Shift-invariant Systems Research Articles

Inversion Algorithm to Calculate Charge Density on Solid Dielectric Surface Based on Surface Potential Measurement

Charge accumulation on a solid dielectric surface is one of the critical concerns for the design and optimization of the insulation system in a high-voltage power equipment, since it will lead to the overstress of electrical insulation. Therefore, it is important to obtain the charge density distribution on a solid dielectric surface with high accuracy. The acquisition of surface charge for insulators requires multipoint potential measurements to establish the inverse calculation for the determination of an unknown charge distribution. Up to now, extensive studies have been conducted on this subject; nevertheless, the methods are either too complicated and time consuming, or only applicable for specific arrangements, or with poor accuracy. In this paper, the problem is divided into two categories, i.e., shift-variant system and shift-invariant system, and the basic principle of an improved inversion algorithm is interpreted to solve the problem. The 2-D Fourier transform and Wiener filter techniques are employed in the algorithm for shift-invariant system thus the relationship between potential and charge density can be processed in spatial frequency domain, which tremendously simplifies the conventional procedure. The accuracy and resolution of the algorithm are discussed in detail with the aid of numerical examples. In the end, experiments are conducted and the effectiveness of the algorithm is verified.

The unitary extension principle on locally compact abelian groups

The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for L 2 ( R s ) based on a given refinable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions; via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.

Generalized shift-invariant systems and approximately dual frames

Dual pairs of frames yield a procedure for obtaining perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force us to apply sequences that do not exactly form dual frames. In this article, we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames. The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.

Investigation of x-ray spectra for iodinated contrast-enhanced dedicated breast CT

Screening for breast cancer with mammography has been very successful, resulting in part to a reduction of breast cancer mortality by approximately 39% since 1990. However, mammography still has limitations in performance, especially for women with dense breast tissue. Iodinated contrast-enhanced, dedicated breast CT (BCT) has been proposed to improve lesion analysis and the accuracy of diagnostic workup for patients suspected of having breast cancer. A mathematical analysis to explore the use of various x-ray filters for iodinated contrast-enhanced BCT is presented. To assess task-based performance, the ideal linear observer signal-to-noise ratio (SNR) is used as a figure-of-merit under the assumptions of a linear, shift-invariant imaging system. To estimate signal and noise propagation through the BCT detector, a parallel-cascade model was used. The lesion model was embedded into a structured background and included a realistic level of iodine uptake. SNR was computed for 84,000 different exposure settings by varying the kV setting, x-ray filter materials and thickness, breast size, and composition and radiation dose. It is shown that some x-ray filter material/thickness combinations can provide up to 75% improvement in the linear ideal observer SNR over a conventionally used x-ray filter for BCT. This improvement in SNR can be traded off for substantial reductions in mean glandular dose.

Linear System Models for Ultrasonic Imaging: Intensity Signal Statistics.

Despite a great deal of work characterizing the statistical properties of radio frequency backscattered ultrasound signals, less is known about the statistical properties of demodulated intensity signals. Analysis of intensity is made more difficult by a strong nonlinearity that arises in the process of demodulation. This limits our ability to characterize the spatial resolution and noise properties of B-mode ultrasound images. In this paper, we generalize earlier results on two-point intensity covariance using a multivariate systems approach. We derive the mean and autocovariance function of the intensity signal under Gaussian assumptions on both the object scattering function and acquisition noise, and with the assumption of a locally shift-invariant pulse-echo system function. We investigate the limiting cases of point statistics and a uniform scattering field with a stationary distribution. Results from validation studies using simulation and data from a real system applied to a uniform scattering phantom are presented. In the simulation studies, we find errors less than 10% between the theoretical mean and variance, and sample estimates of these quantities. Prediction of the intensity power spectrum (PS) in the real system exhibits good qualitative agreement (errors less than 3.5 dB for frequencies between 0.1 and 10 cyc/mm, but with somewhat higher error outside this range that may be due to the use of a window in the PS estimation procedure). We also replicate the common finding that the intensity mean is equal to its standard deviation (i.e., signal-to-noise ratio = 1) for fully developed speckle. We show how the derived statistical properties can be used to characterize the quality of an ultrasound linear array for low-contrast patterns using generalized noise-equivalent quanta directly on the intensity signal.

On Wilson Bases in $L^2(\mathbb{R}^d)$

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. It is well known that, starting from a tight Gabor frame for $L^{2}(\mathbb{R})$ with redundancy 2, one can construct an orthonormal Wilson basis for $L^2(\mathbb{R})$ whose generator is well localized in the time-frequency plane. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. Specifically, we show that one can construct $d$-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$, where $k=1, 2, \hdots, d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.

Explicit constructions and properties of generalized shift-invariant systems in L 2 ( ℝ ) $L^{2}(\mathbb {R})$

Generalized shift-invariant (GSI) systems, originally introduced by Hernandez et al. and Ron and Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calderon sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderon sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hernandez et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).

Spatial resolution, signal-to-noise and information capacity of linear imaging systems

A simple model for image formation in linear shift-invariant systems is considered, in which both the detected signal and the noise variance are varying slowly compared to the point-spread function of the system. It is shown that within the constraints of this model, the square of the signal-to-noise ratio is always proportional to the "volume" of the spatial resolution unit. In the case of Poisson statistics, the ratio of these two quantities divided by the incident density of the imaging particles (e.g. photons) represents a dimensionless invariant of the imaging system, which was previously termed the intrinsic imaging quality. The relationship of this invariant to the notion of information capacity of communication and imaging systems, which was previously considered by Shannon, Gabor and others, is investigated. The results are then applied to a simple generic model of quantitative imaging of weakly scattering objects, leading to an estimate of the upper limit for the amount of information about the sample that can be obtained in such experiments. It is shown that this limit depends only on the total number of imaging particles incident on the sample, the average scattering coefficient, the size of the sample and the number of spatial resolution units.

Evaluation of the validity of the scalar approximation in optical wave propagation using a systems approach and an accurate digital electromagnetic model

The cause and amount of error arising from the use of the scalar approximation in monochromatic optical wave propagation are discussed using a signals and systems formulation. Based on Gauss’s Law, the longitudinal component of an electric field is computed from the transverse components by passing the latter through a two input single output linear shift-invariant system. The system is analytically characterized both in the space and frequency domains. For propagating waves, the large response for the frequencies near the limiting wave number indicates the small angle requirement for the validity of the scalar approximation. Also, a discrete simulator is developed to compute the longitudinal component from the transverse components for monochromatic propagating electric fields. The simulator output helps to evaluate the validity of the scalar approximation when the system output cannot be analytically calculated.

Matrix-free Krylov iteration for implicit convolution of numerically low-rank data

Evaluating the response of a linear shift-invariant system is a problem that occurs frequently in a wide variety of science and engineering problems. Calculating the system response via a convolution may be done efficiently with Fourier transforms. When one must compute the response of one system to m input signals, or the response of m systems to one signal, it may be the case that one may approximate all system responses without having to compute all m Fourier transforms. This can lead to substantial computational savings. Rather than process each point individually, one may only process basis vectors that span the output data space. However, to get a low-error approximation, it is necessary that the output vectors have low numerical rank if they were assembled into a matrix. We develop theory that shows how the singular value decay of a matrix ΦA that is a product of a convolution operator Φ and an arbitrary matrix A depends in a linear fashion on the singular value decays of Φ and A. We propose gap-rank, a measure of the relative numerical rank of a matrix. We show that convolution cannot destroy the numerical low-rank-ness of ΦA data with only modest assumptions. We then develop a new method that exploits low-rank problems with block Golub–Kahan iteration in a Krylov subspace to approximate the low-rank problem. Our method can exploit parallelism in both the individual convolutions and the linear algebra operations in the block Golub–Kahan algorithm. We present numerical examples from signal and image processing that show the low error and scalability of our method.

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramer---Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton---Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, $$\alpha $$ź-scale-spaces, summed $$\alpha $$ź-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.

Shift-Variance and Nonstationarity of Linear Periodically Shift-Variant Systems and Applications to Generalized Sampling-Reconstruction Processes

We present a systematic analysis on shift-variance of linear $T$ -periodically shift-variant ( $T$ -LPSV) systems with continuous-time input and output. We first determine how far a $T$ -LPSV system $H$ is away from being shift-invariant in terms of two measures, namely, the distance and the angle between $H$ and the subspace of linear shift-invariant systems. We then consider the level and the index of shift-variance (SVI) of $H$ via commutator of $H$ and the shift-operator, and examine the amount of shift-variance of $H$ under particular input. All these quantities are characterized in terms of the so-called shift-variant part and the shift-variance kernel of $H$ . Moreover, we study nonstationarity of wide-sense cyclostationary random processes, possibly induced by $T$ -LPSV systems. We define and calculate a measure of nonstationarity (NSt) of the output as the SVI of the autocorrelation operator. The Nst can be used to characterize performance loss when the output is treated with a WSS approach. The expected shift-variance of $H$ subject to particular WSS random process is also calculated. We apply the above results to generalized sampling-reconstruction processes (GSRPs). In addition, we consider the approximation error of the GSRP and relate it to the shift-variance measures. Three sampling schemes (namely, the orthogonal, the consistent, and the minimax regret sampling) are examined in detail. We show that for both deterministic and WSS random inputs, the minimax regret GSRP always introduces less shift-variance than the consistent counterparts. Numerical results for the GSRPs of B-splines of various orders are provided.

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

In this paper we connect the well-established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we letΓj\Gamma _{\!j},j∈Jj \in J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian groupGGand study systems of the form⋃j∈J{gj,p(⋅−γ)}γ∈Γj,p∈Pj\bigcup _{j \in J}\{ g_{j,p}(\cdot - \gamma )\}_{\gamma \in \Gamma _{\!j}, p \in P_j}with generatorsgj,pg_{j,p}inL2(G)L^2(G)and with eachPjP_jbeing a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technicalα\alphalocal integrability condition (α\alpha-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas forL2(G)L^2(G). This generalizes results on generalized shift invariant systems, where eachPjP_jis assumed to be countable and eachΓj\Gamma _{\!j}is a uniform lattice inGG, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform latticesΓj\Gamma _{\! j}, our characterizations improve known results since the class of GTI systems satisfying theα\alpha-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames forL2(G)L^2(G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform inL2(Rn)L^2(\mathbb {R}^n)are special cases of the same general characterizing equations.

Estimating a Repeatable Statistical Law by Requiring Its Stability During Observation

Consider a statistically-repeatable, shift-invariant system obeying an unknown probability law p(x) ≡ q2(x): Amplitude q(x) defines a source effect that is to be found. We show that q(x) may be found by considering the flow of Fisher information J → I from source effect to observer that occurs during macroscopic observation of the system. Such an observation is irreversible and, hence, incurs a general loss I - J of the information. By requiring stability of the law q(x), as well, it is found to obey a principle I − J = min. of “extreme physical information” (EPI). Information I is the same functional of q(x) for any shift-invariant system, and J is a functional defining a physical source effect that must be known at least approximately. The minimum of EPI implies that I ≈ J or received information tends to well-approximate reality. Past applications of EPI to predicting laws of statistical physics, chemistry, biology, economics and social organization are briefly described.

Frame properties of generalized shift-invariant systems in discrete setting

In this paper, we present a simple characterization of those sequences for which purely shift-invariant systems are normalized tight frames for . As an application, a characterization for the Gabor system to be a normalized tight frame can be directly derived. Further, we prove that if the Calderón condition holds, then such purely shift-invariant systems are still normalized tight frames for . At the end of the paper, a sufficient condition for those purely shift-invariant systems to be frames for is established which is a variant of the classical wavelet systems to be frames for .

Singular value decomposition for photon-processing nuclear imaging systems and applications for reconstruction and computing null functions

Recent advances in technology are enabling a new class of nuclear imaging systems consisting of detectors that use real-time maximum-likelihood (ML) methods to estimate the interaction position, deposited energy, and other attributes of each photon-interaction event and store these attributes in a list format. This class of systems, which we refer to as photon-processing (PP) nuclear imaging systems, can be described by a fundamentally different mathematical imaging operator that allows processing of the continuous-valued photon attributes on a per-photon basis. Unlike conventional photon-counting (PC) systems that bin the data into images, PP systems do not have any binning-related information loss. Mathematically, while PC systems have an infinite-dimensional null space due to dimensionality considerations, PP systems do not necessarily suffer from this issue. Therefore, PP systems have the potential to provide improved performance in comparison to PC systems. To study these advantages, we propose a framework to perform the singular-value decomposition (SVD) of the PP imaging operator. We use this framework to perform the SVD of operators that describe a general two-dimensional (2D) planar linear shift-invariant (LSIV) PP system and a hypothetical continuously rotating 2D single-photon emission computed tomography (SPECT) PP system. We then discuss two applications of the SVD framework. The first application is to decompose the object being imaged by the PP imaging system into measurement and null components. We compare these components to the measurement and null components obtained with PC systems. In the process, we also present a procedure to compute the null functions for a PC system. The second application is designing analytical reconstruction algorithms for PP systems. The proposed analytical approach exploits the fact that PP systems acquire data in a continuous domain to estimate a continuous object function. The approach is parallelizable and implemented for graphics processing units (GPUs). Further, this approach leverages another important advantage of PP systems, namely the possibility to perform photon-by-photon real-time reconstruction. We demonstrate the application of the approach to perform reconstruction in a simulated 2D SPECT system. The results help to validate and demonstrate the utility of the proposed method and show that PP systems can help overcome the aliasing artifacts that are otherwise intrinsically present in PC systems.

Aberrations in shift-invariant linear optical imaging systems using partially coherent fields

Here the role and influence of aberrations in optical imaging systems employing partially coherent complex scalar fields is studied. Imaging systems require aberrations to yield contrast in the output image. For linear shift-invariant optical systems, we develop an expression for the output cross-spectral density under the space-frequency formulation of statistically stationary partially coherent fields. We also develop expressions for the output cross-spectral density and associated spectral density for weak-phase, weak-phase–amplitude, and single-material objects in one transverse spatial dimension.

Dual Gramian analysis: Duality principle and unitary extension principle

Abstract. Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames [40] and the unitary extension principle for wavelet frames [38]. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in [38], e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.

Duality for Frames

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

DIRECTIONAL TIME–FREQUENCY ANALYSIS VIA CONTINUOUS FRAMES

Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in $L^{2}(\mathbb{R}^{n})$ based on Gabor systems in $L^{2}(\mathbb{R})$. The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for $L^{2}(\mathbb{R}),$ in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ${\it\epsilon}$-nets.