Two-dimensional Spatial Frequency Research Articles

Modeling the co-assembly of binary nanoparticles

In this work, we present a binary assembly model that can predict the co-assembly structure and spatial frequency spectra of monodispersed nanoparticles with two different particle sizes. The approach relies on an iterative algorithm based on geometric constraints, which can simulate the assembly patterns of particles with two distinct diameters, size distributions, and at various mixture ratios on a planar surface. The two-dimensional spatial-frequency spectra of the modeled assembles can be analyzed using fast Fourier transform analysis to examine their frequency content. The simulated co-assembly structures and spectra are compared with assembled nanoparticles fabricated using transfer coating method are in qualitative agreement with the experimental results. The co-assembly model can also be used to predict the peak spatial frequency and the full-width at half-maximum bandwidth, which can lead to the design of the structure spectra by selection of different monodispersed particles. This work can find applications in fabrication of non-periodic nanostructures for functional surfaces, light extraction structures, and broadband nanophotonics.

The Spatial Frequency Content of Urban and Indoor Environments as a Potential Risk Factor for Myopia Development.

PurposeTo examine the hypothesis that the spatial frequency spectra of urban and indoor environments differ from the natural environment in ways that may promote the development of myopia.MethodsA total of 814 images were analyzed from three datasets; University of California Berkeley (UCB), University of Texas (UT), and Botswana (UPenn). Images were processed in Matlab (Mathworks Inc) to map the camera color characteristics to human cone sensitivities. From the photopic luminance images generated, two-dimensional spatial frequency (SF) spectra were calculated and converted to one-dimensional spectra by rotational averaging. The spatial filtering profile of a 0.4 Bangerter foil, which has been shown to induce myopia experimentally, was also determined.ResultsThe SF slope for natural scenes followed the recognized 1/fα relationship with mean slopes of −1.08, −0.90, and −1.04 for the UCB, UT and UPenn image sets, respectively. Indoor scenes had a significantly steeper slope (−1.48, UCB; −1.52, UT; P < 0.0001). Urban environments showed an intermediate slope (−1.29, UCB; −1.22, UT) that was significantly different from the slopes derived from the natural scenes (P < 0.0001). The change in SF content between natural outdoor scenes and indoors was comparable to that induced by a 0.4 Bangerter foil, which reduced the SF slope of a natural scene from −0.88 to −1.47.ConclusionsCompared to natural outdoor images, man-made outdoor and indoor environments have spatial frequency characteristics similar to those known to induce form-deprivation myopia in animal models. The spatial properties of the man-made environment may be one of the missing drivers of the human myopia epidemic.

Two-dimensional random access multiphoton spatial frequency modulated imaging.

Spatial frequency modulated imaging (SPIFI) enables the use of an extended excitation source for linear and nonlinear imaging with single element detection. To date, SPIFI has only been used with fixed excitation source geometries. Here, we explore the potential for the SPIFI method when a spatial light modulator (SLM) is used to program the excitation source, opening the door to a more versatile, random access imaging environment. In addition, an in-line, quantitative pulse compensation and measurement scheme is demonstrated using a new technique, spectral phase and amplitude retrieval and compensation (SPARC). This enables full characterization of the light exposure conditions at the focal plane of the random access imaging system, an important metric for optimizing, and reporting imaging conditions within specimens.

Super-Bandwidth Two-Step Phase-Shifting Off-Axis Digital Holography by Optimizing Two-Dimensional Spatial Frequency Sampling Scheme

The presence of the auto correlation and twin cross correlation noises restrict the available spatial bandwidth of the holographic microscopy to much less than the available bandwidth of the digital sensor. Therefore, in order to record the same image area as conventional 2D intensity imaging techniques, several images should be taken. We present two-step phase-shifting off-axis digital holography with maximum space-bandwidth product for three-dimensional (3D) digital holography. Removing the autocorrelation term using a two-step phase-shifting technique significantly increases the available bandwidth for off-axis interferometry. An optimizing super-diagonal two-dimensional (2D) spatial frequency sampling scheme at the sub-Nyquist frequency is employed for performing off-axis interferometry in the absence of the autocorrelation term. The spatial bandwidth of the proposed two-step phase-shifting technique is 400% of that in square scheme off-axis digital holography. Experimental results demonstrate the feasibility of this technique in extracting the 3D morphology of transparent microscopic objects with a larger bandwidth.

Adaptive Per-spatial Stream Power Allocation Algorithms for Single-User MIMO-OFDM Systems

This paper presents adaptive per-spatial stream power allocation algorithms for Single User Multiple-Input Multiple-Output Orthogonal Frequency Division Multiplexing (SU MIMO-OFDM) systems. Three efficient and low-complexity Greedy Power Allocation (GPA) algorithms are proposed to maximize the throughput and spectral efficiency of the SU MIMO-OFDM systems. Firstly, the low-complexity pre-coded GPA algorithms are developed for the MIMO systems. The spatial sub-channels are created by applying the so-called Singular Value Decomposition (SVD) technique on the MIMO channel matrix, and then the Pre-GPA algorithms are applied to exploit the multi-path and spatial diversities. Secondly, the spatial and frequency diversities are exploited by adaptively allocating the system sub-carriers to the spatial sub-channels followed by Per-Spatial GPA (PSGPA). Finally, spatial multiplexing-based GPA algorithms are proposed to optimize the spectral efficiency of the SU MIMO-OFDM system. An optimal two-dimensional Spatial-Frequency GPA (SFGPA) algorithm is proposed to efficiently improve the average system spectral efficiency. The high computational complexity of the optimal SFGPA solution is simplified by proposing a low-complexity Per-Spatial GPA with Excess Power Moving down (PSGPA-EPMd) algorithm, which moves the per-spatial excess power downwards to enhance the spectral efficiency of the spatial multiplexing-based SU MIMO-OFDM systems. The proposed algorithms achieve better spectral efficiency and maximize the throughput in comparison with conventional algorithms.

Two-Dimensional Low Spatial Frequency Laser-Induced Periodic Surface Structuring of Sapphire

Two-Dimensional Low Spatial Frequency Laser-Induced Periodic Surface Structuring of Sapphire

Acousto-optic method of spatial frequency filtration based on diffraction of two eigenmodes of a crystal

A method is proposed for two-dimensional spatial frequency filtration based on acousto-optic (AO) diffraction of two eigenmodes of a crystal on a single acoustic wave. It is shown that AO filters, based on the use of such diffraction, ensure the enhancement of the two-dimensional image edge during its optical Fourier processing. The main theoretical conclusions are experimentally confirmed using an AO paratellurite filter.

Adaptive Resource Allocation Algorithms for Multi-user MIMO-OFDM Systems

Radio resource allocation to the system users is the key challenge issue in multi-user orthogonal frequency division multiplexing (MU-OFDM) systems. In this paper, an efficient and low-complexity proportional rate-adaptive radio resource (sub-carrier and power) allocation algorithm for MU-OFDM is proposed to maximize the sum-rate capacity of the system and achieve acceptable users' rates fairness. Three-dimensional (spatial, frequency, and multi-user) greedy power allocation (GPA) algorithms for the MU multi-input multi-output OFDM (MU MIMO-OFDM) systems are proposed. The proposed algorithms start with spatial sub-carrier allocation followed by an optimal two-dimensional spatial-frequency GPA (SFGPA) step, which exploits both the spatial and frequency diversities. Therefore, spatial, frequency, and multi-user diversities are exploited by the proposed three-dimensional GPA algorithms. Several experiments are carried out to test the performance of the proposed three-dimensional GPA algorithms in terms of sum-rate capacity. The optimal solution is achieved by the three-dimensional dynamic sub-carrier-SFGPA with average gain (DS-SFGPA-AG) algorithm, which achieves the best sum-rate capacity performance compared with the other power allocation algorithms. The performance of the DS-SFGPA-AG algorithm is also analyzed and compared for the four OFDM-based systems; single-user (SU) OFDM, SU MIMO-OFDM, MU OFDM, and MU MIMO-OFDM. The MU MIMO-OFDM system with adaptive sub-carrier and power allocation outperforms the other systems.

Low-complexity sparse reconstruction for high-resolution multi-static passive SAR imaging*

Abstract Bi-static passive synthetic aperture radar (SAR) systems using ground broadcast and wireless network signals suffer from a low spatial resolution due to the narrow bandwidths and low carrier frequencies. By exploiting multiple distributed illuminators, multi-static passive radar has the possibility of producing high-resolution SAR images. In this paper, a two-stage image formation approach, which combines the Fourier transform and sparse reconstruction strategies, is proposed to process multi-static SAR data. This method exploits the group sparsity of the sparse scene, i.e., the observations associated with different bi-static pairs share the same support of the sparse scene but correspond to aspect-dependent scattering coefficients. Such observations are described as a number of generally disjoint sub-bands in the two-dimensional spatial frequency domain. In each sub-band, the sampling satisfies the Nyquist criterion, whereas different sub-bands are sparsely distributed. In the proposed approach, Fourier-based reconstruction is applied to the sub-band data to produce the coarse-resolution images, which are then combined to produce a high-resolution image through the exploitation of sparse reconstruction techniques. The proposed approach greatly improves the imaging quality as compared to Fourier-based reconstruction, whereas it exhibits significant reduction of the computational complexity when compared to direct application of sparse reconstruction techniques. The exploitation of block sparsity-based techniques also permits practical treatment of the angle-dependent target scattering characteristics in SAR image reconstruction. The advantages of the proposed approach are delineated using analysis and simulations.

Photoacoustic image reconstruction in an attenuating medium using singular-value decomposition

Attenuation effects can be significant in photoacoustic tomography since the generated pressure signals are broadband, and ignoring them may lead to image artifacts and blurring. La Rivière et al. [Opt. Lett. 31(6), pp. 781-783, (2006)] had previously derived a method for modeling the attenuation effect and correcting for it in the image reconstruction. This was done by relating the ideal, unattenuated pressure signals to the attenuated pressure signals via an integral operator. We derive an integral operator relating the attenuated pressure signals to the absorbed optical energy for a planar measurement geometry. The matrix operator relating the two quantities is a function of the temporal frequency, attenuation coefficient and the two-dimensional spatial frequency. We perform singular-value decomposition (SVD) of this integral operator to study the problem further. We find that the smallest singular values correspond to wavelet-like eigenvectors in which most of the energy is concentrated at times corresponding to greater depths in tissue. This allows us to characterize the ill-posedness of recovering the absorbed optical energy distribution at different depths in an attenuating medium. This integral equation can be inverted using standard SVD methods, and the initial pressure distribution can be recovered. We conduct simulations and derive an algorithm for image reconstruction using SVD for a planar measurement geometry. We also study the noise and resolution properties of this image-reconstruction method.

Super-resolution of active spatial frequency heterodyning using holographic approach

This paper describes a technique of super-resolution that is based on holographic imaging in which three holograms corresponding to one orientation of the fringes are recorded. To recover the two-dimensional object spatial frequency, we take four orientations of the fringes with 45 degrees steps. For each orientation of the fringes, the hologram recording scheme will remain the same. The orientation of the reference beam is fixed throughout the measurements. Once the three holograms are recorded for each orientation of the fringes with a fixed amplitude of the reference beam, an algorithm is applied for each orientation. The algorithm processes the three holograms to construct a synthesized spectrum in a particular orientation; taking the inverse Fourier transform of this synthesized spectrum will give the synthesized image in that particular orientation. Different synthesized spectra are combined to obtain an overall synthesized spectrum and a super-resolved image is formed.

Synthetic Aperture Fourier Holographic Optical Microscopy

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.

Receptive Field Properties of Neurons in the Early Visual Cortex Revealed by Local Spectral Reverse Correlation

We introduce a novel class of white-noise analyses, named local spectral reverse correlation (LSRC), which is capable of revealing various aspects of visual receptive field profiles that were undetectable previously in a single simple measurement. The method is based on spectral analyses in a two-dimensional spatial frequency domain for spatially localized areas within and around their receptive fields. Extracellular single-unit recordings were performed for area 17 and 18 neurons in anesthetized cats. A dynamic dense noise pattern was presented in which the pattern covered an area two to three times larger than the classical receptive field. Spike trains were then cross-correlated with frequency spectra of localized noise pattern to obtain spatially localized selectivity maps in the two-dimensional frequency domain. Our findings are as follows. (1) The new LSRC method allows measurements of two-dimensional frequency tunings and their spatial extent even for cells with substantial nonlinearity. (2) A small subset of neurons shows spatial inhomogeneity in the two-dimensional frequency tunings. (3) In addition to facilitatory response profiles, we can also visualize suppressive profiles localized both in space and spatial frequency domains. Our results suggest that the new analysis technique can be a powerful tool for measuring visual response profiles that contain inhomogeneity in space, as well as for studying neurons with substantial nonlinearities. These features make the method particularly suitable for studying response profiles of neurons in early as well as intermediate extrastriate visual areas.

Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes

This paper presents consistent criteria for evaluating the selection of tuning parameters for an industrial model predictive control of large-scale cross-directional (CD) processes using a two-dimensional (temporal and spatial) frequency analysis technique. The concept of rectangular circulant matrices (RCMs) and their properties are presented. It is shown that large-scale CD processes can be approximated as RCMs and then diagonalized by complex Fourier matrices, allowing analysis in terms of a family of SISO transfer functions across the spatial frequencies. Familiar concepts from control engineering such as bandwidth and stability margin are extended into the two-dimensional frequency domain, providing intuitive measures of closed-loop performance and robustness.

Vernier and contrast discrimination in central and peripheral vision

The present paper asks whether Vernier offset discrimination is limited by the observer’s sensitivity to local contrast change in both central and peripheral vision. To answer this question we compared Vernier discrimination and contrast discrimination thresholds (specified in the same units) for a pair of narrow ribbons of cosine gratings. Because the ribbons are narrow, both the offset information (for Vernier discrimination) and the contrast information (for contrast discrimination) are highly localized. We found that when the stimuli are narrow ribbons, the local contrast cue is the limiting factor in Vernier discrimination. However, our results also show that integration of information along the length of the gratings (the ribbon width) is: (i) different for Vernier and contrast discrimination, and (ii) for Vernier discrimination the integration of information along the length of the gratings differs qualitatively in central and peripheral vision. For narrow ribbons, the peripheral ‘template’ for ribbon Vernier acuity is not as well matched to the stimulus (in two-dimensional spatial frequency space) as the foveal ‘template’.

Optical Pattern Recognition Experiments of Walsh Spatial Frequency Domain Filtering Method

An optical parallel processor for an image recognition system using microoptical devices has been developed. This system expands input images to coefficients of a two-dimensional Walsh spatial frequency domain and discriminates them by performing the inner product to the reference filter. In this paper, we demonstrate parallel optical discrimination of Arabic numerals using this method. First, the mechanism of this recognition method is presented. Next, the reference filter is designed. In order to implement the reference filter by using the microoptical devices, the fabrication problems were solved. Finally, the recognition system is constructed and effects of deformation on the input image are experimentally evaluated.

Two-dimensional spatial frequency response of SQUID planar gradiometers

Planar gradiometers can be modelled as two-dimensional spatial filters, taking into account area, baseline and shape of the coils. We associate a spatial frequency response with each configuration studied and show that planar gradiometers behave as band-pass spatial filters. Also, in order to determine a spatial frequency range for typical magnetic field sources, we calculate the two-dimensional Fourier transform of the field due to a current dipole for several liftoffs. Important issues such as gradiometer spatial cutoff frequencies, bandwidth and symmetry are discussed.

Scale selection using three different representations for images

Humans are capable of zooming in on the right range of scale, but it is not clear, yet, whether this processing involves either spatial variables, or two-dimensional spatial-frequency variables, or both spatial and spatial-frequency variables in the image description. The main motivation of this work is to show that at least it is possible in each domain to formulate algorithms giving solutions to this problem of scale selection. We also propose that in situations where no information is available about appropriate scales for analysis, a reasonable approach is to consider descriptions at the spatial scales for active sensors under a data-driven multisensor organization, with the sensors being orientation and spatial-frequency selective. Of course, the sensor scales are defined as those of patterns with frequency components in the sensor.

Practical approach to separating the pattern generator-induced mask CD errors from the blank/process-induced mask CD errors using conventional market measurements

With the continuing reduction of minimum feature sizes for semiconductor wafers, the manufacturing tolerances for photomask CD errors were reduced proportionately. As a result, it has become increasingly important to separate the major sources of mask CD errors so that they can be quantified and appropriately addressed. In this article we describe a reliable, convenient and inexpensive technique for separating the blank, and process CD errors from the pattern generator CD errors using conventional market measurements on cross arrays and two-dimensional spatial frequency Fourier analyses. By taking advantage of the fact that the blank/process-induced CD error spatial distributions are known to be fixed relative to the mask blank, while the pattern generator CD error distributions are known to be fixed relative to the pattern generator, a spatial frequency domain analysis of multiple sets of CD measurements on the masks, each set printed by displacing the mask relative to the previous set, is shown to allow a simple and unambiguous separation of the two types of CD errors. These measurements also directly verify the expected result that the blank/process CD error contributors tend to have mainly low-spatial-frequency components, while the pattern generator tends to mainly have higher-spatial-frequency components. Furthermore, by utilizing this result, it is shown that a simple low-pass filter applied to a single set of market measurements provides a surprisingly accurate estimate of the relative sizes of these two error contributors. Results are presented from masks printed on two commercial mask-making systems, one optical and one e-beam.

Approach to automatic analysis of Young’s fringes in speckle photography

A new flexible automatic method for analysis of Young's fringes that allows software, electronic hardware, and optoelectronic realizations is proposed. A displacement vector is sought in the two-dimensional spatial frequency domain, the separate slices of which are obtained through multiple use of the one-dimensional Fourier transform of the Radon-transformed fringe image. An estimate of fringe angle and spacing is found from a projection with the principal Fourier spectrum peak. Afterward its position is repeatedly refined until the desired accuracy is reached. Although the method is relevant mainly to speckle photography, our approach is also applicable to fringe patterns that are generated by a computer. The problem of fringe visibility is widely discussed, and a new estimate is proposed.