B-Spline Approximations of the Gaussian,their Gabor Frame Properties,and Approximately Dual Frames

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters \(a,b>0\) with \(ab<1\), the scaled version of \(B_N\) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction” within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the \(L^p\)-spaces, as well in the time-domain as in the frequency domain.     

Theory,implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Exponential B-splines and the partition of unity property

We provide an explicit formula for a large class of exponential B-splines. Also, we characterize the cases where the integer-translates of an exponential B-spline form a partition of unity up to a multiplicative constant. As an application of this result we construct explicitly given pairs of dual Gabor frames.     

Pairs of Explicitly Given Dual Gabor Frames in L2(ℝd)

Given certain compactly supported functions g ≥ L²(ℝ^d) whose ℤ^d-translates form a partition of unity, and real invertible d × d matrices B,C for which ||C^T B|| is sufficiently small, we prove that the Gabor system forms a frame, with a (noncanonical) dual Gabor frame generated by an explicitly given finite linear combination of shifts of g. For functions g of the above type and arbitrary real invertible d × d matrices B,C this result leads to a construction of a multi-Gabor frame , where all the generators g_k are dilated and translated versions of g. Again, the dual generators have a similar form, and are given explicitly. Our concrete examples concern box splines.     

Nonlinear approximation with nonstationary Gabor frames

We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions generalizes to the nonstationary case, providing perfect reconstruction and an FFT based implementation for compactly supported window functions sampled at a certain density. It is also known that for some signal classes, NSGFs with flexible time resolution tend to provide sparser expansions than can be obtained with classical Gabor frames. In this article we show, for the continuous case, that sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an associated decomposition space. In this way we characterize signals with sparse expansions relative to NSGFs with flexible time resolution. Based on this characterization we prove an upper bound on the approximation error occurring when thresholding the coefficients of the corresponding frame expansions. We complement the theoretical results with numerical experiments, estimating the rate of approximation obtained from thresholding the coefficients of both stationary and nonstationary Gabor expansions.     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

Sparse Dual Frames and Dual Gabor Functions of Minimal Time and Frequency Supports

Exploitation of the optimality of (non-exact) frames from a sparse dual point of view is presented. Sparse dual frames and dual Gabor functions of the minimal time and/or frequency supports are studied and constructed through the notion of sparse representations. Conditions on the sparsest dual frames and the dual Gabor functions of the minimal time and/or frequency supports are discussed. Algorithms and examples are provided.     

Dimension invariance of finite frames of translates and Gabor frames

A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented.     

Duality inequalities as a numerical aid

This note re-examines the problem of estimating the minimum value of a convex program. To obtain a lower bound to this value a dual program is formulated. The dual involves only explicitly given functions and only inequality constraints. No nonlinear equality constraints appear. Thus a numerically feasible algorithm is obtained.     

Stability in diffraction tomography and a nonlinear “basic theorem”

The stability problem is studied for reconstruction of the refraction coefficient from boundary measurements of solutions of the Helmholtz equation at a fixed time-frequency. An answer is given in terms of Gabor means of the coefficient. A domain in the phase space is shown where the Gabor means can be stably reconstructed. As a corollary, a rigorous form is given to the basic theorem of diffraction tomography.     

A Note on Reassigned Gabor Spectrograms of Hermite Functions

An explicit form is given for the reassigned Gabor spectrogram of an Hermite function of arbitrary order. It is shown that the energy concentration sharply localizes outside the border of a clearance area limited by the “classical” circle where the Gabor spectrogram attains its maximum value, with a perfect localization that can only be achieved in the limit of infinite order.     

Nonstationary Gabor frames for

Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.     

Approximation of the Fourier Transform and the Dual Gabor Window

Many results and problems in Fourier and Gabor analysis are formulated in the continuous variable case, i.e., for functions on . In contrast, a suitable setting for practical computations is the finite case, dealing with vectors of finite length. We establish fundamental results for the approximation of the continuous case by finite models, namely, the approximation of the Fourier transform and the approximation of the dual Gabor window of a Gabor frame. The appropriate function space for our approach is the Feichtinger space S₀. It is dense in L², much larger than the Schwartz space, and it is a Banach space.     

Compactly supported multiwindow dual Gabor frames of rational sampling density

We consider multiwindow Gabor systems (G _N ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski–Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski–Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a?=?1 and b?=?1. As another application, we show that a multiwindow Gabor system (G _N ; 1, 1) forms an orthonormal basis if and only if it has only one window (N?=?1) which is a sum of characteristic functions whose supports ‘essentially’ form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4)     

非均匀Gabor框架的稳定性

研究了当窗函数变化时非均匀Gabor框架的稳定性.对紧支撑Gabor框架,将均匀情况下关于稳定性的结论推广到了非均匀的情况;对一般的Gabor框架,利用W（L^∞,e^1）范数给出了其稳定的一个充分条件.     

Continuous Gabor Transform for Strong Hypergroups

We define a continuous Gabor transform for strong hypergroups and prove a Plancherel formula, an L ² inversion formula and an uncertainty principle for it. As an example, we show how these techniques apply to the Bessel–Kingman hypergroups and to the dual Jacobi polynomial hypergroups. These examples have an interpretation in the setting of radial functions on R ^d and zonal functions on compact two-point homogeneous spaces, where they provide a new transform which possesses many properties of the classical Gabor transform.     

Characterization of rationally sampled quaternionic dual Gabor frames

This paper addresses quaternionic dual Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time-frequency shift parameters not equal to $\frac{1}{2}$, we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.     

Second-order domain derivative of normal-dependent boundary integrals

Numerous reconstruction tasks in (optical) surface metrology allow for a variational formulation. The occurring boundary integrals may be interpreted as shape functions. The paper is concerned with the second-order analysis of such functions. Shape Hessians of boundary integrals are considered difficult to find analytically because they correspond to third-order derivatives of an, in a sense equivalent, domain integral. We complement previous results by considering cost functions depending explicitly on the surface normal. The correctness and practicability of our calculations are verified in the context of a Newton-type shape reconstruction method.     

Representing polyhedra: faces are better than vertices

This paper investigates the reconstruction of planar-faced polyhedra given their spherical dual representation. The spherical dual representation for any genus 0 polyhedron is shown to be unambiguous and to be uniquely reconstructible in polynomial time. It is also shown that when the degree of the spherical dual representation is at most four, the representation is unambiguous for polyhedra of any genus. The first result extends, in the case of planar-faced polyhedra, the well known result that a vertex or face connectivity graph represents a polyhedron unambiguously when the graph is triconnected and planar. The second result shows that when each face of a polyhedron of arbitrary genus has at most four edges, the polyhedron can be reconstructed uniquely. This extends the previous result that a polyhedron can be uniquely reconstructed when each face of the polyhedron is triangular. As a consequence of this result, faces are a more powerful representation than vertices for polyhedra whose faces have three or four edges. A result of the reconstruction algorithm is that high level features of the polyhedron are naturally extracted. Both results explicitly use the fact that the faces of the polyhedron are planar. It is conjectured that the spherical dual representation is unambiguous for polyhedra of any genus.