Objective quantification of acetylcholine receptor aggregation using fast Fourier transforms

A new approach for objectively analyzing the aggregation of acetylcholine receptors (AChRs) through power spectrum analysis derived from fast Fourier transform (FFT) of images has been developed. Presently, detection of AChR aggregates at neuromuscular junctions is not easily accomplished. Though the formation of AChR clusters results in periodic gray-level variations that differ with time, no study reporting their correlation with frequency information in the Fourier domain for aggregates' detection purposes exists. To this end, we processed time-lapse images of AChR aggregates' formation on murine myotubes to extract peak values of power spectra. To validate interpretation of the Fourier spectra analysis, a computer routine was developed to semi-automatically count AChR aggregates. We found: (1) logarithmic maxima of Fourier spectra correlated significantly with experimentation time; (2) cluster count correlated significantly with time only after clusters were discernable from images, signifying that this method heavily depended on definitive growth data and thresholding values; (3) exponents of Fourier maxima versus time and cluster count versus time profiles during this phase compared favorably, indicating that both methods were analyzing identical cluster growth rates. Our observations suggest that analysis via FFT power spectrum is sensitive and robust enough to automatically quantify AChR aggregates.     

六边形稀疏网格上的FFT算法

稀疏网格是一种具有特殊分层插值性质的非均匀网格形式,稀疏网格上的离散傅立叶变换算法称为Hyperbolic Cross FFT算法.这一算法能够有效降低采样点数量,并将指数时间复杂度的d维DFT算法降低到O(Nlog^dN)^[10].六边形网格是另一种具有特殊性质的网格,具有在采样点数量较少和采样效率较高等优势.本文的研究工作主要集中在将六边形网格和稀疏网格相结合,构造六边形稀疏网格上的FFT算法.通过定义六边形和方形网格下标之间的转换,实现了六边形稀疏网格上的FFT算法,并通过数值实验证明了这一算法的有效性.     

基于二维FFT的图像滤波方法及实现

二维快速傅立叶变换(FFT)在一个传统概念的处理机上实现时,需要芯片具有更多的逻辑资源。本文给出了基于FPGA的自定义处理机(CCM)的二维FFT算法和实现。在CCM的Splash-2平台上实现了二维FFT,计算速度达到180Mflops,最快速度超过Sparc-10工作站的23倍。同时,对于一个N×N图像,这种实现方法可以满足二维FFT所需要的O(N2log2N)次的浮点算术运算。     

六角形阵快速傅立叶变换波束形成算法

在六角形阵阵元数较多时,传统的频域相移求和波束形成方法要求的运算量很大.为此提出一种采用六角形快速傅立叶变换HFFT(Hexagonal Fast Fourier Transform)的波束形成算法.使用六角形傅立叶变换HDFT(Hexagonal Discrete Fourier Transform)完成六角形阵的波束形成,由于HDFT存在快速算法HFFT,因此能够显著降低波束形成的运算量.首先在各个通道上做FFT,将信号变换到频域,然后转角重排,再对各个阵元上相同的频点做HFFT,得到频域常规波束形成输出.理论分析表明,对于窄带信号的六角形阵波束形成,所提出的算法所需的计算量比传统的相移求和方法降低了95%以上.仿真和试验结果表明,提出的算法在不影响阵列处理性能的同时,显著降低了波束形成所需的计算量,易于工程实现.     

可扩展的旋转因子表及FFT算法

该文提出了一个用于快速Fourier变换计算的反写码序的旋转因了表，这种旋转因子表具有可扩展性：本质上，这种旋转因子表的分量与变换的点数无关，当点数改变时，这种旋转因子表无须重新计算或者容易扩展；根据这种旋转因子表，该文设计了一个结构规整的基本基4计算2^n点FFT的算法及软件程序，该程序与FFTW软件包进行了对比实验，文中还以蛋白质序列相似性计算为例，对作者的算法与FFTW软件包中的相庆算法进行了对比实验，结果表明，采用该文的算法可节省计算时间约31．7％。     

Performance Analysis of FFT Algorithms on Multiprocessor Systems

A decimation-in-time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in multiprocessors with shared bus, multistage interconnection network (MIN), and in mesh connected computers. Results are derived for data allocation, interprocessor communication, approximate computation time, and speedup of an N point FFT on any P available processing elements (PE's). Further generalization is obtained for a radix-r FFT algorithm. An N X N point two-dimensional discrete Fourier transform (DFT) implementation is also considered when one or more rows of the input data matrix are allocated to each PE.     

Calculating the n-dimensional fast Fourier transform

The one-dimensional fast Fourier transform (FFT) is the most popular tool for calculating the multidimensional Fourier transform. As a rule, to estimate the n-dimensional FFT, a standard method of combining one-dimensional FFTs, the so-called “by rows and columns” algorithm, is used in the literature. For fast calculations, different researchers try to use parallel calculation tools, the most successful of which are searches for the algorithms related to the computing device architecture: cluster, video card, GPU, etc. [1, 2]. The possibility of paralleling another algorithm for FFT calculation, which is an n-dimensional analog of the Cooley-Tukey algorithm [3, 4], is studied in this paper. The focus is on studying the analog of the Cooley-Tukey algorithm because the number of operations applied to calculate the n-dimensional FFT is considerably less than in the conventional algorithm nN ⁿ log₂ N of addition operations and 1/2N ^{n + 1}log₂ N of multiplication operations of addition operations and $\frac{{2^n - 1}} {{2^n }}N^n \log _2 N$ of multiplication operations against: N ^{n + 1}log₂ N of addition operations and 1/2N ^{n + 1}log₂ N of in combining one-dimensional FFTs.     

Fast Reverse Jacket Transform As an Alternative Representation of the N-Point Fast Fourier Transform

The Reverse Jacket matrix (RJM) is a generalized form of the Hadamard matrix. Thus RJM is closely related to the matrix for fast Fourier transform (FFT). It also has a very interesting structure, i.e. its inverse can be easily obtained and has the reversal form of the original matrix. In this paper, we have shown that a transform based on the RJM offers a simple structure of N-point FFT in terms of the decomposition of the corresponding matrix and that it computes very fast the center weighted Hadamard transform.     

The Partial Fast Fourier Transform

An efficient algorithm for computing the one-dimensional partial fast Fourier transform \(f_j=\sum _{k=0}^{c(j)}e^{2\pi ijk/N} F_k\) is presented. Naive computation of the partial fast Fourier transform requires \({\mathcal O}(N^2)\) arithmetic operations for input data of length N. Unlike the standard fast Fourier transform, the partial fast Fourier transform imposes on the frequency variable k a cutoff function c(j) that depends on the space variable j; this prevents one from directly applying standard FFT algorithms. It is shown that the space–frequency domain can be partitioned into rectangular and trapezoidal subdomains over which efficient algorithms can be developed. As in the previous work of Ying and Fomel (Multiscale Model Simul 8(1):110–124, 2009), the contribution from rectangular regions can be reduced to a series of fractional-phase Fourier transforms over squares, each of which can be reduced to a convolution. In this work, we demonstrate that the partial Fourier transform over trapezoidal domains can also be reduced to a convolution. Since the computational complexity of a dealiased convolution of N inputs is \({\mathcal O}(N\log N)\), a fast algorithm for the partial Fourier transform is achieved, with a lower overall coefficient than obtained by Ying and Fomel.     

An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration

A novel adaptable accurate way for calculating polar FFT and log-polar FFT is developed in this paper, named multilayer fractional Fourier transform (MLFFT). MLFFT is a necessary addition to the pseudo-polar FFT for the following reasons: It has lower interpolation errors in both polar and log-polar Fourier transforms; it reaches better accuracy with the nearly same computing complexity as the pseudo-polar FFT; it provides a mechanism to increase the accuracy by increasing the user-defined computing level. This paper demonstrates both MLFFT itself and its advantages theoretically and experimentally. By emphasizing applications of MLFFT in image registration with rotation and scaling, our experiments suggest two major advantages of MLFFT: 1) scaling up to 5 and arbitrary rotation angles, or scales up to 10 without rotation can be recovered by MLFFT while currently the result recovered by the state-of-the-art algorithms is the maximum scaling of 4; 2) No iteration is needed to obtain large rotation and scaling values of images by MLFFT, hence it is more efficient than the pseudopolar-based FFT methods for image registration.     

Noisy fingerprints classification with directional FFT based features using MLP

A methodology is described for classifying noisy fingerprints directly from raw unprocessed images. The directional properties of fingerprints are exploited as input features by computing one-dimensional fast Fourier transform (FFT) of the images over some selected bands in four and eight directions. The ability of the multilayer perceptron (MLP) for generating complex boundaries is utilised for the purpose of classification. The superiority of the method over some existing ones is established for fingerprints corrupted with various types of distortions, especially random noise.     

Fast extraction of wavelet-based features from JPEG images for joint retrieval with JPEG2000 images

In this paper, some fast feature extraction algorithms are addressed for joint retrieval of images compressed in JPEG and JPEG2000 formats. In order to avoid full decoding, three fast algorithms that convert block-based discrete cosine transform (BDCT) into wavelet transform are developed, so that wavelet-based features can be extracted from JPEG images as in JPEG2000 images. The first algorithm exploits the similarity between the BDCT and the wavelet packet transform. For the second and third algorithms, the first algorithm or an existing algorithm known as multiresolution reordering is first applied to obtain bandpass subbands at fine scales and the lowpass subband. Then for the subbands at the coarse scale, a new filter bank structure is developed to reduce the mismatch in low frequency features. Compared with the extraction based on full decoding, there is more than 72% reduction in computational complexity. Retrieval experiments also show that the three proposed algorithms can achieve higher precision and recall than the multiresolution reordering, especially around the typical range of compression ratio.     

Computing the Fourier transform of real data on a hypercube

There are two ways, other than the standard fast Fourier transform (FFT) algorithm, of computing Fourier transforms of real data, namely, (1)the real fast Fourier transform (RFFT) algorithm, and (2) the fast Hartley transform (FHT) algorithm. On a sequential computer, it has been shown that both the RFFT and the FHT algorithms are faster than the FFT algorithm. However, it is not obvious that the same is true on a parallel machine. The communication requirements of the RFFT and the FHT algorithms, which are critical to the cost of any parallel implementation, are different from those of the FFT algorithm. In this paper we present efficient implementations of the RFFT and the FHT algorithms on a hypercube machine. Experimental results are given for the implementation of the RFFT and the FHT algorithms on the NCUBE machine.     

Fast Fourier Transform on FCC and BCC Lattices with Outputs on FCC and BCC Lattices Respectively

In this paper, we show fast Fourier transform (FFT) algorithms for efficient, non-redundant evaluations of discrete Fourier transforms (DFTs) on face-centered cubic (FCC) and body-centered cubic (BCC) lattices such that the corresponding DFT outputs are on FCC and BCC lattices, respectively. Furthermore, for each of those FFTs, we deduce the structures of its spatial (frequency respectively) domains that are contained in the Voronoi cell centered at 0 with respect to the DFT (inverse DFT respectively) associated sublattice.     

基于变换域和噪声估计的路面图像去噪研究

针对不考虑噪声的统计分布,仅使用傅里叶变换或小波变换对图像进行降噪处理会带来图像的失真（扭曲）的问题,提出基于变换域和噪声估计的图像去噪方法。算法根据傅里叶变换和小波变换对图像的有效表示侧重点不同,以及图像噪声在不同变换域下的统计特性,提出先将图像进行傅里叶变换,根据噪声的统计特性构造传递函数H,使用Wiener滤波器进行降噪处理,得到一次降噪图像;再对图像再进行小波变换,根据噪声在小波的各尺度下,以及同一尺度下的不同特性,分别采用软门限降噪法和MMSE准则的降噪方法,得到二次降噪图像。仿真实验证实,该算法能有效提高降噪效果,降噪后的图像不失真,包含噪声少。     

Reconfigurable Superconducting FFT Processor Using Bit-Slice Block Share Processing Unit

We have proposed a reconfigurable high speed and very economical Rapid Single Flux Quantum (RSFQ) superconducting logic design based on the Fast Fourier Transform (FFT) Processor. We have designed a 256 – point FFT processor with the help of a bit-slicing block sharing unit. RSFQ is one of the superconducting device logics comprises of Josephson Junction. The computation complexity of this superconducting FFT is less when the number of points increased. We have proposed three different designs depending on the split radix FFT, the bit-serial radix 2 FFT, and the mixed radix FFT algorithms. The proposed design will slice the 256 – point FFT into eight 32 – point FFT each and each 32 – point FFT is divided into eight 4 – point FFT each for the reduction in hardware cost. For complex multiplication, the computation complexity of our design will be less than N/2 Log₂ N for the radix 2 algorithm based on the Block share processing Unit (BSPU) and further, it is reduced for split radix & mixed radix algorithms based on BSPU based RSFQ logic. Due to this, the speed of the processor is improvised compared to general FFT algorithm based semiconductor technology. we have computed and calculated the latency at 10 GHz for our designs. The main aim of this proposed design is to reduce the complex computation time and better performance of the processor with less hardware cost. This proposed design can furthermore continue to several N² – point by using synchronous clock tree.     

Convolution theorems for the linear canonical transform and their applications

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.     

Five-step FFT algorithm with reduced computational complexity

We propose a fast Fourier transform algorithm, which removes two steps of twiddle factor multiplications from the conventional five-step FFT algorithm. The proposed FFT algorithm not only reduces the computational complexity of the five-step FFT algorithm by O(n) operations, but also reduces its memory requirement.     

基于FPGA的二维FFT图像边缘增强设计

在处理图像类信息时,图像细节往往能传达更多信息,是人们较为关注部分。针 对在光照不理想的条件下,传感器采集到的图像对比度低、细节难以分辨的问题,提出一种基 于现场可编程门阵列(FPGA)的二维快速傅立叶变换的图像边缘提取及增强方法。通过模块化设 计,完成 4 路并行 512×512 点快速傳里叶变换(FFT)运算处理器设计,并通过 FFT 模块复用减 少 FPGA 内资源消耗,同时实现图像频谱的高通滤波算法及傅立叶逆变换算法。经过仿真与实 验,确定该方法有效可靠,实时性强,可以满足工业上图像处理的需求。     

An FFT Performance Model for Optimizing General-Purpose Processor Architecture

General-purpose processor (GPP) is an important platform for fast Fourier transform (FFT),due to its flexibility,reliability and practicality.FFT is a representative application intensive in both computation and memory access,optimizing the FFT performance of a GPP also benefits the performances of many other applications.To facilitate the analysis of FFT,this paper proposes a theoretical model of the FFT processing.The model gives out a tight lower bound of the runtime of FFT on a GPP,and guides the architecture optimization for GPP as well.Based on the model,two theorems on optimization of architecture parameters are deduced,which refer to the lower bounds of register number and memory bandwidth.Experimental results on different processor architectures (including Intel Core i7 and Godson-3B) validate the performance model.The above investigations were adopted in the development of Godson-3B,which is an industrial GPP.The optimization techniques deduced from our performance model improve the FFT performance by about 40%,while incurring only 0.8% additional area cost.Consequently,Godson-3B solves the 1024-point single-precision complex FFT in 0.368 μs with about 40 Watt power consumption,and has the highest performance-per-watt in complex FFT among processors as far as we know.This work could benefit optimization of other GPPs as well.