## Linear

The linear interpolant enjoys a large popularity because the complexity of its implementation is very low, just above that of the nearest-neighbor; moreover, some consider that it satisfies Occam's razor principle by being the simplest interpolant one can think of that builds a continuous function f out of a sequence of discrete samples {fk}. It is made of the (continuous-signal) convolution of a square pulse with itself, which yields a triangle, sometimes also named a hat or a tent function (Fig. 8). Its support covers two units; it is an interpolant, and its approximation order is 2 (it reproduces straight lines of any finite slope). Its regularity is C0, which expresses that it is continuous but not differentiable. In one dimension, this interpolant requires at most two samples to produce an interpolated value. In two dimensions, also called bilinear interpolation, its separable implementation requires four samples, and six in three dimensions (eight in the nonseparable case, where it is called trilinear interpolation [9,39]). The expression for the ID linear synthesis function is

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