Spherically symmetric volume elements with smooth tapering of the values near their boundaries are alternatives to the more conventional voxels for the construction of volume images in the computer. Their use, instead of voxels, introduces additional parameters which enable the user to control the shape of the volume element (blob) and consequently to control the characteristics of the images produced by iterative methods for reconstruction from projection data. For images composed of blobs, efficient algorithms have been designed for the projection and discrete back-projection operations, which are the crucial parts of iterative reconstruction methods. The authors have investigated the relationship between the values of the blob parameters and the properties of images represented by the blobs. Experiments show that using blobs in iterative reconstruction methods leads to substantial improvement in the reconstruction performance, based on visual quality and on quantitative measures, in comparison with the voxel case. The images reconstructed using appropriately chosen blobs are characterized by less image noise for both noiseless data and noisy data, without loss of image resolution.

Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser-Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.

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Spherically symmetric volume elements are alternatives to the more conventional voxels for the construction of volume images in the computer. The image representation, and the calculation of projections of it, are essential components of iterative algorithms for image reconstruction from projection data. A two-parameter family of spherical volume elements is described that allows control of the smoothness properties of the represented image, whereas conventional voxels are discontinuous. The rotational symmetry of the spherical elements leads to efficient calculation of projections of the represented image, as required in iterative reconstruction algorithms. For volume elements whose shape is ellipsoidal (rather than spherical) it is shown that efficient calculation of the projections is also possible by means of an image space transformation.

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