a group-theoretic construction with spatiotemporalwavelets for the analysis of rotational motion

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Journal of Mathematical Imaging and Vision 17: 207–236, 2002

c 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

A Group-Theoretic Construction with Spatiotemporal Wavelets

for the Analysis of Rotational Motion

JEAN-PIERRE LEDUC

Department of Mathematics, University of Maryland, 1301 Mathematics Bldg, College Park,

MD 20742-4015, USA

[email protected]

Abstract. This paper presents a group-theoretic approach for the analysis of rotational motion in image sequences.

This method relies on Lie algebras, Lie groups and Lie group representations to provide not only the continuous

wavelets but also the related tools of harmonic analysis. This approach can be referred to research works presented

in J.S. Byrnes et al. (Wavelets and their Applications, Kluwer Academic Publishers, 1994) who strongly influenced

this topic. For the purpose of modeling motion transformations, this paper introduces the concepts of Lie algebras

and Lie groups as the actual mathematical foundations of all the observable kinematics embedded in spatio-temporal

signals. Rotationalmotion analysis focuses on the estimationof angular velocity and angular accelerations embedded

in image sequences. Rotational motion is usually carried on a trajectory, the complete problem at hand consists in

estimating not only the angular velocity and its temporal derivatives but also the position, the translational velocity

and its temporal derivatives along the carrier trajectory. The paper starts with the usual affine and Galilei groups

and proceeds by successive extensions and sections to the rotational group. The theory of group representations is

central to provide families of continuous wavelets, special functions, PDE’s, ODE’s and integral transforms as newmathematical tools of motion analysis in image sequence to perform optimal and selective detection, estimations,

tracking, and reconstructions. This paper defines rotational wavelets and proposes a structured approach to perform

estimation and tracking in image sequences which fits to Kalman filters. Simulations on real digital image sequences

are also presented with tracking and estimation.

Keywords: Lie group theory, motion analysis, tracking, continuous wavelet transforms, image sequence analysis

1. Introduction

This paper addresses new applications of Lie group

representation theory for digital image processing.More precisely, the purpose of this work is the anal-

ysis of rotational motion in digital image sequences.

To achieve this goal, a theoretical framework is built

on the following structural steps: construct the ap-

propriate Lie groups and Lie algebras that model the

physics and the kinematics [1, 3] of rotational mo-

tion, construct the related group representations in

the function spaces of signals, define new families of

spatio-temporal continuous wavelets, develop the re-

lated harmonic analysis and design the related track-

ing algorithms. In this context, continuous wavelets

are defined from square-integrable unitary irreducible

representations of Lie groups. Following this vein, this

paper derives rotational wavelets from a Lie group con-

struction that starts with the multidimensional affinegroup and the Galilei group. The multidimensional

affine group is a homogeneous extension of the uni-

dimensional affine group. The Galilei group is the

group of classical mechanics which deals with veloc-

ity transformation. The construction proceeds to the

definition of rotational wavelets as families of con-

tinuous wavelets indexed by the following kinemat-

ical parameters: scale, spatial and temporal position,

translational velocity, angular position and angular ve-

locity. In this construction, any family of continuous

wavelets stems from the action of a Lie group G onto



208 Leduc

operators in the functional space of the signals. This

action is defined in terms of a homomorphism i.e. aone-to-one mapping from the group elements g ∈ G

to linear and invertible operators T g. These operators

are built as Unitary Irreducible Representations (UIRs)

of a Lie group fulfill square integrability under ad-

ditional conditions. Square-integrable representations

define continuous wavelet transforms. These continu-

ous wavelets are endowed with more properties than

those rooted in an usual cross-correlation function or

a matched filter [31]. The continuous wavelet trans-

form is an isometry defined on the Hilbert space of

observation H o = L2(Rn , d x) whose range is a closed

subspace of the Hilbert space of the state parameters

H s = L2(G, dg ). The set of kinematical parameters gdefines a Lie group G and characterizes the state of the

moving system. dg is an invariant Haar measure on G

tobe defined. The continuous wavelet transform is also

optimal to many regards; it performs Minimum-Mean-

Square-Error (MMSE) motion estimation [31], edge

detection, image noise rejection, motion jitter smooth-

ing, frame decompositions, interpolations, predictions,

and motion-selective reconstructions [2, 26, 30].

The definition of rotational motion [3] depends upon

the position of the axis around which the motion takes

place. If the axis is the center of inertia of the object, the

motion refers to a spin. If the object revolves aroundan external axis, the motion refers to an orbit. Fur-

thermore, the rotations are expressed through unitary

matrices of transformation namely

R(θi τ i ) =

cos(θi τ i ) − sin(θi τ i )

sin(θi τ i ) cos(θi τ i )

with x 2

1 + x 22 -invariance for circular rotations and

R(φi τ i ) =

cosh(φi τ i ) −sinh(φi τ i )

sinh(φi τ i ) cosh(φi τ i )

with x 2

1 − x 22 -invariance for hyperbolic rotations. This

paper will focus on both rotational aspects and illus-

trate the first case within image sequences. Rotational

motion composes with translational motion. This pa-

per shows how estimation and tracking of translational

and rotational parameters can be dissociated from each

other.

The approach developed in this paper is based on

Lie group representations for signal processing as orig-

inally developed by Antoine et al. in [2] where the

prime focuswas mathematical physics and the wavelets

on the Galilei, Poincare and de Sitter groups and by

Caelli et al. in [11] where the interest was human vi-sion modeled by the Lorentz group. Segman and Zeevi

in [38, 41, 42] stepped on Caelli approach and recog-

nized the relevance of the group-theoretic methods for

computer vision. Duval-Destin and Murenzi [12] have

also been focusing on perceptive aspects of the hu-

man visual system to design the first spatio-temporal

continuous wavelets parametrized with speed. Let us

also remark that the concept of rotating wavelets has

been mentioned in a different context by O.A. Barut

[4]. Barut’s wavelet functions are solutions of the Dirac

and wave equations. Let us also mention the early work

of Fleet and Jepson [13] who developed without any

group-theoretic support the first orientation and veloc-ity selective filters. This paper proceeds fromAntoine’s

original point of view to develop new extensions of

the Galilei group [33] based on observable kinemat-

ics in the functional spaces of the signals. The author

prefers to start from the Lie algebra. Indeed, Lie al-

gebras provide an easy and general way to define the

kinematic and geometry. The construction proceeds to

the calculation of the Lie group and its representations

[28, 29, 31]. At that stage, all the concepts of harmonic

analysis [27] may be revisited and related to the kine-

matics at hand. As a major property of this approach,

all the relevant concepts can be analytically computedand numerically estimated in the signal. The concepts

of interest in harmonic analysis are as follows: the anal-

ysis of uncertainty relations, ODE’s, PDE’s, special

functions and integral and wavelet transforms, exis-

tence of frames and orthogonal bases. Moreover, con-

tinuous wavelet transform on motion transformations

fits perfectly to important theory like Kalman filtering

[30, 31] and irregular sampling [27]. Moreover, con-

tinuous wavelet transform provides a Wold decomposi-

tion of the signal considered as a wide-sense stationary

stochastic processes [15], maximum likelihood estima-

tions, and links to stochastic calculus and Ito integrals

[27]. From this point of view, the group-theoretic ap-

proach presented in this paper differs fundamentally

from other techniques [32] that have been proposed so

far in the literature [13] such as those based on opti-

cal flow, pel-recursive, block matching and Bayesian

models and addresses the actual physics acting upon

the signal. Continuous wavelet transforms also provide

motion estimations that are robust not only against im-

age noise and blur but also against motion jitter [31].

Moreover, as a result of their spatio-temporal filter-

ing and interpolation properties, the wavelet technique



A Group-Theoretic Construction with Spatiotemporal Wavelets 209

can resolve temporary occlusions problem supply by

interpolation the objects at the occluded positions.This paper presents a tracking algorithm based on a

Kalman filter which relies on the exponentiation of the

Lie algebra to generate the state equation of the sys-

tem. The Galilean and rotational wavelets perform se-

lective motion estimation and selective reconstruction

by means of a gradient-based technique which provides

an optimal search of parameters. The wavelet transform

eventually contributes as a motion-selective filter in the

observation equation.

The structure of the paper is as follows. Lie groups

and algebras for rotational motion are developed in a

constructive way along with the related wavelets, and

admissibility conditions. This constructive approachproceeds by successive extensions and sections on al-

gebras and groups. This technique provides an use-

ful way to create many other variants of interest for

applications (for instance, deformations instead of ro-

tations, accelerations instead of velocity). The UIRs

are constructed using the Mackey’s theory of induced

representations and imprimitivity [34] on semi-direct

products [5, 14], and the Kirillov’s method of orbits. It

is shown how the characters of these new groups gen-

erate special functions similar to Bessel functions [8].

Thespecial functionsdefine thekernel of integraltrans-

forms similar to Hankel transforms and the samplingof rotational motion which is related to the symmetries

of the system. The connection with the harmonic oscil-

lator in the group SU (2) and S L(2,C) is investigated

to relate this approach to some physical concepts. An

alternative Lie algebraic approach is finally proposed

for motion tracking and estimation in order to build

the motion trajectories. Simulations on real image se-

quences illustrate how to build the rotational wavelets

and how to perform the parameter estimation. To help

the interested reader, four appendixes sketch compu-

tational techniques that are classically related to the

group constructions.

2. Lie Group Representations and Continuous

Wavelet Transforms

This section contains the main purpose of this pa-

per: the construction of group representations for the

analysis of both translational and rotational motion.

These representation provides the related continuous

wavelets which support the motion estimation filter-

ing and tracking. To clarify the construction, a brief

description of Lie group representation theory is first

provided with some useful references in this field. In

each case of study, we adopt a similar structure whichproceeds with the construction of Lie groups of inter-

est, their Lie algebras, their group representations and

their associated wavelets. The group construction pro-

gresses by successive extensions and sections. Exten-

sions add new group parameters of interest. Sections

add a constraint on the parameters i.e. diminish the

dimension of the parameter space. For the shake of

clarity, our procedure will start from two well-known

groups: theaf fine group in multidimensional space,and

the extended Galilei group which introduces the trans-

lational velocity. The construction will proceed as fol-

lows. A section on the extended af fine-Galilei group

provides the appropriate representations for image se-quence processing. The next step consists of an new

extension followed by a section to add the angular po-

sition and velocity on the later version of the af fine-

Galilei group. This procedure is fairly general; similar

extension-section procedures enable to incorporate

any additional temporal derivative of translational,

rotational and deformational motion.

2.1. Group Representations

Let S(

x, t ) be the spatio-temporal signal under anal-

ysis and ( x, t ) the analyzing function in the Hilbert

space of observation H o = L2(Rn × R, d n xdt ). The

conditions of existence of such an analyzing func-

tion , called a wavelet, are examined latter in this

section. The spatial dimension is n = 2 when con-

tinuous wavelets are discretized to process image

sequences. The continuous wavelet transform W is

defined [10, 20, 21, 26, 30, 31] as a linear map W :

L2(Rn × R, d n xdt ) → L2(G, dg) as follows

[W S](g) = g, S (1)

= Rn×R¯

g( x, t )S( x, t )d

n

x dt (2)

where ., . is an inner product which expresses the

correlation between two functions defined in H o and

the overbar symbol “ ” stands for the complex con-

jugate. Let us denote g ∈ G as an element of the Lie

group G that models the motion transformations. The

element g ∈ G carries all the motion parameters of in-

terest. Hence, g is precisely the state of the moving sys-

tem. Therefore, H s = L2(G, dg ) is clearly the Hilbert

state space. The family of wavelets g is generated as

an orbit with dense span in H o through the action of an



210 Leduc

operator T g such that g( x, t ) = [T g]( x, t ). The map

g → T g is a group representation i.e. a homomorphismthat maps the group elements g ∈ G to operators T gacting in H o. This mapping is such that T g1◦g2

= T g1T g2

and T e = I H o . I H o defines the unit operator in H o and

e the identity element in G. The symbol ◦ stands for

the group composition in G. In the following, all the

group representations T g are constructed as UIRs and

calculated by the technique of induced representations

that invokes the Mackey’s theory of induced represen-

tations [5, 34, 36, 37] and the Kirillov’s method of

orbits [23]. The existence of continuous wavelet trans-

forms is further derived under the condition that the

UIRs be square-integrable. This condition of square-

integrability is equivalently known as the condition of wavelet admissibility and as the condition for the linear

mapping defined in Eq. (1) to be invertible. This third

point of view is referred as the Calderon reproducing

formula [39]. The inverse transform of Eq. (1) yields

the perfect reconstructions of the signal. This reads in

terms of operators I H o = W −1 W and in functional

forms as

S(x) =

G

T g , S(T g)(x)d µl (g) with x = ( x, t )

(3)

The measure d µl (g) is the left-invariant Haar measure

on G. Let us remark that x = ( x, t ) may alternatively

be defined in the Fourier domain as k = (k , ω) where

the variables k and ω are the dual of the translations

i.e. respectively the spatial and temporal frequencies.

In this brief introduction, we have seen how the linear

mapping definedinEq.(1)definesan isometry between

the Hilbert space H o and a sub-space H sr of the Hilbert

space H s . Let us notice that H sr is a reproducing kernel

space [2]. In fact, if the signal is a wide-sense stationary

stochastic process, the isometry operator provides a

Wold decomposition wherethe signal, an unpredictable

process, is decomposed into a regular process and anorthogonal predictable process [15]. The predictable

belongs to H sr and the former process belongs to the

complementary Hilbert space ( H sr )⊥.

Let us now restart with the construction of left-

regular UIRs. They read in H o

[T g](x) = λ( x, g)1/2(g−1x) with f ∈ H o (4)

The function λ( x , g)1/2 is usually a one-dimensional

representation that contains the characters and the nor-

malization factor that provides unitarity. These UIRs

[T g](x) are usually calculated by the technique of in-

duced representations [23, 34]. The representation T g isindeed induced from the representation L K of a closed

subgroup K of G in the Hilbert space H o. G is a locally

compact group and K ⊂ G. Let us define X as the ho-

mogeneous space of left cosets X = G\K ={g K , g ∈G} and denote µ a quasi-invariant measure in X . The

group representations constructed with the technique

referenced here above are derived in a dual space

(Appendix 1). The dual space ˆ X is a Fourier domain

known as the phase space made of patial and temporal

frequencies, k and ω, as respective duals of the spatio-

temporal translations. When the Lie group admits a

non-trivial central extension [40], the phase space also

contains the associated dual parameter which is de-noted m in the sequel and known as the dual of the

phase φ in the group. Eventually, Eq. (4) reads as the

following mapping

IND ↑GK L K : g → [T (g)](k)

= χ 1/2(g)ρ(b, k) L K (g−1k) with

k = (k , ω, m) ∈ ˆ X and b = (b, τ , φ) ∈ X (5)

which defines an induced representation IND ↑GK of G

by L K in L2( ˆ X , d µ). In this equation, χ (g) = µ(g−1k)

µ(k)is

the Radon-Nikodym derivative continuous on G

×X ,

K is a stability sub-group (also called little group) of

G whose representation is L K , ρ (b, k) are the char-

acters related to the dual group ˆ X , g−1k is the left

group action on X expressed by the co-adjoint action

of the group [23]. Therefore, L2( ˆ X , d µ) = H o. We

also know that H o on space and time corresponds to

the Fourier space. If there exists a system of imprim-

itivity defined as the triple (T g, X , µ) and if K ⊂ G

is a commutative stability subgroup, then for any UIR

L K , (IND ↑GK L K ) is also UIR [34]. It turns out that all

semi-direct products G = N S where N is commu-

tative, have a system of imprimitivity [5]. The group

representations treated in this paper belong to thiscategory.

The additional condition for unitary irreducible rep-

resentations to fulfill square-integrability provides a set

of important properties that are not rooted in standard

unitary irreducible representations [2]. For instance,

the condition of square-integrability the condition for

continuous wavelet admissibility, sets up the isometry

as defined above in Eq. (1) with a reproducing ker-

nel sub-space of H sr = L2(G, d µl (g)). The projection

from L2[G, d µl (g)]) onto H o is the integral operator

with reproducing kernel K (g, g) = g, g, that is,




the autocorrelation function of . For a given non-zeroˆ

∈ H

o

, the condition reads

C (, η) =

G

|T g, η|2d µl (g) < +∞ ∀η ∈ H o

(6)

where d µl (g) is the left invariant Haar measure on G

and C (, η) is some finite real constant. The functions

that fulfill the condition are said to be admissible as

a mother wavelet. As a consequence of irreducibility, g defines an orbit with dense span in H o. The orbit

is defined as S g = {

g = T g | g ∈ G; ,

g ∈ H o}

i.e. a family of continuous wavelets indexed by the

motion parameters in g. Hence, square-integrability isa well-defined property of a pair (T g ,O), where T gis a representation and O is an orbit. The condition

of wavelet admissibility is calculated from Eq. 6 by

integrating over G with dg an invariant Haar measure

for G. This construction yields the state space as a sub-

space of L2(G, dg ) with reproducing kernels [2], and

the map W ψ with all the properties of the isometry i.e.

the perfect reconstruction [2], the MMSE estimations

[30]. In Section 6, the Morlet wavelets will be chosen

as admissible rotational wavelets (k , ω) to perform

the numerical simulations. The subsequent sections are

devoted to the characterization of the parameter setsg, the calculation of the UIRs and the conditions of

wavelet admissibility.

2.2. The Multidimensional Affine Group

The n-dimensional af fine group is similar to the one-

dimensional case. In n-dimensional space, a rotation

matrix can be introduced to provide an additional

parameter for the estimation of spatial orientation.

This generalizes the one dimensional case of reflec-

tion which is never used for the practical purpose of

motion analysis. As a result of the small amount of parameters carried by this group, the af fine group is

a best place to start and illustrate the technique. Let

us also remark that the rotations addressed in this pa-

per primarily consist of circular rotations. Hyperbolic

rotations also fit to this approach and will be briefly

discussed in this paper. To distinguish among the suc-

cessive Lie groups and related Lie algebras to be pre-

sented, we adopt the generic notation Gi and Gi re-

spectively for each group and algebra. The integer

i ≥ 1 determines the position the sequence of group

definitions.

The n-dimensional af fine group, denoted here G1 =SIM(n), is an ordered 3-tuple of elements

g = {b, a, R} (7)

where the parameters b ∈ Rn , a ∈ R+\{0} and R ∈

SO(n) stand respectively for translation in space (the

Cartesian position), dilation (the scale) and rotation

in n-dimensional space (the angular orientation). This

group is a subgroup of GL(n +1,R) with the following

matrix representation

g

= a R b

0 1 (8)

Let g = {b, a, R} and g = {b, a, R} be elements of

G1. Consequently, the associative law of composition

and the inverse element may be immediately identified

from matrix multiplication and inversion. These are as

follows

g ◦ g = {b + a Rb, aa , R R} (9)

and

g−1 = {−a−1 R−1b, a−1, R−1} (10)

Let us notice that the Lie group G1 has the following

semi-direct group structure

G1 = Rn[R∗

+ × SO(n)] (11)

where the symbols × and stand respectively for the

direct and the semi-direct group products.

The left and right Haar invariant measures, respec-

tively d µl (g) and d µr (g) for G1 are easily computed

through infinitesimal variations of volumes in the mul-tidimensional space of group parameters {gi}n

i=1 and

are given for the left-invariant measure by the general

form

d µl (g) = det

∂g ◦ g

∂g

−1

g=e

dg1 ∧ dg2 ∧ · · · ∧ dgn

(12)

where ∧ is the exterior product, and for G1, by

d µl (g) = da d b dm( R)

|an+1| (13)



212 Leduc

The right-invariant measure is given in its general form

as

d µr (g) = det

∂g ◦ g

∂g

−1

g=e

dg1 ∧ dg2 ∧ · · · ∧ dgn

(14)

and for G1 as

d µr (g) = da d b dm( R)

|a| (15)

where dm ( R) is the left and right Haar measure for

SO(n). Equations (12) and (14) are easily demonstrated

see in [5, 21].Let us now derive the Lie algebra related to G1 =

SIM(n). Let { J i}n(n−1)

2

i=1 be the standard generators for

so(n), the Lie algebra of SO(n). Then, the Lie alge-

bra G1 of G1 is generated by the elements {P j , Q, J i}where j = 1, . . . , n and i = 1, . . . , n(n−1)

2and the

commutator relations are given as follows

[Pk , P j ] = 0, [Q, P j ] = P j , [ J i , P j ] = k i j Pk ,

[Q, J i ] = 0 ∀i, j, k (16)

where k

i j

is theantisymmetric tensor. It is easy to verify

that these generators form a Lie algebra i.e. all the com-

mutators fulfill anti-symmetry and Jacobi relations. Let

us recall that the Lie algebra is the vector space tangent

to thegroup at identity andendowedby theLie product.

G1 is generated by the set of infinitesimal generators

{P j , Q, J i }. The Lie product is the commutator prod-

uct [ A, B] = AB − B A where A and B are any two

generators. See [5, 17] to read all the major properties

of Liealgebras. It is clear that theLie algebra hasmatrix

representations (Ado’s theorem [5]). We can now ob-

tain the generic group element g = {b, a, R} ∈ G1 and

the composition law by exponentiating a point Y

∈G1.

It is known that the exponential map is a local diffeo-morphism of G onto G. We may denote an element of

G by

X = b P + s Q + r J = {b, s, r } ∈ G. (17)

Thereafter, with g = E ( X ) and E the exponential map,

we have

E ( X ) = exp(b P) exp(s Q) exp(r J ) (18)

with a = es ∈ R+\{0}, R = exp(r ) ∈ SO(n).

The UIRs T g of G1 in the Hilbert space L2(Rn , d nk )

can be thoroughly calculated using the machinery de-veloped in [4, 6, 23, 24, 37, 43]. This computation

requires to proceed through a set of new definitions.

The procedure is sketched in Appendix 1 for this case

which requires the smallest amount of computations.

The UIRs of G1 = SIM(n) are eventually given in the

forms of Eq. (5) in the Fourier space L2(Rn , d nk ) as

[T g](k ) = an2 eib·k (k ) (19)

with

k

=a R−1

k (20)

By construction (see Appendix 1, Eq. (99)), the param-

eter k stands for the dual parameter of x in the duality

defined in Eq. (99). The parameter k has to be interpre-

tated as a Fourier variable for the spatial frequency [2].

In this representation, it is easy to see that the normaliz-

ing factor an2 provides unitarity, and the characters ei b·k

of Rn define the Fourier kernel. The change of variable

to k stems from the action of the group on ˆ X = {k }dual of X = { x}, this is clearly g−1 ˆ X .

At this stage, the function (k ) ∈ L2(Rn , d nk ) can

be considered as a template to be used as a matched

filter but thefunction (k ) requires more to be admittedas a continuous wavelet. To define a matched filter for

a signal S(k ) ∈ L2(Rn, d nk ), the function T g(k ) has

just to fulfill the following well-known condition =¯S to yield the equality in Cauchy-Schwarz inequality.

In this approach, we have defined the cross-correlation

function on the group as [C S](g) a linear map, i.e. the

following function g

[C S](g) = T g , S (21)

In order to obtain the continuous wavelet transform

from this cross-correlation function, the linear 21 map

has to become an isometry. According to Eq. (6), the

condition of square-integrability reads in this case Rn

+∞

−∞|T g , |2 da d nb

an+1< +∞ (22)

and, eventually after some computations and changes

of variables, continuous wavelets do exist under the

condition ∈ L2(Rn , d nk ) such that +∞

−∞|(ξ )|2 d n ξ

|ξ | = C ψ < +∞ (23)




and the representation T g is bounded for all g. The

variable ξ ∈ Rn

is still a Fourier variable.Let us remark that the Weyl-Heisenberg group is an-

other important group which is not related to motion

analysis but can be treated in a similar way. The inter-

ested reader may find the corresponding calculations

in [2, 20].

2.3. The Af fine-Galilei Group

The Galilei group, referred here as G2 = GAL(n),

is the group of physics that describes the motion of

rigid objects with constant velocity v, in the Euclidean

spacetime. An element, g ofthe Galilei group isdefinedto be an ordered 4-tuple of elements, g = {b, τ, v, R};

R ∈ S O(n); b, v ∈ Rn, τ ∈ R realized a subgroup of

S L(n + 2,R) according to this matrix representation

g = {b, τ, v, R} =

R v b0 1 τ

0 0 1

∈ S L(n + 2, R).

(24)

Let us now combine the multidimensional af fine group

G1 = SIM(n) and the Galilei group G2 = GAL(n)

to define the af fine-Galilei group G3 = aff-GAL(n) asan ordered 6-tuple of elements g = {b, τ, v, a, a0, R}with R ∈ SO(n); b, v ∈ R

n , τ ∈ R, a, a0 ∈R

+\{0}. The associated Lie Algebra G3 is a real vec-

tor space of dimension n(n−1)

2+ 2n + 3 spanned by

the elements {Pi , H , K i , Q, Q0, J j ; i = 1, . . . , n; j =1, . . . , n(n−1)

2} where the infinitesimal generators are

definedin therespectiveorderof theparameters defined

in g. In this case, we have

exp(r ) = (0, 0; 0, 1, 1, R)

expi

bi Pi =(

b, 0;

0, 1, 1, I )

exp

i

vi K i

= (0, 0; v, 1, 1, I ) (25)

exp(τ H ) = (0, τ ; 0, 1, 1, R)

exp(s Q) = (0, 0; 0, a, 1, I )

exp(s0 Q0) = (0, 0; v, 1, a0, I )

where r = i αi J i ∈ so(n), es = a ∈ R+\{0} is

the spatial dilation and es0 = a0 ∈ R+\{0} is the tem-

poral dilation. By matrix multiplication, we can check

that

g = {b, τ, v, a, a0, R} ≡ E ( X )

= exp(b P) exp(τ H ) exp(v K ) exp(s Q)

exp(s0 Q0) exp(r ) (26)

where Y ∈ G3.

The attempt to construct continuous wavelets on

the af fine-Galilei group G3 = aff-Gal(n) fails in the

case n > 1 as a result of the divergence of square-

integrability. Fortunately, the af fine-Galilei group may

still be naturally extended with a central parameter φ.

By definition, this additional parameter is such that

its infinitesimal generator M commutes with all theother generators of the af fine-Galilei algebra. This

extension, called a central extension, is not rooted

in the af fine group. This new property will allow

our construction to solve the problem of wavelet ex-

istence in the n-dimensional af fine-Galilei case. In

fact, a dimension-matching change of variables from

(a0, a, v) to (m, ω, k ) can now be applied to yield the

condition of admissibility (see Theorem 1 below and

Appendix 1). The central parameter φ is a phase on

the group; its dual m can be interpretated exactly as

the uncertainty parameter between position and veloc-

ity sharing a whole theoretical similarity with the un-certainty in Weyl-Heisenberg group (for instance the

construction of the harmonic oscillator).

The structure of the extended af fine-Galilei group,

referred to as G4, is a semi-direct product of the form

G = N S with N = Rn+2, S = R

n[R∗

+ × R∗+ ×

S O(n)] where R∗+ means R+\{0}. A typical element,

x ∈ N is of the form x = (φ, b, τ ; 0; 1, 1, I n ) and a

typical element s ∈ S is s = (0, 0, 0; v; a, a0, R).

The extended af fine-Galilei algebraG4 = {G3, M } is

defined by the infinitesimal generators as follows

[ M , Z ]=

0∀

Z ∈G3 [K

i, P

j

]=

δi j

M [K , Z ]=

0

[K i , P j ] = δi j M [ J i , P j ] = k i j Pk [ J i , K j ] = −k

i j Pk

[ H , K i ] = Pi [Q0, H ] = H [Q0, K i ] = −K i

[Q0, M ] = − M [Q, M ] = 2 M [Q, Pi ] = Pi

[Q, K i ] = K i [ J i , J j ] = k i j J k (27)

Let us remark that the commutator [ H , K i ] = Pi

between time and position generate velocity in formb = vt with an uncertainty generated by the com-

mutator [K i , P j ] = δi j M . This structure will be ex-

ploited below to generate the angular velocity from the



214 Leduc

angular position in whole similarity. All the other com-

mutators not mentioned in this collection are equal tozero. Then, this set of generators defines a Lie algebra.

The af fine-Galilei group G3 consists of all elements

of the form E ( X ) where X ∈ G3. These elements are

g = {φ, b, τ ; v; a, a0, R} with φ the parameter of cen-

tral extension. The law of composition of the group can

be computed from the Lie algebra by exponentiating

the product of two elements X , X ∈ G3 i.e. computing

g ◦g = E ( X ) E ( X ) ∈ G3 and repeatedly applying the

Baker-Campbell-Hausdorff formula [44]. An example

of how to treat this kind of computation is provided in

[33]. A matrix representation for the extended af fine-

Galilei group G4 is still available and reads

g = {b, τ, v, a, a0, R}

=

a2

a0vT a

a0 R v2 1

2a0φ

0 a R v b0 0T a0 τ

0 0 0 1

(28)

and allows the computation of the associative law in

an easier way. The law of composition in the extended

af fine-Galilei group G4 is therefore given as

g ◦ g = φ + a2

a0

φ + vT · vτ 2a0

+ aa0

Rb,

b + a Rb + a0vτ , τ + a0τ ;

v + aa−10 Rv, aa , aa

0, R R

(29)

and the inverse element as

g−1 =−a0

a2φ + a−2v ·

a0vτ

2− b

,

− a−1 R−1(b − a0vτ ), −a−10 τ ;

−a0a−1 R−1v, a−1, a−10 , R−1 (30)

The left and right-invariant Haar measure of G are

given by

d µl = an−10 a−(2n+3)d φ dm( R) d b d v d τ da da0

(31)

and

d µr = a−10 a−1d φ dm( R) d b d v d τ da da0 (32)

We see that the group is not unimodular since the left

and right-invariant measures are not equal. In fact, it iseasy to see that the dilation parameters stretch neigh-

borhoods of the identity differently on the left and on

the right which causes unimodularity to fail.

The UIRs of G4, that is the the extended af fine-

Galilei group, arise in L2(Rn+2, d nkd ωdm) using the

method of orbits and the induced representation tech-

nique in the extended spatio-temporal space of dual

variables ˆ X = (m, k , ω). The variable ω stands for the

temporal frequency dual of the parameter τ . The UIRs

of G4 read as

[T g](m, k , ω) = an+2

2 ei(mφ+b·k +τ ω)(m, k , ω)

(33)

with

m = a2

a0

m

k = a R−1(k + mv) (34)

ω = a0

ω − mv2

2− v · k

for ∈ L2( X , dm d kd ω). Let us remark that the factor

an+2

2 provides unitarity; it arises as a Radon-Nikodym

derivative which can be equivalently calculated fromeither the co-adjunct action or the normalizing factor.

The factor ei (mφ+b·k +τ ω) comes from the monomial rep-

resentation which induces the UIRs and plays the role

of characters. The variable m measures the Heisenberg

uncertainty between velocity and position. m is a non-

zero real constant whose effect canbe actually observed

in image sequences [31].

Theorem 1 . The UIRs of G 4 the extended af fine-

Galilei group, in the space L2(Rn+2, dmd kd ω)

which are given by the mapping g → T g are square-

integrable. For a given non-zero

∈ L2(Rn+2, dm d

k

d ω), the condition of square-integrability is equiva-

lent to the statement that the following two integrals be

finite Rn+2

|(m, k , ω)|2 m[|k |2 + 2mω]2

dm d k d ω < +∞ (35)

and Rn+2

|(m, k , ω)|2 dm d k d ω

(m)n [|k |2 + 2mω]3< +∞ (36)




This theorem is proved in Appendix 2. Eventually, a

section is performed in the dual spaceˆ

X = (m, ω, k )on m fixing uncertainty and on the parameter space on

a0 fixing the time scale as follows

m = m0 = 0 a0 = 1 (37)

This section addresses the applications in image se-

quence processing and fulfills the condition for wavelet

admissibility with a dimensional-matching change of

variables from (a, v) to (ω, k ). Let us now apply on

G4 the section referred in Eq. (37) to define a group G5

that provides representations in L2(Rn+1, d nkd ω) with

n

=2 for image sequences. These UIRs are

T gm0

(k , ω) = a

n2 ei (b·k +τ ω)(k , ω) (38)

with

k = a R−1(k + m0v)(39)

ω =

ω − m0v2

2− v · k

Theorem 2 . The UIRs of G 5 (the group obtained

by sectioning the extended af fine-Galilei group) in

the space L2

(Rn

+1

, d n

kd ω) which are given by themapping g → T g are square-integrable. For a given

non-zero ∈ L2(Rn+1, d kd ω), the condition of

square-integrability is equivalent to the statement that

the following two integrals be finite

c= Rn+1

m0(k , ω)

2 I m0

(k , ω) d k d ω < +∞ (40)

where

I m0(k , ω) =

Rn+1 m0

(k , ω)

2

× [|k |2 + 2m0(ω − ω)]n−1

|k |2n mn−10

d k d ω

< +∞ (41)

This demonstration theorem is sketched at the end

of Appendix 2 as a variant of Theorem 1. At this

stage, a comprehensive construction of Lie groups,

Lie algebras and associated continuous wavelets has

been completed for the description of translational mo-

tion in image sequences. Appendix 3 presents an al-

ternative way of computing the wavelet admissibility

through Calderon reproducing formula. With the tech-

nique developed so far and Appendixes 1–3, we arefully equipped to proceed to the main construction of

this section, i.e., Lie groups that includes rotational

motion.

2.4. Groups for Rotational

and Translational Motion

This section proceeds one step further to extend the

af fine-Galilei group G5 to take into account rotational

motion. In this work, we focus on image process-

ing. Therefore, in the sequel, we set n = 2. The trans-

formation of rotational motion is described as fol-lows in R2 × R applied on a spatial function f [ x]:

R2 → R. We also suppose that the function f ( x , t ) is

C ∞ is compactly supported in the spacetime: f ( x , t ) ∈ L2(Rn ×R, d xdt ). Let us apply the rotational transfor-

mation to the function f ( x , t ) that is characterized by

a spin around the origin at uniform angular velocity

θ1 ∈ R. Potential angular accelerations θn ∈ R; n > 2

can also be taken into account. The function f ( x) trans-

forms as

˜ f ( x, t ) = f [ R(φ) x, t ] (42)

φ ∈ R may be expanded into a temporal Taylor series

as

φ(t ) = θ0 + θ1t +k =∞k =2

θk

k !t k (43)

where θ0 is the orientation in space. The construc-

tion of the Lie group, to be referred to as G6, for

rotational motion with angular position θ0 and uni-

form angular velocity θ1 requires a trick to provide

the correct law of composition with the property

to be associative, to fit within rotations as in ma-trix R ∈ SO(2) and to provide a physical law of

composition for the angular coordinates in the form

θ0 + θ 0 + θ1τ which is similar to the translational

version without scaling. The construction proceeds

from the af fine-Galilei group G3 = aff-GAL(n) in two

steps: a group extension with new parameters θ0

and θ1 different from R() followed by a section

that re-establishes the rotational nature of the new

parameters.

Therotational extension of theaf fine-Galileigroupis

denoted G6 = aff-ROT(2) is defined by ordered 7-tuple



216 Leduc

of elements, g, and the matrix representation

g = {b, τ ; v, θ0, θ1, R[], a}

=

1 0T θ1 θ0

0 aR[] v b0 0 1 τ

0 0 0 1

∈ (S L(5), R). (44)

where ∈ [0, 2π), and θ0, θ1 ∈ R. The associated Lie

algebra is given by

G6 = {P, H ; K ; J 0, J 1, J , S} (45)

where the infinitesimal generators are defined in therespective order of the elements in g. The additional

commutators of the Lie algebra G6 fulfill

[ J 1, H ] = J 0 [ J 0, J 1] = 0 [ H , J 0] = 0(46)

[P, H ] = 0 [ J 0, P1] = +P2[ J 0, P2] = −P1

where the commutator [ J 1, H ] = J 0 acts as the com-

mutator [K , H ] = P to generate the velocity as in the

Galilei group. The angular velocity is related to the an-

gular position according to the usual law θ0 = θ1τ . Ex-

ponentiation from this Lie algebra determines that the

elements in G6 = aff-ROT(2) compose in the followingmanner

g ◦ g ={b + aR[]b + vτ , τ + τ ; v + aR[]v;

θ0+ θ 0 + θ1τ , θ1 + θ

1, R[ + ], aa }.

(47)

The inverse element reads

g−1 = {−a−1R[−](b − a0vτ ), −τ ; a−1R[−]v;

− θ0(1 − θ1τ ), −θ1, R[−], a−1} (48)

The group structure is a semi-direct product N S with N = R

n ×R and S = R×R× S O(2)×R+∗ . After this

extension, a section on the group parameters space

= [θ0 + θ1τ ]mod 2π (49)

leads to the appropriate group of rotational motion. Let

us remark that this construction clearly differs from a

one-parameter subgroup since θ0, θ1 and τ have their

own Lie generator. The one-parameter construction

would consist in generating a trajectory as function of

a dummy variable t out of one single parameter θ0.

A central extension can be defined on the af fine-

rotational group G6 =

aff-ROT(2) to generate an ex-

tended af fine-rotational group, referred to as G7 in thesequence. This extension is totally similar to that from

G3 = aff-GAL(n) to G4, the central extension of aff-

GAL(n). The Lie algebraG7 is given by elements of the

form G7 = { M , P, H ; K ; J 0, J 1, J , S}, and the group

elements G7 compose as

g ◦ g =

φ + a2

a0

φ + vT · vτ

2a0

+ a

a0

R[θ0 + θ1τ ]b

,

b + aR[θ0 + θ1τ ]b + vτ , τ + τ ; v + aR[θ0 + θ1τ ]v;

θ0 + θ 0 + θ1τ , θ1 + θ

1, R[θ0 + θ1τ + θ 0 + θ

1τ ], aa

.

(50)

The left and right-invariant Haar measures on G7 are

given by

d µl = an−10 a−(2n+3)d φ dm(θ0) dm(θ1) d b d v d τ da

(51)

and

d µr = a−10 a−1d φ dm (θ0) dm (θ1) d b d v d τ da (52)

The group is not unimodular.

If we proceed with sections on the extended af fine-

rotational group G7 that are similar to those described

in Eqs. (37) and (49), we get the group G8 with UIRs

in L2(R2 × R, d kd ω) expressed asT (g)m0

(k , ω) = an/2ei [ R−1[θ1τ ]b·k +ωτ ]m0

(k , ω)

n = 2 for image sequence (53)

where k and ω represent the same group actions

as those computed in Eq. (39). As rotations are of

concern, it is natural to rewrite these IURs in po-lar coordinates. In this transformation, we denote

= ωθ1

, w = −θ1τ , k = (k , κ), b = (r , β), α = β − κ

and χ − π2

= w − α, successively. We further let v =0 and m0 = 0, to get

[T (g)](k , κ , ) = aθ1ei (α− π2

)ei [χ+kr sin χ ]

× (ak , κ − θ0, θ1) (54)

with the polar decomposition b = (r , β), k = (k , κ).

The character ei [ R−1(θ1τ )b·k +ωτ ] of the UIRs in Eq. (53)

introduces a special function derived by integration on




τ . In this case, we reach the following form

J (k ) = 1

2π

2π

0

ei [u+k sin u]du (55)

which is not a Bessel function except for ∈ Z. In-

deed, the Bessel function has an analytic extension for

n ∈ C [19, 22, 25, 45] given by

Z n( z) = 1

2π

+π

−π

ei (nθ− z sin θ)d θ

− sin nπ

π

+∞

0

e(−nt − z sinh t ) dt .

For n ∈ Z, J n ( z) coincides to Z n . The complexificationof u → i y in Eq. (55) gives rise to hyperbolic rotations

and the special function

H (k ) = i

2π

+∞

0

e−( y+k sinh y) d y (56)

A theorem of additivity for these special functions can

be deduced from the composition of the translations.

In this case, it reads after some computations

+∞

−∞ J [1−l](kr 1) J [l−2](kr 2)dl

=J [1−2][k (r 1

+r 2)]

(57)

The special functions defined in Eq. (55) further lead to

integral transforms where the special function is taken

as kernel; the first example is

[ H f ](k ) = ∞

0

f (r ) J (k r ) rdr n = 2 (58)

Further properties involving those special functions

and integral transforms will be considered in Section 3.

Let us first formally suppose that [aR−1

(θ1τ )k , ω] = [aR(−θ1τ )k , ω] = −1θ1 [F ](k ). Bysimilar arguments as those developed for the proof of

Theorem 2 in Appendix 2, Eq. (127), we reduce the

condition of square-integrability to showing that there

is some non-zero ∈ L2(R× T, d 2 xdt ) such that R4

1

θ1

F m0

(k )

2da d v dm(θ1)

a3< +∞ (59)

Let us remark that the integration on dm (θ1) requires

that the limit of the integrand be zero when θ1 → 0.

This limit corresponds to → ∞. This requires that

F m0= O(|θ1|1+ ) be of compact support on θ1 to

avoid the integration in Eq. (59) to diverge. This state-ment remains a formal wish. The condition to achieve

convergence [39] is reached if we consider the con-

vergence of an (inner) product of F with a compact

supported function. This leads to a weak topology of

convergenceand reconstruction and therefore to a weak

definition of continuous wavelet transforms.

Theorem 3 . The UIRs of G8 in the space

L2(R2+1, d 2kd ω) which are given by the mapping g →T g are square-integrable. The condition of square-

integrability requires that the following integral be

finite

c = Rn+1

F m0

(k )

2 I m0

(k , )d k d < +∞(60)

where

I m0(k , ) =

Rn+1

F m0

(k )

2

×

[|k |2 + 2m0( − ]n−1

|k |2nmn−10

d k d < +∞ (61)

This demonstration theorem is as sketched in

Appendix 1 as a variant of Theorem 2. A similar dis-

cussion can be conducted for hyperbolic rotations and

the related wavelets. In this case, the convergence of

the integral is neatly obtained through the negative real

exponential function.

2.5. Further Extensions to New Kinematics

Two avenues for further extensions and sections on the

af fine-Galilei group have been opened in this paper.

These potentialities demonstrate the general founda-tion of the approach developed in this paper.

The first direction concerns all the variant of exten-

sions with the triple [R(), θ0, θ1; R ∈ S O(n)] defined

in Eq. (44) where the matrix R is a matrix of spatial

transformation, θ0 and θ1 are destined to correspond

to the 0th and 1st order coef ficient of the transforma-

tion after the appropriate section. The matrix R which

has been considered as the group S O(2) can be re-

placed by another group of spatial transformation. For

instance, let us first mention the complexification of

in R() ∈ S O(2) which leads to hyperbolic rotations



218 Leduc

asdefinedin Section 1. A second example is to consider

a deformation matrix of the form A 0

0 A

(62)

and the section A = a0 +a1τ which describes a scaling

action with temporal evolution. A third example is to

take the matrix of shear of the form1 S

0 1

(63)

andthe section S = s0+s1τ . Eachcase ofstudy leads toits own group representation, special functions related

to the characters, a specific harmonic analysis with

PDE’s and ODE’s, and a complete discussion about

the existence of continuous wavelets suited for image

sequence processing.

The second path for extensions is to add columns in

thematrix of Eq.(44) to introducefurther ordersof time

derivative i.e. the angular accelerations θm , associated

with the translational acceleration γ m , and the temporal

translations of higher order τ m ; m > 1. For instance,

the first acceleration will yield the following matrix

representation

g = {b, τ , τ 2; v, γ 1, θ0, θ1, R[], a, a0}

=

1 0T θ2 θ1 θ0

0 aR[] γ 2 v b0 0 1 0 τ

0 0 2τ 1 τ 2

0 0 0 0 1

∈ S L(n + 4), R).

(64)

Rigorously, according to the extension technique that

has been developed, the parameter τ n; n > 1 shouldbe thought first as an independent parameter η which

is sectioned afterwards as η = τ n where τ is the usual

temporal translation. Once again, each additional or-

der of time derivative leads to its own group represen-

tation, special functions related to the characters, the

associated harmonic analysis for PDE’s and ODE’s,

and, again, a complete discussion about the existence

of continuous wavelets for image sequence process-

ing. Moreover, the first and the second paths can be

combined one with the other in a similar discussion.

Our technique of construction reaches in this manner a

complete description of all the observable kinematics

in the Euclidean spacetime.

3. Special Functions and Symmetries

To get more insight on the special functions related to

motion, let us revisit the Fourier transform of a func-

tion ˜ f ( x, t ), x ∈ Rn rotating with the angular velocity

θ1 in a two-dimensional plane. If we first consider the

group representation in Eq. (53) with v = 0, a = 1,

a0 = 1, θ0 = 0 and extend the functions to distribu-

tions to take the function as a Dirac measure along a

trajectory from time t = 0 to t = 1, the integration on

the character leads to the special function in Eq. (55).Let us further assume without loss of generality

that the functions are separable in terms of magnitude

and orientation i.e. f ( x, t ) = f r (r , t ) f s ( x u , t ) where we

consider the polarcoordinates inRn ·r = | x| isthe mag-

nitude in Rn and x u ∈ Sn−1 belongs to the unit sphere

in Rn, Sn−1 = { x u ∈ Rn : | x u| = 1}. We then observe

that

ˆ f (k , ω) = Rn×T

dt d n x f [ R(θ1t ) x]e−i (k · x+ωt )

= Rn×Tdt d n

x f (

x ) e−i [ R(θ1t )k · x +ωt ]

ˆ f (k , k u , ω) = R

∗+×Sn−1×Tdr d σ ( x

u ) dt r n−1 f r (r )

× f s ( x u ) e−i [kr R(θ1t )k u · x

u+ωt ]

=

Sn−1

d σ ( x u ) f s ( x

u )

+∞

0

dr r n−1 f r (r )

× 1

θ1

2π

0

du e−i

( ω

θ1)u+kr R(u)k u · x

u

ˆ f (k , k u , ) = 1

θ1

Sn−1

d σ ( x u ) f s ( x

u )

+∞

0

dr r n−1

× f r (r ) J (kr

k u

· x

u

)

= 1

θ1

Sn−1

d σ ( x u ) f s ( x

u ) H [ f r , k k u · x u ]

= 1

θ1

F [ f s , f r ; k k u ] (65)

where d σ is the Lebesgue measure on the sphere Sn−1.

In this calculation, we have successively applied a

change of variable x = R(θ1t ) x i.e. x = R(θ1t )T x , achange from CartesianRn ×T to general polar coordi-

nates Sn−1 ×T in the space domain. The spatial Fourier

domain is changed to polar coordinates with k ∈ R+




and k u ∈ Rn, |k u| = 1. We also let = ωθ1

. Then, anin-

tegration is taken on u that generates special functionsof real index J . The integration on variable r modified

the Fourier transform into an integral transform H . In

turn, this transform becomes the kernel of an integra-

tion on the sphere Sn−1. Clearly, f ( x , t ) ∈ C ∞c (Rn ×T)

is compactly supported by assumption and we have

F [ f s , f r ; k ]

≤ 2π n/2

(n/2)sup

x u∈Sn−1

f s ( x u ) H [ f r , k · x

u ] < +∞ (66)

where 2π n/2

(n/2)

=µ(Sn−1). The construction of ODE’s

and PDE’s as equations of the wavelet motion is anadditional topic related to the group representation.

Reference [27] presents one way to derive and com-

pute those equations that play an important role for

motion tracking.

The derivation of new motion-based special func-

tions as substitute of the Fourier kernel requires to

revisit the sampling theorem. It is clear indeed that

the sampling of rotational motion [22] requires to state

some new remarks related to symmetries. Without any

particular symmetry, the critical sampling is obtained

at ω = π i.e. θ1 = π rad/image. At this rate the

object is orbiting at one half-round turn per consec-utive image. If the object is shaped with one or several

axes of symmetry, then the finite group of symmetries

[43] brings its characters on the functions f ( x, t ), J and H and involutions that reduce the periodicity at

sampling. One may easily verify that the first com-

ponent of the expansion of the spectrum along the

ω-axis is located at 2mθ1 with m axes of symmetry.

In the case of a square, the integration in Eq. (65) is

performed from −π4

to +π4

. The critical sampling cor-

responds to an angular velocity of θ1 = π4

rad/image.

This reaches the Shannon sampling bound. Similarly,

a spinning 2m-gons reach the bound at π

2m

rad/image.

If m tends to ∞, the polygon tends to a circle whose

shape is rotational invariant. Hence, any angular ve-

locity is potentially admissible and motion becomes

ambiguous.

4. Rotational Motion and Harmonic Oscillator

It is instructive to examine how the rotational mo-

tion is intimately connected with the harmonic oscilla-

tor [46, 47], the groups SU (2) and S L(2,C). In fact,

the existence of an harmonic oscillator may be evi-

denced in the extended Galilei group when consid-

ering its Lie algebra restricted to position, velocityand central extension. Simple analytic computations

lead then to an analysis which is step by step similar

to the classic theory developed for the harmonic os-

cillator in the context of the Weyl-Heisenberg group

[7]. Numerous Lie groups lead to harmonic oscillators

[46, 47]. In the particular case of performing rotational

motion analysis with angular parameters defined on

the real line R, we show from the Hamiltonian that

SU (2) is appropriate to describe the spinning or or-

biting in a plane around the orthogonal axis. SU (2)

is locally isomorphic to the group S O(3) of the ro-

tations in three-dimensional space (Euler angles not

motion). In fact, both algebras su (2) and so(3) are iso-morphic but the group S O(3) is the quotient of SU (2)

by its center Z 2 = { I , − I } i.e. S O(3) ≡ SU (2)/ Z 2.

SU (2) is the group of the 2 × 2 unimodular, unitary

matrices i.e.

g =

a b

−b a

with |a|2 + |b|2 = 1 (67)

SU (2) is a three-parameter group, a UIR is given below

in Eq. (75).

Let us consider the Hamiltonian of the one-

dimensional harmonic oscillator [3]

H = p2

2m+ 1

2m θ 2

1 q2 (68)

where θ1 =√ k /m, k > 0 is usually known as the

spring constant, m is the mass of the system, and p

and q are momentum and position respectively. This

Hamiltonian gives rise to the equation of motion i.e.

the ODE for the position of the system, q , which

reads

q

+θ 2

1

q

=0 (69)

In the phase space of coordinates (q, p), classi-

cal mechanics shows how the time evolution op-

erator can be obtained from the Poisson brackets

[3]. The standard theory [3] in n dimensions fur-

nishes the correspondence between smooth functions

A( p, q) of p and q and the differential operator as

follows:

A → A =n

i=1

∂ A

∂ pi

∂

∂qi

− ∂ A

∂qi

∂

∂ pi

(70)



220 Leduc

In this case, the evolution operator H related to the

Hamiltonian H in Eq. (68) reads

= −i 2θ1 J 2 with J 2 = −i

x

∂

∂ p− p

∂

∂ x

2.

(71)

Let us restate the dual space as a two-dimensional com-

plex plane ( z1, z2) = (q, p) and define the operators

X i j = zi ( ∂∂ zi

) with i, j ∈ (1, 2). Hence, the operators

defined by the following relations

J 1 = ( X 12 + X 21)/2; J 2 = −( X 12 − X 21)/2;

J 3 = ( X 11 − X 22)/2 (72)

i.e. the standard commutation relations read [ J i J j ] =i jk J k , with i jk the anti-symmetric tensor. The above

Lie algebra generated by J i ; i = 1, 2, 3 is the algebra

of so(3). However, since S O(3) and SU (2) are locally

isomorphic about the identity, we immediately see that

J i ; i = 1, 2, 3 istheLiealgebraof SU (2). This is easily

seen if the J i are re-written in terms of matrices

J 1 = i

2

0 1

1 0

, J 2 = 1

2

0 1

1 0

, J 3 = i

2

1 0

0 −1

(73)

Let us consider the one-parameter subgroup of

SU (2) generated by the flow of the infinitesimal gen-

erator J 3 to represent the rotation about the z-axis or-

thogonal to the sensor plane x– y. The exponential map

yields the sub-group G[3,su (2)] ⊂ SU (2) generated by

J 3 as the set of matrices

exp[i θ1 J 3] =

eiθ1 0

0 e−i θ1

(74)

Hence, the basic harmonic oscillator of interest isa group G[3,SU (2)] generated by J 3 as being the set

{eit J 3 | − ∞ < t < +∞}.

The next problem is to find the appropriate UIRs of

SU (2) in L2(C×R, d µ). The group SU (2) is a subset

of S L(2,C) which is the group of the 2 × 2 matrices

g =

α γ

β δ

∈ S L(2,C)

i.e. with det = 1 and α , β , γ , δ ∈C. The UIRs of

S L(2,C) are indexed by pairs (k , i v) with k ∈ Z and

v ∈ R. The representations in L2(C× R, d µ) is given

by [24]

[T g]( z, t ) = |−β z + δ|−2−iv

−β z + δ

|−β z + δ|−k

×

α z − γ

−β z + δ, t

(75)

with z ∈ C. SU (2) is part of the bilinear transforma-

tion thatdescribesthe stereographic projectionof three-

dimensional rotations in the two-dimensional plane. If

we let α = e j φ , δ = e− jφ , β = γ = 0, we effec-

tively yield the UIRs of G[3,SU (2)] in L2(C × R, d µ).

As a compact group, L2[SU (2)] possesses a countable

orthonormal basis according to Peter-Weyl theorem

[9] made of Bessel functions generated by the char-

acters. Let us remark that the bilinear transformations

in Eq. (75) can be built as self-maps of the com-

plex disk (i.e. a Mobius transformation with a rota-

tion). This link leads to considering Bergman spaces

of complex functions in a spacetime approach. We al-

ready know the existence of continuous wavelets in

these spaces. The harmonic oscillator generalizes in

n-dimensions, n > 2, by taking m ≤ n independent

copies of the preceding version. For example, let us

consider n

=3 involving three orthogonal axes of ro-

tation: Gt = SU (2) × SU (2) × SU (2). The functionspace of these oscillators is the tensor product of the

original space

V n =n

⊗i = 1

V .

The resulting group is still compact and corresponds to

the direct group product

Gt =n

×i = 1

SU (2).

In fact, this mathematical structure corresponds to

the usual addition of angular momenta. The group

representation is the product of the characters of

the different SU (2)’s. Moreover, these multidimen-

sional isotropic harmonic oscillators seem to be rich

in structure since there exist many distinct complete

admissible sets of commuting/non-commuting oper-

ators whose eigenvectors span the tensor product

space.




5. Tracking Rotational Motion

It has been shown in [30, 31] that the Galilean wavelets

perform MMSE motion estimations. This property es-

sential to track the velocity along a trajectory. The same

propertieshold for accelerated waveletson acceleration

parameters under the condition that thevelocity param-

eter be fixed i.e. be previously estimated [27]. In this

approach, translational motion estimation starts with

the spatio-temporal location (i.e. the spacetime trans-

lation parameters in the af fine wavelets) and proceeds

with the velocity (Galilean wavelets). On enough small

spatio-temporal neighborhoods, motion is assumed to

be linear and Galilean wavelets are always appropriate.

The estimation process proceeds with the estimation of the acceleration parameters (accelerated wavelets) on

a larger span of scene. This leads to the construction of

optimal trajectories that can be buit with algorithms

derived from dynamic programming and Bellman’s

principle of optimality (Viterbi algorithms). The as-

pects of this construction dealing with Lagrangians

and dynamic programming have already been exam-

ined in [31]. In this section, we develop the specific

part that performs the separate tracking of the rotational

motion.

The technique developed in this section is based on

Kalman filters, an alternative point of view of Lie alge-bra, and rotational continuous wavelet transforms. The

Kalman filter is used as a recursive minimum variance

estimator which introduces an alternative approach to

the formulation and solution of linear MMSE estima-

tion problems. This alternative is particularly useful to

deal with non-stationary processes and to approach the

estimation of dynamic motion problems. We adopt a

structure of Lie algebras in forms of a set of differen-

tial operators that describe the motion. These opera-

tors are derived from the equation of motion i.e. the

equation that relates the kinematical parameters of po-

sition, velocity and accelerations. In this application,

the motion equation superposes rotational motion on a

translational carrier trajectory.

The Kalman filter is based on twoequations, thestate

equation and the observation equation, along with an

optimization algorithm. The state equation is a linear

predictor which provides a prediction of the state of

the sytem at time t + τ from the state of the sytem at

time t . The state q of the system is composed of all the

relevant motion parameters that exhaustively explain

the local behavior on the trajectory. These parameters

under study are known to be position, velocity, and ac-

celerations for translational and rotational motion. As

a matter of fact, the infinitesimal local evolution of thesystem is linear and described by d qdt

. Expanding the

total derivative into its partial derivatives associated to

the equation of motion leads to a set of differential op-

erators whose closure defines a Lie algebra. The state

equation can be seen as a one-parameter sub-group cal-

culated from the Lie algebra which provides the lo-

cal transformation in terms of an evolution operator.

At each step of this prediction, an optimization takes

place after the prediction to optimally estimate the ac-

tual parameters. The difference between predicted and

estimated parameters provides the innovation of the

system. This step is implemented with either gradient

or dynamic programming algorithms. The observationequation may have two implementations. It is either

a convolutional filtering of the image sequence with

continuous wavelets tuned to the optimum parame-

ters or a frame-based reconstruction technique based

on the inverse wavelet transform (see Appendix 3).

The former technique selectively captures and extracts

the moving pattern of interest moving along its tra-

jectory in form of an image sequence. The later tech-

nique reconstructs a motion compensated sequence

where motion has been frozen or cancelled. or a recon-

struction out of the inverse wavelet transform. Partial

selective reconstruction is achieved from the wavelettransform that performs a motion-selective band-pass

filtering on the entire scene. The wavelet family is

generated by the action of a group representation

that describes the global motion transformation under

estimation.

5.1. State Equation and Lie Algebras

Let us consider the infinitesimal transformation of the

Lie algebra given by

q = d qdt = A q (76)

where A is the matrix representation of a Lie algebra

of differential operators and q is the state vector of

the system. Some examples of matrix A are computed

below in this section and in Appendix 4. One can also

see that the state equation that updates the state of the

Kalman filter is a one-parameter subgroup obtained

by exponentiation of operators in the Lie algebra. The

algebra is of finite dimension, the operator t of the

state space equation is the matrix A in Eq. (76) and is



222 Leduc

defined as

q(t + τ ) = eτ t q(t ) = eτ AT q(t ) (77)

The goal is now to build the Lie algebras, define the

states and construct the representation A from given

equation of motion.

The trajectories can be described in terms of Taylor

expansions. The coef ficients of the expansions are the

parameters of motion. Let x(t + τ ) be the position at

time t + τ , the trajectory can be expressed as

x(t + τ ) =∞

k

=0

τ k

k ! d k x(τ )

d τ k τ

=t

(78)

In this method, we also assume to know how many

orders in the expansion are relevant to describe the lo-

cal motion transformation. Let us remark that the mo-

tion parameter estimation performed by the continuous

wavelets enables the determination of how many rele-

vant parameters exhaust the expansion [27]. Newton’s

equation of classical dynamics limits the expansion to

the first-order of acceleration γ 1. This is no longer true

for motion taking place within two-dimensional signal

generated from the projection onto the sensor plane.

The expansion may need higher orders of acceleration.

These aspects are treated in [28]. Hence, the equationof motion captured in the camera can be stated as

γ n (t ) = F [ x (t ), v(t ), γ 1(t ), . . . , γ n−1(t ); t ] (79)

where v(t ) = dxdt

, γ 1(t ) = d vdt

, and γ n−1(t ) = d γ n−2

dt .

From Eqs. (77) and (78), it becomes possible to char-

acterize the differential operator as an infinitesimal

generator that produces the evolution operator

eτ t =∞

k =0

τ k

k !

k

t (80)

and the operator t is clearly a time differential op-erator whose expansion into the partial derivatives is

calculated from the equation of motion (79)

t = d

dt

= v(t )∂

∂ x+ γ 1(t )

∂

∂v+ · · · + γ n (t )

∂

∂γ n+1

+ ∂

∂t

(81)

It is now possible to break t into a sum of elemen-

tary differential operators and to derive a Lie algebra.

Since any Lie algebra is closed under the Lie multi-

plication, new generators arise to yield the algebraicclosure.

By Ado’s theorem [5], Lie algebras have matrix rep-

resentations in gl(n,R) meaning that it exists a homo-

morphism between any Lie algebra and the matrices

A of n × n dimension. The matrix A of the evo-

lution operator is a representation of the differen-

tial operator t defined in the algebra gl(n,R). Let

us now consider two examples of equation of mo-

tion where a harmonic oscillator is superposed on

a carrier trajectory F 1[ x(t ), v(t ); t ] and F 2[ x(t ), v(t ),

γ 1(t ); t ]

F 1 : γ 1(t ) = a1 x(t ) + a2v(t ) + b1 cos(θ1t ) (82)

F 2 : γ 2(t ) = a1 x(t ) + a2v(t ) + a3γ 1(t )

+ b1 cos(θ1t ) + b2 cos

θ2

2t 2

(83)

These examples are thoroughly treated in Appendix 4.

As an illustration of how to proceed, the appendix

presents the corresponding operators t , the infinites-

imal Lie generators X i j involved in t . The closure

of the Lie algebra of operator t is calculated under

the Lie multiplication. We adopt the operator nota-

tion X i j = qi∂

∂q j. The new closure operators are de-

rived from visiting all the commutation relations in the

Jacobi equations. The non-zero commutators are even-

tually presented in Appendix 4 respectively for tra-

jectory F 1 and F 2. Interesting cyclic properties of the

Lie multiplication can be evidenced in both cases as

a result of the derivation properties: [ X i j , X jk ] = X ik

and [ X i j , X kl ] = 0 for any {i j} = {kl}. As a further

consequence of cyclicity, all the Jacobi equations are

also satisfied. Finally, the matrix A is defined in both

cases F 1 and F 2 and belong respectively to gl (4,R) and

gl (7,R). The corresponding Lie groups are GL(4,R)

and G L(7,R).Let us remark that there is no actual obstacle to con-

sider multidimensional spaces, this just requests an in-

crease of size of the Lie algebra. The differential oper-

ator in Eq. (81) reads now

t = v · ∇ x + γ 1 · ∇v +· · ·+ γ n · ∇γ n+1+ ∂

∂t (84)

and both equations of trajectory 78 and of motion 79

become vector equations.




5.2. Diagonalization and Reducibility

of the State Equation

In this section, we showthe reducibility of the Lie alge-

bra representations presented in the previous section.

It turns out that the representations are reducible into

subspaces one for the carrier trajectory and one for

each rotational parameter. It would be interesting to ex-

amine this property with more details since it enables

splitting estimation and tracking into several dedicated

parts. The splitting results from diagonalization and

reducibility as follows:

1. The diagonalization of matrix A. The matrix sup-

ports a (n + m)-dimensional vector space made of n + m eigen-values and eigen-vectors. This means

that AT = P−1ΛP whereΛ = diag [λ1, . . . , λn+m ].

It is also clear that diagonalizing matrix A leads to

diagonalizing the state equation eτ AT = P−1eτ ΛP

2. The matrix representation AT may be written in

forms of

AT =

A1,1 A2,1 A3,1 . . . Am+1,1

0 A2,2 0 . . . 0

0 0 A3,3 . . . 0

0 0 0 . . . Am+1,m+1

.

where A1,1 is a n ×n matrix, A2,1, A3,1, . . . , Am+1,1are 2 × n matrices, A2,2, A3,3, . . . , Am+1,m+1 are

2 × 2 matrices. The matrix AT is reducible into

m + 1 complementary closed sub-spaces. The first

subspace is of dimension n and corresponds to the

parameters of the carrier trajectory x , v, γ i ; i =1, . . . , m − 2. The others are of dimension 2 and

correspond to rotational motion. The structure of re-

ducibility in the representation provides the motion

analysis with a hierarchical structure of estimation

and tracking. This approach splits the carrier trajec-

tory from its rotational motions (velocity and ac-

celerations). In the following, we will consider the

expansion up to rotational acceleration of order 2.

As a consequence of items 1 and 2, we can decom-

pose the matrix P in Q and R as follows

AT = (QR)−1Λ QR with Q =

U 0 0

0 V 0

0 0 W

;

R =

In C1 C2

0 I2 0

0 0 I2

(85)

where In is the n-dimensional unit matrix. We let P =QR and P−

1

= R−1

Q−1

, then

P =

U UC1 UC2

0 V 0

0 0 W

,

P−1 = U−1 −C1V−1 −C2W−1

0 V−1 0

0 0 W−1

,

Λ =

Λ1 0 0

0 Λ2 0

0 0Λ

3

. (86)

Matrix U contains n eigen-vectors of the sub-matrix

A11 since A11 = U−1Λ1U, Λ1 = diag [λ1, . . . ,λn ].

Matrix V contains two eigen-vectors, those of

sub-matrix A2,2 since A2,2 = V−1Λ2V, with Λ2 =

diag [λn+1, λn+2], λn+1 = i θ1, λn+2 =−i θ1, and

V = 1√ 2

1 i

1 −i

.

Similarly, matrix W contains two eigen-vectors, those

of sub-matrix A3,3 since A3,3

=W

−1Λ3W,

Λ3 = diag [λn+3, λn+4], λn+1 = i θ2/2, λn+2 = −i θ2/2.

Let the state equation of the system be given by

u(t ) =

x (t ), v(t ), γ 1(t ), cos[θ1t ], sin[θ1t ],

cos

θ2

t 2

2

, sin

θ2

t 2

2

T

.

The evolution is given by the prediction equation

u(t

+τ ) = eτ t u(t ) = eτ A

T

u(t ). We have also

u(t + τ ) =

x(t + τ ), v(t + τ ), γ 1(t + τ ),

cos [θ1(t + τ )], sin[θ1(t + τ )],

cos

θ2

(t + τ )2

2

, sin

θ2

(t + τ )2

2

T

.

As a result of the diagonalization of matrix eτ AT

,

we yield an interesting closed form of the prediction

equation that splits carrier trajectory from rotational



224 Leduc

motion. The derivation of the one-parameter subgroup

expression with i = e

τ Λi

; i = 1, 2, 3 and 4 = e

τ 2Λ3

leads to three relations x(t + τ )

γ 1(t + τ )

v(t + τ )

= (U−11U)

x (t )

γ 1(t )

v(t )

+ [(U−11U)C1 − C1(V−12V)]

cos[θ1t ]

sin[θ1t ]

+ [(U−11U)C2 − C2(W−13W)]

cos

θ2

t 2

2

sin

θ2

t 2

2

(87)

and alsocos[θ1(t + τ )]

sin[θ1(t + τ )]

= (V−1

Λ2V)

cos[θ1t ]

sin[θ1t ]

cos

θ2(t +τ )2

2

sin

θ2

(t +τ )2

2

= (W−13W)(W−14W) (88)

×

cos

θ2t 2

2

sin

θ2

t 2

2

For quite evident reasons described in [27], the restora-

tion of the double product in (t + τ )2 comes from acontribution of the exponent of the first order in τ .

The tracking strategy is based on combining both

Kalman filters and wavelet transform. The state of the

system defines the state of the Kalman filter. This state

is given by the set of the wavelet parameters. These

parameters defined in Sections 2.3 and 2.4 are nothing

but the group parameters. Without any major change

in the reasoning, the scale parameter can be added in

the state vector of Eq. (77). A whole tracking algorithm

is described in [31] that exploits the wavelet transform

as state parameter estimator and the Bellman theory

of dynamic programming to build the trajectory on the

span of the scene.

5.3. Observation Equation and Wavelet Transform

The estimation of the motion parameters i.e. the group

elements, and the motion-based feature extraction and

reconstruction are based on two concepts: a variational

principle that provides an optimal estimation of motion

and a motion-based filtering that extracts the pattern

with motion of interest. These processes are part of

the Kalman filter and are iteratively performed along

a trajectory. At step 0, all the inital conditions i.e. the

motion parameters are provided by an external device(RADAR or satellite). This allows the Kalman filter to

start computing.

The observation equation is based on the wavelet

transform where the optimum wavelet T gopt( x, t ) per-

forms the motion-based extraction of the object in mo-

tion. gopt corresponds to the state parameters at step n,

it is estimated by a gradient algorithm in L2(G, d µl (g))

to yield

gopt = argg

max |g | S|2 (89)

or by a dynamic programming approach [31]. The

wavelet gopt tuned to the estimated state parametersact as a motion-oriented band-pass filter. The contin-

uous wavelet transform discretized on the grid of the

signal captures and isolates the selected objects from

the scene S to provide a display I ,

I (b, τ ) = gopt

S + V . (90)

I is the segmented image of the selected object, dis-

played alone at its correct location; S is the original sig-

nal under analysis,and V (b, τ ) isthe noise producedby

the optical sensors. The estimation of state parameters

in the next section is performed with Morlet wavelets.

An alternative way of reconstructing a relevant in-

formation from the inverse wavelet transform consists

of generating a motion-compensated sequence S( x, t )

S( x , t ) =

G

S | T g(T g)( x, t ) d µl (g) (91)

Appendix 3 shows how this technique compensates the

motion to produce a frozen object.

6. Applications and Simulation Results

The applications presented in this paper are based on

Morlet wavelets. An anisotropic Morlet wavelet is ad-

missible as an continuous wavelet in the rotational and

translational family. In this case, the wavelet is struc-

turally a non-separable filter meaning that it can not be

completely factorized in terms of thespatial andtempo-

ral variable. The numerical computation is performed

in the Fourier domain. The Morlet wavelet reads in the

Fourier domain

(k) = |detC |1/2

e− 12(k−k0) | C (k−k0) − e− 1

2k0 | C k0

×e− 12k | C k

(92)




i.e. a Gaussian function shifted to the point of coordi-

nates k0 = (k 0, ω0). Moreover, k = (k , ω) ∈R

2

×R

andC is a positive-definite matrix. For two-dimensional

spatio-temporal signals, we haveC =

k x 0 0

0 k y 0

0 0 ω

where the factors introduce anisotropy in the

wavelet shape. Figures 1–3 presents three configura-

tions of four-symmetrical rotational Morlet wavelets.

TheShannon sampling bound is at π4

rad/image. The in-

terested reader should read [26, 31, 35] to compare thespectrum of the Galilean wavelets in G4 and compare

to the spectrum of pure rotational motion presented in

these figures.

Figure4 presentsthe sequencecalled “Caltrain” with

a ball simultaneously in translation and rotation along

a track. Detection and estimation of the angular ve-

locity and acceleration is made possible by the texture

on the rotating ball made of four white dots acting as

the corner of a square in rotation. Figure 4 presents

the sequence at images 1 and 32. The scene to be

analyzed is made of 32 images, each of 256 × 256

−4

−2

0

2

4

−4

−2

0

2

40

2

4

6

8

10

12

14

16

18

omega (temporal frequency)

SPECTRUM OF ROTATIONAL WAVELET

k1 (horizontal frequency)

Figure 1. Four symmetrical rotational Morlet wavelet functions close to θ1t = (π/4)t , and v = (0, 0) pix/image: square modulus in plane of

the (ω, k x ) axes i.e. k y = 0. This yields critical temporal sampling.

pixels. The ball is traveling from right to left; the cen-

ter is located at x ≈ 150 in image 1 and at x ≈ 95 inimage 32.

Figure 5 presents the global analysis of velocity per-

formed with Galilean wavelets from group G4 ina win-

dow B of size 60 × 60 centered on the ball at image

18. The square modulus of the wavelet transform is

computed on the window as a function of v x , v y

f (v x , v y ) =

b∈B,τ =18

g1

S2

g20, with

g1 = {v, θ0, a, b, τ }, g20 = {a = a0, θ0 = θ00}(93)

provides the analysis with a representation of the lo-

cal spatio-temporal content in terms of its velocity

content. Figure 6 presents the estimation of the mean

angular velocityand thescaleof theball at velocitywith

rotational wavelet tuned to velocity v = (−1.7, 0.1).

Figure 7 proceeds to the estimation of the angular ac-

celeration, the measurement shows a deceleration of

−0.0028 rad/image2. Figure 8 presents the tracking

of the ball in the scene and the carrier trajectory. The

algorithm exploits the Galilean wavelets from group

G4. The square modulus of the wavelet transform is



226 Leduc

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

SPECTRUM CONTOUR OF ROTATIONAL WAVELET

k 2 ( v e r t i c a l f r e q u e n c y )


Figure 2. Four symmetrical rotational Morlet wavelet function at θ1t = (π/10)t , and v = (0, 0) pix/image: contours of the square modulus

in plane of the (ω, k y ) axes i.e. k y = 0. This yields 4 × π/10 on the temporal frequency axis. The Fourier signature of this rotational motion

is a ball of energy located on the frequency axis. The Fourier signature of a translational motion is located on a so-called velocity plane. As

described in [26, 31], this plane is orthogonal to the velocity vector v = (v, 1).

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

SPECTRUM CONTOUR OF ROTATIONAL WAVELET

k 1 ( h o r i z o n t a l f r e q u e n c y )


Figure 3. Four symmetrical rotational Morlet wavelet functions at θ1t = (π/8)t , and v = (1, 0) pix/image: square modulus in plane of the

(ω, k x ) axes i.e. k y = 0. In this sketch of the energy, translational motion and rotational motion compose each to the other. The Fourier signature

is two distinct components. One is odd in the velocity plane around the origin and the other is even on the temporal frequency axis tilded by the

velocity. Each signature may be estimated independently of the other.




Figure 4. Image 1 and 32 in the Caltrain sequence ([256× 256] × 32 images): motion analysis of the ball, more precisely of four white spots,

which arespinning, and moving on a quasi-horizontal trajectory. There is a deceleration fromimage and moving on a quasi-horizontal trajectory:

deceleration from images 1 to 14 followed by acceleration from images 15 to 32.

−3−2

−10

12

3

−3

−2

−1

0

1

2

30

50

100

150

200

250

300

350

400

vx

ENERGY OF GALILEAN WAVELET TRANSFORM

vy

Figure 5. Analysis of translational velocities contained in the ball at image 18 on a window that contains the ball: the square modulus of the

wavelet transform (the energy) is computed using the Galilean wavelet of group G4. The energy density is integrated on a window at image

18. Two signatures are visible; two symmetric peaks result from the rotational motion estimated with group G4 which is suited for translational

motion alone and a “domed wall” testifies of the spreading of translational velocities which results from accelerated motion. This treatment is

instructive for the shape of the Fourier signature but not well suited for precise estimations.

successively maximized on the velocity as processed

in [30, 31]. The gravity center of the ball is computed

from the maxima of the square modulus of the wavelet

transform within a window B of size 60 × 60 which

is sliding according to the velocity estimations. These

maxima are located along the contours of the ball in

translational motion. The corresponding reconstruction

is presented in Fig. 9.



228 Leduc

0

0.02

0.04

0.06

0.08

0.1

1

2

3

4

50

1

2

3

4

5

x 1011

angular velocity: theta (radian/image)

ESTIMATION OF SCALE AND ANGULAR VELOCITY

scale a

Figure 6 . Estimation of the angular velocity and the scale of the rotating ball in Caltrain sequence with rotational wavelets. The diagram

sketches the square modulus of the wavelet transform at v = (−1.7, 0.1) pix/image. The component at θ1 = 0 stands for the non-rotating

structures. The component at θ1 = 0.045 rad/image, a = 3.3 is the actual ball contribution which is observed rotating of π/2 over 32 images.

The component at θ1 = 0.09 rad/image is the first harmonic.

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.0220

22

24

26

28

30

32

34

s q u a r e d m

o d u l u s o f t h e w a v e l e t t r a n s f o r m | < Ψ | s > | 2

angular acceleration θ2

Figure 7 . Estimation of the angular acceleration of the rotating ball in Caltrain sequence on image 4 with rotational wavelets from an extended

version of group G7 which contains θ2. The diagram sketches the square modulus of the wavelet transform at v = {−1.7, 0.1} pixel/image,

a = 3.3. The maximum at θ2 = −0.003 rad/image2 corresponds to the angular deceleration. The ball is indeed observed decreasing speed and

rotational motion from image 1 to 14 in the tracking presented in image 7.




0 5 10 15 20 25 30 3590

100

110

120

130

140

150

160

TRAJECTORY TRACKING OF THE BALL

x − c o o r d i n a t e s

image number

Figure 8. Tracking the moving-ball position in the Caltrain sequence with Galilean wavelets from group G4. At image 15, the ball is pushed

and keeps constant speed. The x -coordinates decrease as a result of the motion steering to the left.

Figure 9. Reconstruction of the rotating ball at image 4 in the Caltrain sequence; the scene is filtered by rotational wavelets from group G7

traveling at velocity v = (−1.7, 0.1) pix/image, the scale a = 0.5 to trace all the contours with fine resolution, t = 10 to be more accurate in

position and θ1 = 0.045 rad/image. As a result of the uncertainty between velocity and angular orientation, the intensity of the ball boundaries

depends upon the orientation. This explains the fading of the edge intensities on the upper and lower arcs. The isolation of the moving object of

interest is eventually performed by a Gaussian filter of standard radius of 50 (aball = 3.3) centered on the maximum intensity.



230 Leduc

7. Conclusions

In this paper, we have developed new families of

continuous wavelets to analyze rotational motions (an-

gular velocity and accelerations) in spatio-temporal

digital signals. It has been shown how rotational pa-

rameters superpose and combine with translational pa-

rameters when travelingalong thecarrier trajectory and

how the motion estimation and tracking of parameters

can be performed quasi-independently. Simulations re-

sults have presented the practical capabilities of these

wavelet estimation process.

Appendix 1

This appendix provides some guidelines and outlined

ideas on how Lie group representationshave been com-

puted in this paper. As an example, the multidimen-

sional af fine group SIM(n) is sketched here below. The

reason of this choice is the small amount of compu-

tations involved for this case. The Galilei and rota-

tional cases proceed exactly on the same sketch but

involve more machinery. For all definitions, theorems,

properties, the interested reader is invited to consult

the following references [4, 6, 9, 14, 23, 24, 34, 37].

The construction of UIRs may proceed from two ap-

proaches namely Mackey’s Methodof Induction[5, 34]

and the Kirillov’s Method of Orbits [23]. Both methods

can also be intertwined.

In the case where G is structured as a semi-direct

product G = H K where K and H are subgroups.

Let φ be a group homorphism from K into AUT( H ).

G is the set of the ordered pairs (k , k ) with h ∈ H

and k ∈ K . We have now the following strategy to

determine all the unitary representations of G.

1. Determine the set H ∗ of all the irreducible charac-

ters of H .

2. Search for the orbitsOK (ω∗) in the dual group H ∗;ω∗ ∈ H ∗. Classify all the orbits.

3. From each orbit, select an element ω0 ∈ OK (ω∗)

and determine the form of the stability subgroup

K (ω0) ⊂ K .

4. Take a representation L K of K (ω0) and induce the

representation T gψ as IND ↑GK L K in Eq. (5).

The interested reader should read Glimm’s theorem

[18]. The importance of this theorem lay in its ability to

guarantee the irreducibility of induced representations.

Basically, Glimm’s theorem allows us an easy way to

check for the existence of a system of imprimitivity for

a representation [5, 34]. Let G = H

K , and let theorbits of G in H ∗ satisfy the conclusions of Glimm’s

theorem. Then

1. For each orbitOK (ω∗) ∈ H ∗, and for each UIR L K

of the stability subgroup, there exists an induced

representation for G defined as IND ↑GK L K in

Eq. (5).

2. The carrier space of the induced UIR IND ↑GK L K

is the Hilbert space H IND↑GK L K

= L2(OK , d µ).

The form of these induced representations has been

presented in Eq. (5) and their computation required to

derive the adjoint and coadjoint actions. Let us treatthe case G1 = SIM(n) for the shake of simplicity. If g,

g ∈ G1, then the conjugation on the group is given by

gg g−1 = (b + a Rb − a R R R−1b, a, R R R−1)

(94)

and, thus, the adjoint action of G on G (computed from∂gg

g−1

∂g |g

=e is given by

Ad g X = X = (b, s , r ), X , X ∈ G (95)

withb = a Rb − (sI + Rr R−1)b (96)

s = s (97)

r = Rr R−1 (98)

We further consider G∗ to be the algebraic dual of G

in the usual manner, and let X ∗ ∈ G∗ to be X ∗ =( p∗, s∗, r ∗) with p∗ = k defined in Eq. (5). The duality

between G and G∗ is expressed by

X , X ∗ = p · p∗ + ss ∗ + tr (rr ∗) (99)

where tr is the trace. We can proceed to the coadjoint

action of G on G∗ which is computed as Ad ∗g = Ad T g

and is given by

Ad ∗g X ∗ = ˜ X ∗ = ( p

, s , r ), X ∗, ˜ X ∈ G∗ (100)

with

p = a R−1 p ∗ (101)

s = s∗ − b · p ∗ (102)

r = R−1r ∗ R + R−1b ⊗ r −1 p ∗ (103)




Let G(k ) = H H (k ) denote the stabilizer of point k

in H ∗ where H (k ) is called the little group (stabilitysubgroup). For G = SIM(n), V ∗ Rn. Let us fix the

point e = (0, 1) in Rn (0 ∈ Rn−1). The stabilizer of eis G(e) = SIM(e) = R

n H (e) = R

n SO(n − 1). In

this example, the little group is H (e) =SO(n − 1) and

then

L H (s) = D[ j ](s), ∀s ∈ S O(n − 1) (104)

where the D[ j ] are the representations of SO(n − 1) [5,

16]. The Radon-Nikodym derivative is computed from

the coadjoint action det

a R−1

= an . For applications

in signal processing, we let D[ j ](s

) = 1 = L H (s

) (nospinor representations are required). Following Eq. (5),

the UIRs T g of G1 in the space L2(Rn , d x) are then

given by

[T g]( p) = an2 ei b· p( p) (105)

with

p = a R−1 p (106)

These UIRs are unitary and irreducible.

Appendix 2

Theorem 1 . The UIRs of G3 in the space L 2(Rn+2,

d x) given by g → T g are square-integrable.

Proof: Let usrecall thatthe UIRs g → T g are square-

integrable if and only if G

|T g, |2d µl (g) < +∞ (107)

Let us denote the element (m, k , ω) by ξ and denote

the measure dm d k d ω by d ξ . Then, we may interpret

ξ as an element of ˆ N , the dual vector space of N , and

we may consider the integration over ˆ N in the inner

product in Eq.(107) to be an inverseFourier Transform.

We let

d x = d φ d b d τ, ds = d v da da0 dm ( R)

a2n+3a−(n−1)0

(108)

so that d µl (g) = d xds.

The integration in Eq. (107) proceeds as follows

G

|T g, |2 dg

=

S

N

ˆ N

[T g](ξ )(ξ )d ξ

2

d xds (109)

=

S

[ I N ] ds (110)

Moreover, we let

V (ξ ) = [T g](ξ )(ξ ) (111)

and calculate I N as follows (the symbols ˆ and ˇ stand

respectively for direct and inverse Fourier versions).

Since V (ξ ) contains the ei x·ξ in T g, the integral on ˆ N

is an actual inverse Fourier transform. Therefore,

I N =

N

|V ( x)|2 d x (112)

= V ( x )2 L 2( N ,dx ) (113)

= V (ξ )2

L 2( ˆ N ,d ξ )(114)

=

ˆ N

|V (ξ )|2 d ξ (115)

= N |[T

g](ξ )

|2

|(ξ )

|2 d ξ (116)

At this point, the Fubini’s theorem applied to Eq. (110)

grants S

[ I N ] ds =

ˆ N

|(ξ )|2

S

|T g](ξ )|2ds

d ξ

(117)

Thus, if there exists a non-zero ∈ L2(Rn+2, d x ) so

that

c

= N |(ξ )

|2 S |

[T g ](ξ )|

2 dsd ξ <+∞

(118)

the theorem is proved. Let us first observe that SO(n) is

a compact group, the integration over R can be pushed

outside of Eq. (118). So as long as the remaining inte-

grals are finite, c will also be finite. Next, Eq. (118)

may be rewritten as

c = Rn+2

|(m, k , ω)|2 I 1 dm d k d ω < +∞,

(119)



232 Leduc

where

I 1 = Rn+2

|(m, k , ω)|2 d v da da0

an+1a−(n−1)0

< +∞.

(120)

At this stage, the change of variable dm d k d ω = J (d v da da0) is required to eliminate the parameters

from S. The Jacobian is given by

J =

det

∂m∂a0

∂k ∂a0

∂ω ∂ a0

∂m∂a

∂k ∂a

∂ω ∂a

∂m∂

v

∂k ∂

v

∂ω ∂

v

(121)

=det

−a2

a20

0T a0ω

2 aa0

[a−1k ]T 0

0 am I n −a0a−1 Rk

(122)

= an−1mn−1

a0

[|k |2 + 2mω]. (123)

Thus, I 1 yields after the change of variables

I 1 = Rn+2

|(m, k , ω)|2 m dm d k d ω

a2n mn

an0

a4[|k |2 + 2mω]

(124)

But

a2 = |k |2 + 2mω

|k |2 + 2mω, m = a2m

a0

(125)

Therefore, Eq. (124) may be rewritten as Rn+2

|(m, k , ω|2 m[|k |2 + 2mω]2

(m)n[|k |2 + 2mω]3dm d k d ω

(126)

Hence, c < +∞ in Eq. (118) if the following pair of integrals is finite Rn+2

|(m, k , ω)|2m[|k |2 + 2mω]2 dm d k d ω < +∞(127)

and Rn+2

|(m, k , ω)|2 dm d k d ω

(m)n [|k |2 + 2mω]3< +∞

(128)

Since there exist non-zero functions in L2(Rn+2, d x)

which fulfill these two conditions, the theorem isproven.

Theorem 2 . The UIRs of G4 in the space

L2(Rn+1, d nkd ω) which are given by the mapping

g → T g are square-integrable.

Proof: In this case, following the same path as in

Theorem 1, we want to examine the boundness of

I 2 = Rn+1

m0

(k , ω)

2 da d van+1

(129)

Therefore, we reach a change of variables from (a, v)to (k , ω) given by d k d ω = J da d v, where

J =det

∂k ∂a

∂ω ∂a

∂k ∂v

∂ω ∂v

(130)

=det

a−1(k )T 0

am 0 R−1 −a−1 Rk

(131)

= an−3mn−10 |k |2 (132)

Thus, Eq. (129) becomes

I 2 = Rn+1

m0(k , ω)

2 d k d ω

a2n−2mn−10 |k |2

(133)

Therefore, we obtain the condition of square-

integrability (129) which requires that the following

integral be finite

c = Rn+1

m0(k , ω)

2 I m0

(k , ω) d k d ω < +∞(134)

where

I m0(k , ω) =

Rn+1

m0(k , ω)

2

×

[|k |2 + 2m0(ω − ω)]n−1

|k |2n mn−10

d k d ω

< +∞(135)

There are many functions for which the integral will

be finite. For > 0, the following ∈ L2(R × R)

will be admissible as Galilean wavelets: (ξ , ω) =




O(|ξ |−(3n+2+)), |ξ | → ∞, ∀ω ∈ R, (ξ , ω) =O(|ξ |

n

+

), |ξ | → 0, ∀ω ∈ R, (ξ , ω) =O(|ω|−(2n−1+)), |ω| → ∞, ∀ξ ∈ Rn, (ξ , ω) =

O(|ω|−1+ ), |ω| → 0, ∀ξ ∈ Rn .

Appendix 3

In this appendix, the condition of wavelet admissibility

is examined with Calderon’s point of view. In this ap-

proach, we look for the condition to yield an invertible

wavelet transform i.e. an isometry. The example to be

considered is the Galilei group with g = {b, τ , v , a}in a one-dimensional spacetime; so we may consider

m0 = 0. To illustrate this technique developed byCalderon, we will refer to the spacetime version of the

group representation in (38). The inner product

F , T g = R×R

F

y

ρ

a−1/2

× 1

a[ y − b − v(ρ − τ )]

ρ − τ

d y d ρ

is the wavelet transform defined in Section 2. The func-

tional version of the operator statement of the isometry

I H o

= W −1

W reads as follows

F , T g = R×R

F

y

ρ

a−1/2

× 1

a[ y − b − v(ρ − τ )]

ρ − τ

d y d ρ

read as follows

F ( x , t ) =

G

F , T g(T g)( X )d µl (g)

= R R R R+

R×R F y

ρ a−1/2

× 1

a[ y − b − v(ρ − τ )]

ρ − τ

d y d ρ

a−1/2

× 1

a[ x − b − v(t − τ )]

t − τ

db d τ d v da

a3

= R

R+

R×R

F

y

ρ

R

R

a−1

× 1

a[ x − b − v(ρ − τ )]

ρ − τ

a−1

×1a

[ y − b − v(t − τ )]

t − τ d y d ρ db d τ × d v da

a2

= R

R+

R×R

F

y

ρ

R

R

a

×

x − y − u − v(t − ρ − w)

t − ρ − w

a

×

u − vw

w

d y d ρ du d w

d v da

a2

= R R+ R×R F y

ρ (a ∗˜

a )

×

( x − y) − v(t − ρ)

t − ρ

d y d ρ

d v da

a2

= R

R+

{F ∗ (a ∗ a)}

x

t

d v da

a2

where we have let τ = ρ + w, b = y + u and also

( x, t ) = (− x , −t ) and a ( x , t ) = a−1( xa

, t ).

Let us also remark that the introduction of the spatio-

temporal convolution denoted ∗. Spatial and temporal

translations are related to each other through the veloc-

ity v along a Galilean frame moving at constant velocityv. In this case, the convolution defined on the Galilei

group is twisted andperformedalong this displacement

(i.e. along the trajectory) in order to reconstruct the

still signal F ( x, t ). This is a reminiscence of the tech-

nique of motion-compensated filtering. Eventually, we

moved to the Fourier domain to retrieve the condition

of admissibility of the Galilean wavelet as described

in [26]. We have introduced in the second equation the

left-invariant Haar measure d µl (g) which satisfies by

definition

G

F (g−1

h) d µl (h) = G

F (h) d µl (h) ∀g, h ∈ G

(136)

The left-invariant Haar measure can be calculated from

the law of composition as an infinitesimal variation of

volume in the parameter space

d µl (g) =det

∂ge ◦ g

e

∂ge

−1

ge=e

db d τ d v da (137)

= 1

a3db d τ d v da (138)



234 Leduc

Proceeding further with F ( x, t ) in the Fourier domain,

we get

F (k , ω) = F (k , ω)

R

R+

|(a k , ω − k v)|2 d v da

a2

(139)

which leads to the usual condition of square-

integrability of the Galilean wavelet in one-

dimensional spacetime R

R

|(k , ω)|2

|k |2dk d ω = 1 (140)

We have illustrated an alternative way of determiningthe condition of wavelet admissibility. Taking into ac-

count the spacetime domain, we have learned that the

motion-based wavelet analysis performs the motion-

compensated filtering.

Appendix 4

This appendix illustrates the way of how to compute the

Lie algebras G and the matrix representation A defined

in Eqs. (76) and (77). We proceed successively from

equations of motion F 1 and F 2.

Equation of motion F 1 is given by

γ 1(t ) = a1 x(t ) + a2v(t ) + b cos[θ1t ] (141)

The time differential operator reads

1 = v∂

∂ x+ a1 x

∂

∂v+ a2v

∂

∂v+ b cos[θ1t ]

∂

∂v+ ∂

∂t (142)

The state of the dynamical system is composed as fol-

lows: q1 = x; q2 = v; q3 = cos θ1t ; q4 = sin θ1t . The

infinitesimal Lie generators in 1 are given by

X 12 = x∂

∂v; X 21 = v

∂

∂ x; X 22 = v

∂

∂v;

X 32 = cos[θ1t ]∂

∂v; X 43 = 1

θ1

∂

∂t

The additional Lie generators for closure are as follows

X 11 = x∂

∂ x; X 31 = cos[θ1t ]

∂

∂ x; X 41 = sin[θ1t ]

∂

∂ x;

X 42 = sin[θ1t ]∂

∂v

A generic element Y ∈ G becomes

Y = a X 11 + b X 21 + c X 22 + d X 32 + e X 43

+ f X 31 + g X 41 + h X 42

The representation in the parameter state reads

A = X 21 + a1 X 12 + a2 X 22 + b X 32 + θ1( X 34 − X 43)

(143)

Eventually, matrix A reads

0 a1 0 0

1 a2 0 0

0 b 0 θ1

0 0 −θ1 0

(144)

The construction of the Lie algebra G and the matrix

representation A from equation of motion F 2 proceeds

as follows. The equation of motion F 2 is given by

γ 2(t ) = a1 x(t ) + a2v(t ) + a3γ (t ) + b1 cos[θ1t ]

+ b2 cos

θ2

t 2

2

(145)

The time differential operator is

2 = v∂

∂ x+ γ

∂

∂v+ a1 x

∂

∂γ + a2v

∂

∂γ + a3γ

∂

∂γ

+ b cos[θ1t ]∂

∂γ + ∂

∂t (146)

The state of the dynamical system is as follows:q1 = x; q2 = v, q3 = γ ; q4 = cos θ1t ; q5 = sin θ1t ; q6 =cos θ2

t 2

2; q7 = sin θ2

t 2

2. The infinitesimal Lie genera-

tors in 2 are given by

X 32 = γ ∂

∂v; X 21 = v

∂

∂ x; X 13 = x

∂

∂γ ; X 23 = v

∂

∂γ ;

X 33 = γ ∂

∂γ ; X 42 = cos[θ1t ]

∂

∂γ ; X 54 = 1

θ1

∂

∂t

X 76 = 2

θ2

∂

∂t 2




The additional Lie generators for closure follow as

X 11 = x∂

∂ x; X 12 = x

∂

∂v; X 22 = v

∂

∂v; X 23 = v

∂

∂γ ;

X 31 = γ ∂

∂ x; X 41 = cos[θ1t ]

∂

∂ x; X 42 = cos[θ1t ]

∂

∂v;

X 51 = sin[θ1t ]∂

∂ x; X 52 = sin[θ1t ]

∂

∂v;

X 53 = sin[θ1t ]∂

∂γ ; X 61 = cos

θ2

2t 2

∂

∂ x;

X 62 = cos

θ2

2t 2

∂

∂v; X 71 = sin

θ2

2t 2

∂

∂ x;

X 72 = sin θ2

2 t

2 ∂

∂v ; X 73 = sinθ2

2 t

2 ∂

∂γ

A general element Y ∈ G reads

Y = a X 11 + b X 12 + cX 22 + d X 23 + e X 31 + f X 41

+ g X 42 + h X 51 + X 52 + j X 53

The representation in the parameter state is given by

A = X 21 + X 32 + a1 X 13 + a2 X 23 + a3 X 33 + b X 42

+ θ1( X 45 − X 54) + θ2

2( X 67 − X 76)

Eventually, matrix A comes as follows

0 0 a1 0 0 0 0

1 0 a2 0 0 0 0

0 1 a3 0 0 0 0

0 0 b1 0 θ1 0 0

0 0 0 −θ1 0 0 0

0 0 b2 0 0 0 θ2

2

0 0 0 0 0 − θ1

20

(147)

Acknowledgments

The author wishesto express his gratitude to the anony-

mous reviewer who provided substantial and construc-

tive details to improve the paper presentation, and to

Prof. B. Blank and J. Corbett, PhD student, both in the

Mathematics Department of Washington University in

Saint Louis for helpful discussions on thetopic. Theau-

thor also thanks Prof. B.K. Ghosh in the Department of

Science Systems and Mathematics, Washington Uni-

versity in Saint Louis for providing the computing

support.

This material is based upon work supported by the

US Air Force under grant F49620-99-1-0068.

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1976.

Jean-Pierre Leduc received the degree of Electrical Engineer at

the Faculte Polytechnique de Mons (Mons, Belgium) in 1978. He re-

ceivedhis Masterdegree at Columbia University (New York)in 1987

and his Doctorate degree at the Universite Catholique de Louvain

(Louvain-la-Neuve, Belgium) in 1993, both in Electrical Engineer-

ing (Signal Processing and Telecommunications). He also earned

a degree in Operations Research and in Theoretical Physics from

the Faculte Polytechnique de Mons and theUniversite Catholique de

Louvain, respectively. He has been successivelyat the Laboratoire de

Telecommunications et de t el´ ed etection of the Universite Catholiquede Louvain, from 1988 until 1993, at the Institut de Recherche en

Informatique et Systemes Aleatoires in Rennes, France, from 1993

until 1996, at theGeorgia Instituteof Technology from 1996to 1997,

and at Washington University in Saint Louis (Math dept.) from 1997

to 2000. Since September 2000, he is with the Mathematics Depart-

ment of the University of Maryland, College Park.

His present research interests include spatio-temporal signal pro-

cessing, discrete and continuous wavelet transforms, group repre-

sentation theory and optimal control. He is also interested in motion

analysis, detection and tracking.

a group-theoretic construction with spatiotemporalwavelets for the analysis of rotational motion

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