Backprojection Operator Research Articles

Reconstruction from CT truncated data based on dual-domain transformer coupled feature learning

To propose a CT truncated data reconstruction model (DDTrans) based on projection and image dualdomain Transformer coupled feature learning for reducing truncation artifacts and image structure distortion caused by insufficient field of view (FOV) in CT scanning. Transformer was adopted to build projection domain and image domain restoration models, and the long-range dependency modeling capability of the Transformer attention module was used to capture global structural features to restore the projection data information and enhance the reconstructed images. We constructed a differentiable Radon back-projection operator layer between the projection domain and image domain networks to enable end-to-end training of DDTrans. Projection consistency loss was introduced to constrain the image forwardprojection results to further improve the accuracy of image reconstruction. The experimental results with Mayo simulation data showed that for both partial truncation and interior scanning data, the proposed DDTrans method showed better performance than the comparison algorithms in removing truncation artifacts at the edges and restoring the external information of the FOV. The DDTrans method can effectively remove CT truncation artifacts to ensure accurate reconstruction of the data within the FOV and achieve approximate reconstruction of data outside the FOV.

ReconU-Net: a direct PET image reconstruction using U-Net architecture with back projection-induced skip connection

Objective. This study aims to introduce a novel back projection-induced U-Net-shaped architecture, called ReconU-Net, based on the original U-Net architecture for deep learning-based direct positron emission tomography (PET) image reconstruction. Additionally, our objective is to visualize the behavior of direct PET image reconstruction by comparing the proposed ReconU-Net architecture with the original U-Net architecture and existing DeepPET encoder–decoder architecture without skip connections. Approach. The proposed ReconU-Net architecture uniquely integrates the physical model of the back projection operation into the skip connection. This distinctive feature facilitates the effective transfer of intrinsic spatial information from the input sinogram to the reconstructed image via an embedded physical model. The proposed ReconU-Net was trained using Monte Carlo simulation data from the Brainweb phantom and tested on both simulated and real Hoffman brain phantom data. Main results. The proposed ReconU-Net method provided better reconstructed image in terms of the peak signal-to-noise ratio and contrast recovery coefficient than the original U-Net and DeepPET methods. Further analysis shows that the proposed ReconU-Net architecture has the ability to transfer features of multiple resolutions, especially non-abstract high-resolution information, through skip connections. Unlike the U-Net and DeepPET methods, the proposed ReconU-Net successfully reconstructed the real Hoffman brain phantom, despite limited training on simulated data. Significance. The proposed ReconU-Net can improve the fidelity of direct PET image reconstruction, even with small training datasets, by leveraging the synergistic relationship between data-driven modeling and the physics model of the imaging process.

TomoCAM: fast model-based iterative reconstruction via GPU acceleration and non-uniform fast Fourier transforms.

X-ray-based computed tomography is a well established technique for determining the three-dimensional structure of an object from its two-dimensional projections. In the past few decades, there have been significant advancements in the brightness and detector technology of tomography instruments at synchrotron sources. These advancements have led to the emergence of new observations and discoveries, with improved capabilities such as faster frame rates, larger fields of view, higher resolution and higher dimensionality. These advancements have enabled the material science community to expand the scope of tomographic measurements towards increasingly in situ and in operando measurements. In these new experiments, samples can be rapidly evolving, have complex geometries and restrictions on the field of view, limiting the number of projections that can be collected. In such cases, standard filtered back-projection often results in poor quality reconstructions. Iterative reconstruction algorithms, such as model-based iterative reconstructions (MBIR), have demonstrated considerable success in producing high-quality reconstructions under such restrictions, but typically require high-performance computing resources with hundreds of compute nodes to solve the problem in a reasonable time. Here, tomoCAM, is introduced, a new GPU-accelerated implementation of model-based iterative reconstruction that leverages non-uniform fast Fourier transforms to efficiently compute Radon and back-projection operators and asynchronous memory transfers to maximize the throughput to the GPU memory. The resulting code is significantly faster than traditional MBIR codes and delivers the reconstructive improvement offered by MBIR with affordable computing time and resources. tomoCAM has a Python front-end, allowing access from Jupyter-based frameworks, providing straightforward integration into existing workflows at synchrotron facilities.

A gradient-based approach to fast and accurate head motion compensation in cone-beam CT.

Cone-beam computed tomography (CBCT) systems, with their flexibility, present a promising avenue for direct point-of-care medical imaging, particularly in critical scenarios such as acute stroke assessment. However, the integration of CBCT into clinical workflows faces challenges, primarily linked to long scan duration resulting in patient motion during scanning and leading to image quality degradation in the reconstructed volumes. This paper introduces a novel approach to CBCT motion estimation using a gradient-based optimization algorithm, which leverages generalized derivatives of the backprojection operator for cone-beam CT geometries. Building on that, a fully differentiable target function is formulated which grades the quality of the current motion estimate in reconstruction space. We drastically accelerate motion estimation yielding a 19-fold speed-up compared to existing methods. Additionally, we investigate the architecture of networks used for quality metric regression and propose predicting voxel-wise quality maps, favoring autoencoder-like architectures over contracting ones. This modification improves gradient flow, leading to more accurate motion estimation. The presented method is evaluated through realistic experiments on head anatomy. It achieves a reduction in reprojection error from an initial average of 3 mm to 0.61 mm after motion compensation and consistently demonstrates superior performance compared to existing approaches. The analytic Jacobian for the backprojection operation, which is at the core of the proposed method, is made publicly available. In summary, this paper contributes to the advancement of CBCT integration into clinical workflows by proposing a robust motion estimation approach that enhances efficiency and accuracy, addressing critical challenges in time-sensitive scenarios.

Asymptotic linearized inversion for acoustic variable-tilt transversely isotropic media through elliptic perturbation analysis

The linearized inverse scattering problem for acoustic variable-tilt transversely isotropic (TTI) media is studied. In depicting the acoustic anisotropy behavior of the TTI medium with the P-wave normal moveout velocity (NMO) Vn, density ρ, and anisotropic parameters δ and η, a single-scattering procedure of NMO pressure wavefields is developed through elliptical perturbation analysis. By high-frequency approximation, the singly scattered pressure field is naturally related with an acoustic TTI generalized Radon transform (GRT), in which the TTI scattering patterns are explicitly expressed. We asymptotically invert for the multiple material parameters of an acoustic TTI medium via constructing the GRT backprojection operator. The effectiveness of the proposed approach is demonstrated on the layered model, and its superiority over the traditional acoustic isotropic and VTI GRT inversion methods is illustrated on the overthrust model.

Alternative fan-beam backprojection and adjoint operators

Abstract We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel–Neumann series in the frequency domain that can be implemented as an O ⁢ ( N 2.3729 ) {O(N^{2.3729})} matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O ⁢ ( N 3 ) {O(N^{3})} backprojection representations showing more robustness when dealing with highly noisy data.

An iterative reconstruction algorithm without system matrix for EPR imaging

Electron paramagnetic resonance (EPR) imaging is an advanced oxygen imaging modality for oxygen-image guided radiation. The iterative reconstruction algorithm is the research hot-point in image reconstruction for EPR imaging (EPRI) for this type of algorithm may incorporate image-prior information to construct advanced optimization model to achieve accurate reconstruction from sparse-view projections and/or noisy projections. However, the system matrix in the iterative algorithm needs complicated calculation and needs huge memory-space if it is stored in memory. In this work, we propose an iterative reconstruction algorithm without system matrix for EPRI to simplify the whole iterative reconstruction process. The function of the system matrix is to calculate the projections, whereas the function of the transpose of the system matrix is to perform backprojection. The existing projection and backprojection methods are all based on the configuration that the imaged-object remains stationary and the scanning device rotates. Here, we implement the projection and backprojection operations by fixing the scanning device and rotating the object. Thus, the core algorithm is only the commonly-used image-rotation algorithm, while the calculation and store of the system matrix are avoided. Based on the idea of image rotation, we design a specific iterative reconstruction algorithm for EPRI, total variation constrained data divergence minimization (TVcDM) algorithm without system matrix, and named it as image-rotation based TVcDM (R-TVcDM). Through a series of comparisons with the original TVcDM via real projection data, we find that the proposed algorithm may achieve similar reconstruction accuracy with the original one. But it avoids the complicated calculation and store of the system matrix. The insights gained in this work may be also applied to other imaging modalities, for example computed tomography and positron emission tomography.

Fast parallel implementation for total variation constrained algebraic reconstruction technique.

In computed tomography (CT), the total variation (TV) constrained algebraic reconstruction technique (ART) can obtain better reconstruction quality when the projection data are sparse and noisy. However, the ART-TV algorithm remains time-consuming since it requires large numbers of iterations, especially for the reconstruction of high-resolution images. In this work, we propose a fast algorithm to calculate the system matrix for line intersection model and apply this algorithm to perform the forward-projection and back-projection operations of the ART. Then, we utilize the parallel computing techniques of multithreading and graphics processing units (GPU) to accelerate the ART iteration and the TV minimization, respectively. Numerical experiments show that our proposed parallel implementation approach is very efficient and accurate. For the reconstruction of a 2048 × 2048 image from 180 projection views of 2048 detector bins, it takes about 2.2 seconds to perform one iteration of the ART-TV algorithm using our proposed approach on a ten-core platform. Experimental results demonstrate that our new approach achieves a speedup of 23 times over the conventional single-threaded CPU implementation that using the Siddon algorithm.

Inpainting of regular textures using ridge functions

Abstract In this paper, we propose a method for inpainting regular textures composed of oriented patterns and completion of images distorted by long stripes. We have developed a model fit to describe a regular texture as a sum of a few ridge functions. The ridge functions are known in tomography as a result of a back-projection operator applied to projections, or samples of the Radon transform. A computational scheme for the singular value decomposition of a regular texture into the sum of informative ridge functions is presented. The numerical experiments on various images demonstrate the efficiency of the techniques proposed.

End-to-End Deep Learning CT Image Reconstruction for Metal Artifact Reduction

Metal artifacts are common in CT-guided interventions due to the presence of metallic instruments. These artifacts often obscure clinically relevant structures, which can complicate the intervention. In this work, we present a deep learning CT reconstruction called iCTU-Net for the reduction of metal artifacts. The network emulates the filtering and back projection steps of the classical filtered back projection (FBP). A U-Net is used as post-processing to refine the back projected image. The reconstruction is trained end-to-end, i.e., the inputs of the iCTU-Net are sinograms and the outputs are reconstructed images. The network does not require a predefined back projection operator or the exact X-ray beam geometry. Supervised training is performed on simulated interventional data of the abdomen. For projection data exhibiting severe artifacts, the iCTU-Net achieved reconstructions with SSIM = 0.970±0.009 and PSNR = 40.7±1.6. The best reference method, an image based post-processing network, only achieved SSIM = 0.944±0.024 and PSNR = 39.8±1.9. Since the whole reconstruction process is learned, the network was able to fully utilize the raw data, which benefited from the removal of metal artifacts. The proposed method was the only studied method that could eliminate the metal streak artifacts.

Enabling Few-View 3D Tomographic Image Reconstruction by Geometry-Informed Deep Learning

<h3>Purpose/Objective(s)</h3> Deep learning affords enormous opportunities to augment the armamentarium of biomedical imaging, albeit its design and implementation have potential flaws. Fundamentally, most deep learning models are driven entirely by data without consideration of any physical priors, which dramatically increases the complexity of neural networks and limits the application scope and generalizability of the resultant models. Here we establish a geometry-informed deep learning framework for ultra-sparse tomographic image reconstruction. We introduce a novel mechanism for the integration of geometric priors of the imaging system. We demonstrate that the seamless inclusion of known geometric priors is essential to enhance the performance of volumetric computed tomography imaging with ultra-sparse sampling. <h3>Materials/Methods</h3> We propose a geometry-informed deep learning framework for 3D tomographic image reconstruction and illustrate the way that the known imaging geometry is integrated into the dual-domain deep learning. Specifically, the proposed framework consists of three modules: a) 2D projection generation network is developed to learn to generate novel-view projections from the given sparse views; b) geometric back-projection operator transforms the 2D projections to 3D images, referred to as geometry preserving images (GPI), which geometrically relates the pixelated 2D input data to the corresponding ray lines in 3D space; and c) 3D image refinement network learns to refine the GPI to generate the final 3D images. To evaluate feasibility of the proposed approach, we conduct experiments on a dataset containing 1018 lung CT images, where 80% and 20% data are used for training and testing. The projection images are digitally produced from CT images using geometry consistent with a clinical on-board cone-beam CT system for radiation therapy. <h3>Results</h3> We deploy the trained model on the held-out testing set for few-view 3D image reconstruction. The reconstructed results are compared with ground truth qualitatively and quantitatively. For single-/two-/three-view reconstruction, the average NRMSE / SSIM / PSNR values over all testing data are 0.368 / 0.734 / 20.770, 0.300 / 0.807 / 22.687 and 0.274 / 0.838 / 23.669, respectively. By visualizing the reconstructed CT images, we observe the proposed model can generate images closely to the targets although the anatomic structures of different patients have a large variance, indicating the potential of the proposed model for volumetric imaging even with few views. Moreover, experiments also show the proposed model can also be generalized to multi-view 3D image reconstruction and outperform deep models without geometry priors. <h3>Conclusion</h3> We present a novel geometry-integrated deep learning model for volumetric imaging with ultra-sparse sampling. The study opens new avenues for data-driven biomedical imaging and promises to provide substantially improved imaging tools for various clinical imaging and image-guided interventions.

Craniofacial Reconstruction via Face Elevation Map Estimation Based on the Deep Convolution Neutral Network

In this study, to achieve the possibility of predicting face by skull automatically, we propose a craniofacial reconstruction method based on the end-to-end deep convolutional neural network. Three-dimensional volume data are obtained from 1447 head CT scans of Chinese people of different ages. The facial and skull surface data are projected onto two-dimensional space to generate a two-dimensional elevation map, and then, use the deep convolution neural network to realize the prediction of skull to face shape in two-dimensional space. The encoder and decoder are composed of first feature extraction through the encoder and then as the input of the decoder to generate the craniofacial restoration image. In order to accurately describe the features of different scales, we adopt an U-shaped codec structure with cross-layer connections. Therefore, the output features are decomposed with the features of the corresponding scales in the encoding stage to achieve the integration of different scales while restoring the feature scales in the compression and decoding stage. Meanwhile, the U-net structures help to avoid the problem of loss of detail features in the downsampling process. We use supervised learning to obtain the prediction model from skull to facial elevation map. Back-projection operation is performed afterwards to generate facial surface data in 3D space. Experiments show that the proposed method in this study can effectively achieve craniofacial reconstruction, and for most part of the face, restoration error is controlled within 2 mm.

An empirical artifact correction for proton computed tomography.

To reduce image artifacts of proton computed tomography (pCT) from a preclinical scanner, for imaging of the relative stopping power (RSP) needed for particle therapy treatment planning using a simple empirical artifact correction method. We adapted and employed a correction method previously used for beam-hardening correction in x-ray CT which makes use of a single scan of a custom-built homogeneous phantom with known RSP. Exploiting the linearity of the filtered backprojection operation, a function was found which corrects water-equivalent path lengths (RSP line integrals) in experimental scans using a prototype pCT scanner. The correction function was applied to projection values of subsequent scans of a homogeneous water phantom, a sensitometric phantom with various inserts and an anthropomorphic head phantom. Data were acquired at two different incident proton energies to test the robustness of the method. Inaccuracies in the detection process caused an offset and known ring artifacts in the water phantom which were considerably reduced using the proposed method. The mean absolute percentage error (MAPE) of mean RSP values of all inserts of the sensitometric phantom and the water phantom was reduced from 0.87% to 0.44% and from 0.86% to 0.48% for the two incident energies respectively. In the head phantom a clear reduction of artifacts was observed. Image artifacts of experimental pCT scans with a prototype scanner could substantially be reduced both in homogeneous, heterogeneous and anthropomorphic phantoms. RSP accuracy was also improved.

Linearized scattering problem in acoustic transversely isotropic media with a vertical symmetry axis

The ray-based generalized Radon transform (GRT) inversion/migration strategy is studied in this paper for acoustic transversely isotropic media with a vertical symmetry axis (VTI media) represented by the P-wave normal moveout velocity, Thomsen parameter δ, anelliptic parameter η, and density. The multi-parameter GRT inversion solutions for multi-shot/multi-offset acquisition geometries are derived from the single-scattering approximation of the wavefields and an elliptically anisotropic background. Results from classical two-dimensional numerical tests show the true amplitude properties of the GRT backprojection operators.

Position coordinates-based iterative reconstruction for robotic CT

Robotic CTs can achieve customized trajectory scanning with x-ray tube and detector held by flexible robotic arms rather than fixed rails or gantry. However, the motion errors of the robotic arms cannot be neglected. Hence, the reconstruction method of Robotic CTs should be suitable for arbitrary trajectory and should take motion errors into full consideration. In this paper, we present an iterative reconstruction method for robotic CT systems. Unlike the analytical algorithms, such as FDK, this method makes no assumption about the scan trajectory. The projection and backprojection operations are modeled by 3D distance-driven algorithm using the coordinates of x-ray source and detector center fed back from the robotic arm’s positioning system directly. Both numerical simulations and practical experiments are conducted to verify the effectiveness of this method in arbitrary trajectory reconstruction and motion errors correction for robotic CT systems. For our non-circular and non-planar trajectory scan, this proposed method could easily handle the reconstruction and obtain a result comparable to reference. In addition, for 0.1% motion errors, using the proposed method could improve the reconstruction quality, and the RMSE could be reduced by 30%. This iterative reconstruction method is suitable for arbitrary trajectory scans and can decrease the degradation of image quality caused by motion errors of robotic arm.

A Dual-Domain CNN-Based Network for CT Reconstruction

Convolutional neural network (CNN)-based deep learning techniques have enjoyed many successful applications in the field of medical imaging. However, the complicated between-manifold projection from the projection domain to the spatial domain hinders the direct application of CNN techniques in computed tomography (CT) reconstruction. In this work, we proposed a novel CT reconstruction framework based on a CNN, i.e., an intelligent back-projection network (iBP-Net). The proposed iBP-Net method fused a pre-CNN, a back-projection layer, and a post-CNN into an end-to-end network. The pre-CNN adopted CNN techniques to model a filtering operation in the projection domain. In the back-projection layer, a back-projection operation was employed to perform between-manifold projection. Based on CNN techniques, the post-CNN worked together with the pre-CNN to recover reconstructed images from the outputs of the back-projection layer in the spatial domain while maintaining high visual sensitivity. The experimental results demonstrate the feasibility of the proposed iBP-Net framework for CT reconstruction.

Mixed Compressive Sensing Back-Projection for SAR Focusing on Geocoded Grid

This article presents a new scheme called 2-D mixed compressive sensing back-projection (CS-BP-2D), for synthetic aperture radar (SAR) imaging on a geocoded grid, in a single measurement vector frame. The back-projection linear operator is derived in matrix form and a patched-based approach is proposed for reducing the dimensions of the dictionary. Spatial compressibility of the radar image is exploited by constructing the sparsity basis using the back-projection focusing framework and fast solving the reconstruction problem through the orthogonal matching pursuit algorithm. An artifact reduction filter inspired by the synthetic point spread function is used in postprocessing. The results are validated for simulated and real-world SAR data. Sentinel-1 C-band raw data in both monostatic and space-borne transmitter/stationary receiver bistatic configurations are tested. We show that CS-BP-2D can focus both monostatic and bistatic SAR images, using fewer measurements than the classical approach, while preserving the amplitude, the phase, and the position of the targets. Furthermore, the SAR image quality is enhanced and also the storage burden is reduced by storing only the recovered complex-valued points and their corresponding locations.

Light Weight IBP Deep Residual Network for Image Super Resolution

Single -image super resolution (SR) is used to reconstruct a high-resolution image with more high-frequency details based on a low-resolution image as input. In recent years, image SR reconstruction based on deep learning methods has shown a considerably better performance than traditional methods. Early deep-learning-based methods deepen convolutional layers and directly reconstruct high-resolution images with complex neural networks. However, with the stacking of modules, network depth and model parameters increase, thereby raising computational resource; hence, it is difficult to apply on low-configuration devices. Furthermore, existing methods ignore the high-frequency details of the image, resulting in unsatisfactory performance. To solve these problems, a lightweight network model that applies the iterative back projection (IBP) mechanism to the network and reduces the dimensionality of the input image features is proposed. The proposed network model consists of three parts, namely, entrance module, main body module, and exit module. It designs a lightweight modular design to reduce the model calculation and control the network depth more easily by adjusting the number of ADB modules. The main part of the model consists of four lightweight accelerating deep residual back projection (ADB) modules. Each ADB module initially decreases the dimensionality of the input image features through a 1×1 convolutional layer to reduce the amount of calculation. Then, IBP is used to back project the image iteratively. Each ADB module only performs the downsampling back projection operation because the image features become larger after upsampling. Through three iterative downsampling back projection units, the high-frequency features of the image are fully explored, and the output image features are then compared with the shallow layer. Image feature fusion is used as input of the next ADB module, and the output part combines the high- and low-frequency image feature output by multiple ADB modules to complete the upsampling by using the PixelShuffle method to generate high-resolution images. Several experiments confirm that the proposed algorithm achieves better SR reconstruction accuracy with faster reconstruction speed than existing image SR methods.

Arbitrarily large tomography with iterative algorithms on multiple GPUs using the TIGRE toolbox

3D tomographic imaging requires the computation of solutions to very large inverse problems. In many applications, iterative algorithms provide superior results, however, memory limits in available computing hardware restrict the size of problems that can be solved. For this reason, iterative methods are not normally used to reconstruct typical data sets acquired with lab based CT systems. We thus use state of the art techniques such as dual buffering to develop an efficient strategy to compute the required operations for iterative reconstruction. This allows the iterative reconstruction of volumetric images of arbitrary size using any number of GPUs, each with arbitrarily small memory. Strategies for both the forward and backprojection operators are presented, along with two regularization approaches that are easily generalized to other projection types or regularizers. The proposed improvement also accelerates reconstruction of smaller images on single or multiple GPU systems, providing faster code for time-critical applications. The resulting algorithm has been added to the TIGRE toolbox, a repository for iterative reconstruction algorithms for general CT, but this memory-saving and problem-splitting strategy can be easily adapted for use with other GPU-based tomographic reconstruction code.

Numerically robust tetrahedron-based tomographic forward and backward projectors on parallel architectures

X-ray tomographic reconstruction typically uses voxel basis functions to represent volumetric images. Due to the structure in voxel basis representations, efficient ray-tracing methods exist allowing fast, GPU accelerated implementations. Tetrahedral mesh basis functions are a valuable alternative to voxel based image representations as they provide flexible, inhomogeneous partitions which can be used to provide reconstructions with reduced numbers of elements or with arbitrarily fine object surface representations. We thus present a robust parallelizable ray-tracing method for volumetric tetrahedral domains developed specifically for Computed Tomography image reconstruction. Tomographic image reconstruction requires algorithms that are robust to numerical errors in floating point arithmetic whilst typical data sizes encountered in tomography require the algorithm to be parallelisable in GPUs which leads to additional constraints on algorithm choices. Based on these considerations, this article presents numerical solutions to the design of efficient ray-tracing algorithms for the projection and backprojection operations. Initial reconstruction results using CAD data to define a triangulation of the domain demonstrate the advantages of our method and contrast tetrahedral mesh based reconstructions to voxel based methods.