Radon Domain Research Articles

Next-Generation Secure and Reversible Watermarking for Medical Images using Hybrid Radon-Slantlet Transform

This research article proposes a security-enhanced watermarking method for medical images using the Radon and Slantlet transforms. The first step involves transforming the cover image from the spatial domain to the Radon domain. This transformation involves rotation, scaling, and translation through the Radon transform, which alters the locations of concealed bits. As a result, identifying the embedded data poses a considerable challenge. The embedded data cannot be identified without employing the inverse Radon transform. Subsequently, the Radon-transformed image is converted to the frequency domain using the Slantlet transform. Secret bits are incorporated into frequency coefficients during this phase through the pixel pair mapping approach. The final watermarked image is generated by inserting side information into the robust watermarked image. Simulation experiments are carried out to evaluate the imperceptibility of watermarks in medical images, employing metrics such as PSNR and SSIM. The results indicate high imperceptibility, with PSNR values exceeding 45 dB and SSIM values surpassing 0.95 for all tested images. Furthermore, the proposed method's robustness and reversibility are assessed by exposing watermarked images to various attacks. Performance is measured through the BER and NCC. Experimental findings reveal a BER of 0.2 % for the watermarked information, indicating strong resilience against attacks. Additionally, the NCC is determined to be 0.96, highlighting a high level of reversibility in extracting embedded data.

A transcranial multiple waves suppression method for plane wave imaging based on Radon transform

Transcranial ultrasound imaging presents a significant challenge due to the intricate interplay between ultrasound waves and the heterogeneous human skull. The skull’s presence induces distortion, refraction, multiple scattering, and reflection of ultrasound signals, thereby complicating the acquisition of high-quality images. Extracting reflections from the entire waveform is crucial yet exceedingly challenging, as intracranial reflections are often obscured by strong amplitude direct waves and multiple scattering. In this paper, a multiple wave suppression method for ultrasound plane wave imaging is proposed to mitigate the impact of skull interference. Drawing upon prior research, we developed an enhanced high-resolution linear Radon transform using the maximum entropy principle and Bayesian method, facilitating wavefield separation. We detailed the process of wave field separation in the Radon domain through simulation of a model with a high velocity layer. When plane waves emitted at any steering angles, both multiple waves and first arrival waves manifested as distinct energy points. In the brain simulation, we contrasted the characteristic differences between skull reflection and brain-internal signal in Radon domain, and demonstrated that multiples suppression method reduces side and grating lobe levels by approximately 30 dB. Finally, we executed in vitro experiments using a monkey skull to separate weak intracranial reflection signals from strong skull reflections, enhancing the contrast-to-noise ratio by 85 % compared to conventional method using full waveform. This study deeply explores the effect of multiples on effective signal separation, addresses the complexity of wavefield separation, and verifies its efficacy through imaging, thereby significantly advancing ultrasound transcranial imaging techniques.

Evaluation of low-dose computed tomography reconstruction using spatial-radon domain total generalized variation regularization

The x-ray radiation dose in computed tomography (CT) examination has been a major concern for patients. Lowing the tube current and exposure time in data acquisition is a straightforward and cost-effective strategy to reduce the x-ray radiation dose. However, this will inevitably increase the noise fluctuations in measured projection data, and the corresponding CT image quality will be severely degraded if noise suppression is not performed during image reconstruction. To reconstruct high-quality low-dose CT image, we present a spatial-radon domain total generalized variation (SRDTGV) regularization for statistical iterative reconstruction based on penalized weighted least-squares (PWLS) principle, which is called PWLS-SRDTGV for simplicity. The presented PWLS-SRDTGV model can simultaneously reconstruct high-quality CT image in space domain and its corresponding projection in radon domain. An efficient split Bregman algorithm was applied to minimize the cost function of the proposed reconstruction model. Qualitative and quantitative studies were performed to evaluate the effectiveness of the PWLS-SRDTGV image reconstruction algorithm using a digital 3D XCAT phantom and an anthropomorphic torso phantom. The experimental results demonstrate that PWLS-SRDTGV algorithm achieves notable gains in noise reduction, streak artifact suppression, and edge preservation compared with competing reconstruction approaches.

3D high-resolution radon transform based on a strong sparseLp–1norm and its applications

AbstractMultiple reflections are among the most challenging noises to suppress in seismic data, as they differ from effective waves only in terms of apparent velocity. Also, radon transform, an essential technique for attenuating multiple reflections, has been widely incorporated into various commercial software packages. Thus, this study introduces a 3D radon transform method based on the ${L_{p - 1}}$ norm to enhance sparsity-constraining capability in the transform domain, leveraging high-resolution radon transform techniques. Specifically, an iteratively reweighted least-squares algorithm is employed to obtain the transformed data in the radon domain. Given that the ${L_{p - 1}}$ norm was first applied to the Radon transform in seismic data processing methods, this paper theoretically demonstrates its powerful sparsity-constraining capability. Indeed, the proposed strategy enhances energy concentration in the radon transform domain, better separating primaries from multiples, and ultimately suppressing the multiples. Both model tests and real data indicate that the 3D radon transform constrained by the ${L_{p - 1}}$ norm outperforms existing high-resolution radon transform methods with sparsity constraints regarding energy concentration and effectiveness in multiple reflection attenuation.

Nonconvex weighted variational metal artifacts removal via convergent primal-dual algorithms

Direct reconstruction through filtered back projection engenders metal artifacts in polychromatic computed tomography images, attributed to highly attenuating implants, which further poses great challenges for subsequent image analysis. Inpainting the metal trace directly in the Radon domain for the extant variational method leads to strong edge diffusion and potential inherent artifacts. With normalization based on pre-segmentation, the inpainted outcome can be notably ameliorated. However, its reconstructive fidelity is heavily contingent on the precision of the pre-segmentation, and highly accurate segmentation of images with metal artifacts is non-trivial in actuality. In this paper, we propose a nonconvex weighted variational approach for metal artifact reduction. Specifically, in lieu of employing a binary function with zeros in the metal trace, an adaptive weight function is designed in the Radon domain, with zeros in the overlapping regions of multiple disjoint metals as well as areas of highly attenuated projections, and the inverse square root of the measured projection in other regions. A nonconvex L1−αL2 regularization term is incorporated to further enhance edge contrast, alongside a box-constraint in the image domain. Efficient first-order primal-dual algorithms, proven to be globally convergent and of low computational cost owing to the closed-form solution of all subproblems, are devised to resolve such a constrained nonconvex model. Both simulated and real experiments are conducted with comparisons to other variational algorithms, validating the superiority of the presented method. Especially in comparison to Reweighted JSR, our proposed algorithm can curtail the total computational cost to at most one-third, and for the case of inaccurate pre-segmentation, the recovery outcomes by the proposed algorithms are notably enhanced.

A robust blind color watermarking algorithm based on the Radon-DCT transform

Digital watermarking is an effective technique for image copyright protection. Many digital image watermarking algorithms are sensitive to geometric distortions. These distortions make it difficult to detect and extract watermarks. In this study, a robust color watermarking algorithm combining the Radon transform and DCT transform is proposed. First, the color carrier image is converted to the YUV color space from RGB, and the U component of YUV is transformed to the Radon domain using the Radon transform. Then, a two-dimensional 8 × 8 discrete cosine transform (2D-DCT) is carried out on the selected blocks of the Radon domain, and some fixed midfrequency coefficients are selected to embed watermark information. Moreover, the Arnold transform is applied to encrypt the watermarks and the random permutation function to scramble the embedding positions. Finally, the watermark embedding and blind extraction processes are completed by modifying the midfrequency coefficients using the presented rules. A series of simulation experiments show that the watermark algorithm has high imperceptibility and good robustness to various attacks.

Enhanced Needle Visualization with Reflection Tuned Apodization based on the Radon Transform for Ultrasound Imaging.

In ultrasound (US) guided interventions, accurately tracking and visualizing needles during in-plane insertions is a significant challenge due to strong directional specular reflections. These reflections violate the geometrical delay and apodization estimations in the conventional delay and sum beamforming (DASB) degrading the visualization of needles. This study proposes a novel reflection tuned apodization to address this issue and facilitate needle enhancement through DASB. The method leverages both temporal and angular information derived from the Radon transforms of the RF data from plane-wave imaging to filter the specular reflections from the needle and their directivity. The directivity information is translated into apodization center maps through time-to-space mapping in the Radon domain which is subsequently integrated into DASB. We assess the influence of needle angulations, projection angles in the Radon transform, needle gauge sizes, and the presence of multiple specular interfaces on the approach. The analysis shows that the method surpasses conventional DASB in enhancing the image quality of needle interfaces while preserving the diffuse scattering from the surrounding tissues without significant computational overhead. The work offers promising prospects for improved outcomes in US guided interventions and better insights into characterizing US reflections with Radon transforms.

Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain

Multiple reflection is a common interference wave in offshore petroleum and gas exploration, and the Radon-based filtering method is a frequently used approach for multiple removal. However, the filtering parameter setting is crucial in multiple suppression and relies heavily on the experience of processors. To reduce the dependence on human intervention, we introduce the geometric mode decomposition (GMD) and develop a novel processing flow that can automatically separate primaries and multiples, and then accomplish the suppression of multiples. GMD leverages the principle of the Wiener filtering to iteratively decompose the data into modes with varying curvature and intercept. By exploiting the differences in curvature, GMD can separate primary modes and multiple modes. Then, we propose a novel sparse Radon transform (RT) constrained with the elastic half (EH) norm. The EH norm contains a l1/2 norm and a scaled l2 norm, which is added to overcome the numerical oscillation problem of the l1/2 norm. With the help of the EH norm, the estimated Radon model can reach a remarkable level of sparsity. To solve the optimization problem of the proposed sparse RT, an efficient alternating multiplier iteration algorithm is employed. Leveraging the high sparsity of the Radon model obtained from the proposed transform, we improve the GMD-based multiple removal framework. The high-sparsity Radon model obtained from the proposed Radon transform can not only simplify the separation of primary and multiple modes but also accelerate the convergence of GMD, thus improving the processing efficiency of the GMD method. The performance of the proposed GMD-based framework in multiple elimination is validated through synthetic and field data tests.

Automatic Probe Localization in Multiaperture Ultrasound by Radon Domain Signal Matching.

In multiaperture ultrasound, several ultrasound probes with different insonification angles are combined to increase the field of view and angular coverage of image structures. A full reconstruction incorporating all possible combinations of transmitting and receiving probes has been shown to improve resolution, contrast, and angular coverage beyond what can be achieved by the registration of single images from different probes. A major challenge in multiaperture imaging is the correct determination of relative probe locations. A registration based on the content of images from different probes is challenging due to the decorrelation of image structures and speckle with increasing angle between the probes. We propose a probe localization method for plane-wave ultrasound that uses solely the receive dataset of a nontransmitting probe. The localization is performed by signal tracking in the Radon domain. To demonstrate that the method does not rely on common structures in the individual images, we show that a satisfying localization can be performed in pure speckle for angles, where the speckle patterns have completely decorrelated. The method shows potential for real-time probe localization in free-hand multiprobe ultrasound imaging or for flexible and wearable multiarray combination of multiple capacitive micromachined (CMUT)-based systems in the future.

Reprocessing 2-D airgun seismic reflection data SALTFLU (salt deformation and sub-salt fluid circulation in the Algero-Balearic abyssal plain) in the Balearic promontory and the Algerian basin

In an ever more challenging context for the acquisition of seismic data in the Mediterranean Sea, reprocessing to improve the quality of legacy data has become increasingly important. This work presents the newly reprocessed, open access dataset SALTFLU acquired in the Algerian basin by the National Institute of Oceanography and Applied Geophysics (OGS) in 2012. We apply a ‘broadband’ reprocessing strategy adapted for offset-limited (3 km streamer for a target 4 km below the sea level) airgun reflection seismic data acquired in deep water settings. We then assess if the reprocessed images provide new geological insights on the Mediterranean sub-surface. The workflow relies on an integrated approach combining geophysics and geological interpretation to iteratively build the velocity model. In this way we aim to tackle some of the challenges linked to imaging deep complex geological structures containing high velocity contrasts with 2-D, offset-limited seismic data. We first broaden the bandwidth of the data through multi-domain de-noising, deghosting and a source designature using an operator derived from the seabed reflection. We then perform iterative migration velocity analysis, pre-stack time migration and multiple attenuation in the Radon domain to obtain time-migrated images. The initial velocity model is derived from the resulting time migration velocities, and geologically driven model updates are generated using a combination of travel-time tomography, seismic interpretation of the major salt horizons and velocity gradient flooding. The gradient flooding aims to reproduce the large scale first-order velocity variations, while the travel-time tomography aims to resolve the smaller second-order velocity variations. The results improve our deep geological knowledge of the under-explored Algerian basin down to the base salt and the pre-salt. Fluid indicators are imaged within the Plio-Quaternary of the Algerian basin, which we interpret as thermogenic or biogenic gas sourced from either the Messinian Upper Unit or from the pre-salt, migrating through a hydro-fractured salt. The reprocessed data image lateral and vertical seismic facies variation within the Messinian units that could shed new light on the tectono-stratigraphic processes acting during the Messinian Salinity Crisis. It also reveals numerous previously unresolved volcanic structures within the Formentera basin.

InDuDoNet+: A deep unfolding dual domain network for metal artifact reduction in CT images.

During the computed tomography (CT) imaging process, metallic implants within patients often cause harmful artifacts, which adversely degrade the visual quality of reconstructed CT images and negatively affect the subsequent clinical diagnosis. For the metal artifact reduction (MAR) task, current deep learning based methods have achieved promising performance. However, most of them share two main common limitations: (1) the CT physical imaging geometry constraint is not comprehensively incorporated into deep network structures; (2) the entire framework has weak interpretability for the specific MAR task; hence, the role of each network module is difficult to be evaluated. To alleviate these issues, in the paper, we construct a novel deep unfolding dual domain network, termed InDuDoNet+, into which CT imaging process is finely embedded. Concretely, we derive a joint spatial and Radon domain reconstruction model and propose an optimization algorithm with only simple operators for solving it. By unfolding the iterative steps involved in the proposed algorithm into the corresponding network modules, we easily build the InDuDoNet+ with clear interpretability. Furthermore, we analyze the CT values among different tissues, and merge the prior observations into a prior network for our InDuDoNet+, which significantly improve its generalization performance. Comprehensive experiments on synthesized data and clinical data substantiate the superiority of the proposed methods as well as the superior generalization performance beyond the current state-of-the-art (SOTA) MAR methods. Code is available at https://github.com/hongwang01/InDuDoNet_plus.

Crosstalk attenuation for imaging of multiples based on angle gather residual moveout analysis

Imaging of multiples, as a supplement to imaging of primaries, can provide a wider range of subsurface illumination. Therefore, it can provide more detailed information on subsurface structures. However, imaging of multiples suffers from crosstalk issues generated by unrelated events. Many strategies have been proposed to attenuate crosstalk, among which the angle domain Radon crosstalk attenuation algorithm achieves good application effect. In the angle domain, the true imaging is flat, while the crosstalk events have moveouts. Therefore, it is convenient to identify the crosstalk in angle gathers using the Radon transform. However, the conventional Radon transform lacks a quantitative description for crosstalk in angle gathers, which would affect the accuracy of crosstalk attenuation. In this paper, residual moveout kernels are derived with a Radon transform to attenuate crosstalk in angle gathers for imaging of multiples. First, two types of residual moveout (RMO) equations are derived based on the causality of crosstalk. A three-layer model is used to verify the correctness of the analytical solutions. Then, based on the derived equations, the two types of crosstalk can be attenuated respectively in the Radon domain. Synthetic experiments demonstrate that the derived RMO equations can effectively attenuate the crosstalk events in imaging of multiples.

Sparse Parabolic Radon Transform with Nonconvex Mixed Regularization for Multiple Attenuation

The existence of multiple reflections brings difficulty to seismic data processing and interpretation in seismic reflection exploration. Parabolic Radon transform is widely used in multiple attenuation because it is easily implemented, highly robust and efficient. However, finite seismic acquisition aperture of seismic data causes energy diffusion in the Radon domain, which leads to multiple residuals. In this paper, we propose a sparse parabolic Radon transform with the nonconvex Lq1-Lq2(0<q1,q2<1) mixed regularization (SPRTLq1-Lq2) that constrains the sparsity of primary and multiple reflections to overcome the energy diffusion and improve the effect of multiple attenuation, respectively. This nonconvex mixed regularization problem is solved approximately by the alternating direction method of multipliers (ADMM) algorithm, and we give the convergence conditions of the ADMM algorithm. The proposed method is compared with least squares parabolic Radon transform (LSPRT) and sparse parabolic Radon transform based on L1 regularization (SPRTL1) for multiple attenuation in the synthetic data and field data. We demonstrate that it improves the sparsity and resolution of the Radon domain data, and better results are obtained.

Study on Ship Kelvin Wake Detection in Numerically Simulated SAR Images

To improve ship safety and increase ship concealment, we introduce a nonconvex regularization with a Cauchy-based penalty for discussing the influence of ship parameters and speckle noise in numerically simulated SAR images. First, the Kelvin wake geometry was modeled based on the classic theory of ship wave generation. Second, the scattering echo of the Kelvin wake was calculated using the two-scale method (TSM). Then, using the range-Doppler algorithm (RDA), the scattering echo data obtained by the TSM were processed to obtain the Kelvin wake in SAR images. Finally, the wake was reconstructed in the Radon domain using the Cauchy proximal splitting based on the forward–backward algorithm. The simulation results showed that Kelvin wakes were more easily detected in HH polarization with a large pitch angle and X-band, based on which the influence of ship parameters and speckle noise on the detection of ship wake in numerically simulated SAR images was discussed at different wind speeds. The research conclusions are of value to the development of ship wake stealth technology and the improvement of ship safety.

A Radon diffraction theorem for plane wave ultrasound imaging.

The rising demand on high frame rate ultrasound imaging applications necessitates the development of fast algorithms for plane wave image reconstruction. We introduce a new class of plane wave reconstructions that relies on a relation between receive data and image data in the Radon domain. This relation is derived for arbitrary dimensions and validated on multiple two-dimensional plane wave data sets. We further present a mathematical relation between conventional delay-and-sum and Fourier domain reconstruction methods and the method proposed. Our analysis shows that they all rely on the same physical model with slight variations in certain filtering steps and, therefore, the new Radon domain reconstruction yields similar results as other methods in terms of image quality. However, we show that our method offers a huge potential to improve computation time by reducing the number of applied projections and to improve image quality by introducing nonlinear operations in the Radon domain, e.g., for edge enhancement. As the Radon transform retains both angular and temporal information, the relation also provides new insights on the fundamentals of plane wave imaging that can be leveraged for optimizing acquisition schemes or for developing novel compounding strategies in the future.

Supervirtual Refraction Interferometry in the Radon Domain

Accurate picking of seismic first arrivals is very important for first arrival travel time tomography, but the first arrivals appearing at far offsets are often more difficult to pick accurately due to the low signal-to-noise ratio (SNR). The conventional supervirtual refraction interferometry (SVI) method can improve the SNR of first arrivals to a certain extent; however, it is not suitable for seismic data that interfered by strong noise. In order to better process the first arrivals at far offsets with serious noise interference, we propose a modified method, in which SVI implemented in the Radon domain (RDSVI) due to the cross-correlation in the Radon domain have a better effect. According to the kinematic characteristics of first arrival refractions, SVI is performed in the linear Radon domain. Both synthetic data and field data demonstrate the proposed method can enhance the effective signal and attenuate the strong noise simultaneously, so as to significantly improve the SNR of the first arrival data. Meanwhile, the RDSVI method is tested on the first arrival data with missing traces, which proves that this method can overcome the influence of abnormal traces and is suitable for the reconstruction of sparsely sampled seismic data.

Joint FWI of Active Source Data and Passive Virtual Source Data Reconstructed Using an Improved Multidimensional Deconvolution

Traditional full waveform inversion (FWI) highly depends on sufficient low-frequency data or a good initial model. Passive seismic data contain rich low-frequency components, and passive seismic FWI using virtual source data by seismic interferometry (SI) is a promising method. However, the distribution of passive sources in the subsurface is always inhomogeneous, which will lead to artifacts in the reconstruction results by SI using cross-correlation (CC). SI by multidimensional deconvolution (MDD) can counteract the source inhomogeneity but requires the separation of the reference wavefields, which is difficult to achieve for noise source data. To mitigate this problem, we propose an improved SI method by linear Radon transform based multidimensional deconvolution (LRTMDD). The interferometric point-spread function can be extracted more accurately and efficiently in the linear Radon domain, thus improving the reconstruction results. The passive virtual source full waveform inversion (PVSFWI) based on LRTMDD is further proposed, which can effectively use the low-frequency information in the virtual source data to invert the macroscopic velocity structures even in the case of inhomogeneous source distributions, and without the need to estimate the virtual source wavelets. A joint multi-source FWI strategy is proposed to solve the problem of missing low-frequency data suffered by active source FWI. Numerical experiments on the Marmousi model and the SEG/EAGE overthrust model show that the proposed methods can fully combine the respective advantages of multisource seismic data to stably achieve high-accuracy velocity models in the case of inhomogeneous passive source distributions and the lack of low-frequency data in active seismic data.

High-Dimensional Seismic Data Reconstruction Based on Linear Radon Transform–Constrained Tensor CANDECOM/PARAFAC Decomposition

Random noise and missing seismic traces are common in field seismic data, which seriously affects the subsequent seismic processing flow. The complete noise-free high-dimensional seismic dataset in the frequency–space (f-x) domain under the local linear assumption are regarded as a low-rank tensor, and each high dimensional seismic dataset containing only one linear event is a rank-1 tensor. The tensor CANDECOM/PARAFAC decomposition (CPD) method estimates complete noise-free seismic signals by characterizing high-dimensional seismic signals as the sum of several rank-1 tensors. In order to improve the stability and effect of the tensor CPD algorithm, this paper proposes a linear Radon transform–constrained tensor CPD method (RCPD) by using the sparsity of factor matrix in the Radon domain after high-dimensional seismic signal tensor CPD and uses alternating direction multiplier method (ADMM) to solve the established optimization problem. This proposed method is an essential realization of the high-dimensional linear Radon transform, and the results of synthetic and field data reconstruction prove the effectiveness of the proposed method.

Pseudo-3D Receiver Deghosting of Seismic Streamer Data Based on l1 Norm Sparse Inversion

The ghost effect in marine seismic data causes low-frequency suppression and frequency notch, resulting in incomplete frequency information for seismic records, which poses challenges for high-resolution imaging. The deghosting effect depends on the approximation of the ghost delay operator. Due to the strict requirements of dense sampling, the 2D deghosting method for a densely sampled inline dataset is still the mainstream method for marine data processing. As the trade-off, inversion-based methods are widely used in the industry to reduce the influence of the inaccurate ghost delay operator. In order to overcome the sampling limits and improve the 2D deghosting effect, we propose a pseudo-3D deghosting framework based on an l1 norm sparse inversion. In the framework, the inversion problem is divided into two subproblems, i.e., pseudo-3D operator building and optimization inversion. Considering the data coherence along the shot direction, we derive a pseudo-3D ghost delay operator to deghost simultaneously for multi-shot gathers. We then introduce a sparse inversion method in the pseudo-3D radon domain (multi-shot gathers) to further improve the inversion accuracy. The proposed framework is easy to implement, is not sensitive to noise, and shows superior performance in synthetic and field examples.

Sparse-View CT Reconstruction Based on a Hybrid Domain Model with Multi-Level Wavelet Transform.

The reconstruction of sparsely sampled projection data will generate obvious streaking artifacts, resulting in image quality degradation and affecting medical diagnosis results. Wavelet transform can effectively decompose directional components of image, so the artifact features and edge details with high directionality can be better detected in the wavelet domain. Therefore, a hybrid domain method based on wavelet transform is proposed in this paper for the sparse-view CT reconstruction. The reconstruction model combines wavelet, spatial, and radon domains to restore the projection consistency and enhance image details. In addition, the global distribution of artifacts requires the network to have a large receptive field, so that a multi-level wavelet transform network (MWCNN) is applied to the hybrid domain model. Wavelet transform is used in the encoding part of the network to reduce the size of feature maps instead of pooling operation and inverse wavelet transform is deployed in the decoding part to recover image details. The proposed method can achieve PSNR of 41.049 dB and SSIM of 0.958 with 120 projections of three angular intervals, and obtain the highest values in this paper. Through the results of numerical analysis and reconstructed images, it shows that the hybrid domain method is superior to the single-domain methods. At the same time, the multi-level wavelet transform model is more suitable for CT reconstruction than the single-level wavelet transform.