The segmentation and classification of carotid plaques in ultrasound images play important roles in the treatment of atherosclerosis and assessment for stroke risk. Although deep learning methods have been used for carotid plaque segmentation and classification, two-stage methods will increase the complexity of the overall analysis and the existing multi-task methods ignored the relationship between the segmentation and classification. These will lead to suboptimal performance as valuable information might not be fully leveraged across all tasks. Therefore, we propose a multi-task learning framework (RCCM-Net) for ultrasound carotid plaque segmentation and classification, which utilizes a region confidence module (RCM) and a sample category confidence module (CCM) to exploit the correlation between these two tasks. The RCM provides knowledge from the probability of plaque regions to the classification task, while the CCM is designed to learn the categorical sample weight for the segmentation task. A total of 1270 2D ultrasound images of carotid plaques were collected from Zhongnan Hospital (Wuhan, China) for our experiments. The results showed that the proposed method can improve both segmentation and classification performance compared to existing single-task networks (i.e., SegNet, Deeplabv3+, UNet++, EfficientNet, Res2Net, RepVGG, DPN) and multi-task algorithms (i.e., HRNet, MTANet), with an accuracy of 85.82% for classification and a Dice-similarity-coefficient of 84.92% for segmentation. In the ablation study, the results demonstrated that both the designed RCM and CCM were beneficial in improving the network's performance. Therefore, we believe that the proposed method could be useful for carotid plaque analysis in clinical practice.

The vessel-wall-volume (VWV) measured based on three-dimensional (3D) carotid artery (CA) ultrasound (US) images can help to assess carotid atherosclerosis and manage patients at risk for stroke. Manual involvement for measurement work is subjective and requires well-trained operators, and fully automatic measurement tools are not yet available. Thereby, we proposed a fully automatic VWV measurement framework (Auto-VWV) using a CA prior-knowledge embedded U-Net (CAP-UNet) to measure the VWV from 3D CA US images without manual intervention. The Auto-VWV framework is designed to improve the repeated VWV measuring consistency, which resulted in the first fully automatic framework for VWV measurement. CAP-UNet is developed to improve segmentation accuracy on the whole CA, which composed of a U-Net type backbone and three additional prior-knowledge learning modules. Specifically, a continuity learning module is used to learn the spatial continuity of the arteries in a sequence of image slices. A voxel evolution learning module was designed to learn the evolution of the artery in adjacent slices, and a topology learning module was used to learn the unique topology of the carotid artery. In two 3D CA US datasets, CAP-UNet architecture achieved state-of-the-art performance compared to eight competing models. Furthermore, CAP-UNet-based Auto-VWV achieved better accuracy and consistency than Auto-VWV based on competing models in the simulated repeated measurement. Finally, using 10 pairs of real repeatedly scanned samples, Auto-VWV achieved better VWV measurement reproducibility than intra- and inter-operator manual measurements. The code is available at https://github.com/Yue9603/Auto-VWV.

Complete tumor coverage by the thermal ablation zone and with a safety margin (5 or 10 mm) is required to achieve the entire tumor eradication in liver tumor ablation procedures. However, 2D ultrasound (US) imaging has limitations in evaluating the tumor coverage by imaging only one or multiple planes, particularly for cases with multiple inserted applicators or irregular tumor shapes. In this paper, we evaluate the intra-procedural tumor coverage using 3D US imaging and investigate whether it can provide clinically needed information. Using data from 14 cases, we employed surface- and volume-based evaluation metrics to provide information on any uncovered tumor region. For cases with incomplete tumor coverage or uneven ablation margin distribution, we also proposed a novel margin uniformity -based approach to provide quantitative applicator adjustment information for optimization of tumor coverage. Both the surface- and volume-based metrics showed that 5 of 14 cases had incomplete tumor coverage according to the estimated ablation zone. After applying our proposed applicator adjustment approach, the simulated results showed that 92.9% (13 of 14) cases achieved 100% tumor coverage and the remaining case can benefit by increasing the ablation time or power. Our proposed method can evaluate the intra-procedural tumor coverage and intuitively provide applicator adjustment information for the physician. Our 3D US-based method is compatible with the constraints of conventional US-guided ablation procedures and can be easily integrated into the clinical workflow.

Thanka murals are important cultural heritages of Tibet, but many precious murals were damaged during history. Thanka mural restoration is very important for the protection of Tibetan cultural heritage. Partial convolution has great potential for Thanka mural restoration due to its outstanding performance for inpainting irregular holes. However, three challenges prevent the existing partial convolution-based methods from solving Thanka restoration problems: 1) the features of multi-scale objects in Thanka murals cannot be extracted correctly because of single-scale partial convolution; 2) the stroke-like Thanka inpainting mode cannot be effectively simulated and learned by existing rectangular or arbitrary masks; and 3) the original content of damaged Thanka murals cannot be restored. To resolve these problems, we propose a Thanka mural inpainting method based on multi-scale adaptive partial convolution and stroke-like masks. The proposed method consists of three parts: 1) a kernel-level multi-scale adaptive partial convolution (MAPConv) to accurately discriminate valid pixels from invalid pixels, and to extract the features of multi-scale objects; 2) a parameter-configurable stroke-like mask generation method to simulate and learn the stroke-like Thanka inpainting mode; and 3) a 2-phase learning framework based on MAPConv Unet and different loss functions to restore the original content of Thanka murals. Experiments on both simulated and real damages of Thanka murals demonstrated that our approach works well on a small dataset (N=2780), generates realistic mural content, and restores the damaged Thanka murals with high speed (600 ms for multiple holes in 512×512 images). The proposed end-to-end method can be applied to other small datasets-based inpainting tasks.

Measurement of total-plaque-area (TPA) is important for determining long term risk for stroke and monitoring carotid plaque progression. Since delineation of carotid plaques is required, a deep learning method can provide automatic plaque segmentations and TPA measurements; however, it requires large datasets and manual annotations for training with unknown performance on new datasets. A UNet++ ensemble algorithm was proposed to segment plaques from 2D carotid ultrasound images, trained on three small datasets (n = 33, 33, 34 subjects) and tested on 44 subjects from the SPARC dataset (n = 144, London, Canada). The ensemble was also trained on the entire SPARC dataset and tested with a different dataset (n = 497, Zhongnan Hospital, China). Algorithm and manual segmentations were compared using Dice-similarity-coefficient (DSC), and TPAs were compared using the difference (ΔTPA), Pearson correlation coefficient (r) and Bland-Altman analyses. Segmentation variability was determined using the intra-class correlation coefficient (ICC) and coefficient-of-variation (CoV). For 44 SPARC subjects, algorithm DSC was 83.3-85.7%, and algorithm TPAs were strongly correlated (r = 0.985-0.988; p <; 0.001) with manual results with marginal biases (0.73-6.75) mm$^2$ using the three training datasets. Algorithm ICC for TPAs (ICC = 0.996) was similar to intra- and inter-observer manual results (ICC = 0.977, 0.995). Algorithm CoV = 6.98% for plaque areas was smaller than the inter-observer manual CoV (7.54%). For the Zhongnan dataset, DSC was 88.6% algorithm and manual TPAs were strongly correlated (r = 0.972, p <; 0.001) with ΔTPA = -0.44±4.05 mm$^2$ and ICC = 0.985. The proposed algorithm trained on small datasets and segmented a different dataset without retraining with accuracy and precision that may be useful clinically and for research.

Radiofrequency ablation (RFA) is now a widely used minimally invasive treatment method for hepatic tumors. Preoperative planning plays a vital role in RFA therapy. With increasing tumor size, multiple overlapping ablations are needed, which are challenging to optimize while considering clinical constraints. In this paper, we present a new automatic RFA planning method. First, a 2-steps set cover-based model is formulated, which can integrate multiple clinical constraints for optimization of overlapping ablations. To ensure that the planning model can be solved in a reasonable time, a search space reducing strategy is then proposed. We also developed an algorithm for automatic RFA electrode selection, which provides a proper electrode ablation zone for the planning model. The proposed method was evaluated with 20 tumors of varying sizes (0.92 cm3to 28.4 cm3). Results showed that the proposed method can generate clinical feasible RFA plans with a minimum number of RFA electrodes and ablations, complete tumor coverage and minimized ablation of normal tissue.

Vessel-wall-volume (VWV) is an important three-dimensional ultrasound (3DUS) metric used in the assessment of carotid plaque burden and monitoring changes in carotid atherosclerosis in response to medical treatment. To generate the VWV measurement, we proposed an approach that combined a voxel-based fully convolution network (Voxel-FCN) and a continuous max-flow module to automatically segment the carotid media-adventitia (MAB) and lumen-intima boundaries (LIB) from 3DUS images. Voxel-FCN includes an encoder consisting of a general 3D CNN and a 3D pyramid pooling module to extract spatial and contextual information, and a decoder using a concatenating module with an attention mechanism to fuse multi-level features extracted by the encoder. A continuous max-flow algorithm is used to improve the coarse segmentation provided by the Voxel-FCN. Using 1007 3DUS images, our approach yielded a Dice-similarity-coefficient (DSC) of 93.2±3.0% for the MAB in the common carotid artery (CCA), and 91.9±5.0% in the bifurcation by comparing algorithm and expert manual segmentations. We achieved a DSC of 89.5±6.7% and 89.3±6.8% for the LIB in the CCA and the bifurcation respectively. The mean errors between the algorithm-and manually-generated VWVs were 0.2±51.2 mm3for the CCA and -4.0±98.2 mm3for the bifurcation. The algorithm segmentation accuracy was comparable to intra-observer manual segmentation but our approach required less than 1s, which will not alter the clinical work-flow as 10s is required to image one side of the neck. Therefore, we believe that the proposed method could be used clinically for generating VWV to monitor progression and regression of carotid plaques.

Preterm neonates with a very low birth weight of less than 1,500 grams are at increased risk for developing intraventricular hemorrhage (IVH), which is a major cause of brain injury in preterm neonates. Quantitative measurements of ventricular dilatation or shrinkage play an important role in monitoring patients and evaluating treatment options. 3D ultrasound (US) has been developed to monitor ventricle volume as a biomarker for ventricular changes. However, ventricle volume as a global indicator does not allow for precise analysis of local ventricular changes, which could be linked to specific neurological problems often seen in the patient population later in life. In this work, a 3D+t spatial-temporal deformable registration approachis proposed, which is applied to the analysis of the detailed local changes of preterm IVH neonatal ventricles from 3D US images. In particular, a novel sequential convex/dual optimization algorithm is introduced to extract the optimal 3D+t spatial-temporal deformable field, which simultaneously optimizes the sequence of 3D deformation fieldswhile enjoying both efficiencyand simplicity in numerics. The developed registration technique was evaluated by comparing two manually extracted ventricle surfaces from the baseline and the registered follow-up images using the metrics of Dice similarity coefficient (DSC), mean absolute surface distance (MAD), and maximum absolute surface distance (MAXD). The performed experiments using 14 patients with 5 time-point images per patient show that the proposed 3D+t registration approach accurately recovered the longitudinal deformation of ventricle surfaces from 3D US images. The proposed approach may be potentially used to analyse the change pattern of cerebral ventricles of IVH patients, their response to different treatment options, and to elucidate the deficiencies that a patient could have later in life. To the best of our knowledge, this paper reports the first study on the longitudinalanalysis of neonatal ventricular system from 3D US images.

In magnetic resonance (MR)-targeted, 3-D transrectal ultrasound (TRUS)-guided biopsy, prostate motion during the procedure increases the needle targeting error and limits the ability to accurately sample MR-suspicious tumor volumes. The robustness of the 2-D-3-D registration methods for prostate motion compensation is impacted by local optima in the search space. In this paper, we analyzed the prostate motion characteristics and investigated methods to incorporate such knowledge into the registration optimization framework to improve robustness against local optima. Rigid motion of the prostate was analyzed adopting a mixture-of-Gaussian (MoG) model using 3-D TRUS images acquired at bilateral sextant probe positions with a mechanically assisted biopsy system. The learned motion characteristics were incorporated into Powell's direction set method by devising multiple initial search positions and initial search directions. Experiments were performed on data sets acquired during clinical biopsy procedures, and registration error was evaluated using target registration error (TRE) and converged image similarity metric values after optimization. After incorporating the learned initialization positions and directions in Powell's method, 2-D-3-D registration to compensate for motion during prostate biopsy was performed with rms ± std TRE of 2.33 ± 1.09 mm with ~3 s mean execution time per registration. This was an improvement over 3.12 ± 1.70 mm observed in Powell's standard approach. For the data acquired under clinical protocols, the converged image similarity metric value improved in ≥8% of the registrations whereas it degraded only ≤1% of the registrations. The reported improvements in optimization indicate useful advancements in robustness to ensure smooth clinical integration of a registration solution for motion compensation that facilitates accurate sampling of the smallest clinically significant tumors.

Recent studies highlight temporal ultrasound data as highly promising in differentiating between malignant and benign tissues in prostate cancer patients. Since Hidden Markov Models can be used for capturing order and patterns in time varying signals, we employ them to model temporal aspects of ultrasound data that are typically not incorporated in existing models. By comparing order-preserving and order-altering models, we demonstrate that the order encoded in the series is necessary to model the variability in ultrasound data of prostate tissues. In future studies, we will investigate the influence of order on the differentiation between malignant and benign tissues.

UNLABELLED: This paper presents the results of a computer-aided intervention solution to demonstrate the application of RF time series for characterization of prostate cancer, in vivo. METHODS: We pre-process RF time series features extracted from 14 patients using hierarchical clustering to remove possible outliers. Then, we demonstrate that the mean central frequency and wavelet features extracted from a group of patients can be used to build a nonlinear classifier which can be applied successfully to differentiate between cancerous and normal tissue regions of an unseen patient. RESULTS: In a cross-validation strategy, we show an average area under receiver operating characteristic curve (AUC) of 0.93 and classification accuracy of 80%. To validate our results, we present a detailed ultrasound to histology registration framework. CONCLUSION: Ultrasound RF time series results in differentiation of cancerous and normal tissue with high AUC.

A common challenge when performing surface-based registration of images is ensuring that the surfaces accurately represent consistent anatomical boundaries. Image segmentation may be difficult in some regions due to either poor contrast, low slice resolution, or tissue ambiguities. To address this, we present a novel non-rigid surface registration method designed to register two partial surfaces, capable of ignoring regions where the anatomical boundary is unclear. Our probabilistic approach incorporates prior geometric information in the form of a statistical shape model (SSM), and physical knowledge in the form of a finite element model (FEM). We validate results in the context of prostate interventions by registering pre-operative magnetic resonance imaging (MRI) to 3D transrectal ultrasound (TRUS). We show that both the geometric and physical priors significantly decrease net target registration error (TRE), leading to TREs of 2.35 ± 0.81 mm and 2.81 ± 0.66 mm when applied to full and partial surfaces, respectively. We investigate robustness in response to errors in segmentation, varying levels of missing data, and adjusting the tunable parameters. Results demonstrate that the proposed surface registration method is an efficient, robust, and effective solution for fusing data from multiple modalities.

In surface-based registration for image-guided interventions, the presence of missing data can be a significant issue. This often arises with real-time imaging modalities such as ultrasound, where poor contrast can make tissue boundaries difficult to distinguish from surrounding tissue. Missing data poses two challenges: ambiguity in establishing correspondences; and extrapolation of the deformation field to those missing regions. To address these, we present a novel non-rigid registration method. For establishing correspondences, we use a probabilistic framework based on a Gaussian mixture model (GMM) that treats one surface as a potentially partial observation. To extrapolate and constrain the deformation field, we incorporate biomechanical prior knowledge in the form of a finite element model (FEM). We validate the algorithm, referred to as GMM-FEM, in the context of prostate interventions. Our method leads to a significant reduction in target registration error (TRE) compared to similar state-of-the-art registration algorithms in the case of missing data up to 30%, with a mean TRE of 2.6 mm. The method also performs well when full segmentations are available, leading to TREs that are comparable to or better than other surface-based techniques. We also analyze robustness of our approach, showing that GMM-FEM is a practical and reliable solution for surface-based registration.

In this study, we proposed an efficient nonrigid magnetic resonance (MR) to transrectal ultrasound (TRUS) deformable registration method in order to improve the accuracy of targeting suspicious regions during a three dimensional (3-D) TRUS guided prostate biopsy. The proposed deformable registration approach employs the multi-channel modality independent neighborhood descriptor (MIND) as the local similarity feature across the two modalities of MR and TRUS, and a novel and efficient duality-based convex optimization-based algorithmic scheme was introduced to extract the deformations and align the two MIND descriptors. The registration accuracy was evaluated using 20 patient images by calculating the TRE using manually identified corresponding intrinsic fiducials in the whole gland and peripheral zone. Additional performance metrics [Dice similarity coefficient (DSC), mean absolute surface distance (MAD), and maximum absolute surface distance (MAXD)] were also calculated by comparing the MR and TRUS manually segmented prostate surfaces in the registered images. Experimental results showed that the proposed method yielded an overall median TRE of 1.76 mm. The results obtained in terms of DSC showed an average of 80.8±7.8% for the apex of the prostate, 92.0±3.4% for the mid-gland, 81.7±6.4% for the base and 85.7±4.7% for the whole gland. The surface distance calculations showed an overall average of 1.84±0.52 mm for MAD and 6.90±2.07 mm for MAXD.

We propose a novel global optimization-based approach to segmentation of 3-D prostate transrectal ultrasound (TRUS) and T2 weighted magnetic resonance (MR) images, enforcing inherent axial symmetry of prostate shapes to simultaneously adjust a series of 2-D slice-wise segmentations in a "global" 3-D sense. We show that the introduced challenging combinatorial optimization problem can be solved globally and exactly by means of convex relaxation. In this regard, we propose a novel coherent continuous max-flow model (CCMFM), which derives a new and efficient duality-based algorithm, leading to a GPU-based implementation to achieve high computational speeds. Experiments with 25 3-D TRUS images and 30 3-D T2w MR images from our dataset, and 50 3-D T2w MR images from a public dataset, demonstrate that the proposed approach can segment a 3-D prostate TRUS/MR image within 5-6 s including 4-5 s for initialization, yielding a mean Dice similarity coefficient of 93.2%±2.0% for 3-D TRUS images and 88.5%±3.5% for 3-D MR images. The proposed method also yields relatively low intra- and inter-observer variability introduced by user manual initialization, suggesting a high reproducibility, independent of observers.

Segmenting 3D end firing transrectal ultrasound (TRUS) prostate images efficiently and accurately is of utmost importance for the planning and guiding 3D TRUS guided prostate biopsy. Poor image quality and imaging artifacts of 3D TRUS images often introduce a challenging task in computation to directly extract the 3D prostate surface. In this work, we propose a novel global optimization approach to delineate 3D prostate boundaries using its rotational resliced images around a specified axis, which properly enforces the inherent rotational symmetry of prostate shapes to jointly adjust a series of 2D slice wise segmentations in the global 3D sense. We show that the introduced challenging combinatorial optimization problem can be solved globally and exactly by means of convex relaxation. In this regard, we propose a novel coupled continuous max-flow model, which not only provides a powerful mathematical tool to analyze the proposed optimization problem but also amounts to a new and efficient duality-based algorithm. Extensive experiments demonstrate that the proposed method significantly outperforms the state-of-art methods in terms of efficiency, accuracy, reliability and less user-interactions, and reduces the execution time by a factor of 100.

Radical prostatectomy is performed on approximately 40% of men with organ-confined prostate cancer. Pathologic information obtained from the prostatectomy specimen provides important prognostic information and guides recommendations for adjuvant treatment. The current pathology protocol in most centers involves primarily qualitative assessment. In this paper, we describe and evaluate our system for automatic prostate cancer detection and grading on hematoxylin & eosin-stained tissue images. Our approach is intended to address the dual challenges of large data size and the need for high-level tissue information about the locations and grades of tumors. Our system uses two stages of AdaBoost-based classification. The first provides high-level tissue component labeling of a superpixel image partitioning. The second uses the tissue component labeling to provide a classification of cancer versus noncancer, and low-grade versus high-grade cancer. We evaluated our system using 991 sub-images extracted from digital pathology images of 50 whole-mount tissue sections from 15 prostatectomy patients. We measured accuracies of 90% and 85% for the cancer versus noncancer and high-grade versus low-grade classification tasks, respectively. This system represents a first step toward automated cancer quantification on prostate digital histopathology imaging, which could pave the way for more accurately informed postprostatectomy patient care.

In this paper, we propose a novel global optimization based 3-D multi-region segmentation algorithm for T1-weighted black-blood carotid magnetic resonance (MR) images. The proposed algorithm partitions a 3-D carotid MR image into three regions: wall, lumen, and background. The algorithm performs such partitioning by simultaneously evolving two coupled 3-D surfaces of carotid artery adventitia boundary (AB) and lumen-intima boundary (LIB) while preserving their anatomical inter-surface consistency such that the LIB is always located within the AB. In particular, we show that the proposed algorithm results in a fully time implicit scheme that propagates the two linearly ordered surfaces of the AB and LIB to their globally optimal positions during each discrete time frame by convex relaxation. In this regard, we introduce the continuous max-flow model and prove its duality/equivalence to the convex relaxed optimization problem with respect to each evolution step. We then propose a fully parallelized continuous max-flow-based algorithm, which can be readily implemented on a GPU to achieve high computational efficiency. Extensive experiments, with four users using 12 3T MR and 26 1.5T MR images, demonstrate that the proposed algorithm yields high accuracy and low operator variability in computing vessel wall volume. In addition, we show the algorithm outperforms previous methods in terms of high computational efficiency and robustness with fewer user interactions.

An integrated system for minimally invasive robot-assisted image-guided lung brachytherapy has been developed. The system incorporates an experimental setup for accurate radioactive seed placement with commercially available dosimetry planning software. The end result is a complete system that allows planning and executing a brachytherapy procedure with increased accuracy. The results of the in vitro seed placement evaluation show that seed misplacement has a significant effect on the volume receiving more than 200% of the dose (V200), and the minimum dosage received by 90% of the volume (D90).

Large core needle biopsy is a common procedure used to obtain histological samples when cancer is suspected in diagnostic breast images. The procedure is typically performed under image guidance, with freehand ultrasound and stereotactic mammography (SM) being the most common modalities used. To utilize the advantages of both modalities, a biopsy device combining three-dimensional ultrasound (3DUS) and digital SM imaging with computer-aided needle guidance was developed. An implementation of a stereo camera method was applied to SM calibration, providing a target localization error of 0.35 mm. The 3-D transformation between the two imaging modalities was then derived, with a target registration error of 0.52 mm. Finally, the needle guidance error of the device was evaluated using tissue-mimicking phantoms, showing a sample mean and standard deviation of 0.44 +/- 0.22 and 0.49 +/- 0.27 mm for targets planned from 3DUS and SM images, respectively. These results suggest that a biopsy procedure guided using this device would successfully sample breast lesions at a size greater than or equal to the smallest typically detected in mammographic screening (approximately 2 mm).

Atherosclerosis at the carotid bifurcation resulting in cerebral emboli is a major cause of ischemic stroke. Most strokes associated with carotid atherosclerosis can be prevented by lifestyle/dietary changes and pharmacological treatments if identified early by monitoring carotid plaque changes. Registration of 3-D ultrasound (US) images of carotid plaque obtained at different time points is essential for sensitive monitoring of plaque changes in volume and surface morphology. This registration technique should be nonrigid, since different head positions during image acquisition sessions cause relative bending and torsion in the neck, producing nonlinear deformations between the images. We modeled the movement of the neck using a "twisting and bending" model with only six parameters for nonrigid registration. We evaluated the algorithm using 3-D US carotid images acquired at two different head positions to simulate images acquired at different times. We calculated the mean registration error (MRE) between the segmented vessel surfaces in the target image and the registered image using a distance-based error metric after applying our "twisting and bending" model-based nonrigid registration algorithm. We achieved an average registration error of 0.80 +/-0.26 mm using our nonrigid registration technique, which was a significant improvement in registration accuracy over rigid registration, even with reduced degrees-of-freedom compared to the other nonrigid registration algorithms.

Summary form only given. This paper introduces a novel segmentation scheme for carotid artery ultrasound images. The proposed scheme is based on watershed segmentation algorithm. It consists of four major stages. These stages are preprocessing, watershed segmentation, region merging and finally boundary extraction. The proposed scheme is tested using a set of carotid artery ultrasound images. The experimental results show that the proposed scheme can produce accurate contours.

A linearly scanned three-dimensional (3-D) ultrasound imaging system is considered. The transducer array is initially oriented along the x axis and aimed in the y direction. After being tilted by an angle theta about the x axis, and then swiveled by an angle phi about the y axis, it is translated in the z direction, in steps of size d, to acquire a series of parallel two-dimendional (2-D) images. From these, the 3-D image is reconstructed, using the nominal values of the parameters (phi, theta, d). Thus, any systematic or random errors in these, relative to their actual values (phi0, theta0, d0), will respectively cause distortions or variances in length, area, and volume in the reconstructed 3-D image, relative to the 3-D object. Here, we analyze these effects. Compact linear approximations are derived for the relative distortions as functions of the parameter errors, and hence, for the relative variances as functions of the parameter variances. Also, exact matrix formulas for the relative distortions are derived for arbitrary values of (phi, theta, d) and (phi0, theta0, d0). These were numerically compared to the linear approximations and to measurements from simulated 3-D images of a cubical object and real 3-D images of a wire phantom. In every case tested, the theory was confirmed within experimental error (0.5%).