# Linear Operation

Let **O***[***f ***]* denote an operator applied to a signal **f ***(u)* of a d-dimensional variable, *u = [***u**_{1}*, ***u**_{2}*,..., ***u*** _{d}]^{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* R^{d}, and **f ***(?)* and **g***(-)* are real functions: R^{d} ! R, the convolution of**f ***(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 **/***(***x**_{1}*, ***x**_{2}*, ***x**_{3}*)* (1 < **x***i <* Wi, **i ***= 1,2,3** )* and let a filter be denoted by

**G***(*

**x***]_,*

**x**

_{2}*,*

**x**

_{3}*)*

*where*

**,**

**G***()*is a real function: Z

^{3}! R. Assuming that the value of the filter

**G***(*

**x**

_{1}*,*

**x**

_{2}*,*

**x**

_{3}*)*is equal to zero when

*x = (*

**x**

_{1}*,*

**x**

_{2}*,*

**x**

_{3}*)*is outside of a bounded region, V=f(x

_{1},

**x**

_{2}*,*

**x**

_{3}*)*

^{T}—

**V**

_{s}<

**x**

_{s}<

**V**

_{s},

**s***=*1,2, 3}. Then, following

- 2 Fundamental Theories and Techniques
- 47

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 =* {(x_{b} *x _{2}, x_{3})^{T}*|1 <

*x*1,2,3} denote the domain of the image. The outside of

_{s}< W_{s}, s =*R*can be filled with zeros or

*I*(x

_{out}) = I(x*

_{n}) can be set, where x

_{out}^

*R*and x*

_{n}denotes the inside voxel closest

^{to x}out.