Abstract-We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree . A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor . For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio. The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives.

The continuous wavelet transform (CWT) is a common signal-processing tool for the analysis of nonstationary signals. We propose here a new B-spline-based method that allows the CWT computation at any scale. A nice property of the algorithm is that the computational cost is independent of the scale value. Its complexity is of the same order as that of the fastest published methods, without being restricted to dyadic or integer scales. The method reduces to the ÿltering of an auxiliary (pre-integrated) signal with an expanded mask that acts as a kind of modiÿed 'Â a trous' ÿlter. The algorithm is well-suited for a parallel implementation. ?

Abstract-We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the -sense (not restricted to the usual = 2), where can take noninteger values. The underlying image model is specified using shiftinvariant basis functions, such as B-splines. The solution is well-defined and determined by an iterative optimization algorithm based on digital filtering. Its convergence is accelerated by the use of first and second order derivatives. For close to 1, we show that the ringing is reduced and that the histogram of the detail image is sparse as compared with the standard case, where = 2.

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).

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