Neural field algorithms for computed tomography image reconstruction

Abstract

Computed tomography image reconstruction uses neural field algorithms to reconstruct images for breast cancer screening and diagnosis. An Attenuation Field Network (AFN) uses x-ray liner attenuation coefficients that form the basis for CT images as a continuous function of imaging volume coordinates. The AFN uses a fully connected network trained on attenuation measurements, captured using a limited number of X-ray projections through an image volume, to learn a continuous function that maps each volumetric coordinate within the image volume to its respective attenuation coefficient. The AFN renders the entire attenuation field or synthesizes high-fidelity projections that are acquired or unacquired. The synthesized projections may be combined with existing projections to yield denser or complete projection data that may be used by other image reconstruction methods

Description

NEURAL FIELD ALGORITHMS FOR COMPUTED TOMOGRAPHY IMAGE RECONSTRUCTION CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional 63 / 522,658 filed June 22, 2023, which is hereby incorporated by reference as submitted in its entirety. FEDERAL FUNDING

[0002] This invention was made with government support under grant numbers CA241709 and CA199044 awarded by the National Institute of Health. The government has certain rights in the invention. BACKGROUND 1. Field of the Invention

[0003] The present invention generally relates to the field of image reconstruction, and more particularly to the field of computed tomography image reconstruction using neural field algorithms. 2. Background

[0004] Tomographic imaging is an imaging technique that provides cross-sectional ("tomographic") images (virtual "slices") of the object. In this approach, electromagnetic radiation is generated from a source, traverses through the object, and the transmitted electromagnetic radiation is captured ("measured") by a detector. The measurements are obtained at various orientations around the object, by translating or rotating the source around the object, and are referred to as "projections." One common implementation is Computed Tomography (CT), which uses an X-ray radiation source and radiation detectors to measure the attenuation by different tissues inside the body. The radiation source and radiation detectors are typically rotated around the subject to project X-rays from different angles and obtain attenuation measurements at each of those angles. The radiation detector comprises a multitude of X-ray sensitive detection elements ("pixels"). The attenuation measurements through each angle are then processed using tomographic reconstruction algorithms to estimate the attenuation coefficient of each sub volume within the subject, which are then used to produce cross-sectional ("tomographic ") images (virtual "slices") of the subject.

[0005] To reduce the radiation dose delivered during CT scans, there is a desire to reduce the number of X-ray projections that are captured. That is referred to as "sparse-view sampling." In analytical reconstruction, sparse-view sampling leads to streaking artifacts. Another approach to reduce the radiation dose delivered during CT scans is to limit the range of angles from which X- ray projections are captured (referred to as "limited-angular sampling"). Depending on the angular range, that is referred to as "short-scan acquisition " or "tomosynthesis." In analytical reconstruction, "short-scan acquisition" leads to inhomogeneous intensity and "tomosynthesis" leads to substantial artifacts.

[0006] Various prior art methods exist to overcome the drawbacks of capturing a limited number of X-ray projections. For example, fully supervised deep learning neural networks have been trained using previously captured projections and / or previously reconstructed images to recover or denoise projection data and / or denoise image data. However, those prior art methods have drawbacks. For example, training those fully supervised deep learning networks is a very memory intensive process, as they need to be trained using full sets of X- ray projections (and / or images reconstructed using full sets of X-ray projections) captured from a large and diverse set of subjects. Meanwhile, because of the variations from subject to subject, the functions inferred by networks trained using projections from other subjects (and / or images reconstructed using those projections) may not accurately recover or denoise the projection data or denoise the image data of different subject. SUMMARY

[0007] In the present disclosure, systems and methods for implementing neural field algorithms for computed tomography image reconstruction are provided. The disclosed technology includes systems and methods for memory-efficient deep learning algorithms referred to as an attenuation field network (AFN). The AFN uses a fully connected network trained on attenuation measurements, captured using a limited number of X-ray projections through an image volume, to learn a continuous function that maps each volumetric coordinate within the image volume to its respective attenuation coefficient. BRIEF DESCRIPTION OF THE FIGURES

[0008] For a detailed description of various examples, reference will now be made to the accompanying drawings.

[0009] FIG.1 shows an exemplary attenuation field network (AFN) process for mapping a vector of three-dimensional coordinates in accordance with one or more embodiments of the disclosed technology.

[0010] FIG.2 shows exemplary animal sinograms in accordance with one or more embodiments of the disclosed technology.

[0011] FIG.3 shows exemplary animal projection views in accordance with one or more embodiments of the disclosed technology.

[0012] FIG.4 shows exemplary axial views of animal reconstructions in accordance with one or more embodiments of the disclosed technology.

[0013] FIG.5 shows exemplary coronal views of animal reconstructions in accordance with one or more embodiments of the disclosed technology.

[0014] FIG.6 shows exemplary sagittal views of animal reconstructions in accordance with one or more embodiments of the disclosed technology.

[0015] FIG.7 shows an exemplary pipeline for cone beam breast CT data in partial scan using shifted-detector geometry (PSSD) in accordance with one or more embodiments of the disclosed technology.

[0016] FIG.8 shows an exemplary attenuation field network (AFN) training procedure in accordance with one or more embodiments of the disclosed technology.

[0017] FIG.9 shows an exemplary sinogram prediction using AFN in accordance with one or more embodiments of the disclosed technology.

[0018] FIG.10 shows exemplary reconstructions of a medium-size breast in accordance with one or more embodiments of the disclosed technology.

[0019] FIG.11 shows exemplary reconstructions of a large-size breast in accordance with one or more embodiments of the disclosed technology.

[0020] FIG.12 shows an exemplary process for segmentation of a 3D volumetric sample in accordance with one or more embodiments of the disclosed technology.

[0021] FIG.13 shows an exemplary scheme of generation of 8 ms and 4 ms X-ray pulse width projections in accordance with one or more embodiments of the disclosed technology.

[0022] FIG.14 shows an exemplary visualizations of FDK images in accordance with one or more embodiments of the disclosed technology.

[0023] FIG.15 shows an exemplary reconstruction pipeline for short-scan and offset- detector geometry in cone-beam breast CT in accordance with one or more embodiments of the disclosed technology.

[0024] FIG.16 shows exemplary AFN synthesized projection data in accordance with one or more embodiments of the disclosed technology.

[0025] FIG.17 shows exemplary breast CT image reconstructions in accordance with one or more embodiments of the disclosed technology.

[0026] FIG.18 shows exemplary additional image reconstructions in accordance with one or more embodiments of the disclosed technology.

[0027] FIG.19 shows image reconstruction of a median-size breast in accordance with one or more embodiments of the disclosed technology.

[0028] FIG.20 shows the FDK reference image in accordance with one or more embodiments of the disclosed technology.

[0029] FIGs.21A and 21B show an exemplary scheme for AFN synthesized projection data with centered or laterally-shifted detector in accordance with one or more embodiments of the disclosed technology.

[0030] FIG.22 shows an exemplary multifunction user device in accordance with one or more embodiments of the disclosed technology. DETAILED DESCRIPTION

[0031] The present disclosure relates to, inter alia, systems and methods for implementing neural field algorithms for computed tomography image reconstruction. The disclosed technology includes systems and methods for a memory-efficient deep learning algorithm referred to as an attenuation field network (AFN). The AFN uses a fully connected network trained on attenuation measurements, captured using a limited number of X-ray projections through an image volume, to learn a continuous function that maps each volumetric coordinate within the image volume to its respective attenuation coefficient.

[0032] In some embodiments of the disclosed technology, the AFN training follows similar principles of conventional ray tracing algorithms which makes the deep learning algorithm readily explainable. As a minimum training procedure of the AFN, an X-ray trajectory is formed from the X-ray source at s = (sx, sy, sz) to a detector pixel at d = (dx, dy,dz). Along the ray^^^^^^^^ , multiple points denoted asti= s + ai(d – s), 0 < ai< 1, are randomly sampled. Estimated attenuation coefficients of those sampled points are gathered (for example, according to Beer-Lambert’s law for X-ray imaging) to render a projection intensity at the detector pixel. The error between the projection intensity renderedby the AFN and the projection intensity captured by the detector pixel is minimized to optimize the weights of AFN. The AFN is trained by repeating that procedure (forming an X- ray trajectory, randomly sampling points along that trajectory, evaluating the attenuation coefficients of those points, aggregating those attenuation coefficients to estimate the projection intensity at the detector pixel, minimizing the error between the captured projection intensity and the rendered projection intensity, and optimizing the weights of AFN) at least once for each of the captured X-ray trajectories.

[0033] Once trained, the disclosed AFN can output the respective attenuation coefficient µ(r) for each continuous volume coordinate r = (x, y, z). The disclosed AFN can then be used to re-project the attenuation field through additional angles (where X-ray projections were not acquired) and estimate additional attenuation measurements through those additional angles. Both the attenuation measurements captured by the X-ray detector and additional attenuation measurements estimated by the AFN can then be used to produce tomographic images of the subject, minimizing the various drawbacks of capturing fewer X-ray projections described above. In some embodiments, for instance, the AFN may provide both the attenuation measurements captured by the X-ray detector and the additional attenuation measurements estimated by the AFN to a reconstruction algorithm. Additionally, or alternatively, the AFN may construct tomographic images of the object by outputting the estimated attenuation coefficients at spatial coordinates within the image volume (e.g., a three-dimensional array or one or more two-dimensional arrays) calculated according to the continuous function.

[0034] Using the process described above, the disclosed AFN leverages the redundancy inherent in even a limited number of attenuation measurements to estimate additional attenuation measurements through additional angles. Accordingly, the disclosed AFN reduces the number of X-ray projections that need to be captured and, by extension, the radiation dose that needs to be delivered to provide the analytical reconstruction algorithm with a dataset that is large enough to produce tomographic images with minimal artifacts. For example, if X-ray projections are captured from a limited range of angles, the disclosed AFN can estimate the attenuation measurements that would have been captured from angles encircling the entire image volume. In another example, if X-ray projections are captured with a low sampling rate (e.g., from angles encircling the entire image volume but with large angles between each projection), the disclosed AFN can estimate the attenuation measurements that would have been captured with a higher sampling rate. In yet another example, if, at a source position (i.e., projection angle), the detector is not sufficiently large enough to capture X-rayprojections of the entire object (i.e., only a portion of the object volume is imaged), the disclosed AFN can estimate the attenuation measurements that would have been captured by additional X-ray projections through additional portions of the image volume.

[0035] Unlike fully supervised neural networks known in the prior art, the disclosed AFN may be trained to learn the continuous function for each image volume using only the attenuations through the image volume. Therefore, using the disclosed AFN, large datasets of reference data do not need to be stored or even acquired. Meanwhile, the disclosed attentuation field is impervious to variations from subject to subject because the continuous function for each image volume is learned using attenuations that are specific to that image volume.

[0036] In some embodiments of the disclosed technology, an AFN is provided that includes a cone beam CT application. In some embodiments, the disclosed AFN may be applied to any imaging application including optical imaging, ultrasound, or the like. Additionally, the disclosed AFN may have a transmission geometry where an image volume is between one or more sources and one or more detectors.

[0037] In some embodiments of the disclosed technology, an upright breast computed tomography (CT) using short scan acquisition is provided. Prior art systems are limited to scanning one breast at a time, which necessitates repositioning of the patient for contrast enhancement studies. The disclosed technology provides a breast CT system designed to image two breasts simultaneously. In some embodiments, a self-supervised learning algorithm, attenuation field network (AFN), can inpaint a sinogram acquired by an incomplete offset detector in a short-scan. For bilateral imaging, a detector with a larger field of view (FOV) (e.g., 38 cm X 30 cm) may be laterally shifted for a larger FOV. Additionally, Monte Carlo simulations of bilateral breast CT data from 1 participant generated projections using a 78 kV 0.7mm CPU X-ray spectrum. The angular scanning range, angular sampling rate, and detector offset were empirically tuned. In some embodiments, an FDK algorithm with appropriate weighting schemes (FDK-W) may be used to reconstruct the sinograms.

[0038] In some embodiments of the disclosed technology, simultaneous reduction of radiation dose and X-ray scatter in cone-beam breast CT (CBBCT) using deep learning image reconstruction is provided. To translate dedicated CBBCT to a breast cancer screening modality, it is necessary to reduce both radiation and scatter contamination. The disclosed method may be used to simultaneously reduce radiation dose and scatter effects. In some embodiments, a flat planet panel detector which is nearly half size of the FOV may be used.The detector may have a narrower X-ray beam to reduce radiation dose to the patients and scatter radiation. Images may be reconstructed by the FDK algorithm with an appropriate weighting scheme and a self-supervised deep learning algorithm, attenuation field network (AFN). Radiation dose and scatter to primary ratio may be estimated via Monte Carlo simulations using breast-specific models with various detector sizes. 1. Attenuation Field Network (AFN) for Cone Beam CT Reconstruction

[0039] In some embodiments of the disclosed technology, the radiation dose in cone beam computed tomography, or CBCT, may be reduced with less acquisition of X-ray projections. That reduction in projection views may be translated into sparse view or limited angle sampling. In analytical reconstructions, sparse-view sampling leads to streaking artifacts, and limited angle sampling leads to inhomogeneous intensity. Iterative reconstruction methods leverage sparsity or prior image can suppress those aliasing artifacts yet suffer from intensive computation and data-dependent hyperparameter tuning.

[0040] Deep-learning based methods have gained increased popularity in CT for their promising reconstruction performance. Supervised neural networks can be generally divided into three categories: image domain, sensor domain, and hybrid domain networks. Image domain networks are typically trained using a pair of an aliased image and a reference image and are used for post-denoising. Sensor domain networks aim to recover or denoising projection data prior to reconstruction, but the performance is constrained by the fact that any small error in projection data can be amplified as severe artifacts in image after reconstruction. Networks operated in hybrid sensor and image domains are typically learned in an unrolled manner where each roll interleaves the network forward propagation and the forward model of imaging physics. The inclusion of data consistency improves reconstruction robustness at the cost of an increased memory footprint. Common characteristics of these supervised learning-based approaches are the need for large diverse and labeled training data and the poor generalization ability to unseen data. Moreover, the backbone of these networks is convolution. For tackling high resolution 3-dimensional problems in CBCT, 3D convolution requires hundreds of gigabits of GPU memory and thus limits the feasibility of training a full 3D network. To resolve the memory bottleneck, efforts have been made to decompose the high dimensional problem into subproblems lying in a lower dimension, for instance, converting cone beam data to parallel beam data or learning projection to image transformation on ray beamlets. However, the problem decomposition trades reconstructionperformance for memory efficiency as those methods often require an additional post processing network for image refinement.

[0041] In some embodiments of the disclosed technology, a memory-efficient deep learning algorithm, referred to as an attenuation field network (AFN) is provided. AFN adapts the emerging paradigm in computer vision, which represents a scene as a continuous function of coordinates using a fully connected, or non-convolutional network. In CBCT, AFN maps each physical coordinate of the image volume to its respective attenuation coefficient. Once the continuous mapping is built, AFN renders the attenuation coefficients of the entire image volume using the respective volumetric coordinates. In some embodiments, the disclosed AFN may be used in cone beam micro-CT (mCT) applications.

[0042] As shown in FIG.1, AFN may use a fully connected network whose input is a single continuous volume coordinate r = (x, y, z) and whose output is the respective attenuation coefficient µ(r). The training procedure of AFN coincides with the conventional ray tracing algorithm. A ray trajectory is formed from the X-ray source at s = (sx, sy, sz) to adetector pixel at d = (dx, dy, dz). Along the Ray^^^^^^^^ multiple points denoted as ti = s + ai(d –s), 0 < ai < 1 are randomly sampled. The estimated attenuation coefficients of those sampled points may be gathered according to Beer-Lambert's law to render the projection intensity at the ray end. Since the training process is fully differentiable, the model can be optimized (e.g., network forwarding and projection rendering). Error can be minimized between the acquired projection and the rendered projection. That error serves as the training loss and may be written aswhereis the distance between two adjacent points, β is the view angle, and (u, v) denote the detector column and row indices, respectively. After the network iterates through all the acquired rays, it can query the entire attenuation field by simply inputting the image volume coordinates. Additionally, it may also reproject the attenuation field to obtain projections that are on acquired. That “inpainted” projection from AFN can be further reconstructed by other methods of choice, including, for example, analytic reconstruction methods.

[0043] As shown in FIG.1, the AFN may serve as a continuous function that maps a vector of three-dimensional coordinates 102, through a fully connected network 104 to their respective attenuation coefficient 106. The training procedure of AFN coincides with the conventional ray tracing algorithm where points along a ray are sampled from the X-ray source to a detector pixel. According to Beer-Lambert's law, the attenuations and distances of those points are gathered to estimate the projection intensity at the detector. That volume rendering may be fully differentiable 108 to allow for optimization of attenuation representations by minimizing the error between the estimated and the acquired projection.

[0044] AFN may be evaluated using Monte Carlo generated projections (64 views in 180°) of a numerical mouse phantom. Example parameters of a simulated circular CBCT geometry may be 0.25 mm detector pitch, 600x600 detector size, 150 mm source-to-origin distance, and 250 mm source-to-detector distance. A reference data sampled 128 views in 360° was also simulated and reconstructed using the Feldkamp-Davis-Kress (FDK) algorithm with a matrix size of 512x512x900 (0.1 mm voxel). The network may consist of four fully connected layers with a feature dimension of 64. At the input layer, a hash grid encoding maybe used to enhance the learning of high frequency features. AFN may be trained on the half-scan data using an ADAM optimizer with a learning rate of 1E-3 for 200 epochs. In each iteration, five view angles and 1024 rays per angle may be randomly sampled, yielding a batch size of 5,120 rays. For each ray, 724 points may be uniformly sampled with a local perturbation to roughly match the voxel pitch of 0.1 space millimeters. The training may take about 15 minutes and the rendering of a full scan projection data having 128 views may take about 10 minutes on an NVIDIA RTX A6000 graphic card.

[0045] FIG.2 shows mouse sinograms at detector row 250. Using the acquired 64-view projections in 180°, view 202, AFN may synthesize additional 64-view projections covering the other 180° and yield minor errors compared to the full-scan reference. View 204 shows 128 views in 360° generated by the AFN. View 206 shows 128 views in 360°. View 208 shows an absolute percentage error (from 0% to 10%) between 204 and 206. FIG.3 shows mouse projection views 302, 304, and 306 at 11.25° (acquired) and views 308, 310, and 312 at 191.25° (unacquired). In the AFN generated projections, in view 302 there is no visual difference from the simulated projections 304 as shown in their absolute percentage error map 306. At the unacquired angle of 191.25 degrees, the AFN generates high quality projection in 308 that is visually similar to simulated projection 310 where their absolute percentage error map is shown in 312. AFN renders nearly identical projections at 11.25°,which illustrates the potent representation ability of coordinate-based mapping. AFN may also synthesize unacquired projection views that are visually similar to the reference projection. FIG.4, FIG.5, and FIG.6 show phantom mouse reconstructions in axial, coronal and sagittal views, respectively.

[0046] FIG.4 illustrates the axial view of mouse reconstructions. The simulated projections, 64 views in 180°, were constructed using FDK with Parker weights as shown in view 402, and in view 404 using AFN. In view 406, additional 64 view projections are shown in the other 180° and synthesized using AFN. The synthesized projections along with the original 64-view projections in 180° were spliced to yield 128 views in 360°, which were reconstructed using FDK. In view 408, projections with 128 views in 360° are shown that were generated using Monte Carlo and reconstructed using FDK as the reference. Views 410, 412, and 414 show the absolute difference images of views 402, 404, and 406, respectively.

[0047] FIG.5 shows the coronal view of mouse reconstructions. The simulated projections, 64 views in 180°, were constructed using FDK with Parker weights in view 502 and AFN in view 504. In view 506, additional 64 view projections in the other 180° that were synthesized using AFN are shown. The synthesized projections along with the original 64 projections in 180° were spliced to yield 128 views and 360°, which were constructed reconstructed using FDK. In view 508, projections with 128 views in 360° were generated using Monte Carlo and reconstructed using FDK as the reference. Views 510, 512, and 514 show absolute difference images of view 502, 504, and 506, respectively.

[0048] FIG.6 shows the sagittal view of mouse reconstructions. The simulated projections, 64 views in 180°, were constructed using FDK with Parker weights in view 602 and AFN in view 604. In view 606, additional 64 view projections in the other 180° that were synthesized using AFN are shown. The synthesized projections along with the original 64 projections in 180° were spliced to yield 128 views and 360°, which were constructed reconstructed using FDK. In view 608, projections with 128 views in 360° were generated using Monte Carlo and reconstructed using FDK as the reference. Views 610, 612, and 614 show absolute difference images of 602, 604, and 606, respectively.

[0049] The FDK reconstruction of half-scan data shows shading artifacts on the right although Parker weights were applied. The use of Parker weights has a limited impact in this case since it requires at least 180° plus the fan angle. In contrast, AFN can directly render a blurred yet homogeneous reconstruction using half-scanned data. Moreover, a spliced full- scan projection may be constructed using the original 64 view projections in the first 180°and the AFN-synthesized 64-view projections in the second 180°. The splice projection data may be reconstructed using FDK, yielding reduced error as shown in the absolute difference images.

[0050] In some embodiments of the disclosed technology, AFN may trim down the network training to each geometric ray, which allows a shallow fully connected network to build the mapping from volumetric coordinates to attenuation fields. That addresses the bottleneck of graphic processing memory caused by typical convolutional layers and / or the inclusion of forward physics models used in common deep learning methods. Additionally, AFN is self-supervised, requiring no training data set or data labeling. Numerical cone beam mouse data is shown where AFN can directly render attenuation fields as well as provide synthesized projections that are unacquired. 2. An Attenuation Field Network for Cone-Beam Breast CT with a Laterally-Shifted Detector in Short Scan

[0051] In some embodiments of the disclosed technology, in cone-beam computed tomography (CBCT), asymmetrically mounted detectors can enlarge the imaging field of view compared to centered detectors. That geometry can also reduce radiation dose and photon scatter but acquires truncated projections. The truncated projections may be compensated by weighting functions applied in conventional analytic reconstruction methods. Asymmetric or laterally shifted detector geometry is feasible in a prone breast CBCT system. To enable the transition of a dedicated breast CT system from prone to upright patient positioning, a shifted detector geometry may be used to acquire partial scan (e.g.270°) data. A self-supervised attenuation field network, AFN, may be used to reconstruct incomplete data. The AFN may learn a continuous mapping from the physical coordinates of imaging volume to its respective attenuation coefficients during training and can render attenuation images or projection data by querying volumetric coordinates during inference. The synthesized projections may be further used in any reconstruction methods. In testing, the disclosed methods were evaluated using 50 clinical breast CBCT datasets. AFN, when combined with analytical reconstruction methods, obtains visually similar reconstructions and yields comparable noise variance, contrast, and spatial resolution compared to reference reconstruction using full-scan data. Additionally, AFN may potentially enable a partial scan shifted detector geometry for dedicated breast CBCT imaging.

[0052] In CBCT systems, the field of view (FOV) is limited by the X-ray detector size. A laterally-shifted detector or offset detector is known to extend the FOV without reducing thesystem magnification. Meanwhile, this extended FOV geometry can also reduce photon scatter and radiation dose. In some embodiments of the disclosed technology, a shifted detector geometry and dedicated breast CBCT may be used. In some cases, partial scan data, or short scan data, may be acquired in conjunction with shifted detector geometry, also referred to as PSSD, to realize a clinical transition from a prone scanner to an upright scanner for dedicated breast CT imaging.

[0053] In some embodiments, the truncated projection data may be acquired using a shifted detector that may be compensated using weighting functions along the fan angle direction. However, these truncation weightings may be based on the data redundancy of fan beam data and thus require full scan data. It has shown that direct use of those weighting functions failed to reconstruct the PSSD data in developed the disclosed attenuation AFN instead to approach the goal. The disclosed AFN adopts the emerging neural field paradigm in computer vision where a scene is represented as a continuous function of coordinates. In the context of CT, neural fields were learned either in the projection domain or in the image domain.

[0054] FIG.7 shows an example pipeline of the disclosed process. For example, in a circular cone beam CT scan 702, a laterally-shifted detector acquires truncated projection data in a partial scan. In 704, the incomplete data is fed to an AFN for self-supervised training. In 706, the trained AFN can synthesize both acquired and non-acquired projection data where the acquired data is shown in 704 and the additional data shown in 706 is the non- acquired data. In 708, the synthesized projections along with the acquired projections are used for post reconstructions using analytical methods, iterative methods, or deep learning methods.

[0055] FIG.8 illustrates an example training procedure of the AFN, a fully connected network whose input is a single continuous three-dimensional coordinate r = (x, y, z) and whose output is the respective attenuation coefficient µ(r). A minimum training sample of AFN is a ray propagating from the X-ray source s = (sx, sy, sz) to a detector pixel d = (dx, dy,dz). Along the ray^^^^^^^^ , multiple 3D coordinates denoted as ti = s + ai(d – s), 0 < ai < 1 may berandomly sampled. The attenuation coefficients of those sampled coordinates may be queried and then integrated, according to Beer-Lambert's law, to render the projection intensity at the ray end. Since the ray propagation is fully differentiable, the model may be optimized by minimizing the error between the acquired projection in the render projection. That error serves as the training loss and is written as ||∑ih(ti)|ti+1 u, v)|| where h(·) denotes thefully connected network, p denotes the acquired projection data after taking the log, β is the view angle, and (u, v) specify a detector pixel. After the network iterates through all the acquired rays, it can query the entire attenuation field by simply inputting the volumetric coordinates of the image. Additionally, or alternatively, it may also reproject the attenuation field to obtain projections that are unrequired.

[0056] As shown in FIG.8, the AFN serves as a continuous function that maps a vector of three-dimensional coordinates 802, through a fully connected network 804, to their respective attenuation coefficient 806. The training procedure of AFN coincides with the conventional ray tracing algorithm where points along a ray may be sampled from the X-ray source to a detector pixel. According to the Beer-Lambert's law, the attenuations of those points may be integrated to estimate the projection intensity at the detector. That volume rendering is fully differentiable 808 to allow for optimization of attenuation representations by minimizing the error between the estimated and the acquired projection.

[0057] In some embodiments, the network may consist of three full connected layers with the feature dimension of 64. Prior to the input layer, a hash grid encoding may be employed to accelerate training as well as to enhance the learning of high frequency features. At the output layer, exponential function may be used with its gradient clipped as the activation to enforce the non-negativeness of attenuations. ReLU activations may be used for other layers. For each breast PSSD data, the AFN may be trained using ADAM optimizer with a learning rate of 1E-3 for 250 epochs. The learning rate decays by 1 / 3 for every 50 epochs. In each iteration, 2048 rays may be randomly selected from an acquired view angle and 512 points per ray may be sampled to roughly match the reconstruction voxel pitch. The effective batch size is 2048 X 512 = 1024 Ki, which amounts to approximately 20 GB GPU memory usage. This training may take about two hours, and the rendering of 300 view full scan data takes about 20 minutes on an NVIDIA RTX A6000 graphic card.

[0058] For training purposes, 50 CT data sets of BIRAD 4 or 5 women acquired by prone, clinical prototype scanner, maybe used. Among the 50 datasets, 27 contain calcifications. The chest wall diameters of the breast range from 8.5 cm to 18.5 cm, with an average of 13.0 cm. The scanner used a center detector with 40 cm x 30 cm FOV in 2 x 2 binning mode, yielding a resolution 0.388 mm x 0.388 mm with its matrix size 1024 x 768. To retrospectively obtain PSSD data, 225 views were selected out of 300 views corresponding to 270° ranging from -135° to 135° where each projection view is truncated 256 out of 1024 on the left this in total results in an under-sampling rate of 3 / 4 by 3 / 4which equals 9 / 16. The full scanned data may be reconstructed using the FDK algorithm (0.273 mm voxel) as reference. Three construction methods were evaluated: 1) AFN was trained using the PSSD data and used to predict missing data such that a full scan full size detector (FSFD) data can be formed along with the acquired PSSD data. This spliced FSFD data was reconstructed using FDK with a shifted detector weight. This additional weighting is applied to suppress the truncation artifacts due to slight inconsistency between AFN synthesized projection and the acquired projection.2) The PSSD data was reconstructed using FDK with a modified Parker weight to only account for the partial scan. The unacquired 1 / 4 data in the fan angle dimension was zero filled.3) The PSSD data was reconstructed using FDK with a shifted detector weight to only account for truncated projections. The reconstructed image was scaled by the ratio of the number of views in the full scan to the number of views in the partial scan. Quantitative metrics including noise variance estimated in the adipose region, signal difference to noise ratio (SDNR) between adipose and fibroglandular tissues, and the full width at half maximum (FWHM) of calcifications along the medial lateral direction and the superior-inferior direction were used.

[0059] FIG.9 shows a breast sinogram upper towards the chest wall. Using PSSD data, the AFN can generate a high fidelity sinogram compared to the full scan reference as shown in 908, other than the large errors outside the sinogram, AFN only makes small errors within the sinogram and marginal errors where the data is acquired (see top right and bottom right regions of view 908). In view 902, a PSSD sinogram is shown. While AFN may be trained on the PSSD sinogram, it may generate a high fidelity full sinogram. In view 904, an AFN sinogram is shown. In view 906, a reference sinogram is shown. In view 908, a 10X absolute error between the AFN sinogram and reference sinogram is shown using the same dynamic range as views 902, 904, and 906. As shown in view 908, the AFN yields minor errors in the top right and bottom right regions where the data is acquired and yields larger errors elsewhere.

[0060] FIG.10 shows image reconstructions of a medium-size breast having a chest wall diameter of 14.5 centimeters in coronal, sagittal, and axial views. The image 1002 reconstructed using FDK with Parker weight exhibits truncation artifacts due to shifted detector, whereas the image 1004 obtained using FDK with shifted detector weight manifests the missing wedge artifacts in the coronal plane due to partial scan and inhomogeneous attenuations across all three planes. In contrast the post FDK reconstruction using AFN compensated projection data 1006 is free of artifacts and yields similar visual quality as theFDK reference 1008. The PSSD data may be used in three competing methods, including FDK with Parker weight, FDK with shifted detector weight, and the disclosed AFN method 1006. FSFD data was reconstructed using FDK to obtain image 1008. FDK with weighting functions can either account for partial scan or shift-detector geometry, leading to truncation artifacts in image 1002 or missing wedge artifacts in image 1004. As expected in contrast, AFN in 1006 predicts the missing projections in the fan angle dimension and the view angle dimension to yield FSFD data for post reconstructions. The disclosed method delivers artifact free and visually similar images as the reference. The display window is [0.15, 0.35] cm-1.

[0061] FIG.11 shows image reconstructions of a medium-size breast having a chest wall diameter of 18.3 centimeters in coronal, sagittal, and axial views. Similar observations may be made except at the breast skin structures near the chest wall due to the larger diameter are not fully reconstructed in 1106 as can be seen in the coronal and axial images. Note that those regions correspond to the area where the projection data is most scarce. Table 1 compares the disclosed AFN and reference FDK methods. Noise variance and SDNR metrics suggest AFN yields comparable noise variance and image contrast as the reference. Two FWHM metrics indicate AFN maintains a similar spatial resolution as the reference. Referring to Figure 11, PSSD data was used in the three competing methods, including FDK with Parker weight 1102, FDK with shifted detector weight 1104, and be disclosed AFN method 1106. FSFD data was constructed using FDK to obtain the reference 1108. FDK with weighting functions can either account for partial scan or shifted detector geometry, leading to truncation artifacts in 1102, or missing artifacts in 1104, as expected. In contrast, AFN predicts the missing projections in the fan angle dimension and the view angle dimension to yield FSFD data for post reconstructions in one 106. The disclosed methods generate visually similar image is as the reference 1108 except that some breast skin structures are not fully reconstructed, which is best seen near the chest wall of the axial view. The display window is [0.15, 0.35] cm-1. Table 1 illustrates a quantitative comparison between the disclosed AFN using PSSD data and the FDK reference using full scan data. In Table 1, 50 breast cases were evaluated, among which 27 cases contain calcifications. The noise variance is estimated in the adipose region. Signal difference to noise ratio, or SDNR, is calculated between the adipose and fibroglandular tissues. The full width at the half maximum (FWHM) of the calcification was computed along two orthogonal directions (ML: mediolateral; SI: super-inferior). AFN yields comparable quantities as the reference.Table 1:

[0062] As described, a neural field-based method for cone-beam breast CT reconstruction with simultaneously partial scan and shifted detector data is provided. The disclosed AFN reduces the learning to each propagating ray such that implicit representations of the attenuations may be learned with high efficacy and memory efficiency. It is shown that the learned AFN may generate high fidelity projections in areas where data is either acquired or unacquired. Nevertheless, it is also observed that the post FDK reconstructions using AFN synthesized projections require further refinement for large-sized breasts. It is believed that this may be achieved by either using additional constraints or regularizations during network training or using advanced post reconstruction methods such as iterative algorithms or deep learning techniques. 3. Radiation Dose Reduction Using Half X-Ray Pulse Width with Unsupervised Deep- Learning Technique for Dedicated Breast CT

[0063] In some embodiments of the disclosed technology, dose reduction strategies using short scan or sparse view acquisitions along with the compressed sensing and fully supervised deep learning algorithms for the dedicated cone-beam breast CT (CBBCT) systems is described. In some embodiments, half X-ray pulse width protocol to reduce radiation dose may be used. Instead of using the compressed sensing (CS) and supervised deep learning methods to reduce noise, a self-deep learning technique to impede the half X- ray pulse width sinograms to the full X-ray pulse width sinograms followed by image reconstruction processes may be used. A self-supervised attenuation field network (AFN) learns continuous mapping from imaging system coordinates to the attenuation coefficients. The disclosed AFN technique was evaluated using 20 CBBCT sample results indicating that the AFN technique may provide image quality similar to the full pulse width protocol regarding image noise and calcification visibility while using only half radiation dose. The application of AFN may potentially be extended to other imaging modalities.

[0064] Dedicated CBBCT provides fully 3D reconstructed tomographic images as a special application of CBCT. The fully 3D images without compressing the breast, providingvisualization in any orientation, can eliminate tissue overlap in digital mammography (DM) and digital breast tomosynthesis (DBT) systems. CBBCT systems use prone-position tables and flat panel detectors with pulsed X-ray sources to acquire 300 views or 500 views in a full scan trajectory (360°). The reconstruction algorithm, FDK, is the default option for image reconstruction in clinical settings owing to its fast reconstruction speed. The report has shown that CBBCT outperforms mammography in sensitivity. However, the radiation dose to the breast in the diagnosis is about twice of a standard to view mammogram.

[0065] Several radiation dose reduction strategies have been developed in past decades. The disclosed CBBCT system uses a prone position table and acquires 300 projections distributed uniformly in a full scan. The CBBCT system is an upright geometry using a short scan aimed to be a screening tool with compatible dose levels with standard two-view screening mammograms has also been developed. Both systems have the need for further dose reduction. It is demonstrated that the short scan, sparse view acquisitions along with the compressed sensing or fully supervised deep learning network methods reduce the radiation dose while improving the image quality for CBBCT in terms of noise, resolution, and visualization of the calcifications. However, implementing sparse view acquisition is not trivial because it simultaneously requires synchronizing the gantry, detector, and source. Thus, shorter X-ray pulse widths may be alternatively used to reduce the dose for its easy implementation on scanners. The default protocol of the disclosed CBBCT system uses an 8 ms pulse width in every projection. Sub-projections may be measured during the first 8 ms followed by the frame average process to produce the final single projection at that specific angle. Shorter pulse width may be considered to reduce the radiation dose. The minimum pulse width may be 4 ms in the disclosed system due to hardware limitations of the scanner. In some embodiments, a 4 ms pulse width protocol may be used instead of an 8 ms pulse width. Like other dose reduction methods, the image quality degrades when the number of photons decreases. To improve the image quality, the disclosed methods include a sinogram inpainting technique. That inpainting technique may use a self-supervised deep learning algorithm, AFN, which may synthesize the missing 4 ms sub-projections and splice them with the measured ones to become a hybrid 8 ms sinogram. For simplification, the sinogram generated by 8 ms from the CBBCT system may be called the default protocol, and the sinogram spliced by the first 4 ms from the CBBCT system and the latter 4 ms from the disclosed AFN technique may be referred to as the hybrid half pulse width protocol. When clinical samples were scanned by the disclosed CBBCT system, only default protocol wasoperated. To illustrate the feasibility of using half pulse width, Monte Carlo (MC) simulations may be used to generate both 8 ms and 4 ms sinograms using clinical data as input. Without losing any generality, FDK may be used for both protocols in the provided examples to prove the concept.

[0066] In some embodiments, AFN, a self-supervised deep learning algorithm, may be used to reduce the radiation dose for CBBCT systems. The algorithm may be used for half X- ray pulse width (4 ms) in a full scan (300 views) to reduce the radiation dose. By giving the coordinate and geometry of the imaging system, the self-trained AFN may serve as a continuous attenuation coefficient field and generate another 4ms / 300 view sinogram. The actual for 4ms / 300 views sinogram acquired by the imaging system may be spliced with the sinogram synthesized by the AFN to become a hybrid 8 ms / 300 views sinogram. The disclosed AFN technique may inpaint the sinogram generated by a shorter pulse width. The hybrid half pulse width protocol outperforms the PSNR and NMSE than the default protocol. The disclosed protocol provides compatible image quality in terms of variance, bias, size and FWHMR of the calcification. The disclosed AFN technique is self-supervised and not dependent on other training samples. The AFN technique may also be used to inpaint incomplete projections from different imaging systems using limited angle acquisitions or truncated detectors. The disclosed AFN algorithm is not limited to a CBBCT systems, and it may be potentially applied to other imaging modalities.

[0067] In some embodiments, this CBBCT system may use a prone position table to acquire 300 projections of angularly uniformed sampling in a full scan. Breasts may be imaged by a tungsten anode 0.3 mm nominal focal spot X-ray source using 49 KV, 1.4 m of AI HVL with 8 ms pulse width. The native detector pixel size may be 0.194 mm, and the system may be operated in a 2 X 2 binning mode, resulting in 0.388 mm pixels. The duration for each projection may be 33 ms (8 ms X-ray pulse width and 25 ms readout), resulting in about 10 seconds for a scan. For each projection view, during the 8 ms pulse, multiple sub- projections may be generated in the frame average to be a final projection for that specific angle. The distance from the source to the access of rotation (AOR) may be 650 mm, and the distance between the source and the detector may be 898 mm.

[0068] Circular cone beam CT, the area detector measures the X-ray flux as the source and the detector rotates about the object. The object may be denoted asDenote focus of the X-ray source as s, and a detector pixel as d. The projection at the detector pixel D may be expressed aswhere t is the variable of integration, I(d) denotes the number of transmitted photons, and I0 denotes the constant incident photon count. This equation only considers the primary signal and assumes a monochromatic X-ray source.

[0069] As shown in FIG.8, the disclosed AFN may adopt a multilayer perceptron (MLP) architecture, whose input is the physical coordinates r of the object in the imaging field and whose output is the respect to linear attenuation coefficient µ(r). The training procedure of AFN strictly follows the acquisition of projections a ray path may be formed from the X-raysource s to the detector pixel d, multiple points ti are sequentially sampled along the ray^^^^^^^^ ,which may be written asThe attenuation coefficients of the sample points may be evaluated by the forward pass of AFN and aggregated to estimate the projection at the right end, for example: where hΘ(·) denotes anfield is fully differentiable, which allows for optimization of the attenuation representation by minimizing the error between the estimated and the acquired projection. Ross function may be defined for training the AFN as followswhere D digits the set of coordinates of all the acquired projection data. Once AFN is trained, it can query the three-dimensional image reconstruction by simply inputting the volumetric coordinates of the imaging field. It may also reproject the render attenuation fields to obtain the projections that are acquired or unrequired.

[0070] Full 360-degree coverage is uniformly spanned by N major angles, and each major angle is uniformly spanned by M sub angles. Suppose βi,j denotes the jth sub angle in the ith major angle, 1 <= i <= N, 1 <= j <= M. βi,j may be calculated as

[0071] Since each 1 pulse width data may be generated as as I(βi,1:4). The pulse width maymay be computed aswhere I(·) Here the acquired tosynthesize

[0072] In some embodiments, the AFN may be implemented as a non-convolutional network composed of four fully connected layers with the feature dimension size of 32. Hash grid encoding may be used for the input fully connected layer to enable the shallow network to learn high frequency features. At the end of the last full connected layer, the exponential function may be used to enforce the non-negative constraint of linear attenuation coefficients. The gradient of the exponential function may be clipped within [-15, 15] to prevent the gradient from vanishing or exploding. ReLU activations may be used for other layers.

[0073] For each short pulse width data, the AFN may be trained using an ADAM optimizer with initial learning rate of 1E-3 for 200 epochs. The learning rate decayed by 1 / 3 every 33 epochs. The three-dimensional projection data may be split into multiple two- dimensional projection views. In each iteration, 2048 detector pixels from a projection view may be randomly sampled. Consequently, the network may sparsely iterate through all the acquired projection views. That data splitting strategy helped the fast convergence within the first few epochs. On each ray formed from the X-ray source to a detector pixel, coordinates may be selected using the stratified sampling approach. The entire ray may first be truncated within the imaging field of view and then partitioned into 512 evenly spaced bins, withinwhich one sample may be uniformly drawn from. That many numbers of samples may be selected to match the reconstruction voxel pitch roughly. Since the input to AFN is the coordinates, the effective batch size amounts to 2048 X 512 = 1024 Ki, or roughly a million. The large batch size greatly accelerates the training yet at an affordable graphic processing unit memory consumption of about 15 gigabytes. The disclosed AFN and FDK algorithm may be implemented in pytorch. The training of an AFN may take about 100 minutes, and the rendering of 1200 projection views takes about 30 minutes on an NVIDIA RTX A6000 graphic card.

[0074] The image quality metrics used here were peak signal-to-noise ratio (PSNR), normalized mean square error (NMSE), bias, the number of voxels of the simulated calcification, and full width half maximum ratio (FWHMR) of the simulated calcification along mediolateral (ML) and super-inferior (SI) directions compared to the ground truth. The PSNR, NMSE, and bias were all computed only in the area of the breast parenchyma (region of interest (ROI)). The air area was not included. The PSNR is computed as where lmaxis thethe subscript “gt” stands for the ground truth.

[0075] The

[0076] The bias may be estimated by the mean absolute error compared with the ground truth. The FWHMR in the reconstructed image may be estimated only inshown in FIG. 12. The number of voxels of the calcification in the digital phantom was known (216 voxels). In the reconstructed image, it can be estimated by counting all the voxels of value higher than a threshold in a 32 x 32 x 32 voxels volume of interest (VOI) center at the known location from the reference. The order of attenuation coefficient of all material inside the breast from large to small is calcification, skin, fibroglandular, and adipose tissue. Thus, the threshold was defined as the maximum value of all fibroglandular tissues excluding the VOI. Again,the locations of all fibro glandular tissues may be determined by the segmented part as shown in FIG.12.

[0077] Referring to FIG.12, segmentation is shown where a 3D volumetric sample was segmented into air, skin, adipose, and fibroglandular tissue followed by assigning the associated attenuation coefficients. Segmented parts assigned with the attenuation coefficients or recombined to be a breast phantom, the input of the Monte Carlo (MC) simulations.

[0078] Referring to FIG.13, a scheme of generation of eight millisecond and four millisecond X-ray pulse width projections and a single view is shown. Both circles indicate positions of the X-ray source. Straight rays are both essential rays of the cone beam on the fan angle plane. The section (a) pulse width was generated by MC simulations, eight rays with .036° sampling may be exposed to the digital phantom to produce 8 sub-projections followed by the frame average process to become a single projection at that specific angle. In section (b), the first 4 rays are from section (a) and another four rays from AFN with 0.036° angular sampling were spliced and followed the same procedure to produce a hybrid 8 ms frame averaged projection at the same angle.

[0079] Referring to FIG.14, visualization of FDK images is shown. The arrows indicate locations of the calcification. In the left column, a centigram generated by MC simulations using default protocol (8 milliseconds / 300 views) followed by FDK image reconstruction is shown. In the right column, a sinogram generated by MC simulations using half pulse width protocol (4 milliseconds / 300 views) spliced with a sinogram generated by AFN technique (4 milliseconds / 300 views) is shown. From top to bottom the views shown are coronal, tangential, and sagittal. The display window is [0.2, 0.3] cm-1.

[0080] Monte Carlo (MC) simulated sinograms for the default protocol and the disclosed hybrid half pulse width protocol of each sample maybe reconstructed by the FDK algorithm. FIG.14 shows the FDK images and calcifications, and the disclosed protocol shows very similar visualization as the default protocol. The references here were the ground truth digital breast phantom segmented from clinical samples. The peak signal-to-noise ratio (PSNR) in units of dB of the default protocol and hybrid half pulse split protocol were 29.53 (28.40 - 30.42) and 29.63 (28.47-30.14), respectively. The normalized mean square error (NMSE) for the two protocols was 17.19 (16.1-17.81) and 17.36 (16.17-17.58) dB. The variance (x 10-6mm-2) was 5.0850 + / - 1.4154 for the default and 5.1588 + / - 1.353 for the hybrid half pulse width protocol. The bias (x 10-3mm-1) was 2.772 (2.515 - 3.209) and 2.852 (2.631 - 3.218)for the default and the proposed hybrid half pulse width protocol, respectively. For the estimate of the size of the calcification in the reconstructed images, the number of voxels increased by 52.43% for the default protocol and 55.25% for the disclosed protocol, respectively. The full width half maximum ratio (FWHMR) in the medial lateral direction was 1.0694 + / - 0.1437 and 1.0833 + / - 0.1734 for the default and the hybrid half pulse width protocol, respectively. Along the super-inferior direction, FWHMR was 1.0694 + / - 0.1265 for the default protocol and 1.0833 + / - 0.1273 for the hybrid half pulse width protocol. The image quality is summarized in Table 2. Table 2: default protocol hybrid half pulse width protocol PSNR (dB) 29.53 ( 28.40-30.42) 29.63 (28.24-30.14) NMSE (dB) 17.19 (16.10-17.81) 17.36 (16.17-17.58) Va-6 -1Bi Inc vo FW FW [00 geo trymay employ high resolution, low noise detectors that extends the imaging field of view (FOV) and may provide a higher spatial resolution which could improve microcalcification conspicuity. For full scan acquisition with offset detector geometry, data redundancy from complementary views may be exploited during the image reconstruction period envisioning an upright breast CT system. The disclosed method acquires short scan data in conjunction with offset detector geometry to tackle the resulting incomplete data, as self-supervised attenuation field network (AFN) may be used. AFN leverages the inherent redundancy of cone beam CT data through coordinate-based representation and known imaging physics. A trained AFN may query attenuation coefficients using their respective coordinates or synthesized projection data including the missing projections. The AFN may be evaluated using clinical cone-beam breast CT datasets, for example where n = 50. While conventional analytical and iterative reconstruction methods fail to reconstruct the incomplete data, AFN reconstruction is not statistically different from the reference reconstruction obtained using full scan, full detector data in terms of image noise, image contrast, and the full width at halfmaximum of calcifications. The disclosed method indicates the feasibility of a simultaneous short scan and offset detector geometry for dedicated breast CT imaging. The disclosed AFN technique may be potentially expanded to other cone beam CT applications.

[0082] Cone-beam computed tomography (CT) dedicated to x-ray imaging of the breast is an emerging tool for breast cancer screening and diagnosis. Dedicated breast CT (bCT) provides a true tomographic or three-dimensional visualization of the uncompressed breast and thus eliminates the tissue superposition in existing breast imaging modalities including mammography and digital breast tomosynthesis. However, compared to these “gold- standard” imaging tools, bCT suffers from lower in-plane resolution. When the mean glandular dose (MGD) is made comparable to that of standard two-view mammography, the increased image noise and the lower spatial resolution reduce the conspicuity of microcalcifications.

[0083] FIG.15 illustrates a reconstruction pipeline (d-f) for a short scan and offset detector geometry of (a-b) in cone-beam breast CT. In part (a) half cone-beam geometry and cone angle views are shown. Patients are imaged in a prone position in this scenario. In part (b), the fan angle view is shown where the short scan covers from -α to +α (α < π), which is symmetric about the head feet direction. Additionally, data may be acquired using an offset detector in each view. In part (c), geometry results in acquisitions of incomplete sinogram is shown. In part (d), incomplete sinogram data may be used to train the disclosed AFN and may be reused with the AFN synthesized sinogram to form the spliced full sinogram in part (e). After that, reconstruction methods including analytical ones to obtain a three-dimensional image reconstruction are shown in part (f).

[0084] In some embodiments, cone-beam bCT systems may use complementary metal- oxide-semiconductor (CMOS) detectors with a smaller pixel pitch (75-150 µm) and exhibit an order of magnitude lower electronic noise than the amorphous silicon flat-panel detectors used prior. The largest CMOS detector has an active area of 30 x 302, which is 10 cm narrower in the fan-angle direction than the 40 x 302 amorphous-silicon flat-panel detector. To accommodate the same imaging FOV, system magnification may be reduced by positioning the CMOS detector closer to the axis of rotation (AOR), resulting in a smaller air gap between the breast and the detector, which can increase x-ray scatter and compromise quantitative accuracy. Alternatively, the laterally-shifted (i.e., offset) detector geometry, shown in FIG.15, parts (a) and (b), may be used to maintain the FOV without modifying the system magnification. In full-scan, the truncated projection data acquired using an offsetdetector may be compensated using weighting functions owing to the inherent data redundancy of fan-beam data. In some embodiments, an offset detector geometry in full-scan cone-beam bCT using the FDK algorithm and a compressed sensing-based iterative reconstruction algorithm may be provided.

[0085] While offset-geometry CMOS detectors may improve spatial resolution and preserve the imaging FOV as well as an optimized air gap, prone-positioning requires a breast CT system with a larger footprint that is difficult to accommodate in typical mammography rooms. Additionally, ensuring proper positioning to image the posterior aspects of the breast is time-consuming. In contrast, upright positioning with mammography or tomosynthesis has significantly better coverage of the axilla and is easier to position. Thus, an upright bCT that acquires multiple projection views via a short scan, namely, less than 360 degrees, is disclosed. Like the truncated projection data acquired with offset detectors, short- scan data may be compensated using weighting functions, such as Parker weights or the like.

[0086] In some embodiments, data may be acquired that is incomplete both on the detector plane and on the trajectory, as shown in FIG.15, part (b). Note that the compensation weights may only attend to data being incomplete on one dimension. The use of weighting functions used to compensate for either the truncated projection or the short- scan projection fails to reconstruct the incomplete data. As set forth herein, an attenuation field network (AFN) is used to assist with image reconstruction. In some cases, AFN may adopt the emerging neural field paradigm in computer vision, where a scene is represented as a continuous function of coordinates using a multi-layer perceptron (MLP). In the context of CT, neural fields can represent quantities either in the projection domain or in the image domain. In some embodiments, a memory-efficient training method is used. Additionally, or alternatively, the training method may include splicing of the acquired projections and network synthesized projections, as depicted in FIG.15, parts (c) and (d). Spliced projections ma be complete and may be used for posterior reconstruction methods of user’s choice, as shown in FIG.15, parts (e) and (f). Additionally, or alternatively, hash grid encoding may be included in the network to enhance the learning of high frequency features.

[0087] Referring to FIG.16, AFN synthesized projection data is provided for evaluation in the forms of sinogram images. As shown in part (a), data may be missing in two dimensions, angle dimension and detector dimension. In part (b), acquired projection views are shown demonstrating that the data may be incomplete in the detector dimension. In part (c), an unacquired projection view in between 135° to 225° is depicted. Projection at 180° isshown here (see FIG.15) demonstrating that the data may also be incomplete in the view angle dimension. AFN may fully inpaint a sinogram from the incomplete sinogram and only yields high uncertainties outside the anatomical region. In the acquired projection view b, data may be partially truncated on the left. AFN outputs visually similar projections as the reference. No data is acquired at 180°. In addition, that projection view yields the highest uncertainty since data may be most scarce here. Nevertheless, AFN can generate a high- fidelity projection view except yielding large errors outside the anatomical region.

[0088] In some embodiments, a trained AFN may query the respective attenuation coefficient given a volumetric coordinate. The projection data at any detector pixel may be obtained as a line integral along the ray path from the x-ray source to the detector pixel. The AFN may synthesize projections for an arbitrary imaging geometry including the underlying geometry of the data itself. FIG.16 depicts AFN synthesized three-dimensional projections from different view aspects. In a sinogram, shown in FIG 16, part (a), the short-scan and offset-detector data are incomplete in both the view angle and detector width dimensions. The AFN may inpaint the sinogram with high fidelity and may only yield large errors outside of the sinogram. In part (b) of FIG.16, an acquired projection view is depicted where AFN accurately synthesizes 75% of the acquired projections and recovers the missing 25% projections. Note that, in the error image, no transition artifacts are visible at the truncation except for higher errors towards the chest wall and air on the left. FIG.16, part (c) shows that AFN synthesizes a projection view at 180 degrees, where no data is acquired. Note that this unacquired projection view is located at the center of the unacquired region, as shown in FIG. 15, part (b), which implies that the data is most scarce here. Therefore, AFN is expected to yield the highest uncertainties in this projection view. When the synthesis errors (near chest wall in FIG.16, parts (b) and (c)) or uncertainties (air area in FIG.16) are located outside of FOV, the corresponding image reconstruction is free of artifacts within FOV.

[0089] Referring to FIG.17, breast CT image reconstructions are shown. In this example, the reconstructions have an 11.8 centimeter chest wall diameter and the reconstruction is provided with AFN and or FDK using incomplete data in columns 1 through 4 and compared to the FDK reference using full scan and full detector data in column 5. The trained AFN may either directly output the image volume, column 1, or synthesize projection data for subsequent reconstructions, columns 2 through 4. When the AFN is used to synthesize data for FDK reconstruction, column 2, it produces images visually similar to the AFN only reconstruction, showing a loss of resolution. When the acquired projection data is reused andspliced with AFN synthesized data, column 3, a subsequent FDK reconstruction recovers the lost resolution yet exhibits truncation-like artifacts, see arrows, due to the slight inconsistency between AFN synthesized projections and the acquired projections. Offset detector weight may be incorporated into the FDK algorithm, denoted as FDK-M. This weighted FDK reconstruction uses AFN inpainted data, column four, and effectively eliminates the line artifacts. The display window is [0.15, 0.35] cm-1.

[0090] FIG.17 compares the utilization of AFN in the image domain or the projection domain for a representative breast image reconstruction using incomplete data. Column 1 of FIG.17 shows the three-dimensional AFN attenuation coefficient map, which was directly queried using a canonical volumetric coordinate grid. Column 2 of FIG.17 shows the posterior FDK reconstruction using AFN synthesized data. Both reconstructions illustrate a loss of resolution as well as residual streaks and are of similar visual quality, which in turn validates the efficacy of AFN in learning attenuation representations. In column 3 of FIG.17, the acquired incomplete projection data is reused and spliced with AFN synthesized projections to form complete data, namely a process of AFN inpainting. A subsequent FDK reconstruction greatly improved the image resolution yet creates truncation-like artifacts in all three planes as indicated by the arrows. These artifacts may be due to the non-smooth transition between AFN synthesized projections and the acquired projections since no such artifacts were observed in the images of columns 1 and 2. An offset-detector weighting function may be used with FDK reconstruction, as denoted as FDK-M. As shown in column 4 of FIG.17, the inclusion of offset-detector weights eliminates the truncation artifacts in all three planes and further enhances fine structures in the coronal plane (see arrows), compared to the FDK reference reconstruction from complete data in column 5 of FIG.17.

[0091] Referring to FIG.18, image reconstruction of (a) median-size breast (14.5 cm chest wall diameter) and (b) a large-size breast (18.3 cm chest wall diameter) is shown. Incomplete projection data may be used in the five competing methods including FDK with Parker weight, FDK with offset detector weight, FRIST (Fast, Regularized, Iterative Statistical Technique) initialized using FDK with Parker weight, FRIST initialized using FDK-M, and FDK-M with AFN inpainted data. Full projection data may be reconstructed using FDK to obtain the reference as shown in column six. In both cases, FDK with weighting functions can either account for the short span or the offset detector geometry, leading to residual artifacts and or inhomogeneous attenuation coefficients as expected. The two FRIST methods further exacerbate the artifacts and / or inhomogeneities despite denoisingthe breast images. Lastly, FDK-M with AFN inpainting delivers visually similar images as the reference. The display window is [0.15, 0.35] cm-1.

[0092] FIG.18, part (a), shows the image reconstructions of a medium-sized breast (14.5 cm chest wall diameter). The images obtained using FDK with (modified) Parker weight exhibit truncation artifacts due to the offset-detector geometry. The images reconstructed using FDK with the offset-detector weight (e.g., FDK-M), manifest inhomogeneous intensities locally and globally due to the short scan. In column 2 of FIG.18, the arrow indicates the artifacts appearing as elongated structures, which are most profound in the coronal plane. FDK with both weights applied resolves neither the artifacts nor inhomogeneities since either of the two weight is wrongly modulated by the other reduced amount of data. In contrast, the FDK-M reconstructions using AFN inpainted projection data are free of the truncation artifacts or the inhomogeneous attenuations and appear less noisy than the two weighted FDK reconstructions, owing to the addition of AFN synthesized projection data.

[0093] FIG.18, part (b), shows the image reconstructions of a large-size breast (18.3 cm chest wall diameter). The compressed sensing method, Fast, total variation-Regularized, Iterative, Statistical Technique (FRIST), suppresses the image noise in the central glandular tissue region. However, FRIST, using either of the two weighted FDK reconstructions as an initialization, further exacerbates the artifacts or inhomogeneities appearing in the weighted FDK reconstructions (columns 1 and 2) as a result of severe data inconsistency and inefficacy of total variation regularization on image artifacts. In this large-size breast, one side of the breast skin near the chest wall is not fully reconstructed using the disclosed method, as indicated by the arrows in the coronal and axial images. The side of the breast in the axial view is the most under-scanned for this simulated short-scan and offset-detector geometry, and the reconstruction of this region is more vulnerable to artifacts.

[0094] FIG.19 shows image reconstructions of a median-size breast (14.5 cm chest wall diameter). Incomplete projection data was reconstructed using FDK with Parker weight, FDK with offset-detector weight (denoted as FDKM), and two fully supervised learning (FSL) methods, whose network inputs were either of the two weighted FDK reconstructions. The incomplete data was inpainted by AFN and further reconstructed by FDK-M. Complete projection data was reconstructed using FDK to obtain the reference (last column). Using the images reconstructed by FDK w / Parker weight (column 1), FSL (column 2) alleviates the truncation artifacts indicated by the arrows in column 1 yet creates additional artifacts asindicated by the arrows in column 2. Similarly, using the images reconstructed by FDK w / offset-detector weight (column 3), FSL (column 4) addresses the non-homogeneity indicated by the arrows in column 3 yet generates severe artifacts as indicated by the arrows in column 4. In contrast, reconstructions using AFN inpainted data (column 5) are visually similar to the reference in column 6. The display window is [0.15, 0.35] cm−1.

[0095] FIG.19 shows the image reconstructions of a medium-size breast (14.5 cm chest wall diameter). Using the images reconstructed by FDK w / Parker weight (column 1), the fully supervised learning (FSL) method (column 2) alleviates the truncation artifacts indicated by the red arrows in column 1 yet creates additional artifacts as indicated by the yellow arrows in column 2. Similarly, using the images reconstructed by FDK w / offset- detector weight (column 3), FSL (column 4) addresses the non-homogeneities indicated by the arrows in column 3 yet generates severe artifacts as indicated by the yellow arrows in column 4. Notably, the “calcification-like” artifact in the sagittal plane can be detrimental to breast cancer diagnosis. In contrast, reconstructions using AFN inpainted data (column 5) are visually similar to the FDK reference (column 6).

[0096] Table 3 compares the proposed FDK-M reconstruction with AFN inpainting and the reference FDK method using four metrics including the noise variance, the signal difference to noise ratio (SDNR), and the full width at half maximum (FWHM) of calcifications along the mediolateral (ML) direction and the superior-inferior (SI) direction. All four metrics except the SDNR metric (P = 0.194 , Shapiro–Wilk’s test) did not satisfy the normality assumption. There was no significant difference between our proposed method and the reference FDK method for the SDNR metric (P = 0.886 , paired t-test). We performed the non-parametric Wilcoxon-signed rank test for the other three metrics and failed to reject the null hypothesis that the median difference between the two methods is zero due to the p- values being above the significance level of 0.05. All four metrics suggest that our proposed method using incomplete data yields comparable image noise, image contrast, and spatial resolution of calcifications as the FDK reference using complete data. Table 3

[0097] As shown in FIG.20, part (a) the FDK reference in three planes of the same large- size breast shown in FIG.18 (b–d) the AFN assisted image reconstructions at different undersampling rates in the coronal (b), sagittal (c), and axial (d) planes, respectively. The undersampling rates in percentage are reported in (c) for each combination of detector offset (vertical) and angular coverage (horizontal). AFN reconstructions with a 7.5 cm detector offset (37.5% truncation of a 40 cm detector) yield prominent artifacts in the coronal and axial planes even for the 270° angular coverage. AFN reconstructions with a 5 cm detector offset (25% detector truncation) yield not fully reconstructed breast skin structures near the chest wall, which are less pronounced as the view angle increases. Lastly, AFN reconstructions with a 2.5 cm detector offset (12.5% detector truncation) yield no visible artifacts even for the minimum view angle 204°. The display window is [0.15, 0.35] cm−1.

[0098] FIG.20 illustrates the FDK reference in three planes of the same large-size breast shown in Fig.18. FIG.20, parts (b–d) show the AFN-assisted image reconstructions at different undersampling rates in the coronal (b), sagittal (c), and axial (d) planes, respectively. The undersampling rates in percentage are reported in Fig.7c for each combination of detector offset (vertical) and angular coverage (horizontal). In each panel of (b)–(d), the image on the top-right corner is associated with the highest rate (65.6%) whereas the image on the bottom-left corner is associated with the lowest rate (35.4%). AFN reconstructions with a 7.5 cm detector offset (37.5% truncation of a 40 cm detector) yield prominent artifacts in the coronal and axial planes even for the 270° angular coverage. AFN reconstructions with a 5 cm detector offset yield not fully reconstruction breast skin structures near the chest wall, which are less pronounced as the view angle increases. Lastly, AFN reconstructions with a 2.5 cm detector offset (requiring a 35 cm wide detector) yield no visible artifacts even for the minimum view angle 204°. This can be promising since the combination of 25 cm offset and 204° angular coverage has a lower sampling rate than the current combination of 5 cm offset and 270° angular coverage and will allow for more operation room for the envisioned upright CT system. This data sufficiency analysis indicates that the current cone-beam geometry using AFN-assisted reconstruction is mostly constrained by the detector width.

[0099] In some embodiments, a self-supervised image reconstruction technique, AFN, for cone-beam bCT data using a simultaneous short-scan and offset-detector geometry is disclosed. The resulting incomplete data posed challenges for FDK and compressed sensing methods. Conventional analytical reconstruction methods exploit the projection dataredundancy in the view angle and detector width dimensions (as represented in Eq.3), which requires at least one of the two dimensions to be complete. In contrast, AFN can exploit the inherent redundancy of projection data through coordinate-based representation (aligned with the physical imaging coordinates), known imaging physics (the Beer-Lambert law based training loss), and high-resolution representation ability (from the hash grid encoding). A trained AFN serves as a continuous attenuation coefficient field and thus can be used to generate novel projection data by integrating all the attenuation coefficients along a ray path. AFN can inpaint the short-scan and offset-detector projection data to yield complete data, which can be further reconstructed using the FDK algorithm or its variants with weighting functions. The resulting reconstruction yielded comparable image quality as the reference FDK in terms of image noise, image contrast, and calcification resolution. It is worth noting that the synthesis of AFN is fundamentally different from that of generative adversarial networks (GANs), whose predictions may suffer from hallucinations especially for unseen data. While GANs can synthesize arbitrary projection images from random noise, AFN intends to infer the underlying attenuation coefficients of a breast by leveraging the inherent correlation or redundancy of the acquired projections of that breast. AFN is self-supervised and is independently trained for each breast case, i.e., without the need for a training dataset or data labeling. This also eliminates the generalization problem of supervised networks. For instance, a fully-supervised network trained using sparse view breast CT data showed reduced performance on calcifications due to being a minority class in the training data. In contrast, unimpaired calcification resolution in the AFN-assisted reconstruction. Moreover, training an AFN is memory efficient and versatile since a minimal training example of AFN is a ray propagation from the x-ray source to a detector pixel. AFN can be suited to learn incomplete CT in other forms, such as low-resolution projection data. The work can potentially be expanded to other cone-beam CT applications with different imaging geometries.

[0100] In another embodiment, the field of view (FOV) can be extended by using AFN to synthesize the rays outside of the FOV by using either a centered detector or a laterally shifted detector geometry. Referring to FIG.21A, a centered detector is used to acquire projection data of an object, wherein the lateral dimension of the centered detector is insufficient to fully image the object. The missing projection data through the object that is beyond the lateral dimension of the detector and on both sides beyond the detector are synthesized using AFN. The synthesized projection data in conjunction with the acquiredprojection data are then used to reconstruct the object using image reconstruction algorithms. This allows extending the FOV. The method can be used with both short-scan and full-scan acquisitions. Referring to FIG.21B, the detector is laterally shifted and projection data are acquired. The missing projection data through the object that is beyond the lateral dimension of the detector and on one side beyond the detector is synthesized using AFN. The synthesized projection data in conjunction with the acquired projection data are then used to reconstruct the object using image reconstruction algorithms. This allows extending the FOV. In FIGs.21A and 21B, the object can be any body part, anatomical region, or organ for medical imaging and any other object for other applications. Exemplary Multifunction Client Computing Device

[0101] Referring now to FIG.22, a simplified functional block diagram of illustrative multifunction device 2200 is shown according to one embodiment. Multifunction electronic device 2200 may include processor 2205, display 2210, user interface 2215, graphics hardware 2220, device sensors 2225 (e.g., proximity sensor / ambient light sensor, accelerometer and / or gyroscope), microphone 2230, audio codec(s) 2235, speaker(s) 2240, communications circuitry 2245, digital image capture circuitry 2250 (e.g., including camera system or X-ray image capture systems), video codec(s) 2255 (e.g., in support of digital image capture unit), memory 2260, storage device 2265, and communications bus 2270. Multifunction electronic device 2200 may be, for example, a standalone PC or a personal electronic device, CT scanner device, or the like.

[0102] Processor 2205 may execute instructions necessary to conduct or control the operation of many functions performed by device 2200. Processor 2205 may, for instance, drive display 2210 and receive user input from user interface 2215. User interface 2215 may allow a user to interact with device 2200. For example, user interface 2215 can take a variety of forms, such as a button, keypad, dial, a click wheel, keyboard, display screen and / or a touch screen. Processor 2205 may also, for example, be a system-on-chip such as those found in mobile devices and include a dedicated graphics processing unit (GPU). Processor 2205 may be based on reduced instruction-set computer (RISC) or complex instruction-set computer (CISC) architectures or any other suitable architecture and may include one or more processing cores. Graphics hardware 2220 may be special purpose computational hardware for processing graphics and / or assisting processor 2205 to process graphics information. In one embodiment, graphics hardware 2220 may include a programmable GPU.

[0103] Image capture circuitry 2250 may include one or more lens sensor assemblies to capture still, x-ray and / or video images. Output from image capture circuitry 2250 may be processed, at least in part, by video codec(s) 2255 and / or processor 2205 and / or graphics hardware 2220, and / or a dedicated image processing unit or pipeline incorporated within circuitry 2250. Images so captured may be stored in memory 2260 and / or storage 2265.

[0104] Sensor and camera circuitry 2250 may capture still and video images that may be processed in accordance with this disclosure, at least in part, by video codec(s) 2255 and / or processor 2205 and / or graphics hardware 2220, and / or a dedicated image processing unit incorporated within circuitry 2250. Images so captured may be stored in memory 2260 and / or storage 2265. Memory 2260 may include one or more different types of media used by processor 2205 and graphics hardware 2220 to perform device functions. For example, memory 2260 may include memory cache, read-only memory (ROM), and / or random-access memory (RAM). Storage 2265 may store media (e.g., audio, image and video files), computer program instructions or software, preference information, device profile information, and any other suitable data. Storage 2265 may include one more non-transitory computer-readable storage mediums including, for example, magnetic disks (fixed, floppy, and removable) and tape, optical media such as CD-ROMs and digital video disks (DVDs), and semiconductor memory devices such as Electrically Programmable Read-Only Memory (EPROM), and Electrically Erasable Programmable Read-Only Memory (EEPROM). Memory 2260 and storage 2265 may be used to tangibly retain computer program instructions or code organized into one or more modules and written in any desired computer programming language. When executed by, for example, processor 2205 such computer program code may implement one or more of the methods described herein. Additional Considerations

[0105] According to some embodiments, a processor or a processing element may be trained using supervised machine learning and / or unsupervised machine learning, and the machine learning may employ an artificial neural network, which, for example, may be a convolutional neural network, a recurrent neural network, a deep learning neural network, a reinforcement learning module or program, or a combined learning module or program that learns in two or more fields or areas of interest. Machine learning may involve identifying and recognizing patterns in existing data to facilitate making predictions for subsequent data. Models may be created based upon example inputs to make valid and reliable predictions for novel inputs.

[0106] According to certain embodiments, machine learning programs may be trained by inputting sample data sets or certain data into the programs, such as images, object statistics and information, historical estimates, and / or image / video / audio classification data. The machine learning programs may utilize deep learning algorithms that may be primarily focused on pattern recognition and may be trained after processing multiple examples. The machine learning programs may include Bayesian Program Learning (BPL), voice recognition and synthesis, image or object recognition, optical character recognition, and / or natural language processing. The machine learning programs may also include natural language processing, semantic analysis, automatic reasoning, and / or other types of machine learning.

[0107] According to some embodiments, supervised machine learning techniques and / or unsupervised machine learning techniques may be used. In supervised machine learning, a processing element may be provided with example inputs and their associated outputs and may seek to discover a general rule that maps inputs to outputs, so that when subsequent novel inputs are provided the processing element may, based upon the discovered rule, accurately predict the correct output. In unsupervised machine learning, the processing element may need to find its own structure in unlabeled example inputs.

[0108] The scope of the disclosed subject matter should be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled. In the appended claims, the terms “including” and “in which” are used as the plain- English equivalents of the respective terms “comprising” and “wherein.”

Claims

CLAIMS 1. A system for reconstructing images, the system comprising: a display; a processor coupled to the display; and a memory communicatively-coupled to the processor, the memory comprising instructions that when executed by the processor are configured to implement: receiving attenuation measurements, each attenuation measurement being indicative of the attenuation of radiation from a source to a detector through an image volume at a respective angle; training a fully connected network, using the received attenuation measurements, to learn a continuous function that maps each volumetric coordinate within the image volume to a respective attenuation coefficient; using the continuous function to estimate additional attenuation measurements through additional angles; and using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume.

2. The system of claim 1, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume comprises providing the received attenuation measurements and the estimated attenuation measurements to a reconstruction algorithm.

3. The system of claim 1, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume comprises using the continuous function to calculate attenuation coefficients at spatial coordinates within the image volume.

4. The system of claim 1, wherein the received attenuation measurements are captured by a computed tomography (CT) scanner.

5. The system of claim 4, wherein the CT scanner is a cone beam CT (CBCT) scanner.

6. The system of claim 5, wherein the CBCT scanner employs a flat panel detector or an area detector employing energy-integrating or photon-counting technology.

7. The system of claim 4, wherein the received attenuation measurements are captured by an energy-integrating or photon-counting detector of the CT scanner.

8. The system of claim 1, wherein the received attenuation measurements are ultrasound measurements.

9. The system of claim 1, wherein the received attenuation measurements are captured using optical imaging.

10. A method for reconstructing images, the method comprising: receiving attenuation measurements, each attenuation measurement being indicative of the attenuation of radiation from a source to a detector through an image volume at a respective angle; training a fully connected network, using the received attenuation measurements, to learn a continuous function that maps each volumetric coordinate within the image volume to a respective attenuation coefficient; using the continuous function to estimate additional attenuation measurements through additional angles; and using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume.

11. The method of claim 10, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume comprises providing the received attenuation measurements and the estimated attenuation measurements to a reconstruction algorithm.

12. The method of claim 10, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the imagevolume comprises using the continuous function to calculate attenuation coefficients at spatial coordinates within the image volume.

13. The method of claim 10, wherein the received attenuation measurements are captured by a computed tomography (CT) scanner.

14. The method of claim 13, wherein the CT scanner is a cone beam CT (CBCT) scanner.

15. The method of claim 14, wherein the CBCT scanner employs a flat panel detector or an area detector employing energy-integrating or photon-counting technology.

16. The method of claim 13, wherein the received attenuation measurements are captured by an energy-integrating or photon-counting detector of the CT scanner.

17. The method of claim 10, wherein the received attenuation measurements are ultrasound measurements.

18. The method of claim 10, wherein the received attenuation measurements are captured using optical imaging.

19. A non-transitory computer readable medium comprising instructions that when implemented by a processor implement: receiving attenuation measurements, each attenuation measurement being indicative of the attenuation of radiation from a source to a detector through an image volume at a respective angle; training a fully connected network, using the received attenuation measurements, to learn a continuous function that maps each volumetric coordinate within the image volume to a respective attenuation coefficient; using the continuous function to estimate additional attenuation measurements through additional angles; and using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume.

20. The medium of claim 19, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume comprises providing the received attenuation measurements and the estimated attenuation measurements to a reconstruction algorithm.

21. The medium of claim 19, wherein using the received attenuation measurements and the estimated attenuation measurements to produce tomographic images of the image volume comprises using the continuous function to calculate attenuation coefficients at spatial coordinates within the image volume.

22. The medium of claim 19, wherein the received attenuation measurements are captured by a computed tomography (CT) scanner.

23. The medium of claim 22, wherein the CT scanner is a cone beam CT (CBCT) scanner.

24. The medium of claim 23, wherein the CBCT scanner employs a flat panel detector or an area detector employing energy-integrating or photon-counting technology.

25. The medium of claim 22, wherein the received attenuation measurements are captured by an energy-integrating or photon-counting detector of the CT scanner.

26. The medium of claim 19, wherein the received attenuation measurements are ultrasound measurements.

27. The medium of claim 19, wherein the received attenuation measurements are captured using optical imaging.

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