# Linear Operation

Let O[f ] denote an operator applied to a signal f (u) of a d-dimensional variable, u = [u1, u2,..., ud]T. It is said to be a linear operator if it satisfies the following conditions for any pair of signals, f (u) and g(u):

and

i

where a is a scalar [30].

# Convolution

Convolution between a target function and a filter function represents a linear filtering operation. Let f (u) denote a target (input) function and let g(u) be functions defined over a three-dimensional space, where u e Rd, and f (?) and g(-) are real functions: Rd ! R, the convolution off (u) and g(u) is defined as follows:

The convolution defined above is linear because it satisfies Eqs. (2.2) and (2.3).

Let an image be denoted by /(x1, x2, x3) (1 < xi < Wi, i = 1,2,3) and let a filter be denoted by G(x]_, x2, x3), where G() is a real function: Z3 ! R. Assuming that the value of the filter G(x1, x2, x3) is equal to zero when x = (x1, x2, x3) is outside of a bounded region, V=f(x1, x2, x3)TVs < xs < Vs, s = 1,2, 3}. Then, following

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Eq. (2.4), the convolution of the image and the Alter is defined as

For computing H(x) in (2.4) at all voxels in the image, one needs values of voxels outside of the image regions. Let R = {(xb x2, x3)T|1 < xs < Ws, s = 1,2,3} denote the domain of the image. The outside of R can be filled with zeros or I(xout) = I(x*n) can be set, where xout ^ R and x*n denotes the inside voxel closest

to xout.