## Classical Interpolation

Although many methods have been designed to perform interpolation, we concentrate here on linear algorithms of the form f (x) = 52 -fc^int (x " k) Vx =(*!> •••, Xq) E ^ (l)

k e zq where an interpolated value f (x) at some (perhaps noninteger) coordinate x in a space of dimension q is expressed as a linear combination of the samples fk evaluated at integer coordinates k =(k1, k2, •••, kq) eZq, the weights being given by the values of the function (x — k). Typical values of the space dimension correspond to bidimensional images (2D), with q = 2, and tridimensional volumes (3D), with q = 3. Without loss of generality, we assume that the regular sampling step is unity. Since we restrict this discussion to exact interpolation, we ask the function to satisfy the interpolation property— as we shall soon see, it must vanish for all integer arguments except at the origin, where it must take a unit value. A classical example of the synthesis function is the sine function, in which case all synthesized functions are band-limited.

As expressed in (1), the summation is performed over all integer coordinates k e Zq, covering the whole of the Cartesian grid of sampling locations, irrespective of whether or not there actually exists a physically acquired sample f^ at some specific k0. In practice however, the number of known samples is always finite; thus, in order to satisfy the formal convention in (1), we have to extend this finite number to infinity by setting suitable boundary conditions over the interpolated function, for example using mirror symmetries (see Appendix). Now that the value of any sample /k is defined (that is, either measured or determined), we can carry the summation all the way to — and from — infinity.

The only remaining freedom lies in the choice of the synthesis function ^int. This restriction is but apparent. Numerous candidates have been proposed: Appledorn [21], B-spline [22], Dodgson [23], Gauss, Hermite, Keys [24, 25], Lagrange, linear, Newton, NURBS [26], o-Moms [27], Rom-Catmull, sinc, Schaum [28], Thiele, and more. In addition to them, a large palette of apodization windows have been proposed for the practical realization of the sinc function, which at first sight is the most natural function for interpolation. Their naming convention sounds like a pantheon of mathematicians [29]: Abel, Barcilon-Temes, Bartlet, Blackman, Blackman-Harris, Bochner, Bohman, Cauchy, Dirichlet, Dolph-Chebyshev, Fejér, Gaussian, Hamming, Hanning, Hanning-Poisson, Jackson, Kaiser-Bessel, Parzen, Poisson, Riemann, Riesz, Tukey, de la Vallée-Poussin, Weierstrass, and more.

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