Linear Shift-invariant Systems Research Articles

Sufficient criterion for instability of 2-D linear discrete systems

In this paper, a theorem for the instability checking of linear shift-invariant two-dimensional discrete systems is proposed. This is based on the results for the location of the roots of polynomials presented by Mori and the criterion for the stability of 2-D discrete systems presented by Murray and Delsarte et al. The theorem provides a new sufficient criterion which can be applied easily for the 2-D instability checking. This is demonstrated by detailed examples.

Improved Classification of Blurred Images with Deep-Learning Networks Using Lucy-Richardson-Rosen Algorithm

Pattern recognition techniques form the heart of most, if not all, incoherent linear shift-invariant systems. When an object is recorded using a camera, the object information is sampled by the point spread function (PSF) of the system, replacing every object point with the PSF in the sensor. The PSF is a sharp Kronecker Delta-like function when the numerical aperture (NA) is large with no aberrations. When the NA is small, and the system has aberrations, the PSF appears blurred. In the case of aberrations, if the PSF is known, then the blurred object image can be deblurred by scanning the PSF over the recorded object intensity pattern and looking for pattern matching conditions through a mathematical process called correlation. Deep learning-based image classification for computer vision applications gained attention in recent years. The classification probability is highly dependent on the quality of images as even a minor blur can significantly alter the image classification results. In this study, a recently developed deblurring method, the Lucy-Richardson-Rosen algorithm (LR2A), was implemented to computationally refocus images recorded in the presence of spatio-spectral aberrations. The performance of LR2A was compared against the parent techniques: Lucy-Richardson algorithm and non-linear reconstruction. LR2A exhibited a superior deblurring capability even in extreme cases of spatio-spectral aberrations. Experimental results of deblurring a picture recorded using high-resolution smartphone cameras are presented. LR2A was implemented to significantly improve the performances of the widely used deep convolutional neural networks for image classification.

CGAN-assisted imaging through stationary scattering media.

Analyzing images taken through scattering media is challenging, owing to speckle decorrelations from perturbations in the media. For in-line imaging modalities, which are appealing because they are compact, require no moving parts, and are robust, negating the effects of such scattering becomes particularly challenging. Here we explore the use of conditional generative adversarial networks (cGANs) to mitigate the effects of the additional scatterers in in-line geometries, including digital holographic microscopy. Using light scattering simulations and experiments on objects of interest with and without additional scatterers, we find that cGANs can be quickly trained with minuscule datasets and can also efficiently learn the one-to-one statistical mapping between the cross-domain input-output image pairs. Importantly, the output images are faithful enough to enable quantitative feature extraction. We also show that with rapid training using only 20 image pairs, it is possible to negate this undesired scattering to accurately localize diffraction-limited impulses with high spatial accuracy, therefore transforming a shift variant system to a linear shift invariant (LSI) system.

Shortest-support multi-spline bases for generalized sampling

Generalized sampling consists in the recovery of a function f, from the samples of the responses of a collection of linear shift-invariant systems to the input f. The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M. While this property allows for an approximation power of order (M+1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M+1). Following this result, we introduce the notion of shortest basis of degree M, which is motivated by our desire to minimize computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.

A New Sufficient Criterion for the Stability of 2-D Discrete Systems

During the past few decades, two and higher dimensional systems have been extensively applied in many areas of research. The representation of the 2-D systems in the frequency domain is usually given by its transfer function. The bounded-input bounded-output (BIBO) stability of the two dimensional discrete systems depends on the zeros of the characteristic polynomial which is the denominator of this transfer function. In this paper, a new sufficient criterion for the stability of two-dimensional linear shift-invariant discrete systems is presented. The new criterion is based on the sufficient condition for stable polynomials with complex coefficients and the stability criterion for 2-D discrete systems proposed by Murray and Delsarte et al. . The new criterion is non-conservative for the stability testing of 2-D discrete systems. It is shown that the proposed sufficient criterion is simple enough to be applied for the stability checking of the 2-D discrete systems. The utility of the proposed criterion is demonstrated by examples.

State Consensus for Discrete-Time Multiagent Systems Over Time-Varying Graphs

We study the state consensus problem for linear shift-invariant discrete-time homogeneous multiagent systems (MASs) over time-varying graphs. A novel approach based on the small gain theorem is proposed to design the consensus control protocols for both neutrally stable and neutrally unstable MASs. It is shown that the state consensus can be achieved for neutrally stable MASs under the uniform observability condition; for neutrally unstable MASs, the state consensus entails more than the uniform observability. Two numerical examples are worked out to illustrate our consensus results.

A Note on BIBO Stability

The statements on the BIBO stability of continuous-time convolution systems found in engineering textbooks are often either too vague (because of lack of hypotheses) or mathematically incorrect. What is more troubling is that they usually exclude the identity operator. The purpose of this note is to clarify the issue while presenting some fixes. In particular, we show that a linear shift-invariant system is BIBO-stable in the $L_\infty$-sense if and only if its impulse response is included in the space of bounded Radon measures, which is a superset of $L_1(\mathbb{R})$ (Lebesgue's space of absolutely integrable functions). As we restrict the scope of this characterization to the convolution operators whose impulse response is a measurable function, we recover the classical statement.

Deterministic Design of Toeplitz Matrices With Small Coherence Based on Weyl Sums

The design of deterministic measurement matrices has been the focus of research in compressed sensing from the early stages. In particular, structured measurement matrices are of great interest as they could be efficiently stored. Our focus in this letter is on Toeplitz structure, which naturally arises in linear shift-invariant systems (convolution operator). We design complex-valued Toeplitz matrices with unit modulus elements that have small coherence. The complex phase of the matrix elements are determined by certain polynomials. We provide upper bounds for the coherence of the resulting matrix using tools from analytic number theory, namely, the Weyl sum theorem. Simulation results confirm that the proposed matrices perform similar to the Gaussian Toeplitz matrices of the same size.

Aberrated dark-field imaging systems

We study generalized dark-field imaging systems. These are a subset of linear shift-invariant optical imaging systems, that exhibit arbitrary aberrations, and for which normally-incident plane-wave input yields zero output. We write down the theory for the forward problem of imaging coherent scalar optical fields using such arbitrarily-aberrated dark-field systems, and give numerical examples. The associated images may be viewed as a form of dark-field Gabor holography, utilizing arbitrary outgoing Green functions as generalized Huygens-type wavelets, and with the Young-type boundary wave forming the holographic reference.

Comparison of optimum reconstruction filters for discrete and continuous-discrete linear observation models

In this paper, the expressions derivation for the frequency response of a reconstructing linear shift-invariant system and restoration error is performed for a continuously-discrete linear observation model. We analyzed three cases – two special ones and the general one. The first special case assumes the absence of dynamic degradations; the second one assumes the absence of additive noise. We presented results of a comparison of the restoration error values of the optimal filters with the same signal parameters, dynamic distortions and additive noise for the selected observation models.

Phase retrieval for arbitrary Fresnel-like linear shift-invariant imaging systems suitable for tomography.

We present a generalization of the non-iterative phase retrieval in X-ray phase contrast imaging applicable for an arbitrary linear shift-invariant (LSI) imaging system with a non-negligible amount of free space propagation (termed as Fresnel-like). Our novel approach poses no restrictions on the propagation distance between optical elements of the system. In turn, the requirements are only demanded for the transfer function of the optical elements, which should be approximable by second-order Taylor polynomials. Furthermore, we show that the method can be conveniently used as an initial guess for iterative phase retrieval, resulting in faster convergence. The proposed approach is tested on synthetic and experimentally measured holograms obtained using a Bragg magnifier microscope – a representative of Fresnel-like LSI imaging systems. Finally, the algorithm is applied to a whole micro-tomographic scan of a biological specimen of a tardigrade, revealing morphological details at the spatial resolution of 300 nm – limiting resolution of the actual imaging system.

Power Spectrum Equalized Scalar Representation of Wide-Angle Optical Field Propagation

It may be desirable to represent optical fields using scalar approximations, due to its simplicity. Since the optical field is an electromagnetic wave, in order to implement an optical setup, a mapping from such a scalar field to the vector electromagnetic field is needed. In the conventional scalar-to-vector field mapping, a large error in power spectrum arises in wide-angle fields due to the neglected large longitudinal component of the electric field. This error could be severe in wide-angle or off-axis imaging setups. In order to find another scalar-to-vector field mapping that compensates for this large magnitude error, first, a general constraint on monochromatic electromagnetic fields to appropriately represent them by a scalar wave in free space is developed. The development of the general constraint begins by formulating the computations of the components of the magnetic field as the outputs of linear-shift invariant (LSI) systems, where the inputs to the LSI systems are the transverse components of the electric field. Furthermore, if one of the transverse components of the electric field can be computed from the other one using a LSI operation, a scalar field, which is related to the transverse components through another LSI system, can be used to fully describe the electromagnetic field. Under this constraint, the required condition on the filters which relates the scalar field to the electric field is presented by taking into consideration the longitudinal component, such that the power spectra of the scalar field and the corresponding electric field are equal. The filters are specified for the electric fields with zero longitudinal component and simple polarization features, as well. Moreover, for the electric fields with simple polarization features, some discrete simulations are performed to compare the scalar field intensity pattern and the corresponding electric field intensity patterns for the conventional and proposed mapping cases. The simulation results show that the excessive amplification of the large frequency components is compensated by the proposed filters, and hence, the undesired effects of the filters used in the computation of the longitudinal component disappear. In this respect, if equality of the power spectra of the scalar field and the corresponding electric field is of concern in an application, the proposed scalar-to-vector wave field mapping should be used.

Propagation of fully coherent and partially coherent complex scalar fields in aberration space

We consider the propagation of both fully coherent and partially coherent complex scalar fields, through linear shift-invariant imaging systems. The state of such imaging systems is characterized by a countable infinity of aberration coefficients, the values for which can be viewed as coordinates for an infinity of orthogonal axes that span the "aberration space" into which the output propagates. For fully coherent complex scalar disturbances, we study the propagation of the field through the imaging system, while for partially coherent disturbances it is the two-point correlation functions whose propagation we study. For both systems we write down generalized propagators in both real and Fourier space, differential equations for evolution through aberration space, transport equations, and Hamilton-Jacobi equations. A generalized form of the Wolf equations is an important special case of our formalism.

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.

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.