A hybrid echocardiography-CFD framework for ventricular flow simulations.

Image-based CFD is a powerful tool to study cardiovascular flows while 2D echocardiography (echo) is the most widely used noninvasive imaging modality for the diagnosis of heart disease. Here, echo is combined with CFD, that is, an echo-CFD framework, to study ventricular flows. To achieve this, the previous 3D reconstruction from multiple 2D echo at standard cross sections is extended by: (a) reconstructing aortic and mitral valves from 2D echo and closing the left-ventricle (LV) geometry by approximating a superior wall; (b) incorporating the physiological assumption of the fixed apex as a reference (fixed) point in the 3D reconstruction; and (c) incorporating several smoothing algorithms to remove the nonphysical oscillations (ringing) near the basal section. The method is applied to echo from a baseline LV and one after inducing acute myocardial ischemia (AMI). The 3D reconstruction is validated by comparing it against a reference reconstruction from many echo sections while flow simulations are validated against the Doppler ultrasound velocity measurements. The sensitivity study shows that the choice of the smoothing algorithm does not change the flow pattern inside the LV. However, the presence of the mitral valve can significantly change the flow pattern during the diastole phase. In addition, the abnormal shape of a LV with AMI can drastically change the flow during diastole. Furthermore, the hemodynamic energy loss, as an indicator of the LV pumping performance, for different test cases is calculated, which shows a larger energy loss for a LV with AMI compared to the baseline one.

Automated Three-Dimensional Reconstruction of the Left Ventricle From Multiple-Axis Echocardiography.

Two-dimensional echocardiography (echo) is the method of choice for noninvasive evaluation of the left ventricle (LV) function owing to its low cost, fast acquisition time, and high temporal resolution. However, it only provides the LV boundaries in discrete 2D planes, and the 3D LV geometry needs to be reconstructed from those planes to quantify LV wall motion, acceleration, and strain, or to carry out flow simulations. An automated method is developed for the reconstruction of the 3D LV endocardial surface using echo from a few standard cross sections, in contrast with the previous work that has used a series of 2D scans in a linear or rotational manner for 3D reconstruction. The concept is based on a generalized approach so that the number or type (long-axis (LA) or short-axis (SA)) of sectional data is not constrained. The location of the cross sections is optimized to minimize the difference between the reconstructed and measured cross sections, and the reconstructed LV surface is meshed in a standard format. Temporal smoothing is implemented to smooth the motion of the LV and the flow rate. This software tool can be used with existing clinical 2D echo systems to reconstruct the 3D LV geometry and motion to quantify the regional akinesis/dyskinesis, 3D strain, acceleration, and velocities, or to be used in ventricular flow simulations.

Is Radiologic Assessment of Alveolar Crest Height Useful to Monitor Periodontal Disease Activity?

The mainstay of periodontal assessment is clinical probing. Radiographic assessment provides quantitative information on the status of tooth-supporting bone. This article reviews methods to assess periodontal structures, including basic radiograph acquisition, assessment of alveolar crest levels, and typical patterns of bone loss. Computer technology to objectively assess loss of alveolar crest from radiographs is reviewed. Developments in computer-assisted quantitation of alveolar crest height are described. Although probing measurements continue to be viewed as more practical than radiographic measurements, radiographic assessment can be made quantitative and is likely easier and more precise than probing for routine assessment of periodontal disease activity.

Filtered region of interest cone-beam rotational angiography.

PURPOSE: Cone-beam rotational angiography (CBRA) is widely used in the modern clinical settings. In a number of procedures, the area of interest is often considerably smaller than the field of view (FOV) of the detector, subjecting the patient to potentially unnecessary x-ray dose. The authors therefore propose a filter-based method to reduce the dose in the regions of low interest, while supplying high image quality in the region of interest (ROI). METHODS: For such procedures, the authors propose a method of filtered region of interest (FROI)-CBRA. In the authors' approach, a gadolinium filter with a circular central opening is placed into the x-ray beam during image acquisition. The central region is imaged with high contrast, while peripheral regions are subjected to a substantial lower intensity and dose through beam filtering. The resulting images contain a high contrast/intensity ROI, as well as a low contrast/intensity peripheral region, and a transition region in between. To equalize the two regions' intensities, the first projection of the acquisition is performed with and without the filter in place. The equalization relationship, based on Beer's law, is established through linear regression using corresponding filtered and nonfiltered data. The transition region is equalized based on radial profiles. RESULTS: Evaluations in 2D and 3D show no visible difference between conventional FROI-CBRA projection images and reconstructions in the ROI. CNR evaluations show similar image quality in the ROI, with a reduced CNR in the reconstructed peripheral region. In all filtered projection images, the scatter fraction inside the ROI was reduced. Theoretical and experimental dose evaluations show a considerable dose reduction; using a ROI half the original FOV reduces the dose by 60% for the filter thickness of 1.29 mm. CONCLUSIONS: These results indicate the potential of FROI-CBRA to reduce the dose to the patient while supplying the physician with the desired image detail inside the ROI.

Accurate MTF measurement in digital radiography using noise response.

PURPOSE: The authors describe a new technique to determine the system presampled modulation transfer function (MTF) in digital radiography using only the detector noise response. METHODS: A cascaded-linear systems analysis was used to develop an exact relationship between the two-dimensional noise power spectrum (NPS) and the presampled MTF for a generalized detector system. This relationship was then utilized to determine the two-dimensional presampled MTF. For simplicity, aliasing of the correlated noise component of the NPS was assumed to be negligible. Accuracy of this method was investigated using simulated images from a simple detector model in which the "true" MTF was known exactly. Measurements were also performed on three detector technologies (an x-ray image intensifier, an indirect flat panel detector, and a solid state x-ray image intensifier), and the results were compared using the standard edge-response method. Flat-field and edge images were acquired and analyzed according to guidelines set forth by the International Electrotechnical Commission, using the RQA 5 spectrum. RESULTS: The presampled MTF determined using the noise-response method for the simulated detector system was in close agreement with the true MTF with an averaged percent difference of 0.3% and a maximum difference of 1.1% observed at the Nyquist frequency (fN). The edge-response method of the simulated detector system also showed very good agreement at lower spatial frequencies (less than 0.5 fN) with an averaged percent difference of 1.6% but showed significant discrepancies at higher spatial frequencies (greater than 0.5 fN) with an averaged percent difference of 17%. Discrepancies were in part a result of noise in the edge image and phasing errors. For all three detector systems, the MTFs obtained using the two methods were found to be in good agreement at spatial frequencies less than 0.5 fN with an averaged percent difference of 3.4%. Above 0.5 fN, differences increased to an average of 20%. Deviations of the experimental results largely followed the trend seen in the simulation results, suggesting that differences between the two methods could be explained as resulting from the inherent inaccuracies of the edge-response measurement technique used in this study. Aliasing of the correlated noise component was shown to have a minimal effect on the measured MTF for the three detectors studied. Systems with significant aliasing of the correlated noise component (e.g., a-Se based detectors) would likely require a more sophisticated fitting scheme to provide accurate results. CONCLUSIONS: Results indicate that the noise-response method, a simple technique, can be used to accurately measure the MTF of digital x-ray detectors, while alleviating the problems and inaccuracies associated with use of precision test objects, such as a slit or an edge.

GPU-based cone beam computed tomography.

The use of cone beam computed tomography (CBCT) is growing in the clinical arena due to its ability to provide 3D information during interventions, its high diagnostic quality (sub-millimeter resolution), and its short scanning times (60 s). In many situations, the short scanning time of CBCT is followed by a time-consuming 3D reconstruction. The standard reconstruction algorithm for CBCT data is the filtered backprojection, which for a volume of size 256(3) takes up to 25 min on a standard system. Recent developments in the area of Graphic Processing Units (GPUs) make it possible to have access to high-performance computing solutions at a low cost, allowing their use in many scientific problems. We have implemented an algorithm for 3D reconstruction of CBCT data using the Compute Unified Device Architecture (CUDA) provided by NVIDIA (NVIDIA Corporation, Santa Clara, California), which was executed on a NVIDIA GeForce GTX 280. Our implementation results in improved reconstruction times from minutes, and perhaps hours, to a matter of seconds, while also giving the clinician the ability to view 3D volumetric data at higher resolutions. We evaluated our implementation on ten clinical data sets and one phantom data set to observe if differences occur between CPU and GPU-based reconstructions. By using our approach, the computation time for 256(3) is reduced from 25 min on the CPU to 3.2 s on the GPU. The GPU reconstruction time for 512(3) volumes is 8.5 s.

Real-time endovascular guidewire position simulation using shortest path algorithms.

PURPOSE: Treatment of vascular disease often involves endovascular interventions which use the vascular system for delivering treatment devices via a previously inserted guidewire to the diseased site. Previous studies show relative reproducibility of guidewire position after insertion, indicating that the guidewire position is constrained and could be represented by an energy minimization approach. Such representation would support the surgeon's decision process in guidewire selection. METHODS: In this paper, we determine the guidewire position using a k-level graph based on 3D vessel information. Guidewire properties are incorporated into the graph as edge weights given by the local bending energy related to the local bending angle. The optimal path through this weighted directed graph is determined using a shortest path algorithm. Volumetric data of two different internal carotid artery phantoms (Ø 3.5-4.6 mm) was acquired. Two guidewires (Ø 0.33 mm) of different material properties (stainless steel, plastic-coated steel core) were inserted into the phantoms. RESULTS: The average RMS distance between actual and simulated guidewire positions varies from 0.9 mm (plastic coated) to 1.3 mm (stainless steel); the computation time to determine the position was <2s. CONCLUSIONS: The results indicate that the proposed technique yields reproducible and accurate guidewire positions within a short, clinically relevant time frame. These calculated positions may be useful in facilitating neurovascular interventions.

Clinical evaluation of angiographic multiple-view 3D reconstruction.

PURPOSE: Endovascular treatment of stroke, a leading cause of death in the United States, involves guidance of devices to the intervention site often through tortuous vessels. Typically, these interventions are performed under two- dimensional (2D) fluoroscopy. To facilitate these procedures, we developed and previously presented a multiple-view self-calibration method involving two steps: (1) calibration of the imaging geometry, and (2) reconstruction of the 3D vessel centerline. Only those 2D angiograms obtained during the procedure are used for reconstruction. In this manuscript, we evaluate this technique on a large set (117 cases) of clinical data obtained over a 12-month period. METHODS: We evaluated the technique using (1) the RMS difference between the calculated 3D centerlines and the average centerline (before and after application of our method), (2) the difference between the projected 3D centerlines and the 2D indicated centerlines, (3) the translations and rotations calculated by our technique, and (4) intra- and inter-user variations. RESULTS: Our approach (1) reduces the RMS 3D differences by a factor of 10, (2) increases the number of projected 3D centerline points lying within 1 mm of the indicated 2D centerline points by over a factor of 2 (from 28 to 71%), (3) provides an assessment of the variations in the gantry geometry as provided by the imaging system, and (4) is insensitive to user variations in indication (<1 mm differences in 3D are seen). CONCLUSIONS: These results indicate that this technique will provide more reliable vessel centerlines in the clinical setting without requiring additional acquisitions or increasing dose to the patient.

The asymmetric vascular stent: efficacy in a rabbit aneurysm model.

BACKGROUND AND PURPOSE: Development of hemodynamic modifying devices to treat intracranial aneurysms is an active area of research. The asymmetrical vascular stent (AVS), a stent containing a low-porosity patch, is such device. We evaluate AVS efficacy in an in vivo intracranial aneurysm model. METHODS: We created 24 elastase rabbit model aneurysms: 13 treated with the AVS, 5 treated with standard coronary stents, and 6 untreated controls. Four weeks after treatment, aneurysms underwent follow-up angiography, cone-beam micro-CT, histological evaluation, and selective electron microscopy scanning. RESULTS: Four rabbits died early in the study: 3 during AVS treatment and 1 control (secondary to intraprocedural vessel injury and an unrelated tumor, respectively). AVS-treated aneurysms exhibited very weak or no aneurysm flow immediately after treatment and no flow in all aneurysms at follow-up. Standard stent-treated aneurysms showed flow both after treatment (5/5) and at follow-up (3/5). All control aneurysms remained patent during the study. Micro-CT scans showed: 9 of 9 scanned AVS aneurysms were occluded, 6 of 9 AVS were ideally placed, and 3 of 9 low-porosity region partially covered the aneurysm neck; standard stent-treated aneurysms were 1 of 5 occluded, 2 of 5 patent, and 2 of 5 partially patent. Histology results demonstrated: for AVS-treated aneurysms, advanced thrombus organization in the (9/9); for standard stent-treated aneurysms, (1/4) no thrombus, (2/4) partially thrombosed, and (1/4) fully thrombosed; for control aneurysms (4/4), no thrombus. CONCLUSIONS: The use of AVS shows promise as a viable new therapeutic in intracranial aneurysm treatment. These data encourage further investigation and provide substantial support to the AVS concept.

Intracranial atherosclerotic disease: medical, biomechanical, imaging, and flow dynamics perspective.

INTRODUCTION: Intracranial stenosis is one of the most common disease entities encountered by neurointerventionists. The physical and hemodynamic properties of the lesion are determined by the experience of the clinician. Computerized analysis may provide more accurate values of these parameters which can be used during the procedure for improving patient care. MATERIALS AND METHODS: We present a case of a 55-year-old man who was having ischemic symptoms related to severe middle cerebral artery stenosis. Intralesional stent placement normalized the diameter of the vessel leading to cessation of the episodes of neurological events. DISCUSSION: Discussants include a neurointerventionist, a neurovascular image analyst, and a neurovascular flow and hemodynamic analyst. Questions discussed included qualitative and quantitative analysis of stenosis; comparison of angiographic perfusion studies including analysis of delay in opacification, transit times and time density curves; role of automation in analysis of "emboli tolerance" and "no re-flow" phenomenon; and relationship of compliance and recoil in normal and diseased segments of a vessel during angioplasty or stent placement.

Distortion, Orientation, and Translation Corrections of Tiled EMCCD Detectors for the New Solid State X-ray Image Intensifier (SSXII).

We report on the technology of imaging corrections for a new solid state x-ray image intensifier (SSXII) with enhanced resolution and fluoroscopic imaging capabilities, made of a mosaic of modules (tiled-array) each consisting of CsI(Tl) phosphor coupled using a fiber-optic taper or minifier to an electron multiplier charge coupled device (EMCCD). Generating high quality images using this EMCCD tiled-array system requires the determination and correction of the individual EMCCD sub-images with respect to relative rotations and translations as well as optical distortions due to the fiber optic tapers. The image corrections procedure is based on comparison of resulting (distorted) images with the known square pattern of a wire mesh phantom. The mesh crossing point positions in each sub-image are automatically identified. With the crossing points identified, the mapping between distorted and an undistorted array is determined. For each pixel in a distorted sub-image, the corresponding location in the corrected sub-image is calculated using bilinear interpolation. For the rotation corrections between sub-images, the orientation of the vectors between respective mesh crossing points in the various sub-images are determined and each sub-image is appropriately rotated with the pixel values again determined using bilinear interpolation. Image translation corrections are performed using reference structures at known locations. According to our estimations, the distortion corrections are accurate to within 1%; the rotations are determined to within 0.1 degree, and translation corrections are accurate to well within 1 pixel. This technology will provide the basis for generating single composite images from tiled-image configurations of the SSXII regardless of how many modules are used to form the images.

Evaluation of guidewire path reproducibility.

The number of minimally invasive vascular interventions is increasing. In these interventions, a variety of devices are directed to and placed at the site of intervention. The device used in almost all of these interventions is the guidewire, acting as a monorail for all devices which are delivered to the intervention site. However, even with the guidewire in place, clinicians still experience difficulties during the interventions. As a first step toward understanding these difficulties and facilitating guidewire and device guidance, we have investigated the reproducibility of the final paths of the guidewire in vessel phantom models on different factors: user, materials and geometry. Three vessel phantoms (vessel diameters approximately 4 mm) were constructed having tortuousity similar to the internal carotid artery from silicon tubing and encased in Sylgard elastomer. Several trained users repeatedly passed two guidewires of different flexibility through the phantoms under pulsatile flow conditions. After the guidewire had been placed, rotational c-arm image sequences were acquired (9 in. II mode, 0.185 mm pixel size), and the phantom and guidewire were reconstructed (512(3), 0.288 mm voxel size). The reconstructed volumes were aligned. The centerlines of the guidewire and the phantom vessel were then determined using region-growing techniques. Guidewire paths appear similar across users but not across materials. The average root mean square difference of the repeated placement was 0.17 +/- 0.02 mm (plastic-coated guidewire), 0.73 +/- 0.55 mm (steel guidewire) and 1.15 +/- 0.65 mm (steel versus plastic-coated). For a given guidewire, these results indicate that the guidewire path is relatively reproducible in shape and position.

Asymmetric vascular stent: feasibility study of a new low-porosity patch-containing stent.

BACKGROUND AND PURPOSE: Intracranial aneurysm (IA) treatment through hemodynamic modification with novel stent designs is a burgeoning area of research. We present a feasibility study for a new low-porosity patch-containing stent designed to treat intracranial aneurysms. The device is deployed so the patch covers the aneurysm neck ensuring strong flow diversion away from the aneurysm while keeping a low probability of occlusion of perforating vessels. METHODS: We created 17 side-wall aneurysms in 6 dogs, 2 per carotid artery if animal size permitted. Twelve proximal aneurysms were treated with AVSs: 5 distal aneurysms were untreated, serving as controls against self-thrombosis; 7 treated aneurysms were fully-covered; and 5 were partially-covered. After 4 weeks, a final angiogram was performed and aneurysms were explanted. Angiograms acquired pre- and posttreatment and at 4-week follow-up were analyzed quantitatively using normalized time-density curves (NTDC). Cone-beam micro-CT and histological specimen analysis were then performed. RESULTS: Posttreatment, NTDC average peaks dropped to 45% of initial values for the partially-covered aneurysms and 78% for the fully-covered aneurysms. Cone-beam micro-CT imaging performed at 4 weeks posttreatment showed partial thrombosis in 4 of 5 partially-covered aneurysms and complete thrombosis in all fully-covered aneurysms. Histology revealed neointimal coverage of all asymmetrical patch regions and thrombus formation in both fully- and partially-covered aneurysms. Four-week follow-up was not done for 1 animal (2 controls, 2 treated) that expired because of groin hemorrhage and for another animal (1 aneurysm) with an occluded carotid. CONCLUSIONS: We demonstrate aneurysmal blood flow diversion using a new low-porosity patch-containing asymmetrical vascular stent in a canine side-wall aneurysm model. Overall results are encouraging and support continued AVS development.

Endovascular image-guided interventions (EIGIs).

Minimally invasive interventions are rapidly replacing invasive surgical procedures for the most prevalent human disease conditions. X-ray image-guided interventions carried out using the insertion and navigation of catheters through the vasculature are increasing in number and sophistication. In this article, we offer our vision for the future of this dynamic field of endovascular image-guided interventions in the form of predictions about (1) improvements in high-resolution detectors for more accurate guidance, (2) the implementation of high-resolution region of interest computed tomography for evaluation and planning, (3) the implementation of dose tracking systems to control patient radiation risk, (4) the development of increasingly sophisticated interventional devices, (5) the use of quantitative treatment planning with patient-specific computer fluid dynamic simulations, and (6) the new expanding role of the medical physicist. We discuss how we envision our predictions will come to fruition and result in the universal goal of improved patient care.

Limited view CT reconstruction and segmentation via constrained metric labeling.

This paper proposes a new discrete optimization framework for tomographic reconstruction and segmentation of CT volumes when only a few projection views are available. The problem has important clinical applications in coronary angiographic imaging. We first show that the limited view reconstruction and segmentation problem can be formulated as a "constrained" version of the metric labeling problem. This lays the groundwork for a linear programming framework that brings metric labeling classification and classical algebraic tomographic reconstruction (ART) together in a unified model. If the imaged volume is known to be comprised of a finite set of attenuation coefficients (a realistic assumption), given a regular limited view reconstruction, we view it as a task of voxels reassignment subject to maximally maintaining consistency with the input reconstruction and the objective of ART simultaneously. The approach can reliably reconstruct (or segment) volumes with several multiple contrast objects. We present evaluations using experiments on cone beam computed tomography.

An algorithm for tracking microcatheters in fluoroscopy.

Currently, a large number of endovascular interventions are performed for treatment of intracranial aneurysms. For these treatments, correct positioning of microcatheter tips, microguide wire tips, or coils is essential. Techniques to detect such devices may facilitate endovascular interventions. In this paper, we describe an algorithm for tracking of microcatheter tips during fluoroscopically guided neuroendovascular interventions. A sequence of fluoroscopic images (1,024 x 1,024 x 12 bits) was acquired using a C-arm angiography system as a microcatheter was passed through a carotid phantom which was on top of a head phantom. The carotid phantom was a silicone cylinder containing a simulated vessel with the shape and curvatures of the internal carotid artery. The head phantom consisted of a human skull and tissue-equivalent material. To detect the microcatheter in a given fluoroscopic frame, a background image consisting of an average of the four previous frames is subtracted from the current frame, the resulting image is filtered using a matched filter, and the position of maximum intensity in the filtered image is taken as the catheter tip position in the current frame. The distance between the tracked position and the correct position (error distance) was measured in each of the fluoroscopic images. The mean and standard deviation of the error distance values were 0.277 mm (1.59 pixels) and 0.26 mm (1.5 pixels), respectively. The error distance was less than 3 pixels in the 93.0% frames. Although the algorithm intermittently failed to correctly detect the catheter, the algorithm recovered the catheter in subsequent frames.

Brachytherapy seed localization using geometric and linear programming techniques.

We propose an optimization algorithm to solve the brachytherapy seed localization problem in prostate brachytherapy. Our algorithm is based on novel geometric approaches to exploit the special structure of the problem and relies on a number of key observations which help us formulate the optimization problem as a minimization integer program (IP). Our IP model precisely defines the feasibility polyhedron for this problem using a polynomial number of half-spaces; the solution to its corresponding linear program is rounded to yield an integral solution to our task of determining correspondences between seeds in multiple projection images. The algorithm is efficient in theory as well as in practice and performs well on simulation data (approximately 98% accuracy) and real X-ray images (approximately 95% accuracy). We present in detail the underlying ideas and an extensive set of performance evaluations based on our implementation.

Evaluating tool-artery interaction force during endovascular neurosurgery for developing haptic engine.

Endovascular neurosurgery has gained acceptance as the best method of treatment of vascular abnormalities like cerebral aneurysms. However, the procedure is associated with difficulties in tool/tissue manipulation. Navigation of stent, catheter and, guide wire through complex arteries without any force information often causes stent snagging, plaque dislocations and formation of thrombosis caused by the damage of the arterial wall. Currently, there is no haptic device available which can provide the surgeons with the force information, related to stent placement procedure. The goal of this work is to create a data base for a fast synthetic endovascular force simulator, which will provide the surgeon with force information during tool-artery interaction, based on the various combinations of tool sizes and vessel complexity, [1, 2] to facilitate better preoperative planning, safer interventions, and advanced training of new surgeons.

Microcatheter tip enhancement in fluoroscopy: a comparison of techniques.

We compared three techniques for enhancement of microcatheter tips in fluoroscopic images: conventional subtraction technique (CST); averaged image subtraction technique (AIST), which we have developed; and double average filtering (DAF) technique, which uses nonlinear background estimates. A pulsed fluoroscopic image sequence was obtained as a microcatheter was passed through a carotid phantom that was on top of a head phantom. The carotid phantom was a silicone cylinder containing a simulated vessel with the shape and curvatures of the internal carotid artery. The three techniques were applied to the images of the sequence, then the catheter tip was manually identified in each image, and 100 x 100 pixel images, centered at the indicated microcatheter tip positions, were extracted for the evaluations. The signal-to-noise ratio (SNR) was calculated in each of the extracted images from which the mean value of the SNR and its standard deviation (SD) were calculated for each technique. The mean values and the standard deviations were 4.36 (SD 3.40) for CST, 6.34 (SD 3.62) for AIST, and 3.55 (SD 1.27) for DAF. AIST had a higher SNR compared to CST in almost all frames. Although DAF yielded the smallest mean SNR value, it yielded the best SNR in those frames in which the microcatheter tip did not move between frames. We conclude that AIST provides the best SNR for a moving microcatheter tip and that DAF is optimal for a temporarily stationary microcatheter tip.

Towards a theory of a solution space for the biplane imaging geometry problem.

Biplane angiographic imaging is a primary method for visual and quantitative assessment of the vasculature. In order to reliably reconstruct the three-dimensional (3D) position, orientation, and shape of the vessel structure, a key problem is to determine the rotation matrix R and the translation vector t which relate the two coordinate systems. This so-called Imaging Geometry Determination problem is well studied in the medical imaging and computer vision communities and a number of interesting approaches have been reported. Each such technique determines a solution which yields 3D vasculature reconstructions with errors comparable to other techniques. From the literature, we see that different techniques with different optimization strategies yield reconstructions with equivalent errors. We have investigated this behavior, and it appears that the error in the input data leads to this equivalence effectively yielding what we call the solution space of feasible geometries, i.e., geometries which could be solutions given the error or uncertainty in the input image data. In this paper, we lay the theoretical framework for this concept of a solution space of feasible geometries using simple schematic constructions, deriving the underlying mathematical relationships, presenting implementation details, and discussing implications and applications of the proposed idea. Because the solution space of feasible geometries encompasses equivalent solutions given the input error, the solution space approach can be used to evaluate the precision of calculated geometries or 3D data based on known or estimated uncertainties in the input image data. We also use the solution space approach to calculate an imaging geometry, i.e., a solution.