class classification (kernel density estimates). Support-vector domain description could be used in a similar way.

14 Questions

15 Properties of operators Below, the following terms will be used to describe the properties of an operator Ψ for binary images X and Y : idempotence : Ψ(Ψ(X)) = Ψ(X) (9) increasingness : X Y Ψ(X) Ψ(Y ) (10) anti-extensiveness : Ψ(X) X. (11) An idempotent operator is also known as a filter. Operators that are anti-extensive, idempotent, and increasing are called openings. Operators that are anti-extensive, idempotent, but not necessarily increasing are called thinnings. Attribute thinning Pattern spectrum

16 Max-tree A peak component P h of image f is a connected region in which f(x) h for all x P h, and all neighbours of P h have gray level smaller than h. P 0 3 P 0 2 P 1 2 P 0 1 P C 0 3 C 0 2 C 0 1 C 0 0 C 1 2 Peak components Attributes Max-tree Filtering using Max-tree with 4 different rules: P 0 1 P 0 0 P 0 3 P 0 2 P 0 1 P 0 0 P 0 3 P 0 1 P 0 0 P 0 2 P 0 1 P 0 0 Min Max Direct Subtractive Back

17 Hu s moment invariants (Central) moments up to some order (p + q) are computed: Moments: m pq = x p y q f(x, y) dx dy (12) R 2 Central moments: µ pq = (x x) p (y ȳ) q f(x, y) dx dy R 2 (13) Normalized central moments: where x = m 10 m 00 and ȳ = m 01 m 00 (14) η pq = µ pq µ γ 00 where γ = p + q 2 (15) + 1 (16) (17) Back

18 Hu s moment invariants Hu s set of seven moment invariants is defined as: φ 1 =η 20 + η 02 φ 2 =(η 20 η 02 ) 2 + 4η 2 11 (18) (19) φ 3 =(η 30 3η 12 ) 2 + (3η 21 η 03 ) 2 (20) φ 4 =(η 30 + η 12 ) 2 + (η 21 + η 03 ) 2 (21) φ 5 =(η 30 3η 12 )(η 30 + η 12 )[(η 30 + η 12 ) 2 3(η 21 + η 03 ) 2 ] + (3η 21 η 03 )(η 21 + η 03 )[3(η 30 + η 12 ) 2 (η 21 + η 03 ) 2 ] (22) φ 6 =(η 20 η 02 )[(η 30 + η 12 ) 2 (η 21 + η 03 ) 2 ] + 4η 11 (η 30 + η 12 )(η 21 + η 03 ) (23) φ 7 =(3η 21 η 03 )(η 30 + η 12 )[(η 30 + η 12 ) 2 3(η 21 + η 03 ) 2 ] + (3η 12 η 30 )(η 21 + η 03 )[3(η 30 + η 12 ) 2 (η 21 + η 03 ) 2 ] (24) Note that these seven moment invariants are computed using central moments up-to(and including) order 3. Back

19 Krawtchouk moment invariants ν nm = N 1 x=0 N 1 y=0 N 2 2M 00 f(x, y) (25) [(x x) cos θ + (y (y) N sin θ 2 /2 + N M 00 2 [(x x) cos θ + (y (y) N sin θ 2 /2 + N M 00 2 n n where N is the width of the input image and θ = 1 2 tan( 1) 2µ 11 µ 20 µ 02 (26) Back

20 Krawtchouk moment invariants K n (x P, N) = N a k,n,p x k = 2 F 1 ( n, x, N; 1 p ) (27) k=0 where x, n = 0, 1, 2...N, N > 0 and p (0, 1) 2F 1 (a, b, c; z) = k=0 (a) k (b) k (c) k z k k! (28) (a) k = a(a + 1)...(a + k 1) (29) The Krawtchouk moment invariants can now be defined by: Q nm = [ρ(n), ρ(m)] 1 2 n i=0 m a i,n,p1 a j,m,p2 ν ij (30) j=0 where a k,n,p are coefficients determined by equation 27. Back