1. Field of the Invention
The present invention relates to a method of reconstruction and processing of a three-dimensional object. More particularly, the invention is intended to permit reconstruction in three-dimensional space of an image of the spatial distribution of the radiological density of a vascular tree from a few two-dimensional projections of said tree acquired by radiology experiments. Radiology experiments are tomodensitometry experiments. In the general case, processing is concerned with display.
2. Description of the Prior Art
Digitized tomodensitometry is a conventional technique which makes it possible to reconstruct the spatial distribution of a characteristic physical quantity in a cross-section of an object illuminated by a given radiation (x-radiation, ultrasonic radiation, microwaves, etc.). To this end, the measurements employed are carried out on a segment of a straight line or of a curve contained in the plane to be reconstructed. One well-known example of devices of this type for carrying out this technique is the digitized x-ray tomodensitometer which has been in use for a few years in the medical imaging field. This technique can be extended in two ways for reconstructing the three-dimensional (3D) object. A first solution consists in producing a number of closely-spaced parallel cross-sections and then in regrouping them in a 3D space. A second solution consists in distributing in respect of each given radiation source position the points of measurement in space on a plane, for example, and no longer on a straight line. In the case just mentioned, the use of a two-dimensional detector makes it possible to carry out simultaneously all the measurements for a given position of the source and therefore to restrict the data acquisition time to a considerable extent. The invention relates to this second approach of the three-dimensional reconstruction.
In the case of x-ray imaging, the application of the principles set forth makes it necessary to reconstruct the 3D distribution of the linear attenuation coefficient of an object from a set of two-dimensional x-ray images taken at different angles of incidence. The measurements can be performed by means of an x-ray image amplifier or any other type of 2D detector or even 1D with scanning provided that it is displaced in such a manner as to acquire a 2D image at each position of the x-ray source. The relative position of the object and of the source-detector assembly is modified between each radiography. This can be achieved either by displacement of the source detector assembly by means of a device similar to that of digitized tomodensitometers or by means of vascular x-ray arches when the object is bulky in medical imaging or by displacement of the object itself when this latter is of small size. The second solution is more particularly suitable for use in nondestructive testing.
A number of methods are known for reconstructing a 3D object from x-ray images and can be classed in two main categories. A first category is concerned with direct analytical methods. These methods are derived from conventional 2D reconstruction methods by filtered back-projection. They consist in directly reversing the integral equation which relates the measurements to the object by utilizing an approximation of the 3D Radon transform in respect of small apex angles of the projection cone or by employing the exact expression of this transform.
These analytical methods are subject to a number of disadvantages. In the first place, they require a large number of projections distributed around the entire object in order to fill the "Radon domain" to a sufficient extent and to avoid reconstruction artifacts. In the second place, the truncation of projections, which is inevitable with the majority of objects of medical interest, introduces major artifacts. Finally, they offer low resistance to measurement noise.
The other methods are of algebraic type. They operate after discretization of the initial integral equation and consider the reconstruction problem as an estimation problem. They have been widely applied in the 2D case since the publication of the article by R. Gordon, R. Bender and G. T. Herman entitled "Algebraic reconstruction technic for three dimensional electron microscopy and x-ray photography", Journal Theo. Biol. 29, pages 471 to 481 (1970). These methods of estimation have shown their superiority over the analytical methods when the problem of reconstruction is of the type consisting of missing views (projections in a small angular sector and/or projections in small number) and/or of truncated views. These algebraic methods make it possible in fact to introduce a priori data on the solution, which are necessary for its stabilization. The drawback of this flexibility is a volume of calculations which may be prohibitive in the 3D case. These methods are therefore usually directed to making the most profitable use of a given a priori item of information. They are therefore specialized in the reconstruction of the single class of objects characterized by this a priori information.
The vascular trees opacified by injection of a contrast-enhancing product form a class of object of this type. This class of objects is in this case characterized by homogeneous a priori information, the 3D reconstruction of which is clearly advantageous. For physiological reasons and in the present state of opacification techniques, the acquisition of data can be carried out only in a narrow time range. Reconstruction must therefore be performed by means of a small number of projections. The projections of the vascular tree alone are obtained either by conventional techniques of logarithmic subtraction of radiographs taken at the same incidence before and after injection of a contrast enhancing product or by techniques (also conventional) of combination of radiographs taken at different energies. In all cases, the object is characterized by its positivity, its high contrast, its convexity, its "sparse" character as well as the sparse character of its projections. Positivity means that the value of the linear attenuation coefficient to be reconstructed is always positive. High contrast is brought about by the technique of opacification and subtraction. The convexity is induced by the treelike structure of the vessels under study. Finally, the "sparse" character is essential. When, an angiographic study is to be undertaken, vascularized tissues are eliminated by the aforementioned subtraction techniques. The projections of the vascular tree are then said to be "sparse" by analogy with a hollow or sparse matrix when it contains a majority of null elements since the points corresponding to the projection of the opacified tree correspond only to a few per cent of all the points of the subtracted image. Similarly, within the 3D space, a few per cent of the total volume are occupied by the vascular tree and the object can therefore also be qualified as "sparse". Other structures of medical interest (bones, etc.) can be characterized by the same a priori information and can therefore be processed by the invention.
Different approaches have already been proposed for reconstructing a 3D vascular tree. They can be classed in three categories. The first category is concerned with stereovision methods. These methods are derived from techniques developed in computer vision. They consist in putting in correspondence, in two views taken at neighboring incidences (angular separation of 6.degree. to 15.degree.) similar elements which are homologous to the object being observed. Knowing the geometry of acquisition and the coordinates of one element in each view, it is possible to deduce the coordinates of said element within the 3D space. These methods have the disadvantage of being sensitive to noise and providing low geometrical accuracy on the reconstructed object.
In order to improve this point, researches have been undertaken in order to work with two or three views taken in orthogonal directions. These extensions often involve other techniques, in particular artificial intelligence techniques for resolving ambiguities of putting in correspondence the significance of measurements belonging to different views but containing information about the same voxels in the body. But recourse even to blurred models of the tree to be reconstructed restricts the application of these methods to pathologies which do not depart too much from normal.
A second family of methods concerns the parametric algebraic methods. In order to reduce the volume of calculations, they seek to restrict the number of parameters to be estimated and utilize for this purpose a parametric model of the object which necessarily relies on a priori assumptions. Methods proposed thus make the assumption, for example, that the vessels have an elliptical cross-section. The problem of 3D reconstruction of a vessel having an elliptical cross-section basically consists in that case in estimating from measured projections the few parameters which define this ellipse in space: coordinates of its center, length of the major and minor axes, internal radiological density. These methods have the disadvantage of introducing a priori information which is too rigid. It is in fact not always possible, for example, to postulate that the cross-section of a blood vessel is elliptical. This assumption can accordingly play a disproportionate role with respect to the measurements in the determination of the solution. Thus the assumption of the elliptical cross-section, which is realistic in the general case, may prove to be false in pathological cases. And since the method of reconstruction is capable of providing only objects having an elliptical cross-section, it will therefore be conducive to errors of diagnosis.
The third methods contemplated are non-parametric algebraic methods. They do not utilize any parametric model of the object. This object is represented simply in the form in which it is sampled by a matrix of volume elements or voxels. Reduction of the number of calculations accordingly takes place by simplification of the method of estimation of the value of each voxel. For example, one technique consists of assigning to one voxel of the volume the minimum value of the measurements corresponding to the projections of said voxel in all the views. This is the so-called extreme value technique (EVT) which explicitly makes use of the fact that a vascular tree already occupies a very small part of the volume which surrounds it and that its projections also occupy a very small part of the radiographs. But this method is also based on the assumption that, for any voxel which does not belong to the tree, there exists at least one view in which this latter is projected with part of the tree, without superposition. This simple method and its variants thus have the disadvantage of resting on a flimsy hypothesis. Furthermore, they make it possible to reconstruct only the convex envelope of the objects. In fact, if these objects are concave, the regions of space which are not included in the object but contained in the concave portion of said object are always projected in conjunction with regions of the object. They are therefore always interpreted as belonging to the object. Finally, they attribute erroneous density values to the voxels of the tree. In point of fact, they substitute for the value of radiological density at a point the value of the integral of said density along the straight line of projection. The radiological density thus reconstructed will not be constant for example within a vessel having a circular cross-section even if the concentration of contrast-enhancing product is uniform.