Fast Computation of Exponential B-splines Takeshi Asahi, Koichi Ichige and Rokuya Ishii Division of Electrical and Computer Engineering, Yokohama National University Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan Tel: +81-45-339-4138, Fax: +81-45-338-1157 E-mail: takeshiasahi@netscape.net, koichi@dnj.ynu.ac.jp, ishii@dnj.ynu.ac.jp Abstract: 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 ex- amples. 1. Introduction Splines consist of piecewise functions, which have a cer- tain continuity at some joint points called knots. Un- til now, in the family of splines, the polynomial ones have been extensively developed [1]. However, a more general formulation of these splines are the exponential ones [2, 3], where polynomial splines can be considered as a specific case. In the present work, a fast algorithm is proposed in order to calculate discrete exponential B-splines at equally spaced knots. This algorithm is an extension of a fast algorithm for calculating the polynomial splines [4], where its calculation is mainly based on the appli- cation of IIR filters in the decomposition, as well as the reconstruction of the interpolated signal. Therefore, the calculation of discrete polynomial B-splines is a partic- ular case, when the parameters of the exponents are set to be zero. Besides, the use of a complex parameter in the exponent leads to one kind of trigonometric splines. In this way, less band-limted functions can be obtained in order to carry out an interpolation. This fact may prove advantageous for the processing of digital images. 2. Exponential B-splines and cone splines A family of continuous piecewise basic spline functions can be obtained by multifold convolution of functions w i (x),i =1,...,n: γ n (x)= w 1  ∗ ··· ∗ w n    n (x), where  ∗ denotes the convolution integral operation. These weights are defined in the domain x ∈ [-1/2, 1/2) and being 0 otherwise for the case of the centered basic splines and the domain x ∈ [0, 1) for the shifted ones. In the case of polynomial B-splines, w i (·) is a rectangu- lar function of height 1. An example of a polynomial B-spline (order n = 2) can be seen in fig.1(a). In order to obtain the exponential B-splines, the weights will be exponential functions w µ i (x) = w µ i 1 exp(µ i x), with µ i ∈ C being the parameter of the exponential function and w µ i 1 ∈ C a normalization fac- tor. So, by defining the vector  µ r =(µ 1 ,...,µ r ) T , the exponential B-spline can also be expressed as [2]: T  µ n (x)=  k∈Z d  µ n [k] σ  µ n (x - k), where the single and multiple discrete difference func- tions are given by d µ r [k] = δ[k] - e µ r δ[k - 1] d  µ n [k] = d µ 1 ∗ d µ 2 ∗···∗ d µ n    n [k] respectively, with δ[k] denoting the Kronecker’s delta function. Also, σ  µ n (·) is the continuous exponential truncated power or exponential cone spline. The lat- ter can be defined by σ µ 1 (x)=  w 1 (x - 1/2) x ∈ [0, ∞) 0 otherwise and then recursively calculated by σ  µ r (x)=  ∞ 0 w r (x 1 - 1/2) σ  µ (r-1) (x - x 1 ) dx 1 . It is convenient to introduce the notation for the ex- panded exponential B-spline: T  µ n m (x)= T  µ n (x/m) = λ  µ n m  k∈Z d  µ n [k] σ  µ n /m (x - mk) (1) where m is a positive integer and λ  µ n m = m n  r=1 sinh(µ r /2m) sinh(µ r /2) exp  - m - 1 2m µ r  is a scalar factor necessary for the expansion. The continuous exponential cone spline σ  µ n (·) can be related to the discrete exponential cone spline s  µ n [·] by the following equation [3]: σ  µ n (x)=  ℓ≥0 s  µ n [ℓ] T  µ n (x - ℓ),