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# 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 = (Xb X2, X3)T denote a location in a real three-dimensional bounded space, Q 2 R3, and let a physical quantity distribution in Q measured by an imaging system be represented by a function, f (X): R3 ! 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 = [A1x1, A2x2, A3x3]T, where (s = 1,2, 3) is the regular

interval between two neighboring lattice points along the s-th axis line and where xs (s = 1,2,3) is an integer. Let x = [x1,x2,x3]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 xs is bounded as 1 < xs < Ws and that the image size is W1 x W2 x W3. 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 = W1 x W2 x W3.

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