Shift-invariant Systems Research Articles

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups

The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

A nonparametric measure of contrast in x-ray images

Objective. We propose a nonparametric figure of merit, the contrast equivalent distance CED, to measure contrast directly from clinical images. Approach. A relative brightness distance δ is calculated by making use of the order statistic of the pixel values. By multiplying δ with the grey value range R, the mean brightness distance MBD is obtained. From the MBD, the CED and the distance-to-noise ratio DNR can be derived. The latter is the ratio of the MBD and a previously suggested nonparametric measure τ for the noise.Since the order statistic is independent of the spatial arrangement of the pixel values, the measures can be obtained directly from clinical images. We apply the new measures to mammography images of an anthropomorphic phantom and of a phantom with a step wedge as well as to CT images of a head phantom. Main results. For low-noise images of a step wedge, the MBD is equivalent to the conventional grey value distance. While this measure permits the evaluation of clinical images, it is sensitive to noise. Therefore, noise has to be quantified at the same time. When the ratio σ/τ of the noise standard deviation σ to τ is available, validity limits for the CED as a measure of contrast can be established. The new figures of merit can be calculated for entire images as well as on regions of interest (ROI) with an edge length not smaller than 32 px.Significance. The new figures of merit are suited to quantify the quality of clinical images without relying on the assumption of a linear, shift-invariant system. They can be used for any kind of greyscale image, provided the ratio σ/τ can be estimated. This will hopefully help to achieve the optimisation of image quality vs dose required by radioprotection laws.

Can processed images be used to determine the modulation transfer function and detective quantum efficiency?

The modulation transfer function (MTF) and detective quantum efficiency (DQE) of x-ray detectors are key Fourier metrics of performance, valid only for linear and shift-invariant (LSI) systems and generally measured following IEC guidelines requiring the use of raw (unprocessed) image data. However, many detectors incorporate processing in the imaging chain that is difficult or impossible to disable, raising questions about the practical relevance of MTF and DQE testing. We investigate the impact of convolution-based embedded processing on MTF and DQE measurements. We use an impulse-sampled notation, consistent with a cascaded-systems analysis in spatial and spatial-frequency domains to determine the impact of discrete convolution (DC) on measured MTF and DQE following IEC guidelines. We show that digital systems remain LSI if we acknowledge both image pixel values and convolution kernels represent scaled Dirac -functions with an implied sinc convolution of image data. This enables use of the Fourier transform (FT) to determine impact on presampling MTF and DQE measurements. It is concluded that: (i)the MTF of DC is always an unbounded cosine series; (ii)the slanted-edge method yields the true presampling MTF, even when using processed images, with processing appearing as an analytic filter with cosine-series MTF applied to raw presampling image data; (iii)the DQE is unaffected by discrete-convolution-based processing with a possible exception near zero-points in the presampling MTF; and (iv)the FT of the impulse-sampled notation is equivalent to the transform of image data.

Reconstruction methods for super-resolution imaging with PSF modulation

Recently, a super-resolution imaging method called the PSF (point spread function) modulation method was developed (Lu, IEEE TUFFC 2024). In this method, the amplitude, phase, or both of the PSF of a linear shift-invariant (LSI) imaging system is modulated so that the modulated PSF has a higher spatial frequency than that of the original PSF to reconstruct super-resolution images. The modulator can be produced and manipulated remotely by methods such as radiation force or it can be a physical particle such as micro- or nano-particle manipulated by an external force such as electrical and electromagnetic force. In principle, the super-resolution imaging method can be applied to any LSI imaging system, such as ultrasound, optical, photoacoustic, electromagnetic, underwater, nondestructive evaluation (NDE), and magnetic resonance imaging (MRI) system. These include pulse-echo ultrasound imaging, transmission imaging, wave source/field imaging, acoustical camera, and optical bright-field microscope. To optimize the quality of the images, methods for the reconstruction of pulse-echo and wave source/field super-resolution images are studied and the results will be presented. These methods include the uses of an analytic envelope of radio-frequency (RF) signals with and without windowing, “I” (in-phase) and “Q” (quadrature) signals, and the DC (direct current) component removal.

Truncations of generalized shift-invariant systems

AbstractWe provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.

Generalization of the two-point-source resolution criterion in the presence of noise.

The two-point-source resolution criterion is widely used to quantify the performance of imaging systems. The two main approaches for the computation of the two-point-source resolution are the detection theoretic and visual analyses. The first assumes a shift-invariant system and lacks the ability to incorporate two different point spread functions (PSFs), which may be required in certain situations like computing axial resolution. The latter approach, which includes the Rayleigh criterion, relies on the peak-to-valley ratio and does not properly account for the presence of noise. We present a heuristic generalization of the visual two-point-source resolution criterion using Gaussian processes (GP). This heuristic criterion is applicable to both shift-invariant and shift-variant imaging modalities. This criterion can also incorporate different definitions of resolution expressed in terms of varying peak-to-valley ratios. Our approach implicitly incorporates information about noise statistics such as the variance or signal-to-noise ratio by making assumptions about the spatial correlation of PSFs in the form of kernel functions. Also, it does not rely on an analytic form of the PSF.

Limits of realizing irradiance distributions with shift-invariant illumination systems and finite étendue sources.

When redistributing the light emitted by a source into a prescribed irradiance distribution, it is not guaranteed that, given the source and optical constraints, the desired irradiance distribution can be achieved. We analyze the problem by assuming an optical black box that is shift-invariant, meaning that a change in source position does not change the shape of the irradiance distribution, only its position. The irradiance distribution we can obtain is then governed by deconvolution. Using positive-definite functions and Bochner's theorem, we provide conditions such that the irradiance distribution can be realized for finite étendue sources. We also analyze the problem using optimization, showing that the result heavily depends on the chosen source distribution.

TWISTED SHIFT-INVARIANT SYSTEM IN

AbstractWe consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$ , consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates.

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.

Different channels to transmit information in scattering media

A communication channel should be built to transmit information from one place to another. Imaging is 2 or higher dimensional information communication. Conventionally, an imaging channel comprises a lens with free space at its both sides, whose transfer function is usually known and hence the response of the imaging channel can be well defined. Replacing the lens with a thin scattering medium, the image can still be extracted from the detected optical field, suggesting that the scattering medium retains or reconstructs not only energy but also information transmission channels. Aided by deep learning, we find that unlike the lens system, there are different channels in a scattering medium: the same scattering medium can construct different channels to match the manners of source coding. Moreover, it is found that without a valid channel, the convolution law for a spatial shift-invariant system (the output is the convolution of the point spread function and the input object) is broken, and in this scenario, information cannot be transmitted onto the detection plane. Therefore, valid channels are essential to transmit information through even a spatial shift-invariant system. These findings may intrigue new adventures in imaging through scattering media and reevaluation of the known spatial shift-invariance in various areas.

A nonparametric measure of noise in x-ray diagnostic images—mammography

Objective. In x-ray diagnostics, modern image reconstruction or image processing methods may render established methods of image quality assessment inadequate. Task specific quality assessment by using model observers has the disadvantage of being very labour-intensive. Therefore, it appears highly desirable to develop novel image quality parameters that neither rely on the linearity and the shift-invariace of the imaging system nor require the acquisition of hundreds of images as is necessary for the application of model observers, and which can be derived directly from diagnostic images. Approach. A new measure for the noise based on non-maximum-suppression images is defined and its properties are explored using simulated images before it is applied to an exposure series of mammograms of a homogeneous phantom and a 3D-printed breast phantom to demonstrate its usefulness under realistic conditions. Main results. The new noise parameter cannot only be derived from images with a homogeneous background but it can be extracted directly from images containing anatomic structures and is proportional to the standard deviation of the noise. At present, the applicability is restricted to mammography, which satisfies the assumption of short covariance length of the noise. Significance. The new measure of the noise is but a first step of the development of a set of parameters that are required to quantify image quality directly from diagnostic images without relying on the assumption of a linear, shift-invariant system, e.g. by providing measures of sharpness, contrast and structural complexity, in addition to the noise measure. For mammography, a convenient method is now available to quantify noise in processed diagnostic images.

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.

Fast CBCT Reconstruction using Convolutional Neural Networks for Arbitrary Robotic C-arm Orbits.

Cone-beam CT (CBCT) with non-circular acquisition orbits has the potential to improve image quality, increase the field-of view, and facilitate minimal interference within an interventional imaging setting. Because time is of the essence in interventional imaging scenarios, rapid reconstruction methods are advantageous. Model-Based Iterative Reconstruction (MBIR) techniques implicitly handle arbitrary geometries; however, the computational burden for these approaches is particularly high. The aim of this work is to extend a previously proposed framework for fast reconstruction of non-circular CBCT trajectories. The pipeline combines a deconvolution operation on the backprojected measurements using an approximate, shift-invariant system response prior to processing with a Convolutional Neural Network (CNN). We trained and evaluated the CNN for this approach using 1800 randomized arbitrary orbits. Noisy projection data were formed from 1000 procedurally generated tetrahedral phantoms as well as anthropomorphic data in the form of 800 CT and CBCT images from the Lung Image Database Consortium Image Collection (LIDC). Using this proposed reconstruction pipeline, computation time was reduced by 90% as compared to MBIR with only minor differences in performance. Quantitative comparisons of nRMSE, FSIM and SSIM are reported. Performance was consistent for projection data simulated with acquisition orbits the network has not previously been trained on. These results suggest the potential for fast processing of arbitrary CBCT trajectory data with reconstruction times that are clinically relevant and applicable - facilitating the application of non-circular orbits in CT image-guided interventions and intraoperative imaging.

A New Statistical Reconstruction Method for the Computed Tomography Using an X-Ray Tube with Flying Focal Spot

Abstract This paper presents a new image reconstruction method for spiral cone- beam tomography scanners in which an X-ray tube with a flying focal spot is used. The method is based on principles related to the statistical model-based iterative reconstruction (MBIR) methodology. The proposed approach is a continuous-to-continuous data model approach, and the forward model is formulated as a shift-invariant system. This allows for avoiding a nutating reconstruction-based approach, e.g. the advanced single slice rebinning methodology (ASSR) that is usually applied in computed tomography (CT) scanners with X-ray tubes with a flying focal spot. In turn, the proposed approach allows for significantly accelerating the reconstruction processing and, generally, for greatly simplifying the entire reconstruction procedure. Additionally, it improves the quality of the reconstructed images in comparison to the traditional algorithms, as confirmed by extensive simulations. It is worth noting that the main purpose of introducing statistical reconstruction methods to medical CT scanners is the reduction of the impact of measurement noise on the quality of tomography images and, consequently, the dose reduction of X-ray radiation absorbed by a patient. A series of computer simulations followed by doctor’s assessments have been performed, which indicate how great a reduction of the absorbed dose can be achieved using the reconstruction approach presented here.

Undecidability in quantum thermalization

The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

Frames associated with shift invariant spaces on positive half line

Abstract In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Preconditioned deconvolution method for high-resolution ghost imaging

Ghost imaging (GI) can nonlocally image objects by exploiting the fluctuation characteristics of light fields, where the spatial resolution is determined by the normalized second-order correlation function g ( 2 ) . However, the spatial shift-invariant property of g ( 2 ) is distorted when the number of samples is limited, which hinders the deconvolution methods from improving the spatial resolution of GI. In this paper, based on prior imaging systems, we propose a preconditioned deconvolution method to improve the imaging resolution of GI by refining the mutual coherence of a sampling matrix in GI. Our theoretical analysis shows that the preconditioned deconvolution method actually extends the deconvolution technique to GI and regresses into the classical deconvolution technique for the conventional imaging system. The imaging resolution of GI after preconditioning is restricted to the detection noise. Both simulation and experimental results show that the spatial resolution of the reconstructed image is obviously enhanced by using the preconditioned deconvolution method. In the experiment, 1.4-fold resolution enhancement over Rayleigh criterion is achieved via the preconditioned deconvolution. Our results extend the deconvolution technique that is only applicable to spatial shift-invariant imaging systems to all linear imaging systems, and will promote their applications in biological imaging and remote sensing for high-resolution imaging demands.

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.

Explicit construction of frames and pairs of dual frames on locally compact abelian groups

Locally compact abelian (LCA) groups form a natural setting for frame theory in signal processing: while classical frame theory was developed in the setting of \( L^2({\mathbb {R}}),\) concrete applications have to be implemented on \({\mathbb {R}}^d,\) two scenarios that are covered under the setup of LCA groups. Due to rapid progress over the past few years, it is known within the mathematical community that large parts of the standard frame theory for \( L^2({\mathbb {R}})\) can be carried over to the setting of LCA groups. We show that this also holds on the level of explicit constructions, which is a crucial issue for applications within signal processing; in fact we provide explicit constructions of frames and pairs of dual frames on LCA groups. Special attention will be given to the so-called elementary LCA groups, i.e., groups that are tensor products of \({\mathbb {R}},\) \({\mathbb {Z}},\) the torus group \({{\mathbb {T}}},\) and the finite group \({\mathbb {Z}}_N.\) The frames will have the form of a Fourier-like system, i.e., the Fourier transform of a generalized shift-invariant system. The constructions are based on a partition of unity; in particular we provide a general and flexible approach for constructing a partition of unity that makes it easy to control key features of the underlying functions, e.g., support size and continuity. The results apply to arbitrary LCA groups with a supply of “sufficiently fine” lattices, a condition that is satisfied for all elementary LCA groups. Concrete constructions of pairs of dual frames for several elementary LCA groups are provided as illustration. In the case of a system of functions generated by a single lattice, such as a Gabor system, the general results become particularly transparent.