Regularity

Consider sampling a smooth function f. From the samples f (hk), and by interpolation, reconstruct a function fh that is an approximation of f. Since fis smooth, intuition tells us that it is desirable that fh be smooth as well; in turn, intuition dictates that the only way to ensure this smoothness is that ^ be smooth, too. These considerations could lead one to the following syllogism:

(1) The order of approximation L requires the reproduction of monomials of degree L — 1.

(2) Monomials of degree N = L — 1 are functions that are at least N-times differentiable.

(3) An N-times differentiable synthesis function is required to reach the Lth order of approximation.

Intuition is sometimes misleading.

A function that is at least «-times continuously differentiable is said to have regularity C". A continuous but otherwise nondifferentiable function is labeled C0, while a discontinuous function is said to possess no regularity. Some authors insist that the regularity of the synthesis function is an important issue [38]. This may be true when the differentiation of fh is needed, but differentiating data more than, say, once or twice, is uncommon in everyday applications. Often, at most the gradient of an image is needed; thus, it is not really necessary to limit the choice of synthesis functions to those that have a high degree of regularity. In fact, the conclusion of the syllogism above is incorrect, and a synthesis function does not need to be N-times differentiable to have an N + 1 order of approximation.

For example, Schaum [28] has proposed a family of interpolants inspired by Lagrangian interpolation. A member of this family is shown in Fig. 6; it is made of pieces of quadratic polynomials patched together. Despite the fact that this

!\nmulirai I rcqiLciiuy u

FIGURE 7 Approximation kernel in the Fourier domain. Solid line: nearest-neighbor. Dotted line: sinc.

!\nmulirai I rcqiLciiuy u

FIGURE 7 Approximation kernel in the Fourier domain. Solid line: nearest-neighbor. Dotted line: sinc.

function is discontinuous, it possesses an approximation order L = 3. Thus, a linear sum of those shifted functions with well-chosen coefficients is able to exactly reproduce a constant, a ramp, and a quadratic as well. In this specific case, the synthesis function is interpolating; thus, we have that

(fk = 1 ^ f (x) = 1 Vx e R Vkez\ fk = k ^ f(x) = x VxeR

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