XXXV Konferencja Statystyka Matematyczna -7-11/ XII 2009

XXXV Konferencja Statystyka Matematyczna -7-11/ XII 2009 1

Statistical Inference for Image Symmetries Mirek Pawlak pawlak@ee.umanitoba.ca 2

OUTLINE I Problem Statement II Image Representation in the Radial Basis Domain III Semiparametric Inference for Image Symmetry 3

IV Testing for Image Symmetries* Testing Rotational Symmetries* Testing Radiality Testing Reflection and Joint Symmetries V References 4

I Problem Statement 5

Z ij = f ( x i, y ) j + ε ij 6

D p ij Δ Z ij = f ( x i, y ) j + ε ij # of data points n 2 ; Δ n 1 7

Problem 1: Let f L 2 ( D). Given and knowing that ( ) = f x cos 2β f x, y Z ij = f ( x i, y ) j + ε ij, 1 i, j n ( ( ) + ysin( 2β ), xsin( 2β ) y cos( 2β )) some β [ 0,π ), estimate β. 8

Problem 1: Let f L 2 ( D). Given and knowing that ( ) = f x cos 2β f x, y Z ij = f ( x i, y ) j + ε ij, 1 i, j n ( ( ) + ysin( 2β ), xsin( 2β ) y cos( 2β )) some β [ 0,π ), estimate β. 9

Problem 2: Let f L 2 ( D). Given Z ij = f ( x i, y ) j + ε ij, 1 i, j n verify whether the null hypothesis is true or not H S : f ( x, y) ( Sf )( x, y) ( Sf )( x, y): Reflectional Symmetry, Rotational Symmetry 10

d = 2 d = 3 d = 4 d = 8 An image becomes invariant under rotations through an angle 2π d d = Radially Symmetric Objects f ( x, y) = g( x 2 + y 2 ) 11

Symmetry Hermann Weyl, Symmetry, Princenton Univ. Press, 1952. J.Rosen, Symmetry in Science: An Introduction to the General Theory, Springer, 1995. J.H. Conway, H. Burgiel, and C. Goodman-Strauss, The Symmetry of Things, A K Peters, 2008. M. Livio, The Equation That Couldn t Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, 2006....Livio writes passionately about the role of symmetry in human perception, arts,... 12

13

14

15

II Image Representation in the Radial Basis Domain 16

Radial Moments and Expansions (Zernike Basis) A pq ( f ) = f x, y D ( ) V pq ( x, y)dxdy = 2π 1 0 0 ( ) f ρ,θ R pq ρ Degree ( ) e iqθ ρdρdθ Angular dependence Bhatia & Born: On circle polynomials of Zernike and related orthogonal sets. Proc. Cambr. Phil. Soc. 1954 L 2 ( D) f ( x, y) p=0 q p p +1 π A pq f ( )V pq x, y ( ) 17

Invariant Properties Reflection f ( x, y) ( τ β f )( x, y) A pq ( f ) A ( pq τ β f ) = A pq i2 qβ ( f )e 18

Rotation f ( x, y) ( r α f )( x, y) A pq ( f ) A pq ( r α f ) = A pq ( f )e iqα 19

A pq ( f ) from noisy data D p ij Δ Z ij = f ( x i, y ) j + ε ij  pq ( f ) = w pq x i, y j ( x i,y j ) D ( ) Z ij w pq ( x i, y ) j = V pq ( x, y)dxdy Δ 2 V pq p ij ( x i, y ) j 20

EÂpq = A pq ( f ) + D pq ( Δ) + G pq ( Δ) D pq G pq ( Δ) = O( Δ) ( Δ) = O( Δ γ ) 1 < γ = 285 208 *Gauss lattice points problem of a circle * 21

III Semiparametric Inference for Image Symmetry Estimation of symmetry parameters: the axis of symmetry( β), the degree of rotational symmetry (d) of nonparametric image function: θ = θ ( f ), f L 2 ( D). β d 22

Semiparametric Inference on the Axis of Reflectional Symmetry A pq ( τ β f ) = e 2iqβ A p, q ( f ) β 23

Contrast Function M N ( β, f ) = N p=0 p +1 π p q= p N A pq ( f ) e 2iqβ A p, q ( f ) 2 M ( β, f ) = p=0 p +1 π p q= p A pq ( f ) e 2iqβ A p, q ( f ) 2 = f τ β f 2 24

Assumption Suppose that f L 2 τ β ( D) is invariant under some unique reflection Hence, if τ β f = f then we have or M ( β, f ) = 0 M N ( β, f ) = 0 for all N 25

Uniqueness of β Lemma U: Under Assumption the solution of M N ( β, f ) = 0 is uniquely determined and must equal β large that M N ( β, f ) contains A pq ( f ) 0 q s is 1. A p1 q 1 if we choose N so { } for which the gcd of { ( f ) 0,..., A pr q r ( f ) 0}, p i N,i = 1,..,r gcd( q 1,...,q r ) = 1. 26

β * = 120 27

M N ( β, f ) based only on A 42 ( f ), A 86 ( f ) 28

M N ( β, f ) based only on A 51 ( f ), A 86 ( f ) 29

M 14 ( β, f ) = 14 p=0 p +1 π p q= p A pq ( f ) e 2iqβ A p, q ( f ) 2 30

A pq ( f ) = A pq ( f ) e ir pq ( f ) M N β : ( β, f ) = N p=0 p +1 π p q= p 4 A pq f ( ( )) ( ) 2 1 cos 2r pq ( f ) + 2qβ r pq ( f ) + qβ = lπ for all ( p,q) with A pq ( f ) 0. 31

Estimated Contrast Function ˆM N ( β) = N p=0 p +1 π p q= p  pq e 2iqβ  p, q 2 = N p=0 p +1 π p 4 Âpq q= p 2 ( 1 cos ( 2 ˆr pq + 2qβ) ) ˆβ Δ,N = arg min β [ 0,π ) ˆM N ( β) van der Vaart: Asymptotic Statistics, 1998. 32

Accuracy of ˆβ Δ,N Theorem 3.1 ( ) and be reflection invariant w.r.t. a unique axis Let f BV D β [ 0,π ). Let N be so large that β is determined as the unique zero of M N ( β, f ). Then ˆβ Δ,N β (P) as Δ 0 Proof: sup β [ 0,π ) ˆM N ( β) M N ( β, f ) 0 (P) 33

Theorem 3.2 ˆβ Δ,N = β + O P ( Δ) ( n 2 rate ) Proof: 0 = ˆM ( 1) ( ) N ˆβΔ,N = ˆM ( 1) ( N β ) + ˆM ( 2) ( N β )( ˆβ Δ,N β ) ˆβΔ,N β = ( 1) ˆM N ˆM N 2 ( β ) ( ) ( ) β 34

From Theorem 3.1: sup β 0,π [ ) ( 2) ˆM N ( 2) ( β) M N ( β, f ) 0 (P) β β (P) ( 2) ˆM ( N β ) ( 2) M N ( β, f ) = N p p +1 16q 2 A pq ( f ) 2 0 π p=0 q= p 35

( 1) ( β ) Δ 1 ( ˆβΔ,N β ) ˆM Δ 1 N ( 2) M ( N β, f ) ˆM ( 1) N ( β ) = N p=0 p +1 π p 8q Âpq q= p 2 sin ( 2 ˆr pq + 2qβ ) ( f ) + qβ = lπ ( r ) pq = N p=0 p +1 π p 8q Âpq q= p ( ( ( ))) 2 sin 2 ˆr pq r pq f 36

N p=0 p +1 π p q= p 8q A pq ( ( ( ))) ( f ) 2 sin 2 ˆr pq r pq f  pq = A pq ( f ) + O P ( Δ) ˆr pq r pq ( f ) = O P ( Δ) 37

Theorem 3.3 Let f be Lip(1). Eε 4 <. Δ 1 ( ˆβΔ,N β ) N 0, ( ) M N 2 8σ 2 ( β, f ) ( 2) M N ( β, f ) = N p=0 p +1 π p q= p 16q 2 A pq ( f ) 2 38

Confidence Interval for β ˆβ Δ,N Φ 1 1 α ( ) 2 2Δ ˆσ ˆM N 2 ( ), ˆβ Δ,N + Φ 1 1 α ( ) 2 2Δ ˆσ ˆM N 2 ( ) ( 2 ˆM ) N = N p=0 p +1 π p q= p 16q 2 Â pq 2 ˆσ 2 = 1 4C Δ ( ) ( ) D ( x i,y j ),( x i+1,y j ), x i,y j+1 {( Z i, j Z ) 2 i+1, j + ( Z i, j Z ) 2 } i, j+1 39

Remark 1: f is not reflection symmetric ˆβ Δ,N β = argmin β f τ β f 2 40

Remark 2: Estimation d ( { 2, 3, 4,... })for images invariant under rotations through an angle 2π d? 41

42

43

IV Testing for Image Symmetries Testing Rotational Symmetries d = 2 - testing image symmetry w.r.t. rotation through π A pq ( f ) A pq ( r d f ) = e 2πiq /d A pq ( f ) = e πiq A pq ( f ) 44

The orthogonal projection f r d f 2 = p +1 1 e 2πiq/d 2 A pq f π p=0 q p ( ) 2 The test statistic (d=2) T N r 2 = N p +1 2 Â pq π p=0 p=odd q p 45

Theorem 4.1: A: Under H r 2 : r 2 f = f let Then for we have T N r 2 σ 2 Δ 2 a N Δ 2 N, N 7 Δ 0, as Δ 0 T N r 2 = ( ) ( ) a N N p +1 2 Â pq π p=0 p=odd q p N ( 0, 2σ 4 ) a( N ) = N ( N + 2) / 4 N even ( N +1) ( N + 3) / 4 N odd 46

B: Under H r 2 : r 2 f f let f C s ( D),s 2 and let Then we have N, N 3/2 Δ γ 1 0, N 2 s+1 Δ as Δ 0 T N r 2 f r 2 f 2 / 4 Δ N ( 0,σ 2 f r 2 f 2 ) 47

Proof: Under H r 2 r : T 2 N is a quadratic form of iid random variables (*de Jong: A CLT for generalized quadratic forms, Pr.Th.Related Fields, 1987*) Under H r 2 : T N r 2 is dominated by a linear term 48

Similar results exist for d = 4 - testing image symmetry w.r.t. rotation by π / 2 d = - testing image radiality, i.e., f ( x, y) = g ( ρ) Testing image mirror symmetry Joint hypothesis H τ y,r 2 : r 2 f = f AND τ y f = f H τ y,r 2 : r 2 f f OR τ y f f 49

V References Symmetry Hermann Weyl, Symmetry, Princenton Univ. Press, 1952. J.Rosen, Symmetry in Science: An Introduction to the General Theory, Springer, 1995. J.H. Conway, H. Burgiel, and C. Goodman-Strauss, The Symmetry of Things, A K Peters, 2008. M. Livio, The Equation That Couldn t Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, 2006....Livio writes passionately about the role of symmetry in human perception, arts,... 50

Symmetry & Statistics Given iid ( p) sequence Z 1,..., Z n verify whether H 0 : p( z) = p( z) Y i,n = m( x ) i,n + Z i,n H 0 : p( z) = p( z) Fan & Gencay(95), Ahmad & Li (97): m ( ) - linear regression Dette et al. (02) : m ( ) - nonparametric regression 51

Bissantz, Holzmann, and Pawlak, Testing for image symmetries - - with application to confocal microscopy, IEEE Trans. Inform. Theory, 2009. Bissantz, Holzmann, and Pawlak, Estimating the axis of reflectional symmetry of an image, Annals of Applied Statistics, to appear. 52