Didier Saint-Felix, Yves Trousset, Catherine Picard and Anne Rougee
GENERAL ELECFRIG CGR - Engineering and New Products Division 283 Rue de la Miniere - 78530 BUC - FRANCE
Abstract
Reconstructing a three-dimensional (3D) volume from a set of two-dimensional X-ray projections raises theoretical, instrumental and computational difficulties. Focused on high contrast objects, solutions are proposed for the successive steps of a 3D reconstruction procedure, from the raw measurements on an image intensifier up to the reconstruction algorithm based on a multi -scale detection I estimation scheme.
Keywords: 3D reconstruction, computed tomography, prior information, volume segmentation, multi-scale reconstruction algorithm, dual energy 1. Introduction
Reconstructing a three-dimensional (3D) volume from a set of two dimensional (2D) X-ray projections raises difficulties on different grounds:
reconstruction theory, instrumentation, computation and display. This paper presents possible solutions to those various difficulties that enable one to go from the raw data measured on an image intensifier (II) up to the visualization. After a general presentation of the 3D reconstruction in the medical context, the basic principles of our approach are stated. Then a quick overview is given of the calibration procedures and of the measurement pre-processing that provide corrected data suitable to the reconstruction algorithms described subsequently. Results obthined on actual data are presented at the end.
1.1 3D Reconstruction: an Ill-posed Problem
The reconstruction of a 3D object from its 2D X- ray projections cannot be
NATO ASI Series, Vol. F 60 3D Imaging in Medicine Edited by K. H. Hohne et 01.
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reduced to a trivial extension of the 2D reconstruction case. First, in practical situations where a 2D projection is obtained from a single position of the X-ray source assumed to be a point source placed at a finite distance, the projection geometry is conic. A given slice in the volume cannot be reconstructed independently of the surrounding slices and the 3D volume thus cannot be obtained by just stacking 2D reconstructions of parallel cross-sections. Second, the direct inversion of the integral equation describing the cone-beam projection, using the Radon transform for instance, requires certain conditions on the source trajectory [Kirillov 1961, Tuy 1983, Smith 1985] that often cannot be met for practical reasons such as gantry size and limited flux of X photons. Then the reconstruction problem has no solution in the usual sense. Extra difficulties are also added by the limited area of 2D detectors which introduces a truncature of the projections, at least in the direction of the body axis. For these reasons, the 3D problem should be considered, in medical applications, as a reconstruction problem with missing data. This point of view is reinforced when one is interested in physiological systems where dynamic phenomena occur, such as a vascular tree opacified by transient injection of a contrast agent. The number of available projections can then be dramatically low because of the time constraint.
Considering 3D reconstruction as a reconstruction problem with missing data has major consequences. The integral equation of the first kind modelling the cone-beam X-ray projection makes the reconstruction problem an ill-posed one and its intrinsic difficulties are increased by the data incompleteness
[Rangayyan 1985]. The usage of regularization techniques is then a must. Since they explicitly introduce prior information on the solution to stabilize it [N ashed 1981 , Saint-Felix 1988], one must therefore admit that a 3D reconstruction method cannot be universal, and that its efficiency increases with its specificity. This point of view is reinforced when the characteristics of today's 2D detector are considered.
1.2 3D Reconstruction of Highly Contrasted Objects
The II is the only existing device suitable fOF fast digital radiography. It exhibits good spatial resolution and good sensitivity, but it suffers from geometrical and densitometric dis torsions and, above all, from a low dynamic range. We thus focus on highly contrasted objects, such as opacified vasculatures and bone structures, where the morphological information is more relevant than the densitometric information. The a priori informations that may characterize both classes are the non-negativity, their sparseness
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(except for the skull) and their high contrast with respect to the background [Trousset 1989]. The last two points are even more significant if subtraction techniques are used to remove the background in the projections: logarithmic subtraction for vascular trees, dual energy techniques for bone structures.
The ability of reconstruction techniques to incorporate prior information largely differs from one method to another. Let us focus on 3D reconstruction of sparse objects, such as vascular trees. Besides the techniques based on computer vision tools and using two or three projections only, the others can be classified according to the kind of object model they use. A first group use parametric models where the number of unknown parameters is reduced by decomposing the solution on a small set of "basis functions"
carefully chosen. The advantage is double: the solution is well regularized and the amount of computation required by the estimation of few parameters can be dramatically low, specially if fast algorithms can be used when the model enjoys some adequate invariance property. Modelling vascular trees by elementary cylinders with elliptical cross-sections belongs to this group of methods [Bresler 1987, Fessler 1989]. Their main limitations are the high number of parameters that may be required by an accurate modelling of a complex object and the lack of robustness with respect to pathological cases.
The influence of the prior information, incorporated in the choice of the basis functions, may indeed predominate over the measurements and mislead the solution, even if stochastic modelling techniques may bring some flexibility.
The second group of methods emphasizes the robustness of the representation by using non-parametric modelling. The object is simply described by the set of the sampled voxe1s; the difficulty is then computational because of the huge number of parameters to be estimated. This problem has no simple solution, even in the 2D case, since the projection matrix that relates the measurements to the unknown object does not exhibit the structural properties (Toeplitz, circu1ante, ... ) that allow derivation of fast algorithms.
Existing methods therefore use estimation schemes that are simple enough to allow an effective implementation [Haaker 1984, Kruger 1987]. The counterpart of this simplicity of the estimation method is a lack of robustness with respect to the measurement noise [Rougee 1988j.
Thus it appeared interesting to develop a regularizing reconstruction method that has the flexibility of the non-parametric approaches but offers a good robustness to the noise and requires a reasonably small amount of computations. The basic idea is to take full advantage of the sparseness property by restricting to only those voxe1s belonging to the object the reconstruction stages that are computationa1y intensive. This idea is
implemented with a two step reconstruction scheme. In the first step, the region of support of the object is determined given the projections. In the second step, the density value of the voxels within the region of support is estimated. But before applying this classical detection / estimation scheme, the raw data must be calibrated and corrected.
2. Pre-processing of the measurements
Whatever the method used to reconstruct a 3D object, the location of the projection of a given voxel in any view must be known. This requires accurate knowledge of the projection process which includes not only the geometrical conic projection but also the geometrical distorsions introduced by the measurement process. Both points can be modelled and treated separately.
An IT introduces a geometrical distorsion between the ideal projection image formed on a plane and the actual measurement. The reasons are the spherical shape of the input screen and the effect of the earth's magnetic field on the electrons' trajectory. The global distorsion can be modelled by a displacement field mapping the ideal image onto the actual one. This field can then be identified using the image of a known reference grid, and the correction of any image is done by resampling the image pixel by pixel using the interpolated inverse field.
The conic projection is easily modelled using homogenous coordinate systems. The coefficients of the transformation matrix applying the 3D space onto a given projection plane can be identified using the corresponding undistorted image of a reference phantom that presents a sufficient number of fiducial points.
3. 3D Reconstruction Algorithms 3.1 Detection
The purpose of this first stage is to determine, given the projections, which are the voxels that belong to the object of interest. These will be called "full", the others "empty". The detection method must avoid negative false alarms, i.e. declaring a given voxel empty when it is full. This could be done at the possible expense of a higher rate of positive false alarms, i.e. declaring a voxel full when it is empty. One expects that the estimation step will attribute later a null value to these positive false alarms. The computational savings provided by the detection is larger when the object of interest is sparser and the
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detection itself is easier when the projections are also sparse. This property can be obtained by subtraction techniques. Possible subtraction and detection schemes suited to the processing of the two classes of objects of concern:
vascular trees and bones, are considered now.
Subtraction and detection for vascular reconstruction. The logarithmic subtraction between images taken before and after the injection of iodine, commonly used in Digital Subtracted Angiography (DSA), provides a good image of the vessels only when there is no move between the two pictures.
This procedure can be used here for each view angle and the resulting subtracted images turn to be the inputs for the 3D reconstruction. During the detection step, a voxel is declared empty if there is at least one subtracted image in which this voxel does not superimpose with any full voxel. This can be implemented using statistical hypothesis testing. For each voxel and for each view, a decision is taken if the measured values around the location of the projected voxel correspond to background or not. This decision is made with a given error probability knowing the noise statistics [Trousset 1989].
Subtraction and detection for selective bone reconstruction. In this case, the emphasis is on the subtraction stage since a simple segmentation of the bones in the projections cannot provide the required attenuation value. Dual energy (DE) techniques must be used for this purpose.
The principle is to distinguish between two materials, e.g. bone and soft tissues, using the different dependence of their attenuation coefficient on the photon energy. At the energy levels used in medical imaging, the absorption of a given material depicts two main phenomena: the Compton scattering and the Photoelectric absorption; it can therefore be assumed to be a function of the energy in a 2D space. This 2D space can be entirely described by the absorption behaviour of two distinct reference materials, such as aluminum and lucite. The total absorption of an inhomogeneous medium along a given ray path can thus be expressed as the sum of the absorptions through
"equivalent thicknesses" of the reference materials. Each equivalent thickness is approximated by a polynomial expansion of th(f absorption measurements made at two different energies. The coefficients of the expansion depend on the reference materials only; they can be estimated with a calibration procedure where different known thicknesses of the reference materials are superimposed and exposed at both energies [Lehman 1981, Brody 1981].
The response of an II suffers from spatial variations at low frequency which are function of the energy. Rather than correcting these
non-unifonnities in each measurement, we take them into account directly in the DE calibration and combination process. For this purpose, the coefficients of the two polynomial expansions are taken dependent on the location of the pixel. In practice, these coefficients are estimated on a regular mesh covering the image and interpolated for the other locations [Picard 1990]. Then, for the object of interest, two "equivalent thickness images" (Aluminum and Lucite) are generated from two exposures at low and high energy by applying for each pixel the corresponding polynomial expansions. A selective projection of bones only is obtained from the two reference material images by a simple linear combination. Example of such a decomposition is given Fig. I: Fig. I-a and l-b shows respectively the high and low energy exposures of a hand phantom, made with actual bones and lucite emulating soft tissues, and Fig.l-e shows the selective bone image resulting from the dual energy decomposition.
For 3D selective bone reconstruction, two exposures of the object must be taken at low and high energy respectively for each X-ray source position.
They are combined according to the above technique and thus a set of selective projections of bones only is generated. The detection of the region of support of the bones can then be done using the method described before for the vascular trees. Alternatively, a segmentation can be done first in the 20 selective projections to locate the 2D regions of support, then a logical back-projection detennines the region of support in the 3D space.
3.2 Estimation
The purpose of this second step is to estimate the value of each voxel belonging to the region of support. Two different approaches to solve this problem are presented. The first one is based on the well known Algebraic Reconstruction Technique (ART) [Gordon 1970], the second one is a multi scale implementation of an iterative procedure such as ART.
Three-dimensional ART. ART stands on the Kaczmarz method which iteratively computes the generalized inverse of a system of linear equations.
Although the asymptotic solution is unstable wl;1en the problem is ill-posed, regularization is obtained by stopping the iterations according to a given rule and by introducing prior infonnation (in the present case, non-negativity and knowledge of the region of support) on the solution [Saint-Felix 1988]. The support infonnation is used by applying the estimation step to only the voxels belonging to the detected support, the value of the voxels outside the support being fixed to a background value after the detection step.
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Multiscale estimation. When estimation is based on an iterative technique such as ART, extra computational savings can be achieved by a multiscale approach.
The underlying idea is that the first few iterations estimate essentially the low frequencies of the volume, and thus may not need to be performed at full resolution. The 3D object can therefore be described at different levels of resolution '}j, jmin ~ j ~ jmax' and the estimation is performed successively for the increasing values of j. Time saving in the global estimation task is expected because the complexity of ART is linear with respect to the number of processed voxels, and thus one iteration at resolution '}j-1 is 8 times faster than one iteration at resolution '}j.
Let us detail this approach. An initial estimation phase is first performed at the lowest resolutionjmin' by means of one or several iterations of ART (or any other iterative estimation method). Then, for any j, jmin ~j <jmax" the estimation obtained at resolution j is interpolated to a '}j+1 grid and used as initial value for the estimation at resolution j+ 1, which is also achieved by one or more iterations of ART. This multiscale approach can be used either with a constraint of support or for the reconstruction of a whole parallelepipedic volume. In the case where a constraint of support is to be applied, the detection is performed at the highest available resolutionj max' in order not to lose any critical detail, such as for instance small vessel branches in the vascular case. The region of support is then reduced to resolutionjmin before the beginning of the estimation step.
Additional time saving can be achieved by eliminating false alarm voxels from the region of support, i.e. voxels that do not belong to the object of interest but have been erroneously detected. In the multiscale implementation, this can be done by performing at each resolution j a segmentation of the estimated volume and by removing from the support all the voxels which do not belong to the segmented object. Because SNR is higher in the estimated volume than in the measured projections, false alarm voxels due to noise in 2D projections can be removed in this way. Furthermore, "empty" voxels located in concave parts of the object (for instance, the inside of the skull in selective bone reconstruction) can also be removed.
4. RESULTS
The reconstruction methods described in this paper have been tested on sequences of 2D conic X-ray projections acquired on two differents kinds of
devices: an experimental bench [Trousset 1989] and a standard digital angiography system.
4.1 Bone imaging
The abovementioned hand phantom was used to test the selective reconstruction method. A set of 120 dual-energy projections pairs (50 and 100 kVp) was acquired on the experimental bench. A comparison was made between non selective reconstruction of the whole hand from the 50 kVp projections and selective reconstruction of the bone from the dual energy pairs. Both reconstructions were done at resolution 256 (physical voxel size:
0.7 mm) with the mono scale method. The selective method provided a speed-up factor of 5.3 for the estimation step (5.3 being the ratio of the number of voxels detected in each case, see Fig.2-a and 2- b), while producing roughly the same quality for the reconstruction of bone (Fig.2-c and 2-d).
4.2 Vascular imaging
The reconstruction of a cadaver heart was performed with the multiscale estimation method. A set of 31 projections of the heart with post-mortem injection of constrast agent in the coronaries arteries was acquired on an angiography system (Fig.3). As it was not possible to acquire also a mask image (i.e. an image without constrast agent) the whole myocardium had to be reconstructed, and not only the coronary arteries. Results are presented in Fig.4, in the form of surface display of the reconstructed arteries. Fig.4-a (resp. 4-b) presents the results obtained with the mono scale approach after 1 (resp. 2) iterations(s) of ART at resolution 256 (physical voxel size: 0.6 mm).
Fig.4-c was produced with the multi scale method, used with 1 iteration at resolution 128 followed by 1 iteration at resolution 256. Fig.4-d was obtained similarly, but with 2 iterations at 128 resolution. The images presented in Fig.4-b, 4-c and 4-d have approximatively the same quality and are superior to the one of Fig.4-a. The conclusion is then that the multiscale approach saves a processing time corresponding roughly to the time of 1 iteration of ART at the highest resolution. This is specially interesting in the case of sparse objects reconstruction, where the introduction of a strong region of support constraint permits the use of a very limited number of iterations [Rougee 1989].
Another conclusion to be drawn from the results obtained on this phantom is the very great importance of 3D vizualisation tools adapted to angiography. Cut planes of the reconstructed volume (FigS) bring a good
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feeling of the 3D anatomy of the myocardium but carry little 3D information about the coronary network itself. Traditional surface display methods cannot perform a correct segmentation and thus a correct rendering of very small vessels (Fig.6-a). A method which may be more specifically adapted to angiography is maximum voxel ray tracing, first introduced in MR angiography [Laub 1988], in which each ray going from the observer eye to the 3D volume is analysed and only the maximum value along this ray is written to the 2D display image (Fig.6-b).
5. Conclusion
The problem of 3D reconstruction from X-ray projections is an ill posed problem, which implies that the reconstruction methods must depend on the kind of object to be reconstructed. We have introduced a methodological
The problem of 3D reconstruction from X-ray projections is an ill posed problem, which implies that the reconstruction methods must depend on the kind of object to be reconstructed. We have introduced a methodological