## 1. Introduction

[2] A common situation in the study of wave propagation in complex imaging environments consists of the specification of a wave field on a particular plane followed by its propagation forward to a new observation space, where one wishes to compute the field. Examples are abundant in the aperture theory of antennas, GPR applications, optical systems composed of sequences of imaging components, tomographic reconstruction, etc.

[3] Classically this is dealt with by Fourier Transformation [*Goodman*, 1968] (representation of the field by a superposition of plane waves), since the forward propagation of plane waves in free space is relatively easy to describe mathematically. However, this is not always objectively the most efficient way to represent the propagation, since the reconstruction of the plane wave superposition at the observation plane may involve much destructive interference between spectral plane waves to produce what is actually a highly localized field distribution in real space. This is particularly true in the case of high-frequency fields which propagate essentially along well-defined trajectories described by simple geometrical optics. Intuitively it would be much more efficient to describe such situations by superpositions of spectral elements which are themselves strongly localized to the trajectories which dominate the GO description of the field. Thus one is led to consider superpositions of collimated beams rather than of plane waves. In order to describe the effects of truncation to a finite set of beams it is necessary to have a properly formulated mathematical theory. As an additional feature one would like the theory to describe asymptotic behavior as the frequency of the wave field becomes very large.

[4] An early step in this direction was taken by *Gabor* [1946], who used a fundamental representation of the form

for the expansion of an arbitrary function *u*(*x*), *x* ∈ **R** where *x* is a suitably scaled domain variable. The coefficients in the expansion are given by the biorthogonal projection formula

with *C* = 1.599. More recently, the use of the windowed Fourier transform

has been extensively studied, with *U* the coefficient function and *g*:**R** → **R** an arbitrary windowing function which vanishes sufficiently rapidly at *x* → ±∞. When *g* is gaussian, *g*(*x*) = (2ν)^{1/4}exp[−νπ*x*^{2}], this is also called the continuous Gabor transform. Here the phase-space parameters (*m*, *n*) ∈ **R**^{2} are real, not restricted to a discrete grid **Z**^{2} as in the discrete Gabor case.

[5] Several facts are known about the discrete and continuous Gabor transforms. For the discrete case, it is known that the sampling of the phase-space afforded by the lattice of discrete points (*m*, *n*) ∈ **Z**^{2} is optimum (one coefficient *per* unit of phase-space area), but that the series tends to be very unstable in the sense that small perturbations of the coefficients can produce large fluctuations of the represented function. This can be remedied by oversampling. More problematic is the fact that the coefficients in the expansion are obtained from a complicated expression involving inner products of the given function *u* on a dual space of biorthogonal functions which are very discontinuous, given explicitly by equation (2), and the inner products are difficult to calculate in closed form even for relatively simple functions. The continuous Gabor transform does not suffer these problems, but on the other hand is highly redundant due to overcompleteness (i.e., there exist uncountably many different sets of coefficient functions for the same function *u*).

[6] Many cases are known of functions *u* whose Gabor coefficients, both discrete and continuous, are highly localized for parameters (*m*, *n*) near to a lower dimensional manifold in the phase-space; this is particularly true for functions of geometrical optics type

where *a* and *s* are smooth functions and *k* is a large parameter, in which case the manifold is called the Lagrange submanifold Λ and is given by the locus of points (*x*, ξ) satisfying

[7] Localization of the Gabor coefficients near a lower-dimensional manifold should imply that the function *u* is very efficiently computed by restricting the superpositions (1) or (2) to parameter points in phase-space that are close to Λ, and neglecting contributions from points that are far from the Λ [*Arnold*, 1995]. However, there are no systematic rules known for characterizing the asymptotic errors involved in this truncation of the phase-space, nor how to construct a robust sampling scheme for a discrete representation of the superposition which avoids both the instability of the discrete Gabor representation and the redundancy of the continuous Gabor representation.

[8] It is most convenient in describing the theory of these representations to normalize the physical length *x* to some scale length *L* and the wave number ξ to 2π/*L*; this is done by the replacements *x*/*L* → *x*, ξ*L*/2π → ξ, which is equivalent to measuring length in units of *L*. This modifies the GO form of (5) to

in the normalized variables. The scale length is selected so that *L* is intermediate between the scale on which *a* and σ change and that of one local cycle of the function exp[2πiσ(*x*)].

[9] A simple example of the construction of a Lagrange submanifold is furnished by a point source located at (*x*, *z*) = (0, −*a*), for which we require the distribution of field over the observation space *z* = 0. The field of this scalar point source at the observation space is described (asymptotically for large *ka*) by

[10] This has a natural phase-amplitude decomposition, with a phase function proportional to the ray length from the source point (0, −*a*) to the observation point (*x*, 0):

[11] The local spatial frequency for this phase function is

and the Lagrange submanifold, which is the graph of the function ξ(*x*), is an S-shaped curve with asymptotes ξ → ±*k*/2π as *x* → ±∞.

[12] A second example of the construction of a Lagrange submanifold is shown in Figure 1. This particular example is for the Gauss–Hermite function *u*(*x*) = *H*_{q}((2π)^{1/2}*x*)exp(−π*x*^{2}), chosen because it represents generic characteristics of smooth Lagrange manifolds quite well. The Gauss–Hermite function satisfies the ordinary differential equation

with boundary conditions that *u* vanishes exponentially as *x* → ±∞. These boundary conditions can only be satisfied for integer values of *q*. Using the WKB method, the function *u* can be represented for large values of *q* by

with

The last equation can be integrated to give

[13] Equation (12) defines the local frequency ξ = d_{x}σ(*x*) = (*R*^{2} − *x*^{2})^{1/2}, which in turn defines the curve

in the phase-space (*x*, ξ). This curve is the Lagrange submanifold for the Gauss–Hermite function. The total area enclosed by this curve in phase-space is π*R*^{2} = *q* + 1/2; it follows that an increase of the integer *q* by 1 increases the area enclosed by the Lagrange submanifold by 1. By a result of *Daubechies* [1988], if the Lagrange submanifold of some general function *u*(*x*) is completely enclosed by the circle of radius *R*_{max}, then the integer *q*_{max} = π*R*_{max}^{2} − is the order of the last significant term in the orthogonal expansion in Gauss–Hermite functions of the general function *u*(*x*):

[14] The Lagrange submanifold defines a multiple-valued phase function

where the function ξ(*x*) is the assignment of frequency to position specified by the Lagrange submanifold. Integration of (16) once by parts leads to a second function

where *x*(ξ) is the inverse function to ξ(*x*). The identity

is called the *Legendre transform*. In this case the functions *x*(ξ) and ξ(*x*) are multiple-valued, but there is a unique assignment of points on the Lagrange submanifold. Each branch of the multiple-valued function ξ(*x*) joins smoothly onto another branch such that the Lagrange submanifold is the union of all the branches.