Fractal Interpolation Waveforms

Abstract

Fractal interpolation is a method of generating functions that pass through given points. We describe the method through an example, illustrated in Figures 1-4. Figure 1 shows four points, through which we will pass a function. Figure 2 shows linear interpolation through the four points; this is the stage in fractal interpolation, and will be referred to as the first To obtain the second iteration, shown in Figure 3, the whole function of Figure 2 is superimposed on each line segment of Figure 2, with the vertical scale multiplied (in this instance) by 0.7. The third iteration is obtained by copying the function of Figure 3 onto each line segment of Figure 2, and so on. Each iteration is thus obtained as three distorted copies of the previous iteration. (This process of distorting and copying is a so-called shear transformation; the heart of the construction is, therefore, three shear transformations.) The function obtained as the limit of infinitely many iterations is the fractal interpolation function. Figure 4 shows the eighth iteration of our example. The limit function is a continuous function that passes through the originally specified points, so it does interpolate. Regarded as a subset of the plane, the graph of the function has, in general, a (fractional) dimension greater than 1 but less than 2, so it is a fractal. The dimension of the graph of the limit function of Figures 1-4 is 1.675. Fractal interpolation is interesting for musical applications because complicated shapes can be specified with relatively little information, namely, the coordinates of the original points, and for each line segment in the iteration (linear interpolation), a number between -1.0 and 1.0 called the displacement for that segment. If we take x values in the range 0 to 1.0 and y values in the range -1.0 to 1.0, this information for the example of Figures 1-4 is just the four points (0, 0), (0.333, 0.5), (0.667, 0.5) and (1.0, 0) and the three displacements 0.7, 0.7, and 0.7. A negative displacement simply means that the waveform of the previous iteration is inverted before being scaled and copied. Figure 5 shows the eighth iteration of the function through the same points as in Figure 1, but with displacements -0.7, -0.7, and -0.7. Figure 6 again shows the same points, but with displacements 0.2, -0.5, and 0.4. Figures 7 and 8 show the and tenth iterations of an extreme example; there are three original points at (0, 0), (0.7853, 0.5), and (1.0, 0), and the two displacements are 0.9 and -0.9.