Shift-invariant Systems Research Articles

A shift-invariant morphological system for software development cost estimation

Abstract This work presents a shift-invariant morphological system to solve the problem of software development cost estimation (SDCE). It consists of a hybrid morphological model, which is a linear combination between a morphological-rank (MR) operator (nonlinear) and a Finite Impulse Response (FIR) operator (linear), referred to as morphological-rank-linear (MRL) filter. A gradient steepest descent method to adjust the MRL filter parameters (learning process), using the Least Mean Squares (LMS) algorithm, and a systematic approach to overcome the problem of non-differentiability of the morphological-rank operator are used to improve the numerical robustness of the training algorithm. Furthermore, an experimental analysis is conducted with the proposed system using the NASA software project database, and in the experiments, two relevant performance metrics and an evaluation function are used to assess its performance. The results obtained are compared to models recently presented in literature, showing superior performance of this kind of morphological systems for the SDCE problem.

The Solution of a Problem of Coifman, Meyer, and Wickerhauser on Wavelet Packets

Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal bases of L 2(ℝ), each one corresponding to a different splitting of L 2(ℝ) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet packet system can be an orthonormal basis for L 2(ℝ) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This answer to this question affects the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant system machinery developed by Ron and Shen.

Blind phase retrieval for aberrated linear shift-invariant imaging systems

We develop a means to reconstruct an input complex coherent scalar wavefield, given a through focal series (TFS) of three intensity images output from a two-dimensional (2D) linear shift-invariant optical imaging system with unknown aberrations. This blind phase retrieval technique unites two methods, namely (i) TFS phase retrieval and (ii) iterative blind deconvolution. The efficacy of our blind phase retrieval procedure has been demonstrated using simulated data, for a variety of Poisson noise levels.

Fast and Robust Deconvolution-Based Image Reconstruction for Photoacoustic Tomography in Circular Geometry: Experimental Validation

Photoacoustic tomography (PAT) is a fast-developing biomedical imaging technology suitable for in vivo imaging. PAT in spherical or circular geometry gives good image resolution yet is slow or expensive in signal acquisition and image formation. Reducing the number of detection angles can ameliorate such issues usually at the expense of image quality. This paper introduces a deconvolution-based algorithm that models the imaging process as a linear and shift-invariant system. As demonstrated by the in vivo experiment, this algorithm not only runs much faster than the back-projection algorithm, but also shows stronger robustness in that it provides better image quality when detection angles are sparse. Therefore, this algorithm promises to enable real-time PAT in circular geometry.

General formulation of digital in-line holography from correlation with a chirplet function

Digital in-line holography is revisited to propose a mathematical model that describes the recording-reconstruction process as a linear shift-invariant system with a pseudo-point spread function even when the images are out of the optimal plane in the sense of signal processing. A particular case is treated to show that the optimal plane is the best focus plane in the sense of optics. Next, an exact solution of the holographic reconstruction by correlation is given. By means of the previous results, we study the behavior of the result of the correlation function between the diffraction pattern function produced by an opaque disk and a chirplet function and between the diffraction pattern produced by a phase disk and the same chirplet function.

Axial imaging necessitates loss of lateral shift invariance: proof with the Wigner analysis

We present a proof of the conjecture reported in Stein and Barbastathis ["Axial imaging necessitates loss of lateral shift invariance," Appl. Opt.41, 6055-6061 (2002)] that axial imaging necessitates loss of lateral shift invariance. We apply the Wigner distribution function (WDF) to the axial imaging process and compare laterally shift variant and invariant imaging systems. Two conditions for axial imaging are established: (i) objects with spatially variant local spatial frequency content to create sufficient change in the WDF with defocus and (ii) properly designed shift variant imaging kernels to estimate the slope of the sheared WDF. The lateral shift variance is a necessary condition for axial imaging. We use two examples from Stein and Barbastathis to show how axial imaging is interpreted in Wigner space.

On Stability of Finitely Generated Shift-Invariant Systems

We consider the problem of completely characterizing when a system of integer translates in a finitely generated shift-invariant subspace of L2(ℝd) is stable in the sense that rectangular partial sums for the system are norm convergent. We prove that a system of integer translates is stable in L2(ℝd) precisely when its associated Gram matrix satisfies a suitable Muckenhoupt A2 condition.

A New Shift-Invariance of Discrete-Time Systems and Its Application to Discrete Wavelet Transform Analysis

This work is motivated by the search for discrete wavelet transform (DWT) with near shift-invariance. After examining the elements of the property, we introduce a new notion of shift-invariance, which is particularly informative for multirate systems that are not shift-invariant in the strict sense. Briefly speaking, a discrete-time system is mu-shift-invariant if a shift in input results in the output being shifted as well. However, the amount of the shift in output is not necessarily identical to that in input. A fractional shift is also acceptable and can be properly specified in the Fourier domain. The mu-shift-invariance can be interpreted as invariance of magnitude spectrum with linear phase offset of output with respect to shift in input. It is stronger than the shiftability in position, which is equivalent to insensitivity of energy to shift in input. Under this generalized notion, the expander is always mu-shift-invariant. The M-fold decimator is mu-shift-invariant for input with width of frequency support not more than 2pi/M ; equivalently, the output contains no aliasing term in some frequency band with length of 2pi . We generalize the transfer function description of linear shift-invariant systems for mu -shift-invariant systems. We then perform mu-shift-invariance analysis of 2-band orthogonal DWT and of the 2-band dual-tree complex wavelet transform (DT-CWT). The analysis in each case provides clarifications to early understanding of near shift-invariance. We show that the DWT is mu-shift-invariant if and only if the conjugate quadrature filter (CQF) is analytic or antianalytic. For the DT-CWT, the CQFs must have supports included within [-2pi/3, 2pi/3], in addition to the well-know half-sample delay condition at higher levels and the one-sample delay condition at the first level.

Quantitative phase reconstruction from input-output intensities using a shift-invariant linear imaging system

We formulate an iteration scheme for quantitative two-dimensional phase retrieval of complex scalar wave fields from input-output intensity profiles using a coherent rotationally symmetric shift-invariant linear optical imaging system. This method is a generalization of our previous method for quantitative single-image pure phase reconstruction using an imperfect shift-invariant linear imaging system. The method is somewhat analogous to the Gerchberg-Saxton iterative phase-retrieval algorithm, but typically converges much more rapidly. We have demonstrated the efficacy of the method using simulated data in the presence of noise.

Properties of oblique dual frames in shift-invariant systems

Gramian analysis is used to study properties of a shift-invariant system X = { ϕ ( ⋅ − B k ) : ϕ ∈ Φ , k ∈ Z n } , where B is an invertible n × n matrix and Φ a finite or countable subset of L 2 ( R n ) under the assumption that the system forms a frame for the closed subspace M of L 2 ( R n ) . In particular, the relationship between various features of such system, such as being a frame for the whole space L 2 ( R n ) , being a Riesz sequence and having a unique shift-generated dual of type I or II is discussed in details. Several interesting examples are presented.

Computing the [formula omitted] gain for linear periodic continuous-time systems

A method to compute the L 2 gain is developed for the class of linear periodic continuous-time systems that admit a finite-dimensional state-space realisation. A bisection search for the smallest upper bound on the gain is employed, where at each step an equivalent discrete-time problem is considered via the well-known technique of time-domain lifting. The equivalent problem involves testing a bound on the gain of a linear shift-invariant discrete-time system, with the same state dimension as the periodic continuous-time system. It is shown that a state-space realisation of the discrete-time system can be constructed from point solutions to a linear differential equation and two differential Riccati equations, all subject to only single-point boundary conditions. These are well behaved over the corresponding one period intervals of integration, and as such, the required point solutions can be computed via standard methods for ordinary differential equations. A numerical example is presented and comparisons made with alternative techniques.

Quantum-Noise Limited Distance Resolution of Optical Range Imaging Techniques

The most common optical distance imaging methods (triangulation, interferometry and time-of-flight ranging) can all be described in a unified way as linear shift-invariant systems in which the determination of distance corresponds to the measurement of a spatial or temporal phase. Since the ultimate precision of such a phase measurement is limited by quantum noise of the involved photons or photocharges, the eventual distance resolution of the three optical ranging methods depends in the same way on quantum noise. Evidence from literature supports our basic assertion that photon and photocharge numbers have a Poisson distribution under most experimental conditions. This allows us to derive our main result that the quantum-noise limited distance resolution of the three optical ranging methods is proportional to the inverse of the signal's modulation amplitude times the square root of the background level. The equation for the precision of the three optical distance measurement techniques contains the method's experimental parameters in a single factor, from which the optimum distance range of the three techniques can easily be deduced: 1 nm-1 mum for interferometry, 1 mum-10 m for triangulation, > 0.1 m for time-of-flight ranging, if visible or near infrared light is used.

On Engineering Education, Superposition, and Commutativity

Attention is focused on continuous-space shift-invariant systems with continuous system maps and inputs and outputs that are elements of . Results are described showing that infinite superposition can fail in this important setting, and that continuous shift-invariant linear mappings need not commute with the operation of integration (even when the two composite operations are well defined).

Comparison of Time and Frequency Domain Methods for Luminescence Lifetime Measurements

Despite the widespread use of luminescence lifetimes in physical measurements, there seems to be no consensus on the best method for lifetime determination. Researchers in this field split into two camps: those who favor methods that proceed from time-domain data and those who favor methods that proceed from frequency-domain data. System theory provides unique insight into the strengths and limitations of these methods of lifetime determination. First, system theory shows that the impulse response experiment and the frequency sweep experiment can both uniquely specify a shift-invariant linear system. Further, the time and frequency data sets are related by the Fourier transform. A careful comparison of these two methods was carried out with both numerical simulations and experiments using the same sample, light source, and detector. These comparisons were not limited to impulse and sine wave excitation; chirp, square wave, and random excitation sequences were evaluated as well. The results indicate that for the same total excitation energies time-domain sequences with a substantial dark period have the lowest uncertainty in the lifetime parameter estimates.

Fresnel diffraction in the case of an inclined image plane

An extension of the theoretical formalism of Fresnel diffraction to the case of an inclined image plane is proposed. The resulting numerical algorithm speeds up computation times by typically three orders of magnitude, thus opening the possibility of utilizing previously inapplicable image analysis algorithms for this special type of a non shift-invariant imaging system. This is exemplified by adapting an iterative phase retrieval algorithm developed for electron microscopy to the case of hard x-ray imaging with asymmetric Bragg reflection (the so-called "Bragg Magnifier"). Numerical simulations demonstrate the convergence and feasibility of the iterative phase retrieval algorithm for the case of x-ray imaging with the Bragg Magnifier.

Robust analysis of discrete linear shift invariant systems represented by an interval matrix (Poster Presentation)

AbstractIn this paper we describe a new bound in order to guaranteed the robust stability of a discrete linear shift invariant system which is represented by an interval matrix. This bound is based in the bound of Juang. The uncertainties are represented by an interval matrix. The system is represented in state variables with parametric uncertainty in the A matrix. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems

We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.

On the origins of decoherence and extinction contrast in phase-contrast imaging

A theoretical formalism describing the formation of images in a linear shift invariant X-ray optical system is derived within the wave-optical theory. It is applicable to a non-crystalline object consisting of two types of features, with the characteristic sizes which are respectively not smaller and much smaller than the resolution of the imaging system. This formalism is then applied to two phase-contrast imaging techniques, the propagation-based and analyser-based imaging. The obtained formulae for the intensity distribution in the image well explain the “decoherence effect” which is observed in the former technique and the “extinction contrast” which is a characteristic of the latter technique. This formalism is shown to be in good agreement with the results of the accurate numerical simulations, using rigorous wave-optical theory, of the propagation-based and analyser-based phase-contrast images of the model objects.

Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines

We consider systems of functions appearing by letting a class of modulations act on a countable collection of functions. These systems correspond to shift-invariant systems, considered on the Fourier side. We provide sufficient conditions for the system to be a frame, as well as an explicit construction of a class of frames and associated duals. We use the result to construct frames based on B-splines with knot sequences satisfying a natural condition, as well as explicitly given duals.

Computed tomography with linear shift-invariant optical systems

Optical systems capable of three-dimensional transmission imaging are considered; these systems employ a conventional tomographic setup with an added linear shift-invariant optical system between the sample and the detector. A theoretical analysis is presented of image formation and sample reconstruction in such systems, examples of which include diffraction tomography and phase-contrast tomography with the use of analyzer crystals. An example is introduced in which the image is obtained by scanning the beam along the line orthogonal to the optic axis and to the axis of rotation with a one-dimensional slit or grating parallel to the rotation axis. We show that under certain conditions the proposed system may allow quantitative local (region-of-interest) tomography.