Showing papers on "Fourier transform published in 2006"

Fast Discrete Curvelet Transforms

Abstract: This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.

Compact system for high-speed velocimetry using heterodyne techniques

Abstract: We have built a high-speed velocimeter that has proven to be compact, simple to operate, and fairly inexpensive. This diagnostic is assembled using off-the-shelf components developed for the telecommunications industry. The main components are fiber lasers, high-bandwidth high-sample-rate digitizers, and fiber optic circulators. The laser is a 2W cw fiber laser operating at 1550nm. The digitizers have 8GHz bandwidth and can digitize four channels simultaneously at 20GS∕s. The maximum velocity of this system is ∼5000m∕s and is limited by the bandwidth of the electrical components. For most applications, the recorded beat frequency is analyzed using Fourier transform methods, which determine the time response of the final velocity time history. Using the Fourier transform method of analysis allows multiple velocities to be observed simultaneously. We have obtained high-quality data on many experiments such as explosively driven surfaces and gas gun assemblies.

Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform

Abstract: We introduce a new low-distortion embedding of l2d into lpO(log n) (p=1,2), called the Fast-Johnson-Linden-strauss-Transform. The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the "Heisenberg principle" of the Fourier transform, ie, its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in l1 and l2. We consider the case of approximate nearest neighbors in l2d. We provide a faster algorithm using classical projections, which we then further speed up by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.

Astronomical Image and Data Analysis

Abstract: From the Contents: Filtering * Deconvolution * Detection * Image Compression * Multichannel Data * An Entropic Tour of Astronomical Data Analysis * Astronomical Catalog Analysis * Multiple Resolution in Data Storage and Retrieval * Towards the Virtual Observatory * Appendix A: Picard Iteration * Appendix B: Wavelet Transform Using the Fourier Transform * Appendix C: Derivative Needed for the Minimization * Appendix D: Generalization of the Derivative Needed for Minimization.

k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity

Abstract: M. Lustig, J. M. Santos, D. L. Donoho, J. M. Pauly Electrical Engineering, Stanford University, Stanford, CA, United States, Statistics, Stanford University, Stanford, CA, United States Introduction Recently rapid imaging methods that exploit the spatial sparsity of images using under-sampled randomly perturbed spirals and non-linear reconstruction have been proposed [1,2]. These methods were inspired by theoretical results in sparse signal recovery [1-5] showing that sparse or compressible signals can be recovered from randomly under-sampled frequency data. We propose a method for high frame-rate dynamic imaging based on similar ideas, now exploiting both spatial and temporal sparsity of dynamic MRI image sequences (dynamic scene). We randomly under-sample k-t space by random ordering of the phase encodes in time (Fig. 1). We reconstruct by minimizing the L1 norm of a transformed dynamic scene subject to data fidelity constraints. Unlike previously suggested linear methods [7, 8], our method does not require a known spatio-temporal structure nor a training set, only that the dynamic scene has a sparse representation. We demonstrate a 7-fold frame-rate acceleration both in simulated data and in vivo non-gated Cartesian balanced-SSFP cardiac MRI . Theory Dynamic MR images are highly redundant in space and time. By using linear transformations (such as wavelets, Fourier etc.), we can represent a dynamic scene using only a few sparse transform coefficients. Inadequate sampling of the spatial-frequency -temporal space (k-t space) results in aliasing in the spatial -temporal-frequency space (x-f space). The aliasing artifacts due to random under-sampling are incoherent as opposed to coherent artifacts in equispaced under sampling. More importantly the artifacts are incoherent in the sparse transform domain. By using the non-linear reconstruction scheme in [1-5] we can recover the sparse transform coefficients and as a consequence, recover the dynamic scene. We exploit sparsity by constraining our reconstruction to have a sparse representation and be consistent with the measured data by solving the constrained optimization problem: minimize ||Ψm||1 subject to: ||Fm – y||2 < e. Here m is the dynamic scene, Ψ transforms the scene into a sparse representation, F is randomized phase encode ordering Fourier matrix, y is the measured k-space data and e controls fidelity of the reconstruction to the measured data. e is usually set to the noise level. Methods For dynamic heart imaging, we propose using the wavelet transform in the spatial dimension and the Fourier transform in the temporal. Wavelets sparsify medical images [1] whereas the Fourier transform sparsifies smooth or periodic temporal behavior. Moreover, with random k-t sampling, aliasing is extremely incoherent in this particular transform domain. To validate our approach we considered a simulated dynamic scene with periodic heart-like motion. A random phase-encode ordered Cartesian acquisition (See Fig. 2) was simulated with a TR=4ms, 64 pixels, acquiring a total of 1024 phase encodes (4.096 sec). The data was reconstructed at a frame rate of 15FPS (a 4-fold acceleration factor) using the L1 reconstruction scheme implemented with non-linear conjugate gradients. The result was compared to a sliding window reconstruction (64 phase encodes in length). To further validate our method we considered a Cartesian balanced-SSFP dynamic heart scan (TR=4.4, TE=2.2, α=60°, res=2.5mm, slice=9mm). 1152 randomly ordered phase encodes (5sec) where collected and reconstructed using the L1 scheme at a 7-fold acceleration (25FPS). Result was compared to a sliding window (64 phase encodes) reconstruction. The experiment was performed on a 1.5T GE Signa scanner using a 5inch surface coil. Results and discussion Figs. 2 and 3 illustrate the simulated phantom and actual dynaic heart scan reconstructions. Note, that even at 4 to 7-fold acceleration, the proposed method is able to recover the motion, preserving the spatial frequencies and suppressing aliasing artifacts. This method can be easily extended to arbitrary trajectories and can also be easily integrated with other acceleration methods such as phase constrained partial k-space and SENSE [1]. In the current, Matlab implementation we are able to reconstruct a 64x64x64 scene in an hour. This can be improved by using newly proposed reconstruction techniques [5,6]. Previously proposed linear methods [7,8] exploit known or measured spatio-temporal structure. The advantage of the proposed method is that the signal need not have a known structure, only sparsity, which is a very realistic assumption in dynamic medical images [1,7,8]. Therefore, a training set is not required. References [1] Lustig et al. 13th ISMRM 2004:p605 [2] Lustig et al. ” Rapid MR Angiography ...” Accepted SCMR06’ [3] Candes et al. ”Robust Uncertainty principals". Manuscript. [4] Donoho D. “Compressesed Sensing”. Manuscript. [5] Candes et al. “Practical Signal Recovery from Random Projections” Manuscript. [6] M. Elad, "Why Simple Shrinkage is Still Relevant?" Manuscript. [7] Tsao et al.. Magn Reson Med. 2003 Nov;50(5):1031-42. [8] Madore et al. Magn Reson Med. 1999 Nov;42(5):813-28. Figure 2: Simulated dynamic data. (a) The transform domain of the cross section is truly sparse. (b) Ground truth crosssection. (c) L1 reconstruction from random phase encode ordering, 4-fold acceleration (d) Sliding window (64) reconstruction from random phase encode ordering.. Figure 1: (a) Sequential phase encode ordering. (b) Random Phase encode ordering. The k-t space is randomly sampled, which enables recovery of sparse spatio-temporal dynamic scenes using the L1 reconstruction.

Fully Automatic Registration of 3D Point Clouds

Abstract: We propose a novel technique for the registration of 3D point clouds which makes very few assumptions: we avoid any manual rough alignment or the use of landmarks, displacement can be arbitrarily large, and the two point sets can have very little overlap. Crude alignment is achieved by estimation of the 3D-rotation from two Extended Gaussian Images even when the data sets inducing them have partial overlap. The technique is based on the correlation of the two EGIs in the Fourier domain and makes use of the spherical and rotational harmonic transforms. For pairs with low overlap which fail a critical verification step, the rotational alignment can be obtained by the alignment of constellation images generated from the EGIs. Rotationally aligned sets are matched by correlation using the Fourier transform of volumetric functions. A fine alignment is acquired in the final step by running Iterative Closest Points with just few iterations.

3D interpolation of irregular data with a POCS algorithm

Abstract: Seismic surveys generally have irregular areas where data cannot be acquired. These data should often be interpolated. A projection onto convex sets (POCS) algorithm using Fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high-quality results. The original 2D image restoration method, the Gerchberg-Saxton algorithm, is extended easily to higher dimensions, and the 3D version of the process used here produces much better interpolations than typical 2D methods. The only parameter that makes a substantial difference in the results is the number of iterations used, and this number can be overestimated without degrading the quality of the results. This simplicity is a significant advantage because it relieves the user of extensive parameter testing. Although the cost of the algorithm is several times the cost of typical 2D methods, the method is easily parallelized and still completely practical.

Synthetic aperture fourier holographic optical microscopy.

Abstract: We report a new synthetic aperture optical microscopy in which high-resolution, wide-field amplitude and phase images are synthesized from a set of Fourier holograms. Each hologram records a region of the complex two-dimensional spatial frequency spectrum of an object, determined by the illumination field's spatial and spectral properties and the collection angle and solid angle. We demonstrate synthetic microscopic imaging in which spatial frequencies that are well outside the modulation transfer function of the collection optical system are recorded while maintaining the long working distance and wide field of view.

Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula

Abstract: The numerical calculation of the Rayleigh-Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.

Retrieval of the Green’s Function from Cross Correlation: The Canonical Elastic Problem

Abstract: In realistic materials, multiple scattering takes place and average field intensities or energy densities follow diffusive processes. Multiple P to S energy conversions by the random inhomogeneities result in equipartition of elastic waves, which means that in the phase space the available elastic energy is distributed among all the possible states of P and S waves, with equal amounts in average. In such diffusive regimes, the P to S energy ratio equilibrates in a universal way independent of the particular details of the scattering. We study the canonical problem of isotropic plane waves in an elastic medium and show that the Fourier transform of azimuthal average of the cross correlation of motion between two points within an elastic medium is proportional to the imaginary part of the exact Green’s tensor function between these points, provided the energy ratio ES / EP is the one predicted by equipartition in two and three dimensions, respectively. These results clearly show that equipartition is a necessary condition to retrieve the exact Green’s function from correlations of the elastic field. However, even if there is not an equipartitioned regime and correlations do not allow to retrieve precisely the exact Green’s function, the correlations may provide valuable results of physical significance by reconstructing specific arrivals.

Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms

Abstract: We propose what we believe is a new method for color image encryption by use of wavelength multiplexing based on lensless Fresnel transform holograms. An image is separated into three channels: red, green, and blue, and each channel is independently encrypted. The system parameters of Fresnel transforms and random phase masks in each channel are keys in image encryption and decryption. An optical color image coding configuration with multichannel implementation and an optoelectronic color image encryption architecture with single-channel implementation are presented. The keys can be added by iteratively employing the Fresnel transforms. Computer simulations are given to prove the possibility of the proposed idea.

The Vertical Structure of the Outer Milky Way H I Disk

Abstract: We examine the outer Galactic H i disk for deviations from the b ¼ 0 � plane by constructing maps of disk surface density, mean height, and thickness. We find that the Galactic warp is well described by a vertical offset plus two Fourier modes of frequency m ¼ 1 and 2, all of which grow with galactocentric radius. Adding the m ¼ 2m ode accounts for the large asymmetry between the northern and southern warps. We use a Morlet wavelet transform to investigate the spatial and frequency localization of higher frequency modes; these modes are often referred to as ‘‘scalloping.’’ We find that the m ¼ 10 and 15 scalloping modes are well above the noise, but localized; this suggests that the scalloping does not pervade the whole disk, but only local regions. Subject headingg Galaxy: disk — Galaxy: kinematics and dynamics — Galaxy: structure — ISM: structure — radio lines: general

Fast focus field calculations.

Abstract: We present a fast calculation of the electromagnetic field near the focus of an objective with a high numerical aperture (NA). Instead of direct integration, the vectorial Debye diffraction integral is evaluated with the fast Fourier transform for calculating the electromagnetic field in the entire focal region. We generalize this concept with the chirp z transform for obtaining a flexible sampling grid and an additional gain in computation speed. Under the conditions for the validity of the Debye integral representation, our method yields the amplitude, phase and polarization of the focus field for an arbitrary paraxial input field on the objective. We present two case studies by calculating the focus fields of a 40×1.20 NA water immersion objective for different amplitude distributions of the input field, and a 100×1.45 NA oil immersion objective containing evanescent field contributions for both linearly and radially polarized input fields.

Research progress of the fractional Fourier transform in signal processing

Abstract: The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.

Impedance Models in Time Domain including the Extended Helmholtz Resonator Model

Abstract: The problem of translating a frequency domain impedance boundary condition to time domain involves the Fourier transform of the impedance function. This requires extending the definition of the impedance not only to all real frequencies but to the whole complex frequency plane. Not any extension, however, is physically possible. The problemshouldremain causal, the variables real, andthe wall passive. This leads to necessary conditions for an impedance function. Various methods of extending the impedance that are available in the literature are discussed. A most promising one is the so-called z-transform by Ozyoruk & Long, which is nothing but an impedance that is functionally dependent on a suitable complex exponent e −iωκ . By choosing κ a multiple of the time step of the numerical algorithm, this approach fits very well with the underlying numerics, because the impedance becomes in time domain a delta-comb function and gives thus an exact relation on the grid points. An impedance function is proposed which is based on the Helmholtz resonator model, called Extended Helmholtz Resonator Model. This has the advantage that relatively easily the mentioned necessary conditions can be satisfied in advance. At a given frequency, the impedance is made exactly equal to a given design value. Rules of thumb are derived to produce an impedance which varies only moderately in frequency near design conditions. An explicit solution is given of a pulse reflecting in time domain at a Helmholtz resonator impedance wall that provides some insight into the reflection problem in time domain and at the same time may act as an analytical test case for numerical implementations, like is presented at this conference by the companion paper AIAA-2006-2569 by N. Chevaugeon, J.-F. Remacle and X. Gallez. The problem of the instability, inherent with the Ingard-Myers limit with mean flow, is discussed. It is argued that this instability is not consistent with the assumptions of the Ingard-Myers limit and may well be suppressed.

Extended high-frame rate imaging method with limited-diffraction beams

Abstract: Fast three-dimensional (3-D) ultrasound imaging is a technical challenge. Previously, a high-frame rate (HFR) imaging theory was developed in which a pulsed plane wave was used in transmission, and limited-diffraction array beam weightings were applied to received echo signals to produce a spatial Fourier transform of object function for 3-D image reconstruction. In this paper, the theory is extended to include explicitly various transmission schemes such as multiple limited-diffraction array beams and steered plane waves. A relationship between the limited-diffraction array beam weighting of received echo signals and a 2-D Fourier transform of the same signals over a transducer aperture is established. To verify the extended theory, computer simulations, in vitro experiments on phantoms, and in vivo experiments on the human kidney and heart were performed. Results show that image resolution and contrast are increased over a large field of view as more and more limited-diffraction array beams with different parameters or plane waves steered at different angles are used in transmissions. Thus, the method provides a continuous compromise between image quality and image frame rate that is inversely proportional to the number of transmissions used to obtain a single frame of image. From both simulations and experiments, the extended theory holds a great promise for future HFR 3-D imaging

Optical excitations in organic molecules, clusters, and defects studied by first-principles Green's function methods

Abstract: Spectroscopic and optical properties of nanosystems and point defects are discussed within the framework of Green's function methods. We use an approach based on evaluating the self-energy in the so-called GW approximation and solving the Bethe-Salpeter equation in the space of single-particle transitions. Plasmon-pole models or numerical energy integration, which have been used in most of the previous GW calculations, are not used. Fourier transforms of the dielectric function are also avoided. This approach is applied to benzene, naphthalene, passivated silicon clusters (containing more than one hundred atoms), and the $F$ center in LiCl. In the latter, excitonic effects and the $1s\ensuremath{\rightarrow}2p$ defect line are identified in the energy-resolved dielectric function. We also compare optical spectra obtained by solving the Bethe-Salpeter equation and by using time-dependent density-functional theory in the local, adiabatic approximation. From this comparison, we conclude that both methods give similar predictions for optical excitations in benzene and naphthalene, but they differ in the spectra of small silicon clusters. As cluster size increases, both methods predict very low cross section for photoabsorption in the optical and near ultraviolet ranges. For the larger clusters, the computed cross section shows a slow increase as a function of photon frequency. Ionization potentials and electron affinities of molecules and clusters are also calculated.

Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography

Abstract: High-speed complex full-range Fourier domain optical coherence tomography (FD-OCT) is demonstrated. In this FD-OCT, the phase modulation of a reference beam (M scan) and transversal scanning (B scan) are simultaneously performed. The Fourier transform method is applied along the direction of the B scan to reconstruct complex spectra, and the complex spectra comprise a full-range OCT image. Because of this simultaneous B-M-mode scan, the FD-OCT requires only a single A scan for each single transversal position to obtain a full-range FD-OCT image. A simple but slow version of the FD-OCT visualizes the cross section of a plastic plate. A modified fast version of this FD-OCT investigates a sweat duct in a finger pad in vivo and visualizes it with an acquisition time of 27 ms.

SAR amplitude probability density function estimation based on a generalized Gaussian model

Abstract: In the context of remotely sensed data analysis, an important problem is the development of accurate models for the statistics of the pixel intensities. Focusing on synthetic aperture radar (SAR) data, this modeling process turns out to be a crucial task, for instance, for classification or for denoising purposes. In this paper, an innovative parametric estimation methodology for SAR amplitude data is proposed that adopts a generalized Gaussian (GG) model for the complex SAR backscattered signal. A closed-form expression for the corresponding amplitude probability density function (PDF) is derived and a specific parameter estimation algorithm is developed in order to deal with the proposed model. Specifically, the recently proposed "method-of-log-cumulants" (MoLC) is applied, which stems from the adoption of the Mellin transform (instead of the usual Fourier transform) in the computation of characteristic functions and from the corresponding generalization of the concepts of moment and cumulant. For the developed GG-based amplitude model, the resulting MoLC estimates turn out to be numerically feasible and are also analytically proved to be consistent. The proposed parametric approach was validated by using several real ERS-1, XSAR, E-SAR, and NASA/JPL airborne SAR images, and the experimental results prove that the method models the amplitude PDF better than several previously proposed parametric models for backscattering phenomena.

A comparative study on solution- and bridgman-grown single crystals of benzimidazole by high-resolution X-ray diffractometry, fourier transform infrared, microhardness, laser damage threshold, and second-harmonic generation measurements

Abstract: Nowadays more attention has been paid to the growth of organic nonlinear optical single crystals due to their high nonlinear optical efficiency and fairly good optical damage threshold comparable t...

Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression

Abstract: In regression with a high-dimensional predictor vector, it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response. Using the Fourier transform, we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively. Under the normality assumption of the predictors, explicit estimates of the central and central mean subspaces are derived. Bootstrap procedures are used for determining dimensionality and choosing tuning parameters. Simulation results and an application to a real data are reported. Our methods demonstrate competitive performance compared with SIR, SAVE, and other existing methods. The approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future.

Pulse profiles of millisecond pulsars and their Fourier amplitudes

Abstract: Approximate analytical formulae are derived for the pulse profile produced by small hotspots on a rapidly rotating neutron star. Its Fourier amplitudes and phases are calculated. The proposed formalism takes into account gravitational bending of light, Doppler effect, anisotropy of emission and time delays. Its accuracy is checked with exact numerical calculations.

Convolution theorems for the linear canonical transform and their applications

Abstract: 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.

Quantum Algorithms for Some Hidden Shift Problems

Abstract: Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.