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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