Abstract-We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths and degrees . We apply our results to derive spline-convolution-based algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the least-squares sense. We then consider the reverse problem and introduce a new spline-convolution version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-degree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.Index Terms-B-spline convolution kernel, computer tomography, filtered back-projection, Radon transform.

We present an explicit formula for spline kernels; these are defined as the convolution of several B-splines of variable widths hi and degrees ni. The spline kernels are useful for continuous signal processing algorithms that involve Bspline inner-products or the convolution of several spline basis functions. We apply our results to the derivation of spline-based algorithms for two classes of problems. The first is the resizing of images with arbitrary scaling factors. The second is the computation of the Radon transform and of its inverse; in particular, we present a new spline-based version of the filtered backprojection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-order formulations (e.g., piecewise constant or piecewise linear models).

We propose a novel technique for coding of texture and three-dimensional data which takes the viewer position into account. This allows to transmit only the most visible parts of information that is needed to render a virtual scene. The rendering operation is modelled and studied using filtering and sampling theory. The technique is applied on real data and results are given for an operational encoding system.

We consider using spline interpolation to improve the standard filtered back-projection (FBP) tomographic reconstruction algorithm. In particular, we propose to link the design of the filtering operator with the interpolation model that is applied to the sinogram. The key idea is to combine the ramp filtering and the spline fitting process into a single filtering operation. We consider three different approaches. In the first, we simply adapt the standard FBP for spline interpolation. In the second approach, we replace the interpolation by an oblique projection onto the same spline space; this increases the peak signal-noise ratio by up to 2.5 dB. In the third approach, we perform an explicit discretization by observing that the ramp filter is equivalent to a fractional derivative operator that can be evaluated analytically for splines. This allows for an exact implementation of the ramp filter and improves the image quality by an additional 0.2 dB. This comparison is unique as the first method has been published only for degree n=0, whereas the two other methods are novel. We stress that the modification of the filter improve the reconstruction quality especially at low (faster) interpolation degrees (n = 1); the difference between the methods becomes marginal for cubic or higher degrees (n ≥ 3).

We present a novel technique for efficient coding of texture to be mapped on 3D landscapes. The technique enables to stream the data across the network using a back-channel. The use of Wavelet and Discrete Cosine Transforms is investigated and compared. This technology has been proposed has a tool for the emerging MPEG-4 standard.