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}]^{T}, where (s = 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}]^{T}. Then an (ideal) imaging system captures an image, /(x), such that

Each point represented by the tuple, x, is called a voxel. Let us assume that x_{s} is bounded as 1 < x_{s} < W_{s} and that the image size is W_{1} x W_{2} x 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}.