This problem can be eliminated by generating a cubic convolution function (15,16), which forces the slope of the ends of the interpolation to be zero. The problem with this approach is that the slope discontinuity at the ends of the waveform leads to amplitude ripples in a reconstructed function. Typically, this is done over a four-sample interval.

As mentioned earlier, the sinc function can be approximated by truncating its tails. 4.3-4, that R 3 (x) is continuous in its first and second derivatives at the sample points. It can be shown by direct differentiation of Eq. It is defined mathematically as R 3 ( x) = 1 2 ⎧2 - ⎪ + x 3 – ( x) 2 3 ⎨ ⎪ 1 - ( 2 – x ) 3 ⎩ 6 for for 0 ≤ x ≤ 1 1 < x ≤ 2 (4.3-4a) (4.3-4b) The cubic B-spline is a particularly attractive candidate for image interpolation because of its properties of continuity and smoothness at the sample points. Convolving the bell-shaped waveform with the square function results in a thirdorder polynomial function called a cubic B-spline (13,14). Convolution of the triangle function with the square function yields a bell-shaped interpolation waveform (in Figure 4.3-2d). The triangle function may be considered to be the result of convolving a square function with itself. Figure 4.3-3 illustrates one-dimensional interpolation using sinc, square, and triangle functions. A triangle function, defined as R 1 ( x) = ⎧x + 1 ⎪ ⎨ ⎪ ⎩1 – x for – 1 ≤x ≤ 0 for 0 IMAGE RECONSTRUCTION SYSTEMS 113 provides the first-order linear sample interpolation with trianglar interpolation waveforms.

It is defined mathematically as R 0 ( x) = 1 1 2 1 2 for –- ≤ x ≤ - (4.3-1) and zero otherwise, where for notational simplicity, the sample spacing is assumed to be of unit dimension. The simplest interpolation waveform is the square pulse function, which results in a zero-order interpolation of the samples. It is possible to approximate the sinc function by truncating it and then performing subscanning (Figure 4.3-1). As stated previously, the sinc function, provides an exact reconstruction, but it cannot be physically generated by an incoherent optical filtering system. Interpolation Functions Figure 4.3-2 illustrates several one-dimensional interpolation functions. 112 IMAGE SAMPLING AND RECONSTRUCTION 4.3.2.

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