Nonuniform Discrete Fourier Transform Research Articles

Optimization in the space domain for density compensation with the nonuniform FFT

The non-uniform Discrete Fourier Transform algorithm has shown great utility for reconstructing images from non-uniformly spaced Fourier samples in several imaging modalities. Due to the non-uniform spacing, some correction for the variable density of the samples must be made. Common methods for generating density compensation values are either sub-optimal or only consider a finite set of points in the optimization. This manuscript presents an algorithm for generating density compensation values from a set of Fourier samples that takes into account the point spread function over an entire rectangular region in the image domain. We show that the reconstructed images using the density compensation values of this method are of superior quality when compared to other standard methods. Results are shown with a numerical phantom and with magnetic resonance images of the abdomen and the knee.

Comparison of uniform resampling and nonuniform sampling direct-reconstruction methods in k-space for FD-OCT

The nonuniform distribution of interference spectrum in wavenumber k-space is a key issue to limit the imaging quality of Fourier-domain optical coherence tomography (FD-OCT). At present, the reconstruction quality at different depths among a variety of processing methods in k-space is still uncertain. Using simulated and experimental interference spectra at different depths, the effects of common six processing methods including uniform resampling (linear interpolation (LI), cubic spline interpolation (CSI), time-domain interpolation (TDI), and K-B window convolution) and nonuniform sampling direct-reconstruction (Lomb periodogram (LP) and nonuniform discrete Fourier transform (NDFT)) on the reconstruction quality of FD-OCT were quantitatively analyzed and compared in this work. The results obtained by using simulated and experimental data were coincident. From the experimental results, the averaged peak intensity, axial resolution, and signal-to-noise ratio (SNR) of NDFT at depth from 0.5 to 3.0[Formula: see text]mm were improved by about 1.9[Formula: see text]dB, 1.4 times, and 11.8[Formula: see text]dB, respectively, compared to the averaged indices of all the uniform resampling methods at all depths. Similarly, the improvements of the above three indices of LP were 2.0[Formula: see text]dB, 1.4 times, and 11.7[Formula: see text]dB, respectively. The analysis method and the results obtained in this work are helpful to select an appropriate processing method in k-space, so as to improve the imaging quality of FD-OCT.

Non-Uniform Sparse Fourier Transform and Its Applications

The fast approximation algorithm of non-uniform discrete Fourier transform (NUDFT) is an important issue in signal processing. In this paper, a novel estimation algorithm is constructed for NUDFT-II, which is the general form of the sparse Fourier transform (SFT). Firstly, we propose the cyclic convolution in the non-uniform frequency domain and derive the product and convolution theorem. Secondly, the relationship is deduced between the inverse NUDFT of a uniformly sampled sequence in the frequency domain and that of a non-uniformly sampled sequence. Then, based on the relationship and cyclic convolution, we establish a random permutation operation, a non-uniform flat window filtering operation, and a frequency subsampling operation for NUDFT-II to compress sparse signal length. Meanwhile, a method to estimate the significant frequencies' locations and values is introduced. Finally, we propose the non-uniform sparse Fourier transform (NUSFT). Simulation results demonstrate that the proposed method has low complexity in sample and runtime, and has high robustness. Furthermore, the NUSFT algorithm has been applied to signal detection and reconstruction, which shows that the required signals are obtained accurately. And the NUSFT can detect frequencies that cannot be detected by the SFT.

Generalized Image Reconstruction in Optical Coherence Tomography Using Redundant and Non-Uniformly-Spaced Samples.

In this paper, we use Frame Theory to develop a generalized OCT image reconstruction method using redundant and non-uniformly spaced frequency domain samples that includes using non-redundant and uniformly spaced samples as special cases. We also correct an important theoretical error in the previously reported results related to OCT image reconstruction using the Non-uniform Discrete Fourier Transform (NDFT). Moreover, we describe an efficient method to compute our corrected reconstruction transform, i.e., a scaled NDFT, using the Fast Fourier Transform (FFT). Finally, we demonstrate different advantages of our generalized OCT image reconstruction method by achieving (1) theoretically corrected OCT image reconstruction directly from non-uniformly spaced frequency domain samples; (2) a novel OCT image reconstruction method with a higher signal-to-noise ratio (SNR) using redundant frequency domain samples. Our new image reconstruction method is an improvement of OCT technology, so it could benefit all OCT applications.

An oversampling software-triggering interferogram method for Fourier-transform infrared spectrometers

Fourier-transform spectrometers (FTSs) are widely used for spectroscopic analysis in various fields. Hardware-triggering interferogram sampling is commonly used in commercial FTSs, which performs interferogram acquisition at equal increments of optical path difference (OPD). However, due to the OPD of FTSs over time is non-uniform, the stability of hardware-triggering sampling method strongly suffers from the variation of sampling frequency in the time domain. In this paper, we propose an oversampling software-triggering interferogram method based on the OPD iterative algorithm and the non-uniform discrete Fourier transform (NUDFT). The non-uniform OPD variable can be accurately calculated, which is significant to spectral recovery using the NUDFT. This oversampling method has been effectively adopted in our home-made near-infrared FTS, which features a rotating interferometer. Moreover, it is found that the spectral signal-to-noise ratio (SNR) can be increased by ~ 5.3 times, benefitting from the oversampling software-triggering interferogram acquisition.Published by AIP Publishing.

Research on Signal Processing in φ-OTDR System Based on Single Frequency IIR Trap

In the traditional phase sensitive optical time domain reflectometer (φ-OTDR) system, after locating of the vibration source, the spectrum diagram of the vibration signal can be obtained through the non-uniform Discrete Fourier Transform (NDFT) by extracting the data at this position. However, spectral aliasing noise included in the spectrum of the obtained vibration signal at this time. Because spectrum aliasing noise is generated in the process of NDFT transformation, traditional noise reduction methods have little effect on it. Therefore, this paper proposes a kind of IIR notch device based on single frequency for use in φ-OTDR. The change of standard deviation in discrete frequency domain is used to locate the vibration source, the large frequency components are filtered out from the time domain to solve the small signal components. The performance of IIR notch filter is verified by simulation. The results show that the notch filter can be used to calculate the small signal components of non-uniformly sampled signals successfully. It is convenient for φ-OTDR to use time-frequency images of different vibration types to identify vibration signals, so as to improve the recognition efficiency and monitoring ability of the system.

Digital Metering of Electrical Power Components Using Adaptive Non-Uniform Discrete Short Time Fourier Transform

Abstract This paper presents an adaptive Goertzel filter bank based discrete short time Fourier transform (DSTFT) implementation algorithm, called adaptive non-uniform discrete short time Fourier transform (ANDSTFT) for online measurement of electrical power components using IEEE Standard 1459–2010. The proposed ANDSTFT algorithm utilizes effective combination of non-uniform discrete Fourier transform (NDFT) and window method to detect the spectrum of each individual finite short time segment of power signals at distinct, arbitrarily located frequencies. Compared with the well-established technique such as windowed FFT interpreted DSTFT based approaches, the proposed method offers (i) better accuracy (ii) higher degree of immunity and insensitivity to noise, and (iii) reduced computational complexity per sample interval. The simulation results have been given and its response time and accuracy have been compared with the conventional windowed FFT interpreted DSTFT based techniques. Real-time implementation of the proposed approach has also been presented.

An Improved Imaging Algorithm for High-Resolution Spotlight SAR with Continuous PRI Variation Based on Modified Sinc Interpolation.

This paper focuses on an improved imaging algorithm for spotlight synthetic aperture radar (SAR) with continuous Pulse Repetition Interval (PRI) variation in extremely high-resolution. Conventional SAR systems are limited in that a wide swath cannot be achieved with a high azimuth resolution in the meantime. This limitation can be overcome by Pulse Repetition Frequency (PRF) variation in a SAR system. However, there are problems such as the ambiguities of point targets or extended targets caused by nonuniform sampling. A reconstructive method, Nonuniform Discrete Fourier Transform (NUDFT) has been presented in the current literature, but it is rather computationally expensive. In this paper, a modified sinc interpolation based on NUDFT is proposed, which is used to reconstruct the uniformly sampled echo in time domain. Since the interpolation kernel length is relatively short, it is more computationally efficient. Then, the two-step processing approach combined with the modified sinc interpolation is further presented, which has much better accuracy than that combined with the conventional sinc interpolation. Both the simulated data and the extracted GF-3 data experiment demonstrate the validity and accuracy of the proposed approach.

A Nonuniform Fast Fourier Transform Based on Low Rank Approximation

By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.

Efficient Nonuniform Fourier Reconstruction for Spaceborne/Airborne Bistatic SAR

In this letter, an efficient imaging algorithm is proposed for reconstructing the spaceborne/airborne bistatic synthetic aperture radar (SA-BSAR) data acquired with a high squint angle. By utilizing the Taylor expansion and the chain rule for derivatives of composite function, the expression of the SA-BSAR ideal matching filter is converted to a 2-D nonuniform discrete Fourier transform (NUDFT). Due to neither input nor output data of the NUDFT being equispaced, nonuniform fast Fourier transform of type 3 (NUFFT-3) is implemented to speed up the SA-BSAR image reconstruction. The computation complexity of the proposed imaging algorithm is O(MNlogMN), where MN is the number of the image pixels. Simulation results demonstrate the validity of the proposed algorithm.

Compressive sensing with dispersion compensation on non-linear wavenumber sampled spectral domain optical coherence tomography

We propose a novel compressive sensing (CS) method on spectral domain optical coherence tomography (SDOCT). By replacing the widely used uniform discrete Fourier transform (UDFT) matrix with a new sensing matrix which is a modification of the non-uniform discrete Fourier transform (NUDFT) matrix, it is shown that undersampled non-linear wavenumber spectral data can be used directly in the CS reconstruction. Thus k-space grid filling and k-linear mask calibration which were proposed to obtain linear wavenumber sampling from the non-linear wavenumber interferometric spectra in previous studies of CS in SDOCT (CS-SDOCT) are no longer needed. The NUDFT matrix is modified to promote the sparsity of reconstructed A-scans by making them symmetric while preserving the value of the desired half. In addition, we show that dispersion compensation can be implemented by multiplying the frequency-dependent correcting phase directly to the real spectra, eliminating the need for constructing complex component of the real spectra. This enables the incorporation of dispersion compensation into the CS reconstruction by adding the correcting term to the modified NUDFT matrix. With this new sensing matrix, A-scan with dispersion compensation can be reconstructed from undersampled non-linear wavenumber spectral data by CS reconstruction. Experimental results show that proposed method can achieve high quality imaging with dispersion compensation.

Hybrid-SPRITE MRI

In a FID based frequency encoding MRI experiment the central part of k-space is not generally accessible due to the probe dead time. This portion of k-space is however crucial for image reconstruction. SPRITE (Single Point Ramped Imaging with T1 Enhancement), SPI with a linearly ramped phase encode gradient, has been employed to image short relaxation time systems for many years with great success. It is a robust imaging method in significant measure because it provides acquisition of high quality k-space origin data. We propose a new sampling scheme, termed hybrid-SPRITE, combining phase and frequency encoding to ensure high quality images with reduced acquisition times, reduced gradient duty cycle and increased sensitivity. In hybrid-SPRITE, numerous time domain points are collected to assist image reconstruction. An Inverse Non-uniform Discrete Fourier Transform (INDFT) is employed in 1D applications. A pseudo-polar grid is exploited in 2D hybrid-SPRITE for rapid and accurate image reconstruction.

NDFT-based Image Steganographic Scheme with Discrimination of Tampers

A new and secure image steganographic scheme based on nonuniform discrete Fourier transform (NDFT) is proposed in this paper. First, the chaotic system is introduced to select embedding points randomly in NDFT domain suitable range, and NDFT is implemented on every non-overlapping block of eight consecutive pixels. Second, the secret messages are scrambled by chaotic systems, and embedded into frequency coefficients by quantization method. The stego-image is obtained by inverse NDFT (INDFT). Besides, in order to discriminate tampers, the low frequency wavelet coefficients of 7 most significant bits (MSBs) of the stego-image are converted into the binary sequence after nonuniform scalar quantization. Then the obtained binary sequence is scrambled by the chaotic systems, and embedded into the least significant bit (LSB) of the stego-image. Finally, the watermarked stego-image can be obtained by a new improved LSB steganographic method. The embedded secret messages can be extracted from the watermarked stego-image without the original cover image. Experimental results show the validity of the proposed scheme, and dual statistics attacks are also conducted to indicate the security.

Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT

We implemented fast Gaussian gridding (FGG)-based non-uniform fast Fourier transform (NUFFT) on the graphics processing unit (GPU) architecture for ultrahigh-speed, real-time Fourier-domain optical coherence tomography (FD-OCT). The Vandermonde matrix-based non-uniform discrete Fourier transform (NUDFT) as well as the linear/cubic interpolation with fast Fourier transform (InFFT) methods are also implemented on GPU to compare their performance in terms of image quality and processing speed. The GPU accelerated InFFT/NUDFT/NUFFT methods are applied to process both the standard half-range FD-OCT and complex full-range FD-OCT (C-FD-OCT). GPU-NUFFT provides an accurate approximation to GPU-NUDFT in terms of image quality, but offers >10 times higher processing speed. Compared with the GPU-InFFT methods, GPU-NUFFT has improved sensitivity roll-off, higher local signal-to-noise ratio and immunity to side-lobe artifacts caused by the interpolation error. Using a high speed CMOS line-scan camera, we demonstrated the real-time processing and display of GPU-NUFFT-based C-FD-OCT at a camera-limited rate of 122 k line/s (1024 pixel/A-scan).

Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system

We develop a high speed spectral domain optical coherence tomography (SD-OCT) system based on a custom-built spectrometer and non-uniform discrete Fourier transform (NDFT) to realize minimized depth dependent sensitivity fall-off. After precise spectral calibration of the spectrometer, NDFT of the acquired spectral data is adopted for image reconstruction. The spectrometer is able to measure a wavelength range of about 138 nm with a spectral resolution of 0.0674 nm at central wavelength of 835 nm, corresponding to an axial imaging range of 2.56 mm in air. Zemax simulations and sensitivity fall-off measurements under two alignment states of the spectrometer are given. Both theoretical simulations and experiments are done to study the depth dependent sensitivity of the developed system based on NDFT in contrast to those based on conventional discrete Fourier transform (DFT) with and without interpolation. In vivo imaging on human finger from volunteer is conducted at A-scan rate of 29 kHz and reconstruction is done based on different methods. The comparing results confirm that reconstruction method based on NDFT indeed improves sensitivity especially at large depth while maintaining the coherence-function-limited depth resolution.

Swept source optical coherence tomography based on non-uniform discrete fourier transform

A high-speed high-sensitivity swept source optical coherence tomography (SSOCT) system using a high speed swept laser source is developed. Non-uniform discrete fourier transform (NDFT) method is introduced in the SSOCT system for data processing. Frequency calibration method based on a Mach-Zender interferometer (MZI) and conventional data interpolation method is also adopted in the system for comparison. Optical coherence tomography (OCT) images from SSOCT based on the NDFT method, the MZI method, and the interpolation method are illustrated. The axial resolution of the SSOCT based on the NDFT method is comparable to that of the SSOCT system using MZI calibration method and conventional data interpolation method. The SSOCT system based on the NDFT method can achieve higher signal intensity than that of the system based on the MZI calibration method and conventional data interpolation method because of the better utilization of the power of source.

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Binaural cue coding (BCC) is an efficient technique for spatial audio rendering by using the side information such as interchannel level difference (ICLD), interchannel time difference (ICTD), and interchannel correlation (ICC). Of the side information, the ICTD plays an important role to the auditory spatial image. However, inaccurate estimation of the ICTD may lead to the audio quality degradation. In this paper, we develop a novel ICTD estimation algorithm based on the nonuniform discrete Fourier transform (NDFT) and integrate it with the BCC approach to improve the decoded auditory image. Furthermore, a new subjective assessment method is proposed for the evaluation of auditory image widths of decoded signals. The test results demonstrate that the NDFT-based scheme can achieve much wider and more externalized auditory image than the existing BCC scheme based on the discrete Fourier transform (DFT). It is found that the present technique, regardless of the image width, does not deteriorate the sound quality at the decoder compared to the traditional scheme without ICTD estimation.

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Estimation of frequency offset using warped discrete-Fourier transform

In this paper, the problem of estimating a small frequency offset in a signal with a large carrier frequency is addressed. The warped discrete-Fourier transform (WDFT) [A. Makur, S.K. Mitra, IEEE Trans. Circuits Systems—I: Fundam. Theory Appl. 6 (9) (September 2001) 1086–1093] is used and the accuracy of estimation and computational complexity of this technique is compared with the conventional discrete-Fourier transform (DFT) and the nonuniform discrete-Fourier transform (NDFT). A numerical example is provided to illustrate the comparison.