Papoulis-Gerchberg Research Articles

Fast algorithms for band-limited extrapolation by over sampling and Fourier series

In this paper, fast algorithms for the extrapolation of band-limited signals are presented by the sampling theorem and Fourier series in the case of over sampling. Assume the band-limited signal is known in a finite interval. We update the signal outside the interval by the Shannon sampling theorem in the case of over sampling. Then we obtain a fast algorithm in the form of Fourier series instead of the Fourier transform in the Papoulis–Gerchberg algorithm. Gibbs phenomena is analyzed in the method. An algorithm is presented to control the Gibbs phenomena, and some examples are given in the experimental results.

Generalized Papoulis–Gerchberg algorithm for PA nonlinearity compensation in coded OFDM systems

Iterative subcarrier regularization (ISR) has been recently proposed to mitigate the nonlinear noise caused by the saturation of power amplifiers in orthogonal frequency-division multiplexing systems. The generalized Papoulis–Gerchberg algorithm (GPGA) used in ISR exploits the frequency-domain prior knowledge specified by standards and time-domain prior knowledge reserved by low nonlinearity amplitudes. It is a reconstruction-based method that may cause bit error rate (BER) degradation when coding techniques are applied. In this work, a dual-loop structure of the GPGA is presented to integrate the cancellation-based methods for maintaining the coding gain. By using a well-known cancellation-based method, iterative clipping noise cancellation (ICNC), as an example, simulations based on the IEEE 802.11ah verified that the proposed scheme effectively maintains the contribution of both the reconstruction-based and cancellation-based methods, thus significantly outperforming the ISR or ICNC in terms of coded BER.

Reduction of lipid contamination in MR spectroscopy imaging using signal space projection.

Lipid contamination can complicate the metabolite quantification in MR spectroscopic imaging (MRSI). In addition to various experimental methods demonstrated to be feasible for lipid suppression, the postprocessing method is beneficial in the flexibility of applications. In this study, the signal space projection (SSP) algorithm is proposed to suppress the lipid signal in the MRSI. The performance of lipid suppression using SSP and SSP combined with the Papoulis-Gerchberg (PG) algorithm (PG+SSP) is examined in 2D MRSI data and the results were compared with outer volume saturation (OVS) methods. Up to 10 lipid spatial components were extracted by SSP from lipid signals in the range of 0.8~1.5 ppm. Our results show that most lipid signals were found in the first 4 to 5 components and that lipid signals on the spectra can be suppressed using 4 to 5 components. Metabolites concentrations were quantified using LCModel. Two regions of interest (ROIs) were manually selected on the peripheral and inner brain regions. The quantification of metabolites in terms of fitting reliability (CRLB) and spatial variations within ROIs (SpaVar) is improved using SSP. When 5 to 6 components were used in SSP and PG+SSP, the metabolite concentrations and the associated SpaVar and CRLB are at the same level as those from the OVS. We have demonstrated that the SSP method can be used to suppress the lipid signals of MRSI and SSP with 5 to 6 components is suggested to have a similar suppression performance as the OVS method.

Enhancing 3D range image measurement density via dynamic Papoulis–Gerchberg algorithm

As one of the most popular range detection methods, lidar is commonly used in various robotic applications. Although most robotic platforms easily adopt 2D lidar for range sensing, 3D lidar is rarely used in mobile robots, owing to its high cost. Some methods reported in the literature obtain 3D range information by rotating a single 2D lidar device. However, for most of these methods, there is a trade-off between 3D scan frequency and measurement density. Existing methods discussed in the literature for increasing the measurement density in high-frequency lidar have high time complexity and require certain conditions on data distribution. In a previous work, we showed the usability of an image super-resolution method, the Papoulis–Gerchberg (P–G) algorithm, on range data represented in the form of a greyscale image. However, the low convergence rate of the original P–G algorithm impedes its use for online applications. In this study, we advanced the P–G algorithm to drastically reduce the convergence time and improve performance by utilizing previous range images. The proposed algorithm now supports application on a mobile robot with online measurement density enhancement for 3D range images collected by rotating a 2D lidar device around its pitch axis with a high 3D scan frequency. We show illustrative examples for different scenarios to present the effectiveness of the proposed method on a 3D range sensor mounted on a mobile robot.

Eliminating saturation-type distortions in ultrasonic signals by the least-squares and Papoulis–Gerchberg algorithms

Situations may occur in ultrasonic testing in which the values of recorded echo-signals exceed the dynamic range of the receiving amplifier and defectoscope’s analog-to-digital converter. As a result, high-amplitude signals are subjected to a cutoff (clipping) operation and drop in amplitude, thus introducing error in the estimation of reflector dimensions. A declipping method that is based on the Papoulis–Gerchberg algorithm has been suggested and compared against a declipping technique that uses least squares. In numerical and model experiments, the Papoulis–Gerchberg algorithm has demonstrated more stable operation than the least squares for noisy signals and in the case of coarse signal sampling interval.

Diffusion‐steered super‐resolution method based on the Papoulis–Gerchberg algorithm

Papoulis-Gerchberg (PG) algorithm, a technique to extrapolate signals, has attracted many researchers for its lower complexity, higher computational efficiency, and effectiveness. One field that receives these merits is super-resolution, which fuses multiple band-limited scenes to generate a high-resolution image. Most super-resolution methods based on the PG algorithm, however, underperform when input images are seriously degraded by blur, noise, and sampling. The current study addresses the challenges by embedding the PG algorithm into a super-resolution minimization problem. The proposed method is iterative and incorporates a diffusion-driven smoothness prior that updates its regularisation process according to the local image features. This well-crafted prior, which attempts to overcome the super-resolution ill-posedness, provides an automatic interplay between flat and contour regions, and ensures necessary levels of regularisations to generate sharper and detailed images. Results show that the current method outperforms some state-of-the-art super-resolution approaches including those based on total variation. Even more importantly, the authors' method contains a robust noise suppressor that treats comfortably noisy scenes.

A Hybrid Quantitative Method for Inverse Scattering of Multiple Dielectric Objects

A hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method (LSM). The hybrid method is composed of three steps combining the capabilities of the LSM, the multiple signal classification (MUSIC) algorithm, and the generalized multipole technique (GMT). Initially, a new measure of the noise level of the multistatic response (MSR) matrix is proposed. Subsequently, the LSM is applied using the Tikhonov regularization. Then, the scattered field is back-propagated into the investigation domain using the GMT, where in order to ensure the stability, a MUSIC-based approach is exploited to determine the optimum reconstructing multipole sources. The case of limited scattered field data is also addressed by introducing a modification of the Papoulis–Gerchberg algorithm (PGA). Finally, the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization. The proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner. The effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data.

Image nonlinearity and non-uniformity corrections using Papoulis - Gerchberg algorithm in gamma imaging systems

In this paper, the authors describe a novel technique for image nonlinearity and non-uniformity corrections in imaging systems based on gamma detectors. The limitation of the gamma detector prevents the producing of high quality images due to the radionuclide distribution. This problem causes nonlinearity and non-uniformity distortions in the obtained image. Many techniques have been developed to correct or compensate for these image artifacts using complex calibration processes. The presented method is based on the Papoulis - Gerchberg(PG) iterative algorithm and is obtained without need of detector calibration, tuning process or using any special test phantom.

An iterative algorithm for computed tomography image reconstruction from limited-angle projections

In application of tomography imaging, limited-angle problem is a quite practical and important issue. In this paper, an iterative reprojection-reconstruction (IRR) algorithm using a modified Papoulis-Gerchberg (PG) iterative scheme is developed for reconstruction from limited-angle projections which contain noise. The proposed algorithm has two iterative update processes, one is the extrapolation of unknown data, and the other is the modification of the known noisy observation data. And the algorithm introduces scaling factors to control the two processes, respectively. The convergence of the algorithm is guaranteed, and the method of choosing the scaling factors is given with energy constraints. The simulation result demonstrates our conclusions and indicates that the algorithm proposed in this paper can obviously improve the reconstruction quality.

Unified approach to extrapolation of bandlimited signals in linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for signal processing and optics. Several extrapolation strategies for bandlimited signals in LCT domain have been proposed. The purpose of this paper is to present an approach that unifies a number of different algorithms for the extrapolation of bandlimited signals in LCT domain. This unification is achieved through integral equation and Hilbert space theories. First, the following existing techniques are unified: (1) a continuous signal extrapolation algorithm based on series expansion in terms of generalized prolate spheroidal functions; (2) a generalized Papoulis–Gerchberg iterative algorithm; (3) a two-step extrapolation algorithm for continuous signal from finite samples; and (4) an iterative extrapolation algorithm based on error energy reduction procedure for continuous signal from finite samples. Then, two extrapolation algorithms for discrete bandlimited signals in LCT domain are proposed, which also belongs to the unified framework.

Super-resolution using an enhanced Papoulis–Gerchberg algorithm

Super resolution (SR) is the process of generating a raster image with a higher resolution than its source. The Papoulis–Gerchberg (P–G) algorithm known as one of the most important approaches of signal reconstruction, has been extensively used to reconstruct high-resolution information from low-resolution images by imposing spatial and frequency domain constraints in an iterative process. Although this procedure is easy to implement, it converges slowly and requires pre-knowledge of the image bandwidth. To overcome these problems, a new procedure is proposed, to enhance the original P–G algorithm in terms of speed and performance, by using dynamic properties in the frequency domain. The proposed technique also allows an automatic adjustment of the bandwidth during each iteration. Computer simulations are given to illustrate the effectiveness of the proposed method.

Papoulis-Gerchberg 방법의 개선에 의한 초해상도 영상 화질 향상

This paper proposes super-resolution reconstruction algorithm for image enhancement. Super-resolution reconstruction algorithms reconstruct a high-resolution image from multi-frame low-resolution images of a scene. Conventional super- resolution reconstruction algorithms are iterative back-projection(IBP), robust super-resolution(RS)method and standard Papoulis-Gerchberg(PG)method. However, traditional methods have some problems such as rotation and ringing. So, this paper proposes modified algorithm to improve the problem. Experimental results show that this proposed algorithm solve the problem. As a result, the proposed method showed an increase in the PSNR for traditional super-resolution reconstruction algorithms.

Reconstruction of Drifting Sensor Responses Based on Papoulis–Gerchberg Method

This paper presents a method to reconstruct the drifting sensor responses for electronic-nose (E-nose) systems. The sensor drift often exhibits low-frequency behaviors. Assume that the drift is a kind of missing data during the vapor exposure period, the drifting signal can then be recovered by interpolation or extrapolation using the Papoulis-Gerchberg method under the measurement boundary condition. Using a Lorentzian model to describe the sensor response, a reconstruction algorithm for the sensor response is developed by minimizing the drift recovery and modeling errors. We further adopt the principal component analysis technique and the squared prediction error index to detect the noninformative sensors so that the subsequent classification and estimation performance will not be deteriorated. Using experimental E-nose data, the proposed method is shown to be effective in reconstructing the sensor responses.

Application Of Papoulis-Gerchberg Method In Image Super-Resolution and Inpainting

In this paper, we study the Papoulis–Gerchberg (PG) method and its applications to domains of image restoration such as super-resolution (SR) and inpainting. We show that the method performs well under certain conditions. We then suggest improvements to the method to achieve better SR and inpainting results. The modification applied to the SR process also allows us to apply the method to a larger class of images by doing away with some of the restrictions inherent in the classical PG method. We also present results to demonstrate the performance of the proposed techniques.

Signal Reconstruction with Boundary-Matching via Iterative Algorithm

In various applications of signals transmission and processing, there is always a possibility of loss of samples. The iterative algorithm of Papoulis-Gerchberg (PG) is famous for solving the band-limited lost samples recovery problem. Two problems are known in this domain: (1) a band-limited signal practically is difficult to be obtained and (2) the convergence rate is too slow. By inserting a subtraction by a polynomial in the PG algorithm, using boundary-matched concept, a significant increase in performance and speed of its convergence has been achieved. In this paper, we propose an efficient approach to restore lost samples by adding a preprocess which meets the periodic boundary conditions of Fast Fourier transform in the iteration method. The accuracy of lost samples reconstruction by using the PG algorithm can be greatly improved with the aid of mapping the input data sequence of satisfying the boundary conditions. Further, we also developed another approach that force the signal to meet a new critical boundary conditions in Fourier domain that make the parameters of the preprocessing can be easily obtained. The simulation indicates that the mean square error (MSE) of the recovery and the convergence rate with the preprocess concept is better and faster than the one without preprocess concept. Our both proposed approaches can also be applied to other cases of signal restoration, which allow Cadzow's iterative processing method, with more convenience and flexibility.

On the convergence of wavelet-based iterative signal extrapolation algorithms

A generalized Papoulis-Gerchberg (PG) algorithm for signal extrapolation based on the wavelet representation has been recently proposed by Xia, Kuo and Zhang. In this research, we examine the convergence property and the convergence rate of several signal extrapolation algorithms in wavelet subspaces. We first show that the generalized PG algorithm converges to the minimum norm solution when the wavelet bases are semi-orthogonal (or known as the prewavelet). However, the generalized PG algorithm converges slowly in numerical implementation. To accelerate the convergence rate, we formulate the discrete signal extrapolation problem as a two-step process and apply the steepest descent and conjugate gradient methods for its solution. Numerical experiments are given to illustrate the performance of the proposed algorithms.

Signal Extrapolation in Wavelet Subspaces

The Papoulis–Gerchberg (PG) algorithm is well known for band-limited signal extrapolation. The authors consider the generalization of the PG algorithm to signals in the wavelet subspaces in this research. The uniqueness of the extrapolation for continuous-time signals is examined, and sufficient conditions on signals and wavelet bases for the generalized PG (GPG) algorithm to converge are given. A discrete GPG algorithm is proposed for discrete-time signal extrapolation, and its convergence is investigated. Numerical examples are given to illustrate the performance of the discrete GPG algorithm.

Unified Hilbert space approach to iterative least-squares linear signal restoration

We deal with iterative least-squares solutions of the linear signal-restoration problem g = Af. First, several existing techniques for solving this problem with different underlying models are unified. Specifically, the following are shown to be special cases of a general iterative procedure [ BialyH., Arch. Ration. Mech. Anal.4, 166 ( 1959)] for solving linear operator equations in Hilbert spaces: (1) a Van Cittert-type algorithm for deconvolution of discrete and continuous signals; (2) an iterative procedure for regularization when g is contaminated with noise; (3) a Papoulis–Gerchberg algorithm for extrapolation of continuous signals [ PapoulisA., IEEE Trans. Circuits Syst.CAS-22, 735 ( 1975); GerchbergR. W., Opt. Acta21, 709 ( 1974)]; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals {and the minimum-norm property of the extrapolation obtained by the iteration [ JainA.RanganathS., IEEE Trans. Acoust. Speech Signal Process. ASSP-29, ( 1981)]}; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals [ TomV., IEEE Trans. Acoust. Speech Signal Process.ASSP-29, 1052 ( 1981)]. The Bialy algorithm also generalizes the Papoulis–Gerchberg iteration to cases in which the ideal low-pass operator is replaced by some other operators. In addition a suitable modification of this general iteration is shown. This technique leads us to new iterative algorithms for band-limited signal extrapolation. In numerical simulations some of these algorithms provide a fast reconstruction of the sought signal.