Fourier-based Imaging Research Articles

Multi-scale CLEAN for Fourier-based hard X-ray solar imaging

Abstract Multi-scale deconvolution is an ill-posed inverse problem in imaging, with applications ranging from microscopy, through medical imaging, to astronomical remote sensing. In the case of high-energy space telescopes, multi-scale deconvolution algorithms need to account for the peculiar property of native measurements, which are sparse samples of the Fourier transform of the incoming radiation. The present paper proposes a multi-scale version of CLEAN, which is the most popular iterative deconvolution method in Fourier-based astronomical imaging. Using synthetic data generated according to a simulated but realistic source configuration, we show that this multi-scale version of CLEAN performs better than the original one in terms of accuracy, photometry, and regularization. Further, the application to a data set measured by the NASA Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) shows the ability of multi-scale CLEAN to reconstruct rather complex flaring topographies.

Fourier-Based Anti-Aliasing Azimuth Imaging for Sparsely Optimized MIMO Arc Array

Fourier-Based Anti-Aliasing Azimuth Imaging for Sparsely Optimized MIMO Arc Array

Improved Methods for Fourier-Based Microwave Imaging.

Fourier-based imaging has been widely adopted for microwave imaging thanks to its efficiency in terms of computational complexity without compromising image resolution. Together with other backpropagation imaging algorithms like delay-and-sum (DAS), they are based on a far-field approach to the electromagnetic expression relating to fields and sources. To improve the accuracy of these techniques, this contribution presents a modified version of the well-known Fourier-based algorithm by taking into account the field radiated by the Tx/Rx antennas of the microwave imaging system. The impact on the imaged targets is discussed, providing a quantitative and qualitative analysis. The performance of the proposed method for subsampled microwave imaging scenarios is compared against other well-known aliasing mitigation methods.

Accurate Range Migration for Fast Quantitative Fourier-Based Image Reconstruction With Monostatic Radar

Range migration (or range focusing) techniques are widely used in optical, acoustic, and microwave real-time image reconstruction methods. They have been successfully applied to far-field 3-D imaging where they rely on plane-wave assumptions, which ignore the data amplitude variation over the acquisition aperture. The accuracy of the plane-wave assumption, however, quickly degrades in close-range imaging, where amplitude variations are significant and where the range to the target is on the order of the range sampling step. Here, we present a range-focusing method of improved accuracy, which is applicable to both far-zone and close-range monostatic radar. It refocuses the system point-spread function (PSF) to any range location, taking into account both magnitude and phase changes. The approach can be applied with any Fourier-based imaging algorithm utilizing the Lippmann–Schwinger equation as the underlying scattering model. Here, it is validated through examples based on simulated and measured data where the images are reconstructed with quantitative microwave holography (QMH). QMH employs measured PSFs to achieve quantitative imaging in real-time. The proposed range-migration method is applicable with measured PSFs, too, leading to reduced system-calibration effort and the ability to focus an image at any desired range.

Unified Reciprocal Space Processing for Short-Range Active and Passive Imaging Systems

Large scale sensing systems increasingly operate in short-range geometries, providing three-dimensional imaging capabilities that leverage the large number of degrees of freedom of distributed arrays. The two most powerful modes of operation include active, multiple-input multiple-output arrays, in which an image is processed from signals measured using different pairs of transmitting and receiving antennas; and passive, synthetic aperture interferometric radiometer arrays, which can image using ambient, non-cooperative radiation by processing signals measured at different receiving antenna pairs. In this paper, we explore the analogy between these two imaging modes, and outline a generalized framework for analyzing these systems in the array near field. Focusing on the similarities between the formalisms describing the operation of generic active and passive systems, it is proposed, to our knowledge for the first time in the scientific literature, to adapt a Fourier-based imaging technique optimized for short-range active MIMO systems to passive technologies, interrogating the emissivity of a spatially distributed target. The systematic method presented in this paper provides a framework for characterizing and optimizing short range coherent or incoherent systems used in threat detection and non-destructive testing applications.

Fourier-based Synthetic-aperture Imaging for Arbitrary Transmissions by Cross-correlation of Transmitted and Received Wave-fields.

Investigations into Fourier beamforming for medical ultrasound imaging have largely been limited to plane-wave and single-element transmissions. The main aim of this work is to generalize Fourier beamforming to enable synthetic aperture imaging with arbitrary transmit sequences. When applied to focused transmit beams, the proposed approach yields a full-waveform-based alternative to virtual-source synthetic aperture, which has implications for both coherence imaging and sound speed estimation. When compared to virtual-source synthetic aperture and retrospective encoding for conventional ultrasound sequences (REFoCUS), the proposed imaging technique shows an 8.6 and 3.8 dB improvement in contrast over virtual source synthetic aperture and REFoCUS, respectively, and a 55% improvement in point target resolution over virtual source synthetic aperture. The proposed image reconstruction technique also demonstrates general imaging improvements in vivo, while avoiding limitations seen in prior techniques.

A Systematic Comparison of Near-Field Beamforming and Fourier-Based Backward-Wave Holographic Imaging

In this paper we show the equivalence of near-field beamforming and backward-wave reconstruction algorithm. The proof is carried out analytically with two different approaches, using the principle of stationary phase from a signal processing point of view and the angular spectrum representation as an electromagnetic point of view. A comparison of the time complexity of the near-field beamforming and backward-wave reconstruction algorithm is given. A detailed discussion of the constraints required for a digital implementation is presented, leading to limitations for the chosen system parameters, especially for the backward-wave reconstruction approach. An exemplarily scenario is simulated and processed, confirming the found equivalence between the two very different approaches of image reconstruction. An additional measurement with a 120 GHz radar showcases the capabilities of both algorithms and validates our findings.

Compressive Sensing Imaging of 3-D Object by a Holographic Algorithm

Existing three-dimensional (3-D) compressive sensing-based millimeter-wave (MMW) imaging methods require a large-scale storage of the sensing matrix and immense computations owing to the high dimension matrix-vector model employed in the optimization. To overcome this shortcoming, we propose an efficient compressive sensing (CS) method based on a holographic algorithm for near-field 3-D MMW imaging. An interpolation-free holographic imaging algorithm is developed and used as a sensing operator, in lieu of the nominal sensing matrix typically used in the CS iterative optimization procedure. In so doing, the problem induced by the large-scale sensing matrix is avoided. With no interpolations required, both the computational speed and the image quality can be improved. Simulation and experimental results are provided to demonstrate the performance of the proposed method in comparison with those of the Omega-K based CS and the traditional Fourier-based imaging techniques.

The Empirical Effect of Gaussian Noise in Undersampled MRI Reconstruction

In Fourier-based medical imaging, sampling below the Nyquist rate results in an underdetermined system, in which a linear reconstruction will exhibit artifacts. Another consequence is lower signal-to-noise ratio (SNR) because of fewer acquired measurements. Even if one could obtain information to perfectly disambiguate the underdetermined system, the reconstructed image could still have lower image quality than a corresponding fully sampled acquisition because of reduced measurement time. The coupled effects of low SNR and underdetermined system during reconstruction makes it difficult to isolate the impact of low SNR on image quality. To this end, we present an image quality prediction process that reconstructs fully sampled, fully determined data with noise added to simulate the SNR loss induced by a given undersampling pattern. The resulting prediction image empirically shows the effects of noise in undersampled image reconstruction without any effect from an underdetermined system. We discuss how our image quality prediction process simulates the distribution of noise for a given undersampling pattern, including variable density sampling that produces colored noise in the measurement data. An interesting consequence of our prediction model is that recovery from an underdetermined nonuniform sampling is equivalent to a weighted least squares optimization that accounts for heterogeneous noise levels across measurements. Through experiments with synthetic and in vivo datasets, we demonstrate the efficacy of the image quality prediction process and show that it provides a better estimation of reconstruction image quality than the corresponding fully sampled reference image.

Near‐field multistatic radar reconstruction with stretched‐phase Fourier accelerated multistatic imaging

There exist few rapid reconstruction methods for applications that use multistatic radar for targets in the near field such as vehicle navigation and personnel screening. Fourier accelerated multistatic imaging (FAMI) is a method that combines the rapid reconstruction speed of Fourier-based imaging algorithms with the flexibility of algebraic algorithms that allow arbitrary radiation patterns and antenna placement. Many of these new applications image in the near field of the individual antennas of a multistatic array, however, FAMI contains approximations that require the target to be in the far field of each antenna. Stretched-phase FAMI (SP-FAMI) overcomes this limitation and allows imaging in the Fresnel region of the antennas so that targets may be very close to the array while retaining the computational benefits of Fourier reconstruction. As radar wavelengths decrease with a corresponding increase in resolution, SP-FAMI allows the imaging of nearby targets that would otherwise be too close for FAMI to image accurately.

An Indirect Range-Doppler Algorithm for Multireceiver Synthetic Aperture Sonar Based on Lagrange Inversion Theorem

This paper presents a novel imaging algorithm which has the capacity of processing multireceiver synthetic aperture sonar (SAS) data based on Vu’s spectrum. There are two important issues when developing the fast Fourier-based imaging algorithms. One is the point target reference spectrum (PTRS) with analytical expression in the 2-D frequency domain; the other is the space variance of the PTRS. Vu has solved the first issue based on the Lagrange inversion theorem. However, the second issue has not been solved yet. Our approach in this paper is to apply Vu’s spectral result in an indirect range-Doppler (R-D) imaging algorithm. For each transmitter/receiver pair, the close-form azimuth modulation is derived first, and then, range/azimuth coupling can be obtained via the difference between the PTRS and the azimuth modulation. Unlike traditional R-D algorithm, which utilizes the interpolation operation to eliminate the range cell migration (RCM), the RCM correction (RCMC) in this paper is carried out through two steps, i.e., bulk RCMC and differential RCMC. Bulk RCMC accounts for the RCMC in one reference range. Differential RCMC accomplished with a range-dependent subblock postprocessing method is used to compensate for the space variant deviation phase error. Via the proposed method, the echoed signal corresponding to each transmitter/receiver pair is processed first. Following which, a high-resolution SAS complex image is obtained by coherent superposition of all the coarse subimages. Numerical simulations and real data processing results show that the proposed method has the similar focusing capabilities with the accurate back projection algorithm across the whole swath. Although the presented method has the disadvantage of latency, it is very easy to make a complete phase center approximation correction, allowing for the tandem nature of the collected data. Besides, the proposed algorithm can handle general case with explicit point of stationary phase (PSP) no matter how complicated the PSP is, and it is also suitable for the imagery of tandem synthetic aperture radar.

Fourier-Based Imaging for Subsampled Multistatic Arrays

This contribution focuses on the limitations of Fourier-based imaging when applied to subsampled arrays used in multistatic millimeter-wave radar systems. The aim is to determine the relationship between the size of the object under test (OUT), its position with respect to the radar aperture, and the sampling requirements on the aperture so as to recover aliasing-free images. Based on the analysis results, a method for recovering spectral information is proposed, the idea of which is to replicate the plane wave spectrum, and then apply a band-pass filter defined by a priori knowledge of the OUT size. For simplicity, the analysis is done in two-dimensional (2-D; range and cross-range dimensions). A simulation-based application is presented for validation purposes.

Fourier-Based 2-D Imaging Algorithm for Circular Synthetic Aperture Radar: Analysis and Application

Circular synthetic aperture radar (CSAR) has become a research hotspot in the SAR community. Since the Fourier-based imaging methods of CSAR are underdeveloped, further developments have to be investigated. In this paper, the exact expression and property analysis of the spectrum of CSAR are presented first. It turns out that the challenges of the Fourier-based 2-D imaging processing of CSAR lie in the projection of spectrum from the slant plane into the ground plane. Correspondingly, a spectrum-projection method by matched filtering as well as the practical subaperture-based imaging flow of CSAR are proposed, during which the trajectory deviation has been taken into account. Simulation and experimental tests are carried out to measure the efficiency and accuracy of the proposed method. The corresponding results verify that the proposed signal model is correct and the presented imaging flow is a good candidate to produce well-focused CSAR imagery with much less time consumption.

The Process of Data Formation for the Spectrometer/Telescope for Imaging X-rays (STIX) in Solar Orbiter

The Spectrometer/Telescope for Imaging X-rays (STIX) is a hard X-ray imaging spectroscopy device to be mounted in the Solar Orbiter cluster with the aim of providing images and spectra of solar flaring regions at different photon energies in the range from a few keV to around 150 keV. The imaging modality of this telescope is based on the Moiré pattern concept and utilizes 30 subcollimators, each one containing a pair of co-axial grids. This paper applies Fourier analysis to provide the first rigorous description of the data formation process in STIX. Specifically, we show that, under the fundamental frequency approximation, the integrated counts measured by STIX subcollimators can be interpreted as specific spatial Fourier components of the incoming photon flux, named visibilities. Fourier analysis also allows the quantitative assessment of the reliability of such interpretation. The description of STIX data in terms of visibilities has a notable impact on the image reconstruction process, since it fosters the application of Fourier-based imaging algorithms.

Fourier-Based Imaging for Multistatic Radar Systems

Fourier-based methods for monostatic and bistatic setups have been widely used for high-accuracy radar imaging. However, the multistatic configuration has several characteristics that make Fourier processing more challenging: 1) a nonuniform grid in k-space, which requires multidimensional interpolation methods, and 2) image distortion when the incident spherical wave is approximated by a plane wave. This contribution presents a Fourier-based imaging method for multistatic systems, solving the aforementioned limitations: the first, by using k-space partitioning and applying interpolation in each domain; the second, by approximating the spherical wave with multiple plane waves. Both solutions are fully parallelizable, thus allowing calculation time savings. Validation and benchmarking with a synthetic aperture radar backpropagation algorithm have been performed through 2-D and 3-D simulation-based examples. Imaging results from radar measurements have been assessed.

Analysis of Local and Global Topographic Order in Mouse Retinocollicular Maps

We introduce the Lattice Method for the quantitative assessment of the topographic order within the pattern of connections between two structures. We apply this method to published visuocollicular mapping data obtained by Fourier-based intrinsic imaging of mouse colliculus. We find that, in maps from wild types and β2 knock-outs, at least 150 points on the colliculus are represented in the visual field in the correct relative order. In maps from animals with knock-out of the three ephrinA ligands (TKO), thought to specify the rostrocaudal axis of the map, the projection on the colliculus of each small circular area of visual field is elongated approximately rostrocaudally. Of these projections, 9% are made up of two distinct regions lying along the direction of ingrowth of retinal fibers. These are similar to the ectopic projections found in other ephrinA knock-out data. Coexisting with the ectopic projections, each TKO map contains a submap where neighbor-neighbor relations are preserved, which is ordered along both rostrocaudal and mediolateral axes, in the orientation found in wild-type maps. The submaps vary in size with order well above chance level, which can approach the order in wild-type maps. Knock-out of both β2 and two of the three ephrinAs yields maps with some order. The ordered TKO maps cannot be produced by correlated neural activity acting alone, as this mechanism is unable to specify map orientation. These results invite reassessment of the role of molecular signaling, particularly that of ephrinAs, in the formation of ordered nerve connections.

Detection of mitochondrial fission with orientation‐dependent optical Fourier filters

We utilize a recently developed optical imaging method based on Fourier processing with Gabor-like filters to detect changes in light scattering resulting from alterations in mitochondrial structure in endothelial cells undergoing apoptosis. Imaging based on Gabor filters shows a significant decrease in the orientation of subcellular organelles at 60 to 100 minutes following apoptosis induction and concomitant with mitochondrial fragmentation observed by fluorescence. The optical scatter changes can be detected at low resolution at the whole cell level. At high resolution, we combine fluorescence imaging of the mitochondria with optical Fourier-based imaging to demonstrate that the dynamic decrease in organelle orientation measured by optical Gabor filtering is spatially associated with fluorescent mitochondria and remains largely absent from nonfluorescent subcellular regions. These results provide strong evidence that the optical Gabor responses track mitochondrial fission during apoptosis and can be used to provide label-free, rapid monitoring of this morphological process within single cells.

Quantitative breast tomosynthesis: From detectability to estimability

This work aimed to extend Fourier-based imaging metrics for modeling and predicting quantitative imaging performance. The new methodology was applied to the platform of breast tomosynthesis for investigating the influence of acquisition parameters (e.g., acquisition angle and dose) on quantitative imaging performance. Two quantitative imaging tasks were considered: Area estimation and volume estimation of a 4 mm diameter spherical target. The maximum likelihood estimator yielded training data to generate a size estimation task function, which was combined with the MTF and NPS to predict estimation performance by computing an "estimability index" analogous to the detectability index. Estimation performance for the two tasks was computed as a function of acquisition angle and dose. The results were used for system optimization in terms of quantitation performance and further compared to the detectability index for the detection of the same spherical target. The estimability index computed with the size estimation tasks correlated well with precision measurements for area and volume estimation over a broad range of imaging conditions and provided a meaningful figure of merit for quantitative imaging performance and optimization. The results highlighted that optimal breast tomosynthesis acquisition parameters depend significantly on imaging task and dose. At nominal dose (1.5 mGy), mass detection was optimal at an acquisition angle of 85 degrees, while area and volume estimation for the same mass were optimal at approximately 125 degrees and 164 degrees acquisition angles, respectively. These findings provide an initial validation that the Fourier-based metrics extended to estimation tasks can represent a meaningful metric and predictor of quantitative imaging performance. The optimization framework also revealed trade-off between anatomical noise and system noise in volumetric imaging systems potentially identifying different optimal acquisition parameters than currently used in breast tomosynthesis and CT.

Spectroscopic imaging from spatially-encoded single-scan multidimensional MRI data

We have recently proposed a protocol for retrieving multidimensional magnetic resonance images within a single scan, based on a spatial encoding of the spin interactions. This methodology relies on progressively dephasing spin coherences throughout a sample; for instance, by sweeping a radiofrequency pulse in the presence of a magnetic field gradient. When spins are suitably refocused by a second (acquisition) field gradient, this yields a time-domain signal reflecting in its magnitude the spatial distribution of spins throughout the sample. It is hereby shown that whereas the absolute value of the resulting signals conveys such imaging information, the hitherto unutilized phase modulation of the signal encodes the chemical shift offsets of the present speciae. Spectroscopically-resolved multidimensional images can thereby be retrieved in this fashion at no additional expense in either experimental complexity, sensitivity or acquisition time—simply by performing an additional analysis of the collected data. The resulting approach to single-scan spectroscopic imaging can also incorporate “RF shimming” compensating abilities, capable of providing high-resolution spectral and high-definition imaging data even under the presence of substantial magnetic field inhomogeneities. The principles of these methodologies as applied to spectroscopic imaging are briefly reviewed and compared against the background of traditional Fourier-based single-scan spectroscopic imaging protocols. Demonstrations of these new multidimensional spectroscopic MRI experiments on simple phantoms are also given.

Extended radar observations with the frequency radar domain interferometric imaging (FII) technique

In this paper, we present high-resolution observations obtained with the Middle and Upper Atmosphere (MU) radar (Shigaraki, Japan, 34.85°N, 136.10°E) using the frequency radar domain interferometric imaging (FII) technique. This technique has recently been introduced for improving the range resolution capabilities of the mesosphere–stratosphere–troposphere (MST) radars which are limited by their minimum pulse length. The Fourier-based imaging, the Capon method have been performed with 5 equally spaced frequencies between 46.25 and 46.75 MHz and with an initial range resolution of 300 m . These results have been compared firstly to results obtained using the frequency domain interferometry (FDI) technique with Δf=0.5 MHz and, secondly, to results from a classical Doppler beam swinging (DBS) mode applied with a range resolution of 150 m . Thin echoing structures could be tracked owing to the improved radar range resolution and some complex structures possibly related to Kelvin Helmholtz instabilities have been detected. Indeed, these structures appeared within the core of a wind shear and were associated with intense vertical wind fluctuations. Moreover, a well-defined thin echo layer was found in an altitude range located below the height of the wind shear. The radar observations have not been fully interpreted yet because the radar configuration was not adapted for this kind of study and because of the lack of complementary information provided by other techniques when the interesting echoing phenomena occurred. However, the results confirm the high potentialities of the FII technique for the study of atmospheric dynamics at small scales.