AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE

FILTERS

Kai Krajsek

J. W. Goethe University, Visual Sensorics and Information Processing Lab

Robert Mayer Str.2-4, D-60054 Frankfurt am Main, Germany

Rudolf Mester

J. W. Goethe University, Visual Sensorics and Information Processing Lab

Robert Mayer Str.2-4, D-60054 Frankfurt am Main, Germany

Keywords:

steerable ﬁlter, quadrature ﬁlter, Lie group theory.

Abstract:

In this paper, a complete theory of steerable ﬁlters is presented which shows that quadrature ﬁlters are only

a special case of steerable ﬁlters. Although there has been a large number of approaches dealing with the

theory of steerable ﬁlters, none of these gives a complete theory with respect to the transformation groups

which deform the ﬁlter kernel. Michaelis and Sommer (Michaelis and Sommer, 1995) and Hel-Or and Teo

(Teo and Hel-Or, 1996; Teo and Hel-Or, 1998) were the ﬁrst ones who gave a theoretical justiﬁcation for

steerability based on Lie group theory. But the approach of Michaelis and Sommer considers only Abelian Lie

groups. Although the approach of Hel-Or and Teo considers all Lie groups, their method for generating the

basis functions may fail as shown in this paper. We extend these steerable approaches to arbitrary Lie groups,

like the important case of the rotation group SO(3) in three dimensions.

Quadrature ﬁlters serve for computing the local energy and local phase of a signal. Whereas for the one

dimensional case quadrature ﬁlters are theoretically well founded, this is not the case for higher dimensional

signal spaces. The monogenic signal (Felsberg and Sommer, 2001) based on the Riesz transformation has

been shown to be a rotational invariant generalization of the analytic signal. A further generalization of the

monogenic signal, the 2D rotational invariant quadrature ﬁlter (K

¨

othe, 2003), has been shown to capture richer

structures in images as the monogenic signal.

We present a generalization of the rotational invariant quadrature ﬁlter based on our steerable theory. Our

approach includes the important case of 3D rotational invariant quadrature ﬁlters but it is not limited to any

signal dimension and includes all transformation groups that own an unitary group representation.

1 INTRODUCTION

Steerable ﬁlters and quadrature ﬁlters are well estab-

lished methods in signal and image processing. Steer-

ability is at least implicitly used when computing di-

rectional derivatives as this is the central operation in

differential motion estimation. Quadrature ﬁlters are

the choice for computing the local energy and local

phase of a signal. In this paper we present a complete

theory of steerable ﬁlters and derive a group invariant

quadrature ﬁlter approach based on our steerable ﬁlter

theory.

Although a large number of approaches dealing

with the theory of steerable ﬁlters have been pub-

lished (Danielsson, 1980; Freeman and Adelson,

1991; Perona, 1995; Simoncelli et al., 1992; Simon-

celli and Farid, 1996; Michaelis and Sommer, 1995;

Teo and Hel-Or, 1996; Teo and Hel-Or, 1998; Yu

et al., 2001), none of these provides a complete and

closed theory. Such a theory shall describe the gen-

eral requirements which are necessary for a ﬁlter ker-

nel to be a steerable ﬁlter. The beneﬁt from this work

is a deeper understanding of the concepts of steerable

ﬁlters and enables the user to construct steerable ﬁl-

ters for every Lie group transformation. None of the

previously published approaches gives a general so-

lution of the following problem: If one is confronted

with a certain ﬁlter kernel and an arbitrary Lie group,

what are the approbate basis functions to steer the ﬁl-

ter kernel and how are the interpolation functions to

be computed.

The steerable approach of Michaelis and Sommer

(Michaelis and Sommer, 1995) give a solution to this

problem in the case of Abelian Lie groups, whereas

the approach of Teo and Hel-Or (Teo and Hel-Or,

1996; Teo and Hel-Or, 1998) handles all Lie group

transformations. But the latter approach may fail as

we show in section 3.4. In contrast these approaches

48

Krajsek K. and Mester R. (2006).

AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS.

In Proceedings of the First International Conference on Computer Vision Theory and Applications, pages 48-55

DOI: 10.5220/0001377100480055

Copyright

c

SciTePress

(Michaelis and Sommer, 1995; Teo and Hel-Or, 1996;

Teo and Hel-Or, 1998) which either do not cover the

case of non-Abelian groups (Michaelis and Sommer,

1995) or do not work for all ﬁlter kernels (Teo and

Hel-Or, 1996; Teo and Hel-Or, 1998), our approach

uses the full power of Lie group theory to gener-

ate the minimum number of basis functions also for

non-Abelian compact Lie groups. It is a direct ex-

tension of the approach of (Michaelis and Sommer,

1995) in which the basis functions are generated by

the eigenfunctions of the generators of the Lie group.

In our approach a Casimir operator is used to gener-

ate the basis functions also for non-Abelian compact

Lie groups. For non-Abelian, non-compact groups,

we show that polynomials serve as appropriate basis

functions.

The quadrature ﬁlter is a well established method in

signal processing and low-level image processing for

computing the local energy and local phase of a sig-

nal. Whereas the local signal is an estimate of the lo-

cal intensity structure, the local phase provides infor-

mation about the local shape of the signal. In the 1D

case the quadrature ﬁlter is well deﬁned by an even

part, a bandpass ﬁlter, and an odd part, the Hilbert

transformation of the bandpass ﬁlter. The ﬁlter out-

put is the analytic signal, a representation of the sig-

nal from which the local energy and local phase can

easily be computed. Large efforts have been made

to generalize the analytic signal (B

¨

ulow and Som-

mer, 2001; Granlund and Knutsson, 1995) to the 2D

case by projecting the scalar valued Hilbert transfor-

mation in the two dimensional space. The drawback

of all of these methods are that they are not rota-

tional invariant. The Riesz transformation has been

shown to be a rotational invariant generalization of

the Hilbert transformation generalizing the analytic

signal to the monogenic signal (Felsberg and Som-

mer, 2001). A further generalization of the mono-

genic signal is the 2D rotational invariant quadrature

ﬁlter (K

¨

othe, 2003), based on rotated steerable ﬁlters,

which is able to capture richer structures from an im-

age than the monogenic signal.

Many interesting computer vision and image

processing applications, like motion estimation, are

not restricted to the two dimensional case. We present

a generalization of the rotational invariant quadrature

ﬁlter (K

¨

othe, 2003) with respect to the signal dimen-

sion and the transformation group. This includes the

important case of a 3D rotational invariant quadrature

ﬁlter, but it is not limited to any signal dimension.

Also other transformations than the rotation is con-

sidered, like shearing, which is the correct transfor-

mation group when describing motion in space time.

2 STEERABLE FILTERS

Let g denote an element of an arbitrary Lie group

G and x ∈ IR

N

the coordinate vector of an N di-

mensional signal space. We deﬁne a steerable ﬁlter

h

g

(x) as the impulse response whose deformed ver-

sion, with respect to the Lie group, equals the linear

combination of a ﬁnite set of basis ﬁlters {b

j

(x)},

j =1, 2, ..., M. Furthermore, only the coefﬁcients

{a

j

(g)}, denoted as the interpolation functions, de-

pend on the Lie group element

h

g

(x)=

M

j=1

a

j

(g) b

j

(x) . (1)

Applying the deformed ﬁlter to a signal s(x) is equiv-

alent to the linear combination of the individual im-

pulse responses of the basis ﬁlters

h

g

(x) ∗ s(x)=

M

j=1

a

j

(g)(b

j

(x) ∗ s(x)) . (2)

Questions arising with steerable functions are:

• Under which conditions can a given function h

g

(x)

be steered?

• How can the basis functions b

j

(x) be determined?

• How many basis functions are needed to steer the

function h

g

(x)?

• How can the interpolation functions a

j

(g) be deter-

mined?

In the last decade, several steerable ﬁlter approaches

have been developed trying to answer these ques-

tions, but all of them, except for the approach of Teo

and Hel-Or (Teo and Hel-Or, 1996; Teo and Hel-Or,

1998), tackle only a special case, either for the ﬁlter

kernel or for the corresponding transformation group.

3 AN EXTENDED STEERABLE

APPROACH BASED ON LIE

GROUP THEORY

In the following section, we present our steerable ﬁl-

ter approach based on Lie group theory covering all

recent approaches developed so far. It delivers for

Abelian Lie groups and for compact non-Abelian Lie

groups the minimum required number of basis func-

tions and the corresponding interpolation functions.

In order to complete the steerable approach the case of

non-Abelian, non-compact Lie groups has to be con-

sidered separately. After presenting our concept, its

relation to recent approaches is discussed and some

examples are presented.

AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS

49

3.1 Conditions for Steerability

In the following we show the steerability of all ﬁl-

ter kernels h :IR

N

→ IC which are expand-

able according to a ﬁnite number M of basis func-

tion B = {b

j

(x)} of a subspace V :=span{B} ⊂ L

2

of all quadratic integrable functions. Since every el-

ement of L

2

can arbitrary exactly be approximated

by a ﬁnite number of basis functions, we consider, at

least approximately, all quadratic integrable ﬁlter ker-

nels. The problem of approximating such a function

by a smaller number of basis functions when allowing

a certain error has been examined in (Perona, 1995)

and is not topic of this paper. With the notation of the

inner product ·, · in L

2

and the Fourier coefﬁcients

c

j

= h(x),b

j

(x) the expansion of h(x) reads

h(x)=

M

j=1

c

j

b

j

(x) . (3)

Furthermore, every basis function b

j

(x) ∈ V shall

belong to an invariant subspace U ⊆ V with respect

to a Lie group G transformation.

Then, h(x) is steerable with respect to G.

We have assigned the preconditions such that this

statement can be easily veriﬁed. Let D(g) denote the

representation of g ∈Gin the function space V and

D(g) the representation of G in the N-dimensional

signal space. It is easy to verify that the transformed

function D(g)h(x) equals the linear combination of

the transformed basis functions

D(g)h (x)=h

D(g)

−1

x

(4)

=

M

j=1

c

j

b

j

(D(g)

−1

x) (5)

=

M

j=1

c

j

D(g)b

j

(x) . (6)

Since every basis function b

j

(x) is, per deﬁnition,

part of an invariant subspace, the transformed version

D(g)b

j

(x) can be expressed by a linear combination

of the subspace basis. Let denote m(j) the mapping

of the index j of the basis function b

j

(x) onto the

lowest index of the basis function belonging to the

same subspace and d(j) the mapping of the index of

the basis function b

j

(x) onto the dimension d

j

of its

invariant subspace. The transformed basis function

D(g)b

j

(x) can be expressed, with the previous def-

inition of m(j) and d(j), and the coefﬁcients of the

linear combination w

jk

(g) as

D(g)b

j

(x)=

m(j)+d(j)−1

k=m(j)

w

jk

(g)b

k

(x) . (7)

Inserting equation (7) into equation (6) yields

D(g)h(x)=

M

i=1

c

i

m(j)+d(j)−1

k=m(j)

w

jk

(g)b

k

(x) . (8)

The double sum can be written such that all co-

efﬁcients belonging to the same basis function are

grouped together, where L denotes the number of in-

variant subspaces in V

D(g)h(x)=

b

k

∈U

1

b

k

(x)

w

jk

∈U

1

c

j

w

jk

(g) (9)

+

b

k

∈U

2

b

k

(x)

w

jk

∈U

2

c

j

w

jk

(g)+...

+

b

k

∈U

L

b

k

(x)

w

jk

∈U

L

c

j

w

jk

(g) .

Thus, in order to steer the function h we have to con-

sider all basis functions spanning the L subspaces.

3.2 The Basis Functions

The next question arising is how to obtain the appro-

priate basis functions. We require the basis functions

to span ﬁnite dimensional invariant subspaces. Fur-

thermore, the invariant subspaces are desired to be

as small as possible in order to lower computational

costs. Group theory provides the solution of this prob-

lem and the functions fulﬁlling these requirements

are, per deﬁnition, the basis of an irreducible repre-

sentation of the Lie group. This has already pointed

out by Michaelis and Sommer (Michaelis and Som-

mer, 1995) and a method for generating such a basis

for Abelian Lie groups has been proposed. We ex-

tend this method for the case of non-Abelian, compact

Lie groups. The case of non-Abelian, non-compact

groups is discussed in subsection 3.2.2.

3.2.1 Basis Functions for Compact Lie Groups

The invariant space spanned by an irreducible ba-

sis cannot be decomposed further into invariant sub-

spaces and thus, forming a minimum number of ba-

sis functions for the steerable function. Michaelis

and Sommer showed that such a basis is given by the

eigenfunctions of the generators in case of Abelian

Lie groups. Since the generators of a non-Abelian

groups do not commutate and thus have no simultane-

ous eigenfunctions, the method does not work in this

case any more. But their framework can be extended

with a slight change to compact non-Abelian groups.

Instead of constructing the basis functions from the

simultaneous eigenfunctions of the generators of the

group, the basis function can also be constructed by

the eigenfunctions of a Casimir operator C of the cor-

responding Lie group. In order to deﬁne the Casimir

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

50

operator we ﬁrst have to introduce the Lie bracket, or

commutator, of two operators

[D(a), D(b)] = D(a)D(b) −D(b)D(a) . (10)

Operators commutating with all representations of the

group elements are denoted as Casimir operators

[C, D(g)] = 0 ∀g ∈G . (11)

Let {b

m

(x)},m =1, ..., d

α

denote the set of eigen-

functions of C corresponding to the same eigenvalue

α. Then, every transformed basis function D(g)b

i

(x)

is an eigenfunction with the same eigenvalue α

CD(g)b

i

(x)=D(g)Cb

i

(x) (12)

= D(g)αb

i

(x)

= αD(g)b

i

(x) .

Thus, {b

m

(x)} forms a basis of a d

α

dimensional in-

variant subspace U

α

. Any transformed element of this

subspace can be expressed by a linear combination of

basis functions of this subspace

D(u)b

m

(x)=

d

α

j=1

w

mj

b

j

(x) . (13)

Thus, we have found a method for constructing in-

variant subspaces also for non-Abelian groups. A

Casimir operator is constructed by a linear combina-

tion of products of generators of the corresponding

Lie group where n denotes the number of generators

C =

ij

f

ij

L

i

L

j

,i,j=1, ..., n . (14)

The coefﬁcients f

ij

are solved by the constraints

[C, L

k

]=0,k=1, ..., n . (15)

If the Casimir operator of a compact group is self-

adjoint with a discrete spectrum, the eigenfunctions

constitute a complete orthogonal basis of the corre-

sponding function space. It is a well known fact from

functional analysis that in this case all eigenfunctions

belonging to the same eigenvalue span a ﬁnite dimen-

sional subspace. If furthermore the considered group

the Casimir operator has the symmetry of the group G,

i.e. there exists no operation which does not belong

to the group and under which the Casimir operator

is invariant, then the eigenfunctions are basis func-

tions of irreducible representation (Wigner, 1959).

After computing one eigenfunction b

1

(x) correspond-

ing to the eigenvalue α we can construct all other

basis functions of this invariant subspace by apply-

ing all possible combinations of generators of the Lie

group to b

1

(x). The sequence of generators is stopped

when the resulting function is linear dependent from

the ones which have already been constructed. This

equals to the method for constructing the basis func-

tions proposed by Teo and Hel-Or (Teo and Hel-Or,

1998) except for the fact that they propose to apply

this procedure directly to the steerable function h(x).

3.2.2 Basis Function for Non-Compact Lie

Groups

Since only Abelian Lie groups and compact non-

Abelian Lie groups are proofed to own complete irre-

ducible representations, i.e. the representation space

falls into invariant subspaces, we have to treat the

case of non-Abelian, non-compact groups separately.

Since we do only require an invariant subspace and

not an entirely irreducible representation we can eas-

ily construct such a space from a polynomial basis

of the space of square integrable functions. The or-

der of a polynomial term does not change by an arbi-

trary Lie group transformation and thus the basis of a

polynomial term constitute a basis for a steerable ﬁl-

ter. In order to steer an arbitrary polynomial we have

to determine the terms of different order. The sum

of the basis functions of the corresponding invariant

subspaces are a basis for the steerable polynomial.

We can now construct for every Lie group transfor-

mation the corresponding basis for a steerable ﬁlter.

For Abelian groups and compact groups we chose the

basis from the eigenfunction of the Casimir operator

whereas for all other groups we choose a polynomial

basis. The next section addresses the question how

to combine these basis functions in order to steer the

resulting ﬁlter kernel with respect to any Lie group

transformation.

3.3 The Interpolation Functions

The computation of the interpolation functions

{a

j

(g)} can already be deduced from equ.(9). In or-

der to obtain the interpolation function corresponding

to the basis function b

m

(x) the transformed version

of the original ﬁlter kernel h(x) has to be projected

onto b

m

(x)

a

m

(g)=D(g)h(x),b

m

(x) (16)

=

D(g)

M

n=1

c

n

b

n

(x),b

m

(x)

=

M

n=1

c

n

D(g)b

n

(x),b

m

(x) .

The relation between {c

k

}

M

1

and {a

k

}

L

1

is a linear

map P ∈ IR

M×L

with the matrix elements

(P)

ij

= D(g)b

i

(x),b

j

(x) (17)

of the coefﬁcient vector

c := (c

1

,c

2

, ..., c

M

) (18)

onto the interpolation function vector

a(g):=(a

1

(g),a

2

(g), ..., a

N

(g)) . (19)

AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS

51

As already pointed out by Michaelis and Sommer

(Michaelis and Sommer, 1995), the basis functions

have not to be the transformed versions of the ﬁlter

kernel as assumed in other approaches (Freeman and

Adelson, 1991; Simoncelli and Farid, 1996). It is suf-

ﬁcient that the synthesized function is steerable. If it

is nonetheless desired to design basis functions which

are transformed versions h

g

(x):=D(g)h(x) of the

ﬁlter kernel h(x) a basis change is sufﬁcient

h(x)=

M

j=1

a

j

(g)b

j

(x)=

M

j=1

˜a

j

h

g

j

(x) . (20)

The relation between {a

j

(g)} and {˜a

j

(g)} can be

found by a projection of both sides of equation (20)

on b

m

(x)

a

m

(g)=

M

j=1

a

j

(g)h

g

j

(x),b

m

(x)

(21)

=

M

j=1

˜a

j

(g)

h

g

j

(x),b

m

(x)

B

jm

.

This can be written as a matrix/vector operation with

˜

a

T

:= (˜a

1

, ˜a

2

,...,˜a

n

) and a

T

:= (a

1

,a

2

,...,a

n

)

a = B

˜

a . (22)

The matrix B describing the basis change is invertible

˜

a = B

−1

a (23)

and the steerable basis can be designed as steered ver-

sions of the original ﬁlter kernel.

3.4 Relation to Recent Approaches

We present a steerable ﬁlter approach for comput-

ing the basis functions and interpolation functions for

arbitrary Lie groups. Since two steerable ﬁlter ap-

proaches based on Lie group theory (Michaelis and

Sommer, 1995; Teo and Hel-Or, 1996; Teo and Hel-

Or, 1998) have already been developed, the purpose

of this section is to examine their relation to our ap-

proach.

Freeman and Adelson (Freeman and Adelson,

1991) consider steerable ﬁlters with respect to the

rotation group in 2D and 3D, respectively. For the

2D case they propose a Fourier basis (of the function

space) times a rotational invariant function as well as

a polynomial basis (of the function space) times a ro-

tational invariant function as basis functions of the

steerable ﬁlter. They realized that the minimum re-

quired set of basis functions depend on the kind of ba-

sis itself but their approach failed to explain the reason

for it. Michaelis and Sommer (Michaelis and Som-

mer, 1995) answer this question based on Lie group

theory: the basis of an irreducible group represen-

tation span an invariant subspace of minimum size.

Since the Fourier basis is the basis for an irreducible

representation of the rotation group SO(2), the re-

quired number of basis function is less as for the poly-

nomial basis. Our approach can be considered as an

extension of the approach of Michaelis and Sommer

from Abelian Lie groups to arbitrary Lie group trans-

formation. Whereas the approach of Michaelis and

Sommer construct the basis function from the gener-

ators of the group, our approach uses a Casimir op-

erator. Since the generators of an Abelian Lie group

commutate with each other, their linear combination

constitute a Casimir operator and thus both meth-

ods become equal in this case. But our method also

works for the case of general compact groups, since

in this case, a self-adjoint Casimir operator with a

discrete spectrum delivers ﬁnite dimensional invari-

ant subspaces. For non-compact, non-Abelian groups

we showed that polynomials serve always as basis for

an invariant subspace. The approach of Teo and Hel-

Or signiﬁcantly differs from our approach in the way

how the invariant subspaces are generated. The basis

functions of the invariant subspace are constructed by

applying all combinations of Lie group generators to

the function that is to be made steerable. A certain

sequence of generators, denoted as generator chain

in case of Abelian Lie groups and generator trees in

the case of non-Abelian Lie groups, is stopped if the

resulting function is linearly dependent to the basis

functions which have already been constructed. In the

following, we will show that this approach may fail.

Let us consider the function h(x, y) = exp(−x

2

)

and the rotation in 2D as the group transforma-

tion. Applying the generator chain which is sim-

ply the successive application of the group generator

L = x

∂

∂y

− y

∂

∂x

does not converge since h(x, y) is

not expandable by a ﬁnite number of basis functions

of a representation of the rotation group. In our ap-

proach, h(x, y) is ﬁrst approximated by a ﬁnite num-

ber of basis functions each belonging to a ﬁnite di-

mensional invariant subspace. Such a ﬁlter is always

steerable by construction.

4 GROUP INVARIANT

QUADRATURE FILTERS

Quadrature ﬁlters have become an appropriate tool for

computing the local phase and local energy of one di-

mensional signals. They are obtained by a bandpass

ﬁlter and its Hilbert transformation. The bandpass ﬁl-

ter is applied to reduce the original signal to a signal

with small bandwidth which is necessary to obtain

a reasonable interpretation of the local phase. The

Hilbert transformation is applied to shift the phase

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

52

Table3.1: Several examples of Lie groups, the corresponding operator(s), generator(s), Casimir operator(s) and basis func-

tions. Terminology: T

N

: translation group in the N-dimensional Signal space; SO(N): special orthogonal group; U

N

:

uniform scaling group; S

N

: shear group.

Group Operators Generators Casimir operator basis functions

T

N

D(a)h(x)=h(x − a) {L

i

=

∂

∂x

i

} C =

N

i=1

L

2

i

exp

jn

T

x

SO(2) D(α)h(r, ϕ)=h(r, ϕ − α) L =

∂

∂ϕ

C = L

2

{f

k

(r)exp(jkϕ)}

SO(3) Rh(x)=h(R

−1

x) {L

k

= x

j

∂

∂x

i

− x

i

∂

∂x

j

} C =

3

i=1

L

2

i

{f

k

(r)Y

m

(θ, ϕ) }

U

N

D(α)h(x)=h(e

−α

x) {L

i

= x

i

∂

∂x

i

} C =

N

i=1

L

2

i

{r

k

}

S

N

D(u)h(x,t)=h(x − ut, t) {L

i

= t

∂

∂x

i

} C =

N

i=1

L

2

i

{f

k

(t)exp(jkx/t)}

of the original signal by ninety degrees such that the

squared sum of the output of the bandpass and its

Hilbert transformation results in a phase invariant lo-

cal energy. In order to apply this concept to image

processing, large efforts have been made to general-

ize the Hilbert transform to 2D dimensional signals

(B

¨

ulow and Sommer, 2001; Granlund and Knutsson,

1995). All of these approaches fail to be rotational in-

variant, but rotational invariance is an essential prop-

erty of all feature detection methods. An appropri-

ate 2D generalization of the analytic signal is the

monogenic signal which is based on the vector-valued

Riesz transformation (Felsberg and Sommer, 2001).

The Riesz transformation is valid for all dimensions

and reduces to the Hilbert transformation in the one

dimensional case. A further generalization of 2D ro-

tational invariant quadrature ﬁlters can be done by

steerable ﬁlters which behave under certain condi-

tions like quadrature ﬁlter pairs (K

¨

othe, 2003). The

monogenic signal is included in this approach. We

will go a step further, using the theory of Lie groups

and the steerable ﬁlter approach presented in the last

section to develop a generalization of the rotational

invariant quadrature ﬁlters to quadrature ﬁlters which

are invariant to compact or Abelian Lie groups and is

also valid for arbitrary signal dimensions. In particu-

lar, we are able to design rotational invariant quadra-

ture ﬁlters in 3D. But also feature detection methods

that are invariant with respect to other transformation

groups are important as in the case of motion estima-

tion. The signal in the space time volume is sheared

and not steered by motion and thus ﬁlters for detect-

ing this signal shall be designed invariant with respect

to the shear transformation.

4.1 Properties of a General

Quadrature Filter

We will ﬁrst recall the main properties of a quadrature

ﬁlter. The main idea of a quadrature ﬁlter is to apply

two ﬁlters to a signal such that the sum of the square

ﬁlter responses reﬂect the local energy of the signal.

Also the local phase of the selected frequency band

shall be determined by the two ﬁlter outputs. Further-

more, the local energy shall be group invariant, i.e.

the ﬁlter outputs shall be invariant with respect to the

deformation of the signal by the corresponding group.

In order to achieve group invariance, we construct our

quadrature ﬁlter from the basis of an unitary group

representation. Groups with an unitary representa-

tion are compact groups and Abelian groups (Wigner,

1959). The even h

e

and odd h

o

components of the

quadrature ﬁlter are constructed by a vector valued

impulse response consisting of the basis functions of

an unitary representation of dimension m

e

and m

o

,

respectively.

h

e

=

⎛

⎜

⎜

⎝

h

e1

(x)

h

e2

(x)

.

.

.

h

em

e

(x)

⎞

⎟

⎟

⎠

, h

o

=

⎛

⎜

⎜

⎝

h

o1

(x)

h

o2

(x)

.

.

.

h

om

o

(x)

⎞

⎟

⎟

⎠

. (24)

In the following we show that all basis functions of

an invariant subspace generated by a Casimir opera-

tor which is point symmetric, i.e. commutates with

the mirror group that acts on the coordinate vector

like Px →−x, own the same parity. Since the parity

operator commutates with the Casimir operator, there

exists simultaneous eigenfunctions. Applying P two

times equals the identity operator and thus the eigen-

values of P are λ = ±1. Thus every basis function

has a certain parity, i.e. is either point symmetric or

point anti-symmetric

Pb

j

(x)=±b

j

(x) . (25)

Let us consider one basis function b

1

(x) with positive

parity, i.e. Pb

1

(x)=b

1

(x). All other basis func-

tions of the same subspace can be generated by linear

combinations of generators of G , where k

i

j

denote the

coefﬁcients of the linear combination and n the num-

ber of generators of G

n

j=1

k

i

j

L

j

b

1

(x)=b

i

(x),i=1, 2, ..., m

e

. (26)

Applying the parity operator on both sides of equ.(26)

and considering that P commutates with all genera-

AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS

53

tors yields

P

n

j=1

k

i

j

L

j

b

1

(x)=Pb

i

(x) (27)

⇔

n

j=1

k

i

j

L

j

b

1

(x)=Pb

i

(x)

⇒ Pb

i

(x)=b

i

(x) ∀i.

If we assume Pb

1

(x)=−b

1

(x) we obtain with the

same deduction Pb

i

(x)=−b

i

(x) for all basis func-

tions of this subspace. Thus, all basis functions be-

longing to the same subspace attain the same par-

ity. The ﬁlter responses of h

e

and h

o

are denoted

as the ﬁlter channels c

e

= s(x) ∗ h

e

(x) and

c

o

= s(x) ∗ h

o

(x), respectively. The square of

the ﬁlter response of each channel are denoted as even

and odd energies. Due to the unitary representation,

both energies are invariant under the corresponding

group action

E

s

=(D(g)c

s

)

T

(D(g)c

s

)=c

T

s

c

s

s ∈{e, o} .

Note that the inner product is taken with respect to

the invariant subspace, not with respect to the func-

tion space. The local energy of the signal is given by

the sum of the even and odd energy. In the follow-

ing we will examine the properties of the ﬁlter chan-

nels required to achieve a phase invariant local energy

when applied to bandpass signals. In the ideal case, a

simple

1

bandpass ﬁltered signal consists of only one

wave vector k

0

and its Fourier transform

2

reads with

the Dirac delta distribution δ(k)

S(k)=S

0

δ(k − k

0

)+S

0

δ(k + k

0

) . (28)

We start with examining the Fourier transform of the

even and odd energies

E

s

= c

T

s

c

s

=

m

s

j=1

(s(x) ∗ h

sj

(x))

2

. (29)

Applying the convolution theorem to E

s

reads

F{E

s

}(k)=

m

s

j=1

(S(k)H

sj

(k)) ∗ (S(k)H

sj

(k)) .

Inserting the signal (28) in the equation above, com-

puting the convolution and performing the inverse

1

simple signal: signal with intrinsic dimensionality one.

2

Note that the Fourier transformed entities are labeled

with capital letters.

Fourier transformation reads

E

s

(x)=S

2

0

m

s

i=1

(H

si

(k

0

))

2

e

4πjk

T

0

x

(30)

+ S

2

0

m

s

i=1

(H

si

(−k

0

))

2

e

−4πjk

T

0

x

+ S

2

0

m

s

i=1

H

si

(k

0

)H

si

(−k

0

)

+ S

2

0

m

s

i=1

H

si

(−k

0

)H

si

(k

0

) .

Note that the ﬁrst two terms are phase variant whereas

the last two ones are not. In order to achieve phase in-

variant local energy, the ﬁrst two terms have to cancel

when adding the even and odd energy. This is exactly

the case when all basis functions of one invariant sub-

space are either even or odd and the sum of squared

Fourier transformed ﬁlter components are equal

m

e

i=1

|H

ei

(k

0

)|

2

=

m

o

k=1

|H

ok

(k

0

)|

2

. (31)

All basis functions of one invariant subspace are ei-

ther even (= their Fourier transforms are real and

even), or odd (= their Fourier transforms are imagi-

nary and odd). Thus, the Fourier transformed ﬁlter

components become

m

e

i=1

H

ei

(±k

0

)

2

=

m

e

i=1

|H

ei

(k

0

)|

2

(32)

in the even case and

m

o

i=1

H

oi

(±k

0

)

2

= −

m

o

i=1

|H

oi

(k

0

)|

2

(33)

in the odd case. Since the inner product of the Fourier

transform of both ﬁlter channels are equal, the ﬁrst

two terms cancel out resulting in a phase invariant lo-

cal energy

E =2S

2

0

⎛

⎝

s

j=1

|H

nj

(k

0

)|

2

+

d

k=1

|H

mk

(k

0

)|

2

⎞

⎠

.

The local phase φ of an intrinsic one dimensional sig-

nal is given by

tan(φ)=

m

o

i=1

(h

oi

(x) ∗ s(x))

2

1

2

m

e

i=1

(h

ei

(x) ∗ s(x))

2

1

2

. (34)

In the next section an example of an group invariant

quadrature ﬁlter is presented.

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

54

4.2 An Example: 3D Rotational

Invariant Quadrature Filters

We now apply the approach presented in the last sec-

tion to the 3D rotational invariant quadrature ﬁlter.

The even h

e

and odd h

o

vector valued impulse re-

sponses have to be the basis functions of an unitary

representation of the rotation group SO(3). A possi-

ble basis of an unitary invariant subspaces are the well

known spherical harmonics times an arbitrary radial

function f

n

(|x|) ∈ L

2

b

nm

(x)=f

n

(|x|)Y

m

(

ˆ

x) . (35)

The spherical harmonics are either even or odd, thus

the even vector valued impulse responses can be con-

structed from all spherical harmonics of even or-

der, the odd vector valued impulse response from

all spherical harmonics of odd order. According to

equ.(31), we have to show that the scalar product

of the Fourier transformed vector valued impulse re-

sponses are equal. It is well known that a radial func-

tions times a spherical harmonic is also spherical sep-

arable in the Fourier domain and vise versa. If we

require, like in the 2D case (K

¨

othe, 2003), the radial

function F

nm

(|k

0

|)=F (|k

0

|) in the Fourier domain

to be the same for all transfer functions, the constraint

equ.(31) becomes

odd

m=−

|Y

m

(

ˆ

k

0

)|

2

=

even

m=−

|Y

m

(

ˆ

k

0

)|

2

. (36)

Since the scalar product of the even as well as the odd

spherical harmonics are rotational invariant the right

and the left hand side of equ.(36) is constant. There-

fore, the constraint equation can be always be fulﬁlled

by an appropriate scaling of the spherical harmonics.

5 CONCLUSION

We have presented a theory for steerable ﬁlters and

quadrature ﬁlters based on Lie group theory. Both

approaches are most general with respect to the

signal dimension as well as with respect to the trans-

formation Lie group. For the steerable ﬁlter case, we

provide for every quadratic integrable function (at

least approximately) the method for constructing the

basis functions for every Lie group transformation.

For compact and Abelian groups we even showed

that this is the minimum required number of basis

functions. Furthermore, we generalized the 2D

rotational invariant quadrature ﬁlter approach with

respect to arbitrary dimension of the signal space and

to Lie group transformation which own an unitary

representation. It turned out that the group invariant

quadrature ﬁlter is a special steerable ﬁlter. The

future work will be the integration of the general

quadrature ﬁlter approach into a tensor representation

and its application to motion and orientation estima-

tion in 3D.

This work was supported by DFG ME 1796/5 - 3

and DAAD 313-PPP-SE/05-lk.

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55