Isotropic Multiresolution Analysis: Theory and Applications

Transcription

1 Isotropic Multiresolution Analysis: Theory and Applications Saurabh Jain Department of Mathematics University of Houston March 17th 2009 Banff International Research Station Workshop on Frames from first principles. S. Jain Isotropic Multiresolution Analysis March / 29

2 Collaborators & Acknowledgements This work has been performed in collaboration with Manos Papadakis (UH), Juan R. Romero (U.Puerto Rico), Simon K. Alexander (UH), Shikha Baid (formerly UH). We gratefully acknowledge the contributions and previous related work by Robert Azencott (UH and ENS) and Bernhard G. Bodmann (UH). S. Jain Isotropic Multiresolution Analysis March / 29

3 Outline Motivation Seismic Imaging Biomedical Imaging and Segmentation Isotropic Multiresolution Analysis Definition Characterizations in the theory of IMRA Examples of IMRA Extension Principles Mathematical Framework for Segmentation Feature spaces and feature maps Steerability S. Jain Isotropic Multiresolution Analysis March / 29

4 Example from Seismic Imaging (a) (b) Figure: Vertical cross-sections of volumes produced using radial (left) and non-radial (right) filters. (Courtesy: Total E&P, USA) S. Jain Isotropic Multiresolution Analysis March / 29

5 Example from Biomedical Imaging (a) 2D slice from 3D µct x-ray data (b) Slice from Intravascular Ultra Sound data Figure: Examples of medical 3D data sets. S. Jain Isotropic Multiresolution Analysis March / 29

6 Segmentation flowchart Stack into Feature Vectors analysis clustering Segmentation S. Jain Isotropic Multiresolution Analysis March / 29

7 Problems with tensor product basis Barbara DB8 decomposition S. Jain Isotropic Multiresolution Analysis March / 29

8 Problems with tensor product basis Barbara DB8 high pass S. Jain Isotropic Multiresolution Analysis March / 29

9 Problems with tensor product basis IMRA high pass DB8 high pass S. Jain Isotropic Multiresolution Analysis March / 29

10 Problems with tensor product basis (Zoom) IMRA high pass DB8 high pass S. Jain Isotropic Multiresolution Analysis March / 29

11 Outline Motivation Seismic Imaging Biomedical Imaging and Segmentation Isotropic Multiresolution Analysis Definition Characterizations in the theory of IMRA Examples of IMRA Extension Principles Mathematical Framework for Segmentation Feature spaces and feature maps Steerability S. Jain Isotropic Multiresolution Analysis March / 29

12 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. S. Jain Isotropic Multiresolution Analysis March / 29

13 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. A d d matrix is expansive if it has integer entries and if all of its eigenvalues have absolute value greater than 1. S. Jain Isotropic Multiresolution Analysis March / 29

14 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. A d d matrix is expansive if it has integer entries and if all of its eigenvalues have absolute value greater than 1. An expansive matrix is radially expansive if M = ao, for some fixed a > 1 and a matrix O SO(d). S. Jain Isotropic Multiresolution Analysis March / 29

15 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. A d d matrix is expansive if it has integer entries and if all of its eigenvalues have absolute value greater than 1. An expansive matrix is radially expansive if M = ao, for some fixed a > 1 and a matrix O SO(d). A unitary dilation operator with respect to M is defined by D M f (x) = det M 1/2 f (Mx). S. Jain Isotropic Multiresolution Analysis March / 29

16 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. A d d matrix is expansive if it has integer entries and if all of its eigenvalues have absolute value greater than 1. An expansive matrix is radially expansive if M = ao, for some fixed a > 1 and a matrix O SO(d). A unitary dilation operator with respect to M is defined by D M f (x) = det M 1/2 f (Mx). For y R d, the (unitary) shift operator T y is defined by T y f (x) = f (x y). S. Jain Isotropic Multiresolution Analysis March / 29

17 Notation and Preliminaries We say that Ω R d is radial if for all O SO(d), OΩ = Ω. A d d matrix is expansive if it has integer entries and if all of its eigenvalues have absolute value greater than 1. An expansive matrix is radially expansive if M = ao, for some fixed a > 1 and a matrix O SO(d). A unitary dilation operator with respect to M is defined by D M f (x) = det M 1/2 f (Mx). For y R d, the (unitary) shift operator T y is defined by T y f (x) = f (x y). A function f L 2 (R d ) is said to be isotropic if there exists a y R d and a radial function g L 2 (R d ) such that f = T y g. S. Jain Isotropic Multiresolution Analysis March / 29

18 Definition An IMRA is a sequence {V j } j Z of closed subspaces of L 2 (R d ) satisfying the following conditions: j Z, V j V j+1, (D M ) j V 0 = V j, j Z V j is dense in L 2 (R d ), j Z V j = {0}, S. Jain Isotropic Multiresolution Analysis March / 29

19 Definition An IMRA is a sequence {V j } j Z of closed subspaces of L 2 (R d ) satisfying the following conditions: j Z, V j V j+1, (D M ) j V 0 = V j, j Z V j is dense in L 2 (R d ), j Z V j = {0}, V 0 is invariant under translations by integers, V 0 is invariant under all rotations. S. Jain Isotropic Multiresolution Analysis March / 29

20 Definition An IMRA is a sequence {V j } j Z of closed subspaces of L 2 (R d ) satisfying the following conditions: j Z, V j V j+1, (D M ) j V 0 = V j, j Z V j is dense in L 2 (R d ), j Z V j = {0}, V 0 is invariant under translations by integers, V 0 is invariant under all rotations. If P 0 is the orthogonal projection onto V 0, then OP 0 = P 0 O for all O SO(d), where O is the unitary operator given by Of (x) := f (O T x) S. Jain Isotropic Multiresolution Analysis March / 29

21 Characterization of V 0 Theorem Let V be an invariant subspace of L 2 (R d ) under the action of the translation group induced by Z d. Then V remains invariant under all rotations if and only if V = PW Ω for some radial measurable subset Ω of R d. S. Jain Isotropic Multiresolution Analysis March / 29

22 Characterization of IMRAs Theorem Let M be a radially expansive matrix and C := M. A sequence {V j } j Z is an IMRA with respect to M if and only if V j = PW C j Ω, where Ω is radial and satisfies 1 Ω CΩ. 2 The set-theoretic complement of j=1 C j Ω is null. 3 lim j C j Ω = 0. Moreover the only singly generated IMRAs are precisely V j = PW C j Ω, where Ω is a radial subset of T d satisfying (1), (2) and (3). S. Jain Isotropic Multiresolution Analysis March / 29

23 Characterization of isotropic refinable functions Definition A function φ in L 2 (R d ) is called refinable with respect to dilations induced by an expansive matrix M, if there exists an H L (T d ) such that φ(m ξ) = H(ξ) φ(ξ), for a.e. ξ R d. The function H is called the low-pass filter or mask corresponding to φ. S. Jain Isotropic Multiresolution Analysis March / 29

24 Characterization of isotropic refinable functions Definition A function φ in L 2 (R d ) is called refinable with respect to dilations induced by an expansive matrix M, if there exists an H L (T d ) such that φ(m ξ) = H(ξ) φ(ξ), for a.e. ξ R d. The function H is called the low-pass filter or mask corresponding to φ. Theorem If φ L 2 (R d ) is a refinable function which is also isotropic and lim φ(ξ) = L 0, ξ 0 then φ PW ρ/(ρ+1), where ρ is the dilation factor of M. S. Jain Isotropic Multiresolution Analysis March / 29

25 Consequences Corollary There are no isotropic, refinable functions that are compactly supported in the spatial domain. S. Jain Isotropic Multiresolution Analysis March / 29

26 Consequences Corollary There are no isotropic, refinable functions that are compactly supported in the spatial domain. Corollary If the Fourier transform of φ is supported outside the ball of radius 1/2 centered at the origin then the MRA generated by φ is not an IMRA. S. Jain Isotropic Multiresolution Analysis March / 29

27 Examples of IMRA Example The sequence of closed subspaces V j = PW 2 j B(0,ρ), for any ρ > 0 and j Z is an IMRA. S. Jain Isotropic Multiresolution Analysis March / 29

28 Examples of IMRA Example The sequence of closed subspaces V j = PW 2 j B(0,ρ), for any ρ > 0 and j Z is an IMRA. Example Let B(0, r, s) denote the (2-dimensional) annulus centered at the origin having inner radius r and outer radius s. Define the sets A = B(0, r n, 2 n 1 ), with r n = 2 n 1 (1/16) n, and B = B(0, 1/2). Define Ω := A B. n=1 S. Jain Isotropic Multiresolution Analysis March / 29

29 Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 S. Jain Isotropic Multiresolution Analysis March / 29

30 Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 Let H 0 be the associated low pass filter, i.e. S. Jain Isotropic Multiresolution Analysis March / 29

31 Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 Let H 0 be the associated low pass filter, i.e. D M ˆφ = H 0 ˆφ. S. Jain Isotropic Multiresolution Analysis March / 29

32 Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 Let H 0 be the associated low pass filter, i.e. D M ˆφ = H 0 ˆφ. Assume that there exists a constant B > 0 such that l Z d ˆφ(ξ + l) 2 B a.e. on R d. S. Jain Isotropic Multiresolution Analysis March / 29

33 Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 Let H 0 be the associated low pass filter, i.e. D M ˆφ = H 0 ˆφ. Assume that there exists a constant B > 0 such that l Z d ˆφ(ξ + l) 2 B a.e. on R d. Furthermore, let H i, i = 1,..., m, be Z d -periodic measurable functions and define m wavelets, ψ i via D M ˆψ i = H i ˆφ. S. Jain Isotropic Multiresolution Analysis March / 29

34 Unitary Extension Principle Theorem (Daubechies, Han, Ron and Shen 2003) Assume H i L (T d ) for all i = 0,..., m, then the following two conditions are equivalent: { } 1 The set D j M T kψ i : j Z, k Z d, i = 1,..., m is a Parseval Frame for L 2 (R d ). 2 For all ξ σ(v 0 ), lim j Θ(M j ξ) = 1. If q (M 1 Z d )/Z d \ {0} and ξ + q σ(v 0 ), then Θ(M ξ)h 0 (ξ)h 0 (ξ + q) + m H i (ξ)h i (ξ + q) = 0. i=1 S. Jain Isotropic Multiresolution Analysis March / 29

35 Fundamental function Θ is the so-called fundamental function, defined by m Θ(ξ) = Hi (M j ξ) 2 j 1 H 0 (M l ξ) 2. j=0 i=1 l=0 S. Jain Isotropic Multiresolution Analysis March / 29

36 Fundamental function Θ is the so-called fundamental function, defined by Θ(ξ) = j=0 i=1 m Hi (M j ξ) 2 j 1 H 0 (M l ξ) 2. Θ M is the so-called mixed fundamental function, defined by Θ M (ξ) = m j=0 i=1 l=0 j 1 Hi s (M j ξ)hi a(m j ξ) H0(M s l ξ)h0 a(m l ξ). l=0 S. Jain Isotropic Multiresolution Analysis March / 29

37 Mixed Extension Principle Theorem (Daubechies, Han, Ron and Shen, 2003) Assume Hi a, Hs i L (T d ) for all i = 0,..., m. Then the following two conditions are equivalent, { } 1 The sets Ψ a := D j M T kψi a : j Z, k Z d, i = 1,..., m and { } Ψ s := D j M T kψi s : j Z, k Z d, i = 1,..., m is a pair of dual frames for L 2 (R d ). 2 For all ξ σ(v0 a) σ(v0 s), Ψ a and Ψ s are Bessel families. lim j Θ M (M j ξ) = 1. If q (M 1 Z d )/Z d \ {0} and ξ + q σ(v0 a) σ(v0 s ), then m Θ M (M ξ)h0(ξ)h s 0 a(ξ + q) + Hi s (ξ)hi a (ξ + q) = 0. i=1 S. Jain Isotropic Multiresolution Analysis March / 29

38 Our versions of Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 S. Jain Isotropic Multiresolution Analysis March / 29

39 Our versions of Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 D M ˆφ = H 0 ˆφ. S. Jain Isotropic Multiresolution Analysis March / 29

40 Our versions of Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 D M ˆφ = H 0 ˆφ. D M ˆψ i = H i ˆφ. S. Jain Isotropic Multiresolution Analysis March / 29

41 Our versions of Extension Principles Let φ be a refinable function such that ˆφ is continuous at the origin and, lim ˆφ(ξ) = 1. ξ 0 D M ˆφ = H 0 ˆφ. X φψ := D M ˆψ i = H i ˆφ. { } { D j M T kψ i : j N {0}, k Z d, i = 1,..., m T k φ : k Z d}. S. Jain Isotropic Multiresolution Analysis March / 29

42 Our version of Unitary Extension Principle Theorem Assume H i L (T d ) for all i = 1,..., m then X φψ is a Parseval frame for L 2 (R d ) if and only if for all q (M 1 Z d )/Z d and for a.e. ξ, ξ + q σ(v 0 ), m H i (ξ)h i (ξ + q) = δ q,0. i=0 S. Jain Isotropic Multiresolution Analysis March / 29

43 Our version of Mixed Extension Principle Theorem Let Hi a, Hs i L (T d ) for i = 1,..., m, then Xφψ a and X φψ s dual frames for L 2 (R d ) if and only if 1 Xφψ a and X φψ s are Bessel families, form a pair of 2 For all q (M 1 Z d )/Z d and for a.e. ξ, ξ + q σ(v a 0 ) σ(v s 0 ), m Hi s (ξ)hi a(ξ + q) = δ q,0. i=0 S. Jain Isotropic Multiresolution Analysis March / 29

44 An example of dual isotropic wavelet frames H a 0 = 1 inside the ball of radius b 2 H a 0 = 0 on Td \ B(0, b 1 ) H a 0 B(0,b 1 ) is radial. S. Jain Isotropic Multiresolution Analysis March / 29

45 An example of dual isotropic wavelet frames Let h a be a Z d -periodic function defined by, h a (ξ) = 1 Ha 0 (ξ) det(m) 1/2. S. Jain Isotropic Multiresolution Analysis March / 29

46 An example of dual isotropic wavelet frames Using h a we define the analysis high pass filters H a i (ξ) := e 2πi q i 1,ξ h a (ξ). S. Jain Isotropic Multiresolution Analysis March / 29

47 An example of dual isotropic wavelet frames H s 0 = 1 inside the ball of radius b 1 H s 0 = 0 on Td \ B(0, b 0 ) H s 0 B(0,b 0 ) is radial. S. Jain Isotropic Multiresolution Analysis March / 29

48 An example of dual isotropic wavelet frames h s = 0 inside the ball of radius b 3, for some b 3 > 0 h s 1 = on T d \ B(0, b det(m) 1/2 2 ) h s T d is radial S. Jain Isotropic Multiresolution Analysis March / 29

49 An example of dual isotropic wavelet frames Using h s we define the synthesis high pass filters: H s i (ξ) = e 2πi q i 1,ξ h s (ξ) S. Jain Isotropic Multiresolution Analysis March / 29

50 An example of dual isotropic wavelet frames m Hi s (ξ)hi a(ξ + q) = δ q,0. i=0 S. Jain Isotropic Multiresolution Analysis March / 29

51 2D IMRA scaling function and wavelet (a) Fourier transform of the scaling function ˆφ (b) Fourier transform of the wavelet ˆψ 1(2.) S. Jain Isotropic Multiresolution Analysis March / 29

52 Outline Motivation Seismic Imaging Biomedical Imaging and Segmentation Isotropic Multiresolution Analysis Definition Characterizations in the theory of IMRA Examples of IMRA Extension Principles Mathematical Framework for Segmentation Feature spaces and feature maps Steerability S. Jain Isotropic Multiresolution Analysis March / 29

53 Feature spaces and feature maps An image is a function f in L 2 (R 3 ). S. Jain Isotropic Multiresolution Analysis March / 29

54 Feature spaces and feature maps An image is a function f in L 2 (R 3 ). To each pixel z R 3, we associate a feature vector belonging to a fixed finite dimensional space V, called the feature vector space. S. Jain Isotropic Multiresolution Analysis March / 29

55 Feature spaces and feature maps An image is a function f in L 2 (R 3 ). To each pixel z R 3, we associate a feature vector belonging to a fixed finite dimensional space V, called the feature vector space. The vector space of all V -valued functions h defined on R 3 such that h( ) is square integrable on R 3 is denoted by H. Thus, h = ( R 3 h(z) 2 dz ) 1/2. S. Jain Isotropic Multiresolution Analysis March / 29

56 Feature spaces and feature maps An image is a function f in L 2 (R 3 ). To each pixel z R 3, we associate a feature vector belonging to a fixed finite dimensional space V, called the feature vector space. The vector space of all V -valued functions h defined on R 3 such that h( ) is square integrable on R 3 is denoted by H. Thus, h = ( R 3 h(z) 2 dz ) 1/2. Let A : L 2 (R 3 ) H be a bounded linear transformation that associates to each f the V -valued function Af defined on R 3. This linear transformation is called a feature map. S. Jain Isotropic Multiresolution Analysis March / 29

57 Steerability Definition Consider a bounded linear feature mapping A generating for each image f and each voxel z R 3 a feature vector Af (z) in the Euclidean space V. We shall say that A is a steerable feature mapping if there is a mapping U from the group G of rigid motions into the general linear group GL(V ) of V such that for each rigid motion R of R 3, the invertible transformation U(R) verifies, A[Rf ](z) = U(R) [Af (R(z))] z R 3. Here R is the following transformation induced by R on L 2 (R 3 ): Rf (z) = f (Rz). S. Jain Isotropic Multiresolution Analysis March / 29

58 An IMRA based steerable feature map Using the functions φ and ψ, we define the following feature map: Af (z) = (< f, T z φ( /8) >, < f, T z ψ( /4) >, < f, T z ψ( /2) >). S. Jain Isotropic Multiresolution Analysis March / 29

59 An IMRA based steerable feature map Using the functions φ and ψ, we define the following feature map: Af (z) = (< f, T z φ( /8) >, < f, T z ψ( /4) >, < f, T z ψ( /2) >). This map is both translation and rotation invariant S. Jain Isotropic Multiresolution Analysis March / 29

60 An IMRA based steerable feature map Using the functions φ and ψ, we define the following feature map: Af (z) = (< f, T z φ( /8) >, < f, T z ψ( /4) >, < f, T z ψ( /2) >). This map is both translation and rotation invariant Hence, it is steerable. S. Jain Isotropic Multiresolution Analysis March / 29

61 References J.R. Romero, S.K. Alexander, S. Baid, S. Jain, and M. Papadakis The Geometry and the Analytic Properties of Isotropic Multiresolution Analysis. to appear in the special issue on Mathematical Methods for Image Processing in Advances in Computational Mathematics, Springer, S. Jain, Manos Papadakis, and Eric Dussaud Explicit Schemes in Seismic Migration and Isotropic Multi-scale Representations. in Radon Transforms, Geometry and Wavelets, Contemporary Mathematics, Volume 464, Pages , AMS, S.K. Alexander, R. Azencott, M. Papadakis Isotropic Multiresolution Analysis for 3D-Textures and Applications in Cardiovascular Imaging. in Wavelets XII, SPIE Proceedings Vol. 6701, S. Jain Isotropic Multiresolution Analysis March / 29