This paper presents a fast algorithm for calculating box splines sampled at regular intervals. This algorithm is based on the representation by directional summations, while splines are often represented by convolutions. The summation-based representation leads less computational complexity: the proposed algorithm requires fewer additions and much fewer multiplications than the algorithm based on convolutions. The proposed algorithm is evaluated in the sense of the number of additions and multiplications for three- and four-directional box splines to see how much those operations are reduced.

This paper proposes a fast method for the calculation of exponential B-splines sampled at regular intervals. This algorithm is based on a combination of FIR and IIR filters which enables a fast decomposition and reconstruction of a signal. When complex values are selected for the parameters of the exponentials, complex trigonometric functions are obtained. Only the real part of these functions are used for the interpolation of real signals, leading less bandlimited signals when they are compared with the polynomial B-spline counterparts. These characteristics were verified with 1-D and 2-D examples. This paper also discusses the effectiveness of exponential B-splines, when they are applied to image processing.

This paper proposes a novel fast algorithm for the decomposition and reconstruction of two-dimensional (2-D) signals by box splines. The authors have already proposed an algorithm to calculate the discrete box splines which enables the fast reconstruction of 2-D signals (images) from box spline coefficients. The problem still remains in the decomposition process to derive the box spline coefficients from an input image. This paper first investigates the decomposition algorithm which consists of the truncated geometric series of the inverse filter and the steepest descent method with momentum (SDM). The reconstruction process is also developed to correspond to the enlargement of images. The proposed algorithm is tested for the expansion of several natural images. As a result, the peak signal-to-noise ratio (PSNR) of the reconstructed images became more than 50 dB, which can be considered as enough high level. Moreover, the property of box splines are discussed in comparison with 2-D (the tensor product of) B-splines.