Augustus J. E. M. Janssen

Measuring time-frequency information content using the Renyi entropies

The generalized entropies of Renyi inspire new measures for estimating signal information and complexity in the time-frequency plane. When applied to a time-frequency representation (TFR) from Cohens class or the affine class, the Renyi entropies conform closely to the notion of complexity that we use when visually inspecting time-frequency images. These measures possess several additional interesting and useful properties, such as accounting and cross-component and transformation invariances, that make them natural for time-frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Renyi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined.

The Zak transform and sampling theorems for wavelet subspaces

The Zak transform is used for generalizing a sampling theorem of G. Waiter (see IEEE Trans. Informat. Theory, vol. 38, p. 881-884, 1992) for wavelet subspaces. Cardinal series based on signal samples f(a+n), n in Z with a possibly unequal to 0 (Waiters case) are considered. The condition number of the sampling operator and worst-case aliasing errors are expressed in terms of Zak transforms of scaling function and wavelet. This shows that the stability of the resulting interpolation formula depends critically on a. >

The duality condition for Weyl-Heisenberg frames

We present formulations of the condition of duality for Weyl-Heisenberg systems in the time domain, the frequency domain, the time-frequency domain, and, for rational time-frequency sampling factors, the Zak transform domain, both for the one-dimensional time-continuous case and the one-dimensional time-discrete case. Many of the results we obtain are presented in the more general framework of shift-invariant systems or filter banks, and we establish, for instance, relations with the polyphase matrix approach from filter bank theory. The formulation of the duality condition in various domains is notably useful for the design of perfect reconstructing shift-invariant Weyl-Heisenberg analysis and synthesis systems under restrictions of the constituent filter responses which may be stated in any of the domains just mentioned. In all considered domains we present formulas for frame operators and frame bounds, and we compute and characterize minimal dual systems.

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system

Taking the classical Ignatowsky/Richards and Wolf formulas as our starting point, we present expressions for the electric field components in the focal region in the case of a high-numerical-aperture optical system. The transmission function, the aberrations, and the spatially varying state of polarization of the wave exiting the optical system are represented in terms of a Zernike polynomial expansion over the exit pupil of the system; a set of generally complex coefficients is needed for a full description of the field in the exit pupil. The field components in the focal region are obtained by means of the evaluation of a set of basic integrals that all allow an analytic treatment; the expressions for the field components show an explicit dependence on the complex coefficients that characterize the optical system. The electric energy density and the power flow in the aberrated three-dimensional distribution in the focal region are obtained with the expressions for the electric and magnetic field components. Some examples of aberrated focal distributions are presented, and some basic characteristics are discussed.

Zak Transforms with Few Zeros and the Tie

We consider the difficult problem of deciding whether a triple(g,a,b),with window g ∈ L 2(ℝ) and time shift parameter a and frequency shift parameterbis a Gabor frame from two different points of view. We first identify two classes of non-negative windows g ∈ L 2(ℝ) such that their Zak transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, non-negative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, non-negative, continuous, integrable windowsg that satisfy on [0, ∞) a condition slightly stronger than strict convexity (superconvexity). Accordingly, the members of these two classes generate Gabor frames for integer oversampling factor (ab)-1 ≥ 1 and ≥ 2, respectively. When we weaken the condition of superconvexity into strict convexity, the Zak transformsZg may have as many zeros as one wants, but in all cases (g, a, b) is still a Gabor frame when(ab) -1 is an integer ≥ 2. As a second issue we consider the question for which a, b > 0 the triple (g, a, b) is a Gabor frame, where gis the characteristic function of an interval [0, ∞) with c0 > 0 fixed. It turns out that the answer to the latter question is quite complicated, where irrationality or rationality of abgives rise to quite different situations. A pictorial display, in which the various cases are indicated in the positive (a, b)-quadrant, shows a remarkable resemblance to the design of a low-budget tie.

Assessment of optical systems by means of point-spread functions

Publisher Summary This chapter presents the computation of the point-spread function of optical imaging systems and the characterization of these systems by means of the measured three-dimensional structure of the point-spread function. The point-spread function, accessible in the optical domain only in terms of the energy density or the energy flow, is a nonlinear function of the basic electromagnetic field components in the focal region. That is why the reconstruction of the amplitude and phase of the optical far-field distribution that produced a particular intensity point-spread function is a nonlinear procedure that does not necessarily have a unique solution. Since the 1970s, the quality of optical imaging systems (telescopes, microscope objectives, high-quality projection lenses for optical lithography, space observation cameras) has been pushed to the extreme limits. At this level of perfection, a detailed analysis of the optical point-spread function is necessary to understand the image formation by these instruments, especially when they operate at high numerical aperture. In terms of imaging defects, it allowed to suppose that the wavefront aberration of such instruments is not substantially larger than the wavelength λ of the light. In most cases, the aberration even has to be reduced to a minute fraction of the wavelength of the light to satisfy the extreme specifications of these imaging systems. The past work on point-spread function analysis and its application to the assessment of imaging systems is presented in the chapter. This includes discussions on: the theory of point-spread function formation, energy density and power flow in the focal region, quality assessment by inverse problem solution, and quality assessment using the extended Nijboer–Zernike diffraction theory.

Two theorems on lattice expansions

It is shown that there is a tradeoff between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if g and g are dual, then (1) at least one of H/sup 1/2/ g and H/sup 1/2/ g is n in L/sup 2/(R), and (2) at least one of Hg and g is not in L/sup 2/(R). Here, H is the operator -1/(4 pi /sup 2/)d/sup 2//(dt/sup 2/)+t/sup 2/. The first result is a generalization of a theorem first stated by R.C. Balian (1987). The second result is new and relies heavily on the fact that, when G in W/sup 2,2/(S) with S=(-1/2, 1/2)*(-1/2, 1/2) and G(0), than 1/G not in L/sup 2/(S). >

Positivity of Weighted Wigner Distributions

In [4] a number of inequalities involving Wigner distributions and their moments are given. The present paper gives theorems on the positivity of weighted Wigner distributions, where the weight function is assumed to be radially symmetric. The main tool is a formula expressing weighted Wigner distributions of a function in terms of its Hermite coefficients and certain integrals involving Laguerre polynomials.

On positivity of time-frequency distributions

This correspondence addresses the problem of how to regard the fundamental impossibility with time-frequency energy distributions of Cohens class always to be nonnegative and, at the same time, to have correct marginal distributions. It is shown that the Wigner distribution is the only member of a large class of bilinear time-frequency distributions that becomes nonnegative after smoothing in the time-frequency plane by means of Gaussian weight functions with BT product equal to unity.