Nonuniform Fourier Research Articles

MRI-based ventilation and perfusion imaging to predict radiation-induced pneumonitis in lung tumor patients at a 0.35 T MR-Linac

Background and purpose:Radiation-induced pneumonitis (RP), diagnosed 6–12 weeks after treatment, is a complication of lung tumor radiotherapy. So far, clinical and dosimetric parameters have not been reliable in predicting RP. We propose using non-contrast enhanced magnetic resonance imaging (MRI) based functional parameters acquired over the treatment course for patient stratification for improved follow-up. Materials and methods:23 lung tumor patients received MR-guided hypofractionated stereotactic body radiation therapy at a 0.35T MR-Linac. Ventilation- and perfusion-maps were generated from 2D-cine MRI-scans acquired after the first and last treatment fraction (Fx) using non-uniform Fourier decomposition. The relative differences in ventilation and perfusion between last and first Fx in three regions (planning target volume (PTV), lung volume receiving more than 20Gy (V20) excluding PTV, whole tumor-bearing lung excluding PTV) and three dosimetric parameters (mean lung dose, V20, mean dose to the gross tumor volume) were investigated. Univariate receiver operating characteristic curve - area under the curve (ROC–AUC) analysis was performed (endpoint RP grade≥1) using 5000 bootstrapping samples. Differences between RP and non-RP patients were tested for statistical significance with the non-parametric Mann–Whitney U test (α=0.05). Results:14/23 patients developed RP of grade≥1 within 3 months. The dosimetric parameters showed no significant differences between RP and non-RP patients. In contrast, the functional parameters, especially the relative ventilation difference in the PTV, achieved a p-value<0.05 and an AUC value of 0.84. Conclusion:MRI-based functional parameters extracted from 2D-cine MRI-scans were found to be predictive of RP development in lung tumor patients.

High Resolution Ranging with Small Sample Number under Low SNR Utilizing RIP-OMCS Strategy and AHRC l1 Minimization for Laser Radar

This manuscript presents a novel scheme to achieve high-resolution laser-radar ranging with a small sample number under low signal-to-noise ratio (SNR) conditions. To reduce the sample number, the Restricted Isometry Property-based optimal multi-channel coprime-sampling (RIP-OMCS) strategy is established. In the RIP-OMCS strategy, the data collected across multiple channels with very low coprime-sampling rates can record accurate range information on each target. Further, the asynchronous problem caused by channel sampling-time errors is considered. The sampling-time errors are estimated using the cross-correlation function. After canceling the asynchronous problem, the data collected by multiple channels are then merged into non-uniform sampled signals. Using data combination, target-range estimation is converted into an optimization problem of sparse representation consisting of a non-uniform Fourier dictionary. This optimization problem is solved using adaptive hybrid re-weighted constraint (AHRC) l1 minimization. Two constraints are formed from statistical attributes of the targets and clutter. Moreover, as the detailed characteristics of the target, clutter, and noise are unknown before the solution, the two constraints can be adaptively modified, which guarantees that l1 minimization obtains the high-resolution range profile and accurate distance of all targets under a low SNR. Our experiments confirmed the effectiveness of the proposed method.

Effects of prospective motion correction on perivascular spaces at 7T MRI evaluated using motion artifact simulation.

The effectiveness of prospective motion correction (PMC) is often evaluated by comparing artifacts in images acquired with and without PMC (NoPMC). However, such an approach is not applicable in clinical setting due to unavailability of NoPMC images. We aim to develop a simulation approach for demonstrating the ability of fat-navigator-based PMC in improving perivascular space (PVS) visibility in T2-weighted MRI. MRI datasets from two earlier studies were used for motion artifact simulation and evaluating PMC, including T2-weighted NoPMC and PMC images. To simulate motion artifacts, k-space data at motion-perturbed positions were calculated from artifact-freeimages using nonuniform Fourier transform and misplaced onto the Cartesian grid before inverse Fourier transform. The simulation's ability to reproduce motion-induced blurring, ringing, and ghosting artifacts was evaluated using sharpness at lateral ventricle/white matter boundary, ringing artifact magnitude in the Fourier spectrum, and background noise, respectively. PVS volume fraction in white matter was employed to reflect its visibility. In simulation, sharpness, PVS volume fraction, and background noise exhibited significant negative correlations with motion score. Significant correlations were found in sharpness, ringing artifact magnitude, and PVS volume fraction between simulated and real NoPMC images (p ≤ 0.006). In contrast, such correlations were reduced and nonsignificant between simulated and real PMC images (p ≥ 0.48), suggesting reduction of motion effects with PMC. The proposed simulation approach is an effective tool to study the effects of motion and PMC on PVS visibility. PMC may reduce the systematic bias of PVS volume fraction caused by motion artifacts.

Mutual Coupling Compensation-based Nonuniform Fourier Transform Technique for Accurate and Efficient Pattern Evaluation and its Application to Synthesis of Aperiodic Arrays

A rapid solution for evaluating the radiation pattern of aperiodic arrays, taking into account mutual coupling, is presented in this paper. The evaluation is achieved by eliminating the anisotropy of the active element pattern of the array through the use of the mutual coupling compensation matrix (MCCM) technique, in conjunction with the non-uniform fast Fourier transform (NuFFT). In order to eliminate the impact of mutual coupling on array pattern calculation, the MCCM is utilized to convert the active element pattern (AEP) of each element into a shared uniform term and make the NuFFT technique suitable for the array pattern calculation. The proposed solution is validated by evaluating the radiation pattern of a 64-element planar aperiodic array. In addition, the proposed solution is integrated into the Particle Swarm Optimization (PSO) to realize a pattern synthesis method. Two synthesized patterns, including a pencil beam with low sidelobe level and a flat-top beam pattern, are executed to validate. Compared with several reported methods, the proposed method can improve the synthesis efficiency and maintain good accuracy simultaneously.

Improved method for modulation transfer function measurement of Bayer color cameras

Aiming at the difficulty of detecting the modulation transfer function (MTF) of Bayer color cameras during focusing, the traditional scheme is modified in this paper. First, through analysis, it is found that the MTF of the Bayer detector decreased substantially compared to the monochromatic detector, which was caused by the loss of a large amount of light from the Bayer detector. When the MTF is calculated for the color image after de-mosaicing, the degradation of image quality leads to the inability to obtain the full-frequency MTF curve. Therefore, this paper proposes a solution to detect the MTFs of the three primary colors in the Bayer image individually, and then weight the MTFs of each primary color to obtain the MTF of the color image based on the calibration results of the light source. Then, considering the steps of differentiation and Fourier transform in the slanted-edge method, this paper solves the problem of non-uniform sampling of edge spread function by using the modification of the slanted-edge method with non-uniform Fourier transform. Finally, the optimal imaging position of a certain type of Bayer color camera is determined using the methods proposed in this paper. Simulation and test results verify the effectiveness and accuracy of the methods in this paper.

Application of the non-uniform Fourier transform to non-uniformly sampled Fourier transform spectrometers

Resampling by interpolation is the traditional method to process interferograms from non-uniformly sampled Fourier transform spectrometers. The non-uniform fast Fourier transform (NUFFT) is an alternative approach that has been mostly overlooked. With the aid of experiments on a high-resolution interferometer with a variety of optical sources, these two methods are compared. It is found that the NUFFT is comparable to interpolation in spectral profile shape and spectral noise levels and is better in spectral amplitude and computer performance. A significant advantage is also found in the case of under-sampling and noise performance by NUFFT due to the unique non-periodic nature of non-uniform sampling. In addition, a novel implementation of NUFFT is presented and analysed.

Ventilation and perfusion MRI at a 0.35 T MR-Linac: feasibility and reproducibility study

BackgroundHybrid devices that combine radiation therapy and MR-imaging have been introduced in the clinical routine for the treatment of lung cancer. This opened up not only possibilities in terms of accurate tumor tracking, dose delivery and adapted treatment planning, but also functional lung imaging. The aim of this study was to show the feasibility of Non-uniform Fourier Decomposition (NuFD) MRI at a 0.35 T MR-Linac as a potential treatment response assessment tool, and propose two signal normalization strategies for enhancing the reproducibility of the results.MethodsTen healthy volunteers (median age 28 ± 8 years, five female, five male) were repeatedly scanned at a 0.35 T MR-Linac using an optimized 2D+t balanced steady-state free precession (bSSFP) sequence for two coronal slice positions. Image series were acquired in normal free breathing with breaks inside and outside the scanner as well as deep and shallow breathing. Ventilation- and perfusion-weighted maps were generated for each image series using NuFD. For intra-volunteer ventilation map reproducibility, a normalization factor was defined based on the linear correlation of the ventilation signal and diaphragm position of each scan as well as the diaphragm motion amplitude of a reference scan. This allowed for the correction of signal dependency on the diaphragm motion amplitude, which varies with breathing patterns. The second strategy, which can be used for ventilation and perfusion, eliminates the dependency on the signal amplitude by normalizing the ventilation/perfusion maps with the average ventilation/perfusion signal within a selected region-of-interest (ROI). The position and size dependency of this ROI was analyzed. To evaluate the performance of both approaches, the normalized ventilation/perfusion-weighted maps were compared and the deviation of the mean ventilation/perfusion signal from the reference was calculated for each scan. Wilcoxon signed-rank tests were performed to test whether the normalization methods can significantly improve the reproducibility of the ventilation/perfusion maps.ResultsThe ventilation- and perfusion-weighted maps generated with the NuFD algorithm demonstrated a mostly homogenous distribution of signal intensity as expected for healthy volunteers regardless of the breathing maneuver and slice position. Evaluation of the ROI’s size and position dependency showed small differences in the performance. Applying both normalization strategies improved the reproducibility of the ventilation by reducing the median deviation of all scans to 9.1%, 5.7% and 8.6% for the diaphragm-based, the best and worst performing ROI-based normalization, respectively, compared to 29.5% for the non-normalized scans. The significance of this improvement was confirmed by the Wilcoxon signed rank test with p, <, 0.01 at alpha , =, 0.05. A comparison of the techniques against each other revealed a significant difference in the performance between best ROI-based normalization and worst ROI (p, =, 0.01) and between best ROI-based normalization and scaling factor (p, =, 0.02), but not between scaling factor and worst ROI (p, =, 0.71). Using the ROI-based approach for the perfusion-maps, the uncorrected deviation of 10.2% was reduced to 5.3%, which was shown to be significant (p, <, 0.01).ConclusionsUsing NuFD for non-contrast enhanced functional lung MRI at a 0.35 T MR-Linac is feasible and produces plausible ventilation- and perfusion-weighted maps for volunteers without history of chronic pulmonary diseases utilizing different breathing patterns. The reproducibility of the results in repeated scans significantly benefits from the introduction of the two normalization strategies, making NuFD a potential candidate for fast and robust early treatment response assessment of lung cancer patients during MR-guided radiotherapy.

A fast MR fingerprinting simulator for direct error estimation and sequence optimization

Magnetic resonance fingerprinting (MRF) is a novel quantitative MR technique that simultaneously provides multiple tissue property maps. When optimizing MRF scans, modeling undersampling errors and field imperfections in cost functions for direct measurement of quantitative errors will make the optimization results more practical and robust. However, optimizing such cost function is computationally expensive and impractical for MRF optimization with tens of thousands of iterations. Here, we introduce a fast MRF simulator to simulate aliased images from actual scan scenarios including undersampling and system imperfections, which substantially reduces computational time and allows for direct error estimation of the quantitative maps and efficient sequence optimization. We evaluate the performance and computational speed of the proposed approach by simulations and in vivo experiments. The simulations from the proposed method closely approximate the signals and MRF maps from in vivo scans, with 158 times shorter processing time than the conventional simulation method using Non-uniform Fourier transform. We also demonstrate the power of applying the fast MRF simulator in MRF sequence optimization. The optimized sequences are validated with in vivo scans to assess the image quality and accuracy. The optimized sequences produce artifact-free T1 and T2 maps in 2D and 3D scans with equivalent mapping accuracy as the human-designed sequence but at shorter scan times. Incorporating the proposed simulator in the MRF optimization framework makes direct estimation of undersampling errors during the optimization process feasible, and provide optimized MRF sequences that are robust against undersampling artifacts and field inhomogeneity.

Imaging Algorithm for Inverse Synthetic Aperture Radar in Condition of Non-Uniform Data Rate

Inverse synthetic aperture radar imaging is an important technique for radar detection and recognition of non-cooperative targets. However, for most of the existing imaging methods, pulse signals are transmitted and received at equal intervals, without considering the possible problems under condition of non-uniform data rate, such as accuracy decline of echo's range alignment and high side lobes in Doppler spectrum. The resulting vertical and horizontal defocusing can reduce the imaging quality and affect the feature extraction of the target. Therefore, an inverse synthetic aperture radar imaging method for non-uniform data rate is proposed. First, by designing the weight to simulate the alignment situation of uniform data rate, the correlation method with weighted sliding window is used to solve the problems of declined range alignment accuracy and vertical defocusing. Then, based on the linear form of non-uniform Fourier transform, the Fourier base matrix of non-uniform data rate is built, and CLEAN technology is used to complete the projection of echo signal on the bases iteratively. Finally, the Fourier basis matrix with uniform data rate is used to reconstruct the echo signal, which solves the problem of horizontal defocusing and makes the data rate of echo signal uniform. The effectiveness of the proposed method is verified by experiments using measured data.

Coherent Dynamics of Quantum Systems with Non-Uniform Fourier Space Excited by Laser Radiation

Coherent Dynamics of Quantum Systems with Non-Uniform Fourier Space Excited by Laser Radiation

Efficient long-range convolutions for point clouds

The efficient treatment of long-range interactions (LRIs) for point clouds is a challenging problem in many scientific machine learning applications. To extract global information, one usually needs a large window size, a large number of layers, and/or a large number of channels. This can often significantly increase the computational cost. In this work, we present a novel neural network layer that directly incorporates long-range information for a point cloud. This layer, dubbed the long-range convolutional (LRC)-layer, leverages the convolutional theorem coupled with the non-uniform Fourier transform. In a nutshell, the LRC-layer mollifies the point cloud to an adequately sized regular grid, computes its Fourier transform, multiplies the result by a set of trainable Fourier multipliers, computes the inverse Fourier transform, and finally interpolates the result back to the point cloud. The resulting global all-to-all convolution operation can be performed in nearly-linear time asymptotically with respect to the number of input points. The LRC-layer is a particularly powerful tool when combined with local convolution as together they offer efficient and seamless treatment of both short- and long-range interactions. We showcase this framework by introducing a neural network architecture that combines LRC-layers with short-range convolutional layers to accurately learn the energy and force associated with a N-body potential. We also exploit the induced two-level decomposition and propose an efficient strategy to train the combined architecture with a reduced number of samples.

Towards general-purpose representation learning of polygonal geometries

Neural network representation learning for spatial data (e.g., points, polylines, polygons, and networks) is a common need for geographic artificial intelligence (GeoAI) problems. In recent years, many advancements have been made in representation learning for points, polylines, and networks, whereas little progress has been made for polygons, especially complex polygonal geometries. In this work, we focus on developing a general-purpose polygon encoding model, which can encode a polygonal geometry (with or without holes, single or multipolygons) into an embedding space. The result embeddings can be leveraged directly (or finetuned) for downstream tasks such as shape classification, spatial relation prediction, building pattern classification, cartographic building generalization, and so on. To achieve model generalizability guarantees, we identify a few desirable properties that the encoder should satisfy: loop origin invariance, trivial vertex invariance, part permutation invariance, and topology awareness. We explore two different designs for the encoder: one derives all representations in the spatial domain and can naturally capture local structures of polygons; the other leverages spectral domain representations and can easily capture global structures of polygons. For the spatial domain approach we propose ResNet1D, a 1D CNN-based polygon encoder, which uses circular padding to achieve loop origin invariance on simple polygons. For the spectral domain approach we develop NUFTspec based on Non-Uniform Fourier Transformation (NUFT), which naturally satisfies all the desired properties. We conduct experiments on two different tasks: 1) polygon shape classification based on the commonly used MNIST dataset; 2) polygon-based spatial relation prediction based on two new datasets (DBSR-46K and DBSR-cplx46K) constructed from OpenStreetMap and DBpedia. Our results show that NUFTspec and ResNet1D outperform multiple existing baselines with significant margins. While ResNet1D suffers from model performance degradation after shape-invariance geometry modifications, NUFTspec is very robust to these modifications due to the nature of the NUFT representation. NUFTspec is able to jointly consider all parts of a multipolygon and their spatial relations during prediction while ResNet1D can recognize the shape details which are sometimes important for classification. This result points to a promising research direction of combining spatial and spectral representations.

Tunable Bessel Beam Shaping for Robust Atmospheric Optical Communication

As a powerful supplement to the existing wireless communication technology, free-space optical communication helps to achieve ultra-high-speed wireless communication in many important applications such as redundant link, disaster recovery and secure link. However, beam distortion and intensity scintillation caused by atmospheric turbulence severely limits its communication performance through optical links, and there are still many problems in existing turbulent suppression schemes. Here, we propose an easy-implemented anti-turbulence free-space optical communication scheme based on the tunable zero-order Bessel beam, which is able to provide 2 dB to 3 dB power compensation in comparison with the ones employing traditional Gaussian beams. For the worse link environment with water fog, our scheme can still ensure the stability of the communication link when the Gaussian beam link is cut off. In addition, to generate tunable Bessel beams, we propose a method based on non-uniform Fourier transform to achieve accurate and fast light field simulation.

NC-PDNet: A Density-Compensated Unrolled Network for 2D and 3D Non-Cartesian MRI Reconstruction

Deep Learning has become a very promising avenue for magnetic resonance image (MRI) reconstruction. In this work, we explore the potential of unrolled networks for non-Cartesian acquisition settings. We design the NC-PDNet (Non-Cartesian Primal Dual Netwok), the first density-compensated (DCp) unrolled neural network, and validate the need for its key components via an ablation study. Moreover, we conduct some generalizability experiments to test this network in out-of-distribution settings, for example training on knee data and validating on brain data. The results show that NC-PDNet outperforms baseline (U-Net, Deep image prior) models both visually and quantitatively in all settings. In particular, in the 2D multi-coil acquisition scenario, the NC-PDNet provides up to a 1.2 dB improvement in peak signal-to-noise ratio (PSNR) over baseline networks, while also allowing a gain of at least 1dB in PSNR in generalization settings. We provide the open-source implementation of NC-PDNet, and in particular the Non-uniform Fourier Transform in TensorFlow, tested on 2D multi-coil and 3D single-coil k-space data.

Guest editorial: Smart communications and networking: architecture, applications, and future challenges

Guest editorial: Smart communications and networking: architecture, applications, and future challenges

Celestial Spectrum Velocimetry With Non-Linear Fourier Phase Shift and Its CRLB

To solve the problem of the non-linear Fourier phase shift caused by the wavelength shift in the celestial spectrum velocimetry, a celestial spectrum velocimetry method based on non-uniform discrete Fourier transform and compressed sensing is proposed. First, according to the properties of the non-uniform discrete Fourier transform with non-periodic celestial spectra, the non-linear relationship between the spectral wavelength shift and the Fourier phase shift is deduced theoretically. And then, the discrete cosine transform is used to decompose and reconstruct the spectrum to construct the phase dictionary of the non-periodic spectrum. In this dictionary, the spectral wavelength shift has a non-linear relationship with the Fourier phase shift. Finally, the non-uniform Fourier transform matrix is used as the measurement matrix. And the Doppler wavelength shift is estimated by using a non-linear and super-resolution matching algorithm to estimate the velocimetry. Experimental results show that when the spacecraft’s velocity is high, the celestial spectrum velocimetry method with non-linear Fourier phase shift (CSV-NFPS) can effectively measure the velocity from the spectrum, and its accuracy approaches the theoretical Cramer-Rao lower bounds (CRLB). Compared with the traditional Taylor method, the CSV-NFPS has an average accuracy improvement of 16.18% and a computational complexity reduction of 25.07%. In short, our method eliminate the non-linear bias in the celestial spectrum velocimetry and reduce a certain computational complexity.

Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction

&lt;p style='text-indent:20px;'&gt;Spare-view CT imaging is advantageous to decrease the radiation exposure, acquisition time and computational cost, but suffers from severe streak noise in reconstruction if the classical filter back projection method is employed. Although a few compressed sensing based algorithms have recently been proposed to remedy the insufficiency of projections and have achieved remarkable improvement in reconstruction quality, they face computational challenges for large-scale CT images (e.g., larger than 2000℅2000 pixels). In this paper, we present a fast non-uniform Fourier transform based reconstruction method, targeting at under-sampling high resolution Synchrotron-based micro-CT imaging. The proposed method manipulates the Fourier slice theorem to avoid the involvement of large-scale system matrices, and the reconstruction process is performed in the Fourier domain. With a total variation penalty term, the proposed method can be formulated into an unconstrained minimization problem, which is able to be efficiently solved by the limited-memory BFGS algorithm. Moreover, direct non-uniform Fourier transform is computationally costly, so the developed NUFFT algorithm is adopted to approximate it with little loss of quality. Numerical simulation is implemented to compare the proposed method with some other competing approaches, and then real data obtained from the Australia Synchrotron facility are tested to demonstrate the practical applications of the proposed approach. In short, the significance of the proposed approach includes (1) that it can handle high-resolution CT images with millions of pixels while several other contemporary methods fail; (2) that it can achieve much better reconstruction quality than other methods when the projections are insufficient.&lt;/p&gt;

Aliasing Artifacts Suppression in MIMO Millimeter-Wave Imaging Systems Based on K-Space Analysis

This paper focuses on dealing with aliasing artifacts in millimeter-wave imaging systems. To address the challenges and complexities of designing multi-static arrays, sparse arrangements are utilized in practical imaging systems. Depending on the arrangement of antennas, such arrays may present severe aliasing artifacts. Some techniques have already been introduced to tackle such artifacts and enhance the image quality. These techniques may either distort the target image by causing some loss in parts of the image, impose high computational/system costs or suffer from the lack of generalizability. One of these techniques uses non-uniform arrays. Such arrays are designed by optimizing the point spread function (PSF). The PSF is a suitable representative of the system&#x2019;s performance provided that the system is shift-invariant. Since non-uniform arrays are shift-variant, these techniques are unreliable, especially in the case of off-center targets. As a consequence, there are no frameworks to design non-uniform arrays on a reliable basis to suppress aliasing artifacts in the whole region of interest (ROI). To address this issue, based on the k-space analysis of aliasing in multi-static arrays and non-uniform Fourier transform concept, we propose a small random displacement to be applied to antennas&#x2019; location in sparse periodic arrays. This displacement prevents the aliasing components from coherently accumulating. Nevertheless, due to the imposed non-uniformity, it is impossible to use FFT-based algorithms in this case. A subsequent resampling or the use of non-uniform FFT can address this problem. The provided simulations and experimental measurements show that the proposed method can efficiently lower the peak level of aliasing artifacts by around 3dB compared to sparse periodic arrays. So, the proposed method can be used as an effective way to suppress aliasing artifacts in real-time multi-static millimeter-wave imaging systems.

Low pilot overhead channel estimation for CP‐OFDM‐based massive MIMO OTFS system

In high-speed mobile scenarios, due to the high-speed relative motion between transmitter and receiver, the high Doppler frequency shift interferes with the inter-subcarrier orthogonality in orthogonal frequency-division multiplexing (OFDM) systems. Therefore its performance is significantly degraded. Recently, orthogonal time–frequency space (OTFS) is considered as an effective alternative scheme to OFDM for time-varying channels. As with OFDM-massive multiple input multiple output (MIMO) systems, downlink channel estimation is necessary for OTFS-massive MIMO systems to improve the spectral efficiency in frequency-division duplex (FDD) mode without channel reciprocity. Here, first the cyclic prefix -OFDM-based massive MIMO OTFS system channel with antenna directivity pattern is analyzed, and transform the burst sparsity in the angle domain into block sparsity by using non-uniform Fourier transform (NUFT). Furthermore, to solve the problem that the pilot overhead grows linearly with the number of antennas, we propose a three-dimensional (3D) dynamic support detect (DSD) algorithm. Compared with the traditional OMP algorithm, and the 3D-structured orthogonal matching pursuit algorithm, simulation results demonstrate the proposed DSD algorithm has higher channel estimation accuracy, and lower pilot overhead.

A Novel 3-D-NUFFT Method for the Efficient Calculation of the Array Factor of Conformal Arrays

A novel 3-D nonuniform Fourier transformation (3-D-NUFFT) method is proposed for the efficient calculation of the array factors of conformal arrays. Interpolation in a least-squares sense is first used to interpolate the topological structure of conformal arrays into that of uniformly spaced volumetric arrays. By applying the equivalent mathematical relationship between the 3-D discrete Fourier transform (DFT) and the array factor of the uniform volumetric arrays, the 3-D-FFT is then developed to calculate the array factor of uniformly spaced volumetric arrays. Therefore, the array factor of conformal arrays can be easily calculated according to the interpolation relationship between the conformal arrays and the uniform volumetric arrays. Representative numerical examples regarding the array factor calculation of spherical arrays with different excitations are provided. In comparison with the direct calculation method, the obtained results confirm the advantages and effectiveness of the proposed method.