Windowed Radon Transforms, Analytic Signals and the Wave Equation

Appeared in Wavelets: A Tutorial in Theory and Applications

C K Chui, Editor, Academic Press, 1992

G. Kaiser

Department of Mathematics

Univ. of Massachusetts, Lowell, MA 01854, USA

R. F. Streater

Department of Mathematics

King’s College, London WC2R 2LS, England

October 1991

Abstract

The act of measuring a physical signal or field suggests a generalization of the wavelet transform that turns out to be a windowed version of the Radon transform. A reconstruction formula is derived which inverts this transform. A special choice of window yields the “Analytic–Signal transform” (AST), which gives a partially analytic extension of functions from to . For n =1, this reduces to Gabor’s classical definition of “analytic signals.” The AST is applied to the wave equation, giving an expansion of solutions in terms of wavelets specifically adapted to that equation and parametrized by real space and imaginary time coordinates (the Euclidean region).

1. Introduction

The ideas presented here originated in relativistic quantum theory [13, 14, 15], where a method was developed for extending arbitrary functions from to in a semi–analytic way. This gave rise to the “Analytic–Signal transform” (AST) [16]. Later it was realized that the AST has a natural generalization to what we have called a Windowed X–Ray transform [17], and the latter is a special case of a Windowed Radon transform, to be introduced below. For , these transforms reduce to the (continuous) Wavelet transform. In the general case, they retain many of the properties of the Wavelet transform.

In Section 2 we motivate and define the d–dimensional Windowed Radon transform in for and derive reconstruction formulas which can be used to invert it. In Section 3 we define the AST in and give some of its applications. In Section 4 we develop a new application of the AST by generalizing a construction in [16] to the wave equation in . This results in a representation of solutions of the wave equation as combinations of “dedicated” wavelets that are especially customized to that equation. In particular, these wavelets are themselves solutions and represent coherent wave packets, being well-localized in both space (at any particular time) and frequency, within the limitations of the uncertainty principle. The parameters labeling these wavelets (i.e., the variables on which the AST depends) have a direct geometrical significance: They give the initial position, direction of motion and average frequency or color of the wavelets. The representation of a solution in terms of these wavelets therefore gives a geometrical–optics (ray) picture of the solution. It is suggested that this could be of considerable practical value in signal analysis, since many naturally occurring signals (e.g., sound waves, electromagnetic waves) satisfy the wave equation away from sources and the geometrical–optics picture gives a readily accessible display of their informational contents.

2. Windowed Radon Transforms

2.1. The Windowed X–Ray Transform

Suppose we wish to measure a physical field distributed in . This field could be a “signal,” such as an electromagnetic field or the pressure distribution due to a sound wave. For simplicity, we assume to begin with that it is real–valued, such as pressure. (Our considerations easily extend to complex–valued, vector–valued or tensor–valued signals, such as electromagnetic fields; we shall indicate later how this is done.) The given field is therefore a function . We may think of as physical space (so that ), in which case the field is time–independent, or as space–time (so that ), in which case the field may be time–dependent. In the former case, is endowed with a Euclidean metric, while in the latter case the appropriate metric is Lorentzian, as mandated by Relativity theory.

Actual measurements are never instantaneous, nor do they take place at a single point in space. A measurement is performed by reading an instrument, and the instrument necessarily occupies some region in space and must interact with the field for some time–interval before giving a meaningful reading. Let us assume, to begin with, that the spatial extension of the instrument is negligible, so that it can be regarded as being concentrated at a single point at any time. We allow our instrument to be in an arbitrary state of uniform motion, so that its position is given by , where is a “time” parameter and . Note that need not be the physical time. For example, if is space–time, then each “point” represents an event, i.e. a particular location in space at a particular time. In that case, the line is called a world–line and represents the entire history of the point–instrument. The “velocity” vector then has one too many components and may be regarded as a set of homogeneous coordinates for the physical velocity. Note that in this case cannot vanish, since this would correspond to an instrument not subject to the flow of time. Even if is space, the case is not interesting since then the instrument can only measure the field at a single point. We therefore assume that , hence .

Let us assume that the reading registered by the instrument at time gives a weight to the value of the field passed by the instrument at time . (For motivational purposes we note that causality would demand that for ; moreover, should be concentrated in some interval , where is a “response time” or memory characteristic of the instrument. However, the results below do not depend on these assumptions.) Our model for the observed value of the field at the “point” , as measured by the instrument traveling with uniform velocity , is then

To accomodate complex–valued signals, we allow the weight function to be complex–valued. will denote the complex conjugate of . In order to minimize analytical subtleties, we assume that is smooth and bounded, and that is smooth with rapid decay (say, a Schwartz test function).

###### Definition 1

The Windowed X–Ray Transform of is the function given by

Remarks.

1. In the special case and , is known as the (ordinary) X–Ray transform of (Helgason [11]), due to its applications in tomography. We may then regard as being defined on the set of all lines in , independent of their parametrization. In the general case, we think of the function as a window, which explains our terminology.

2. Some work along related lines was recently done by Holschneider [12]. He considers a two–dimensional wavelet transform which is covariant under translations, rotations and and dilations of . When the window function is supported on a line, say , this becomes an X–Ray transform in . His inversion method is less direct than ours in that it involves a limiting process.

3. Note that has the following dilation property for :

where . This may be used to study the behavior of as . For the “forbidden” value , the transform becomes , where is the Fourier transform of . (We shall see that for “admissible” .)

4. For and , a change of variables gives

where is the usual wavelet transform of [5, 7, 23], with playing the role of a dilation factor and the window function playing the role of a basic wavelet.

5. All our considerations extend to vector–valued signals. The cleanest approach is to let the window function assume values in the dual vector space, so that and are scalars. More than one window needs to be used (or rotated versions of a single window), in order to ‘probe’ the different components of . The same applies to tensor–valued signals such as electromagnetic fields, since they may be regarded as being valued in a higher–dimensional vector space. However, a more correct way to measure a vector– or tensor–valued field is to use an instrument which is not rotationally invariant, and that implies that the instrument has some spatial extension. This is done in Section 2.3.

It will be useful to write in another form by substituting the Fourier representation of into . Formally, this gives

where is defined by

so that

(We have adopted the convention used in the physics literature, where complex inner products are linear in the second factor and antilinear in the first factor.)

The functions are –dimensional “wavelets” and will be used in the next subsection to reconstruct the signal . Note that (hence also ) is not square–integrable for , since its modulus is constant along directions orthogonal to . But eq. (5) still makes sense provided is sufficiently well–behaved. (This is one of the reasons we have assumed that is a test function.)

A common method for the construction of –dimensional wavelets consists of taking tensor products of one–dimensional wavelets. However, this means that not all directions in are treated equally, and consequently the set of wavelets does not transform “naturally” (in a sense to be explained below) under the affine group of , which consists of all transformations of the form

with a non–singular matrix and . Each such defines a unitary operator on , given by

where denotes the absolute value of the determinant of . The map forms a representation of on , meaning that it preserves the group structure of under compositions. To see how transforms under , note that the unitarity of implies

Hence

which states that affine transformations take wavelets to wavelets. Thus, for example, translations, rotations and dilations merely translate, rotate and dilate the labels , while the factor preserves unitarity. By contrast, tensor products of one–dimensional wavelets are not transformed into one another by rotations.

2.2. A Reconstruction Formula

A reconstruction consists of a recovery of from or its restriction to some subset. In the one–dimensional case, for example, can be reconstructed using all of or (for certain choices of ) just a discrete subset [2, 6, 18, 21, 22]. For general , the choice of reconstructions becomes even richer since various new possibilities arise. For example, may have symmetries which imply that is determined by its values on some lower–dimensional subsets of , making integration over the whole space unnecessary and, moreover, undesirable since it may lead to a divergent integral. Furthermore, may satisfy some partial differential equation which implies that it is determined by its values on subsets of . For example, if is space–time and represents a pressure wave or an electromagnetic potential, it satisfies the wave equation away from sources, hence is determined by initial data on a Cauchy surface in , and it becomes both unnecessary and undesirable to use all of in the reconstruction (cf. Sections 3.2 and 4).

The reconstruction to be developed in this subsection is “generic” in that it does not assume any particular forms for or . It uses all of , so it breaks down for certain choices of or . Again we emphasize that this is far from the only way to proceed; other types of reconstruction will be discussed below and elsewhere. The present reconstruction formula is interesting in part because it generalizes the one for the ordinary continuous wavelet transform .

To reconstruct , we look for a resolution of unity in terms of the vectors . This means we need a measure on such that

(Such an identity is sometimes called a “Plancherel formula.”) For then the map is an isometry from onto its range , and polarization gives

This shows that in , which is the desired reconstruction formula. (Cf. [16] for background on resolutions of unity, generalized frames and related subjects.)

To obtain a resolution of unity, note that

where denotes the inverse Fourier transform, so by Plancherel’s theorem,

We therefore need a measure on such that

since then has the desired property. The solution is simple: Every can be transformed to by a dilation and rotation of . That is, the orbit of (in Fourier space) under dilations and rotations is . Thus we choose to be invariant under rotations and dilations, which gives

where is a normalization constant, is the Euclidean norm of and is Lebesgue measure in . Then for ,

Now a straightforward computation gives

This shows that the measure gives a resolution of unity if and only if

which is precisely the admissibility condition for the usual (one–dimensional) wavelet transform [5]. (As mentioned above, admissibility implies that .) If is admissible, the normalization constant is given by

and the reconstruction formula is

The sense in which this formula holds depends on the behavior of . The class of possible ’s, in turn, depends on the choice of . Note that in spite of the factor in the denominator, there is no problem at since by the admissibility condition, and a similar analysis can be made for small by using the dilation property (eq.(3)).

2.3. The –Dimensional Windowed Radon Transform

Next, we allow the instrument to extend in spatial dimensions. For example, for a wire antenna whereas for a dish antenna. If is space, then ; if is space–time, then . When moving through space with a uniform velocity, the instrument sweeps out a –dimensional surface in , where if the motion is transverse to its spatial extension and if it is not. If , then the set of non–transversal motions is “non–generic” (has measure zero) and can thus be ignored; we therefore set in that case. If , then necessarily . In either case, we represent the moving instrument by a window function .

The parameter has thus been replaced by . The velocity vector , which may be regarded as a linear map from to , is now replaced by a linear map , which we call a motion of the instrument in . Denote the set of all such maps by . Later, when seeking reconstruction, we shall need to restrict ourselves to subsets of (‘rigid’ motions); but this need not concern us presently.

###### Definition 2

The d–dimensional Windowed Radon Transform of is the function given by

Upon substituting the Fourier representation of , a computation similar to the above yields the expression

where is the map dual to . (For given bases in and , is represented by an matrix; then is the transposed matrix.) In the above equation we have set

which gives the generalized wavelets

Let us now attempt to reconstruct from by generalizing the procedure in Section 2.2. Eq. (15) now becomes

Again we need a measure which is invariant under dilations and rotations of . Now the largest set of maps which can be considered consists of all those with rank . (Otherwise the instrument sweeps out a surface of dimension lower than .) Let us call this set . Then a measure on which is invariant with respect to rotations and dilations of has the form

where is the Haar measure on as an additive group. However, no reconstruction is possible using the measure on , because no admissible window exists in general when . (This can be easily verified when .) Thus is too large, and we return to our imaginary measuring process for inspiration. On physical grounds, we are interested in rigid motions of the instrument. A map corresponding to such a motion must have the form , where is the canonical inclusion map, is a rotation of ( gives the orientation of the instrument as well as its direction of motion), and is the speed. (If is space–time, then “rotations” involving the time axis are actually Lorentz transformations! For the present, assume that is space, so is a true rotation.) We therefore parametrize the set of permissible ’s by , where is the group of unimodular orthogonal matrices, which represent rotations in . This parametrization is redundant because two rotations of which have the same effect on the subspace give the same motion. A non–redundant parametrization of rigid motions is given by . However, we use the redundant one here for simplicity. (We shall need the Haar measure on .) Note that for , is represented by the vector and the set of all maps as above coincides with the set of non–zero velocities considered in Sections 2.1 and 2.2. A measure on which is invariant under rotations and dilations of (i.e., under itself) has the form

where is a normalization constant and is the Haar measure on . Thus for all ,

Now

where is the first row of , which is a unit vector, and is the projection of onto . The admissibility condition therefore reads

For , this reduces to eq. (20). If is admissible, we obtain the reconstruction formula

3. Analytic–Signal Transforms

3.1. Analytic Signals in One Dimension

Suppose we are given a (possibly complex–valued) one–dimensional signal . For simplicity, assume that is smooth with rapid decay. Consider the positive– and negative– frequency parts of , defined by

Then and extend analytically to the upper–half and lower–half complex planes, respectively; i.e.,

since the factor decays rapidly for in the respective integrals. and are just the (inverse) Fourier–Laplace transforms of the restrictions of to the positive and negative frequencies. If is complex–valued, then and are independent and the original signal can be recovered from them as

If is real–valued, then

In that case, and are related by reflection,

and

When is real, the function is known as the analytic signal associated with . Such functions were first introduced and applied extensively by Gabor [8]. A complex–valued signal would have two independent associated analytic signals and . What significance do have? For one thing, they are regularizations of . Eq. (36) states that is jointly a “boundary–value” of the pair and . As such, may actually be quite singular while remaining the boundary–value of analytic functions. Also, provide a kind of “envelope” description of (cf. Born and Wolf [4], Klauder and Sudarshan [19]). For example, if , then .

In order to extend the concept of analytic signals to more than one dimension, let us first of all unify the definitions of and by setting

for arbitrary , where is the unit step function, defined by

Then we have

Although this unification of and may at first appear to be somewhat artificial, it turns out to be quite natural, as will now be seen. Note first of all that for any real , we have

since the contour on the right–hand side may be closed in the upper–half plane when and in the lower–half plane when . For , the equation states that

in agreement with our definition, if we interpret the integral as the limit when of the integral from to . Therefore

If this is substituted into our expression for and the order of integrations on and is exchanged, we obtain

for arbitrary . We have referred to the right–hand side as the Analytic–Signal transform of [16, 17]. It bears a close relation to the Hilbert transform, which is defined by

where PV denotes the principal value of the integral. Consider the complex combination

Similarly,

Hence

which for real–valued reduces to

3.2. Generalization to n Dimensions

We are now ready to generalize the idea of analytic signals to an arbitrary number of dimensions. Again we assume initially that belongs to the space of Schwartz test functions , although this assumption proves to be unnecessary.

###### Definition 3

The Analytic–Signal Transform (AST) of is the function defined by

The same argument as above shows that for ,

where is the half–space

We shall refer to the right–hand side of eq. (53) as the (inverse) Fourier–Laplace transform of in . The integral converges absolutely whenever , since on , defining as a function on , although not an analytic one in general (see below). This shows that can actually be defined for some distributions , not only for test functions.

Note: In spite of the appearance of expressions such as , we have not assumed any particular metric structure in . The Fourier transform naturally takes functions on (considered as an abelian group) to functions on the dual space of linear functionals, and merely denotes the value . (See Rudin [25].) This remark becomes especially important when considering time–dependent signals, so that is space–time, for then the natural structure on is a Lorentzian metric rather than a Euclidean metric (cf. [16], Section 1.1.)

For , was analytic in the upper– and lower–half planes. In more than one dimension, need not be analytic, even though, for brevity, we still write it as a function of rather than and its complex conjugate . However, does in general possess a partial analyticity which reduces to the above when . Consider the partial derivative of with respect to , defined by

Then is analytic at if and only if for all . But using our definition of , we find that

It follows that the complex –derivative in the direction of vanishes, i.e.

if decays for large (for example, if is a test function, as we have assumed). Equivalently, using

we have for

Thus is analytic in the direction . In the one–dimensional case, this reduces to

which states that is analytic in the upper– and lower–half planes. In one dimension, there are only two imaginary directions, whereas in dimensions, every defines an imaginary direction.

The multivariate AST is related to the Hilbert transform in the direction (cf. [26], p. 49), defined as

(Usually, it is assumed that is a unit vector; we do not make this assumption.) Namely, an argument similar to the above shows that

hence

For and , this reduces to the previous relation with the ordinary Hilbert transform.

As in the one–dimensional case, is the boundary–value of in the sense that

For real–valued , these equations become

Two unresolved yet fundamental questions are:

(a) For what classes of ‘functions’ (possibly distributions) can the AST be defined, apart from ; i.e., what is the domain of the AST?

(b) Given a vector space of ‘functions’ on for which the AST is defined, what is the range of the AST on ? That is, given a function on , how can we tell whether is the transform of some ?

A necessary, though probably not sufficient, condition for is that satisfy the directional Cauchy–Riemann equation . Complete answers to the above questions can be given in some specific cases: When is a solution of the Klein–Gordon equation, then must satisfy a certain consistency condition (the reproducing property of the associated wavelets). This condition, when satisfied by , also guarantees that for some (cf. [16], Chapters 1 and 4). A similar result will be obtained in Section 4 for solutions of the wave equation in two space–time dimensions. The comments below apply to the general case and are, consequently, informal.

The most direct way to find if is the AST of some is to construct from and then check that . The first part has already been done formally, since has been shown to be the boundary–value of . Here we suggest an alternative method which can be used to construct instead of . Assume that the Fourier transform is defined on . If for some , then the –dimensional Fourier transform of is seen (formally) to be

where is the AST, in Fourier space, of the Dirac measure . (This suggests that the AST, like the Fourier transform, exhibits some symmetry between space and Fourier space.) can be shown to be invariant under real rotations (, with ) and to transform under dilations as

Given , let be a rotation such that , where , and let , so that