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Differentiation of Discrete Signals

A finite difference operator is applied to a given discrete function to compute approximately the differential coefficients of a continuous function represented by the discrete given function. Let a continuous and smooth function be denoted by f (x) where x e R, and let fn = f (nA) (n 2 Z) denote a discrete function, where A > 0 is a sampling interval.

The Taylor expansion off (x) is given as follows:

where the positive real number, e, denotes some small perturbation.

Assume that x = nA. When e = A, the following difference operation is obtained:

Let a discrete filter be denoted by gn such that

Then, Eq. (2.27) is a convolution between fn and gn:

A different operation for the differentiation can be obtained as follows:

Computing Eq. (2.30) minus Eq. (2.31) obtains Substituting e = A obtains The above equation can be rewritten as where

i

Adding Eqs. (2.30)-(2.31) obtains an operator for approximating the second derivative:

As shown in (2.1), an image, I(x), is a discrete signal obtained by measuring a spatial distribution of some physical quantity, f (X), at the lattice points. The spatial differential coefficients of f (X) can be computed approximately by convolving the given image, I(x), with a filter, g, for computing the finite differentiation. The finite differentiation of an image is used for approximately computing differentials, e.g., the gradient at each location, and is one of the most important operations in medical image analysis. The accuracy of the approximation of the finite differentiation varies depending on the coefficients of the filter. It is known, for example, that one can approximate the direction of the computed gradient vector more accurately by applying a consistent filter [31].

 
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