Data modelization in tomography is a key point for iterative reconstruction. The design of the projector starts with the representation of the object of interest, decomposed on a discrete basis of functions. Standard models of projector such as ray driven, or more advanced models such as distance driven, use simple cubic voxels, which result in modelization errors due to their anisotropic behaviour. Moreover approximations made at the projection step increase these errors. Long, Fessler and Balter reduce approximation errors by projecting the cubic voxels more accurately. However anisotropy errors still hold. Spherically symmetric volume elements (blobs) eradicate them, but at the cost of increased complexity. We propose a compromise between these two approaches by using B-splines as basis functions. Their quasi-isotropic behaviour allows to avoid projection inconsistencies, while conserving local influence. Small approximations transform the exact footprint (projection of the basis function) into a separable function, which does not depend on the angle of projection, and is easier and faster to integrate on detector pixels. We obtain a more accurate projector, with no additional computation cost. Such an improvement is particularly of interest in the case of dynamic gated X-ray CT, which can be considered as a tomographic reconstruction problem with very few projection data, and for which we show some preliminary results, with an original method of iterative reconstruction, using spatio-temporal regularization of the "space + time" sequence, and making no use of motion estimation.

When small dense balls are used in geometric calibration of x-ray cone-beam (CB) scanners, it is vital to accurately identify the CB projections of the centers of the balls, called the "center projections." The detector location of the center projection is usually estimated either from the center of the elliptical projection, or from the centroid of the intensity values inside the ellipse. The ellipse center is easily seen to differ from the center projection, and probably for this reason the centroid has been the preferred estimate.We have demonstrated mathematically that the centroid also differs from the center projection, and we have shown using numerical methods that the centroid lies much closer to the ellipse center than to the center projection. We have established that the centroid is only 20% more accurate than the ellipse center for a very wide range of CB geometries, including all situations that would be encountered in practice.Since both estimates present a systematic bias, it is important to evaluate the magnitude of this geometric error, especially for the purposes of CB calibration. We have proved that this geometric error cannot be ascertained from the elliptical projection alone, and we have selected the source-to-detector distance f (also called the SID) as the easiest parameter to estimate independently. We demonstrate that the absolute ellipse-center error can be estimated by eA/pi f where e and A are the ellipse eccentricity and ellipse area of the CB projection of the ball. The centroid error is 4/5 of the ellipse error.Finally we point out that physical effects such as beam hardening and Compton scatter, which typically generate cupping or capping artifacts, can be treated as undistorted projections of balls of non-uniform density, and we extend our analysis to such cases. The centroid error remains well within 70% to 90% of the ellipse-center error for density variations under 50%.In conclusion, the preprocessing required to obtain good centroid estimates in the presence of noisy data is considerably more than for finding the ellipse center, whereas the geometric benefit is relatively small.

Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions.

In geometric calibration of cone-beam (CB) scanners, point-like marker objects such as small balls are imaged to obtain positioning information from which the unknown geometric parameters are extracted. The procedure is sensitive to errors in the positioning information, and one source of error is a small bias which can occur in estimating the detector locations of the CB projections of the centers of the balls. We call these detector locations the center projections. In general, the CB projection of a ball of uniform density onto a flat detector forms an ellipse. Inside the ellipse lie the center projection M, the ellipse center C and the centroid G of the intensity values inside the ellipse. The center projection is invariably estimated from C or G which are much easier to extract directly from the data. In this work, we quantify the errors incurred in using C or G to estimate M. We prove mathematically that the points C, G, M and O are always distinct and lie on the major axis of the ellipse, where O is the detector origin, defined as the orthogonal projection of the cone vertex onto the detector. (The ellipse can only degenerate to a circle if the ball is along the direct line of sight to O, and in this case all four points coincide.) The points always lie in the same order: O, M, G, C which establishes that the centroid has less geometric bias than the ellipse center for estimating M. However, our numerical studies indicate that the centroid bias is only 20% less than the ellipse center bias so the benefit in using centroid estimates is not substantial. For the purposes of quantifying the bias in practice, we show that the ellipse center bias ||CM|| can be conveniently estimated by where A is the area of the elliptical projection, e is the eccentricity of the ellipse and is an estimate of the focal length of the system. Finally, we discuss how these results are affected by physical factors such as beam hardening, and indicate extensions to balls of non-uniform density.

In order to facilitate cone-beam CT using a mobile x-ray C-arm, accurate geometric calibration must be carried out. Such calibration requires a precise correction of x-ray image intensifier (II) distortion. Regional distortion correction (DC) was performed for the x-ray II using a planar grid phantom. The DC reduced the RMS error of known locations (or points) spanning the II field of view from 1.71 pixels to 0.63 pixels. For each projection, a six-ball calibration phantom was then used to determine the behavior of 9 geometric-calibration parameters during a 360° scan of the phantom, using a direct calibration method in the literature. These parameters were subsequently used in a modified Feldkamp (FDK) filtered back projection algorithm to achieve 3D volume reconstruction. Without the correction of II distortion the geometric calibration parameters showed the addition of incorrect structured behavior attributed to distortion while the reconstructed image was observed to be noisier when compared to the calibration and reconstruction when DC was incorporated. Further, a control was implemented using a distortion free digital flat panel detector. The reconstruction using the control was seen to have improved quality but was comparable to the quality of the distortion corrected x-ray II reconstruction.

En tomographie, les methodes de reconstruction iteratives necessitent une modelisation discrete du processus d'obtention des mesures. La representation de l'objet d'interet est le point de depart a l'elaboration d'un modele de projection tomographique sur le detecteur, precis et rapide. Les modeles conventionnels ray driven et distance driven, construits a partir d'indicatrices de voxels, ont l'inconvenient d'etre fortement anisotropes. Nous proposons un modele utilisant des fonctions de bases B-splines de degre adequat, nous fournissant un projecteur quasiment isotrope. Une approximation de la projection par une B-spline separable sur le detecteur conduit a un modele efficace en geometrie parallele et conique, de qualite superieure aux modeles conventionnels. Nous montrons notamment que l'erreur de modelisation est amelioree d'un facteur 10 par rapport au modele distance driven. Nous illustrons l'amelioration de la qualite de reconstruction apportee par notre modele sur des simulations du fantome de Shepp-Logan, en utilisant une methode iterative de reconstruction regularisee.

This work addresses theoretical advances classical (2D) tomographic image reconstruction. During the past several years, inversion formulas have been established that allow ROI reconstruction from incomplete (yet sufficient) data. Such reconstructions have important consequences in certain practical situations, such as truncated projections. The precise relationship between the largest ROI that can be reconstructed and the incompleteness of the sinogram is a complex question which has still not been completely answered in the 2D case. These relationships are inherent to the system and have consequences for iterative/statistical reconstruction methods, because they describe which part of the reconstructed image is determined completely by the data; the other parts of the image will have been more heavily influenced by the regularization method or by the nature of the objective function. Our understanding of the nature of reconstruction from incomplete yet sufficient data relies mainly on formulas obtained from the virtual fanbeam (VFB) method and from the DBP-Hilbert method. The purpose of this work is to provide a structure in which to examine the inherent differences in these two approaches. Using a common reconstruction problem, we reformulate VFB and DBP-Hilbert reconstruction formulas into weight functions that are applied in the sense of an inner product to the sinogram. A common regularization is used for the Hilbert transform in both methods. Unlike the usual Fourier windows used in analytic methods, the regularization we used is applied locally to the singularity to avoid the regularization obscuring the nature of the reconstruction. The weight functions clearly show how truncated projections are being correctly handled. The dissimilarity in the weight functions of the two methods illustrates fundamental differences in managing incomplete data, and suggests that many other such methods exist.

This paper describes a comprehensive method for determining the geometric alignment parameters for cone-beam scanners (often called calibrating the scanners or performing geometric calibration). The method is applicable to x-ray scanners using area detectors, or to SPECT systems using pinholes or cone-beam converging collimators. Images of an alignment test object (calibration phantom) fixed in the field of view of the scanner are processed to determine the nine geometric parameters for each view. The parameter values are found directly using formulae applied to the projected positions of the test object marker points onto the detector. Each view is treated independently, and no restrictions are made on the position of the cone vertex, or on the position or orientation of the detector. The proposed test object consists of 14 small point-like objects arranged with four points on each of three orthogonal lines, and two points on a diagonal line. This test object is shown to provide unique solutions for all possible scanner geometries, even when partial measurement information is lost by points superimposing in the calibration scan. For the many situations where the cone vertex stays reasonably close to a central plane (for circular, planar, or near-planar trajectories), a simpler version of the test object is appropriate. The simpler object consists of six points, two per orthogonal line, but with some restrictions on the positioning of the test object. This paper focuses on the principles and mathematical justifications for the method. Numerical simulations of the calibration process and reconstructions using estimated parameters are also presented to validate the method and to provide evidence of the robustness of the technique.

We have developed a method of automatic geometric calibration and reconstruction for cone-beam tomography with circular orbits. The reconstruction uses the Feldkamp algorithm with small modifications to accommodate geometric system imperfections. Complete analytic calibration is performed to directly estimate the 9 geometric parameters for each scanner position, without iterative schemes or non-linear optimization techniques. All angular positions are calibrated independently using a common reference frame in the filed-of-view, defined by the calibration phantom. Our simulation studies used a 256 × 256 sq-mm detector with 512 × 512 pixels, an orbit radius of 300 mm and detector orbit radius of 150 mm. The calibration object consisted of six balls all of radius 3 mm and varying densities. All 180 projections were processed automatically, and from the estimated parameters, the best circular orbit was fit to obtain suitable reconstruction parameters. The calibration parameters consistently estimated the cone-vertex position to within 0.5 mm or better. The detector positions were found to within 1 mm and the detector normal vectors were obtained to about 0.1 degrees. The in-plane rotation of the detector was found to within 0.03 degrees. Reconstructions using the automated calibration approach produced errors far smaller than the usual circular cone-beam artefacts.

Plane-based self-calibration aims at the computation of camera intrinsic parameters from homographies relating multiple views of the same unknown planar scene. This paper proposes a straightforward geometric statement of plane-based self-calibration, through the concept of metric rectification of images. A set of constraints is derived from a decomposition of metric rectification in terms of intrinsic parameters and planar scene orientation. These constraints are then solved using an optimization framework based on the minimization of a geometrically motivated cost function. The link with previous approaches is demonstrated and our method appears to be theoretically equivalent but conceptually simpler. Moreover, a solution dealing with radial distortion is introduced. Experimentally, the method is compared with plane-based calibration and very satisfactory results are obtained. Markerless self-calibration is demonstrated using an intensity-based estimation of the inter-image homographies.

Region-of-Interest (ROI) image reconstruction in two-dimensions is broadly based on the virtual fanbeam (VFB) method or on the more familiar method of differentiated backprojection (DBP) with Hilbert filtering. Although the VFB method is more complicated, it has the advantage of potentially handling some interior-type ROI problems that cannot be resolved by DBP-Hilbert methods. We have previously proposed combined filtered-backprojection (FBP) and VFB reconstruction for handling truncated projections, but with the constraint that the object support must lie within a known ellipse. However, this elliptical constraint is too restrictive for some truncation problems. In this work we have extended the FBP-VFB method to handle any convex support. The algorithm applies standard FBP filtering to all non-truncated parallel projections, and applies the VFB method to the truncated parallel projections which involves one-dimensional filtering that moves obliquely across the sinogram. We illustrate our method with a specific truncation problem for a thorax-like object support. Our example can not be resolved by other existing analytic methods.

This article presents a model-based approach to Shape From Silhouette reconstruction. It is formulated as a problem of 3D-2D non-rigid registration: a surface model is deformed until it correctly matches the detected silhouettes in the images. An efficient and reliable solution is proposed, based on a Radial Basis Function deformation driven by control points located on the contour generators of the 3D model. Unlike previous methods relying on non-linear optimization techniques, the proposed method only requires a linear system solving. Another advantage of this model-based approach is to produce a surface representation of the visual hull. Moreover, the introduction of shape priors allows to reduce in a dramatic way the number of views required to obtain a realistic reconstruction. Application to human body modeling is given.

In 2004, Clackdoyle and Noo published a class of inversion formulas for the 2D Radon transform which depends on the known radius of support of the unknown function. In this work, we extend this class of inversion formulas from functions of circular support to functions with any compact and convex support. We point out the potential benefits of these new inversion formulas in the context of reconstruction from truncated projections. A preliminary implementation of these new inversion formulas is also presented.

In this work, we discuss algebraic and analytic approaches for dynamic tomography. We present a framework of dynamic tomography for both algebraic and analytic approaches. We finally present numerical experiments.

In this work we study the estimation error of ball center cone beam projections. This problem arises in the context of cone beam system calibration. The cone beam projection of a ball is generally an ellipse. It is well known that the ellipse center is usually different from the ball center projection. We show that the centroid of the ellipse (gray level gravity center) is also different from the ball center projection, even if closer to the ball center projection than the ellipse center is. However, we numerically show that in usual applications (focal length around 1m and detector size less than 0.4m), the centroid or the ellipse center estimation are good approximations of the ball center projection. We study numerically the ball center estimation error as a function of the ball size, the noise quantity and the x-ray source (or optical center) finite size.

We present a reconstruction formula for classical two-dimensional image reconstruction that is impervious to the effects of lateral truncation. The formula applies to objects lying inside an elliptical region whose area is less than that of a scanner's circular field-of-view. This situation would occur for example for a patient whose arms are truncated in anterior and posterior projection views. This formula differs fundamentally from existing algorithms in that it does not require a two-step processing of a DBP (differentiated backprojection) followed by subsequent Hilbert inversion. The formula represents a true filtered backprojection (FBP) algorithm, although the filtering step sometimes requires certain neighboring projections to also contribute to the current filtered projection. The derivation is based on a result published in 2004 on multiple Radon inversion formulas for functions with circular support. We have changed the support to an ellipse and identified a formula that does not access the truncated part of the sinogram. We implemented this reconstruction approach and tested it using simulated truncated data of a modified Shepp-Logan phantom. We suggest that potential advantages of this method would be the reconstruction of a small region-of-interest, which could be performed at high pixel resolution with only local backprojection and localized filtering required.

We are investigating a direct analytic method of determining all geometric calibration parameters for a cone-beam scanner. Each projection is independently calibrated using a common calibration object that remains fixed in the laboratory frame. We make no assumptions or restrictions on the path of the cone vertex (e.g. the X-ray source) or the movement of the detector but we do assume a known field-of-view common to all views. The method uses a calibration object consisting of 30 small balls lying on 3 orthogonal concentric circles and one additional ball at the center of the calibration object. From parametrizations of the 3 image ellipses it is possible to analytically obtain expressions for all the calibration parameters. Using simulations we present our first results with this method, and by adding uncertainty to the locations of the projected balls, we determine the stability of the method. Our ultimate goal is to develop a completely automated calibration procedure requiring no user intervention whatsoever, and providing a seamless link to the reconstruction procedure

Conics drawing is an important issue in image processing and CAD. Many methods exist, but few are based on linear iterative methods X/sub n+1/ = SX/sub n/ for the computation of points belonging to a conic with equation X/sup T/CX = z. This paper studies these methods in a systematic way. It shows that S and C are linked by S = /spl plusmn/exp(/spl theta/JC), where J is the /spl pi//2 rotation matrix and /spl theta/ controls the points' density. Different linear properties are established, especially a bijective connection with simple processes on the unit circle or unit hyperbola. Moreover, an efficient drawing algorithm for elliptic and hyperbolic arcs is derived.

This paper is about calibration of cone-beam (CB) scanners for both x-ray computed tomography and single-photon emission computed tomography. Scanner calibration refers here to the estimation of a set of parameters which fully describe the geometry of data acquisition. Such parameters are needed for the tomographic reconstruction step. The discussion is limited to the usual case where the cone vertex and planar detector move along a circular path relative to the object. It is also assumed that the detector does not have spatial distortions. We propose a new method which requires a small set of measurements of a simple calibration object consisting of two spherical objects, that can be considered as 'point' objects. This object traces two ellipses on the detector and from the parametric description of these ellipses, the calibration geometry can be determined analytically using explicit formulae. The method is robust and easy to implement. However, it is not fully general as it is assumed that the detector is parallel to the rotation axis of the scanner. Implementation details are given for an experimental x-ray CB scanner.