# Mathematical Foundation Hidekata Hontani and Yasushi Hirano

## Signal Processing

Knowledge of signal processing provides a mathematical foundation for describing the relationships between physical objects and (image) data obtained by measuring the objects by (imaging) sensors.

### Digital Images

In many cases, a medical image is defined over a bounded three-dimensional rectangular lattice. Let a three-vector *X =* (X_{b} *X _{2}, X_{3})^{T}* denote a location in a real three-dimensional bounded space,

*Q*

*2*R

^{3}, and let a physical quantity distribution in

*Q*measured by an imaging system be represented by a function,

*f*(X): R

^{3}! R. For example, one uses a computed tomography (CT) scanner to measure physical quantities,

*f*(X), which, in the case of CT, are the degree of attenuation of the X-ray beam at each location,

*X*. In general, an image is a set of measurements obtained at rectangular lattice points in

*Q*

*Let the coordinates of each of the lattice points be denoted by*

**.***X = [A*

_{1}

**x**

_{1}*, A*

_{2}

**x**

_{2}*, A*

_{3}

**x**

_{3}*]*, where (s

^{T}*=*1,2, 3) is the regular

interval between two neighboring lattice points along the s-th axis line and where **x*** _{s}* (s = 1,2,3) is an integer. Let

*x = [*

**x**

_{1}*,*

**x**

_{2}*,*

**x**

_{3}*. Then an (ideal) imaging system captures an image, /(x), such that*

**]**^{T}

Each point represented by the tuple, x, is called a * voxel.* Let us assume that

**x***is bounded as 1 <*

_{s}

**x**

_{s}<

**W***and that the image size is*

_{s}

**W**

_{1}*x*

**W***x*

_{2}

**W**

_{3}*Concatenating all values, /(x), of all voxels into one column, one obtains a D-vector, I, which is widely used for describing an image, where*

**.**

**D***=*

**W**

_{1}*x*

**W**

_{2}*x*

**W***.*

_{3}