акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

Rank [A]:

Number of linearly independent rows or columns.

Inverse A−1:

A−1A = AA−1 = I

For square matrices only.

Trace:

Tr [A] = a(n, n) is the sum of the diagonal elements.

n

Toeplitz and Circulant Matrices:

A Toeplitz matrix, T, is a matrix that has constant elements along the main diagonal and the subdiagonals. This means that the elements t(m, n) depend only on the difference m – n, that is t (m, n) = tm−n. A matrix is called circulant if each of its rows (or columns) is a circular shift of the previous row (or column).

Orthogonal and Unitary Matrices:

An orthogonal matrix is such that its inverse is equal to its transpose, that is A is orthogonal to

So a real orthogonal matrix is also unitary, but a unitary matrix need not be orthogonal.

3.1.3Fourier Transformation

The purpose of image transformation is to transform the images to digital form in pixels to enable image processing procedures, such as noise filtering, edge enhancement, feature extraction, and so on. We will start with the one-dimensional signal. A one-dimensional signal can be represented by an orthogonal series of basis functions. For continuous functions, an orthogonal series expansion provides series coefficients that can be used for any further processing or analysis of the function. For a signal that is a one-dimension sequence, it is usually represented as a vector u of size N, {u (n) , 0 ≤ n ≤ N − 1}.

A unitary transformation can be written as

k=0

Equation (3.6) is a series representation of the sequence u(n). The columns of A T , that is,

v(k) give a representation of the original sequence u(n) and are useful in filtering, the feature extraction of images, and so on.

A Fourier series is a special case of an orthogonal series. Fourier transformation is the transformation from the spatial domain to the frequency domain.

3.1.3.1 One-Dimensional Discrete Fourier Transform

The discrete Fourier transform (DFT) of a sequence {u (n) , n = 0, . . . . . . . . . . , N − 1} is defined as

Equations (3.5) and (3.7) are not scaled properly to be unitary transformations. In image processing, it is more convenient to consider the unitary DFT, which is defined as

×

The N N unitary DFT matrix N is given by

The DFT is one of the most important transforms in digital signal and image processing.

3.1.3.2Properties of DFT and Unitary DFT

Let u(n) be an arbitrary sequence defined for n = 0,1, . . . , N – 1. A circular shift of u(n) by l, denoted by u(n – l) is defined as u(n – l) modulo N.

The DFT and unitary DFT matrices are symmetric. By definition the matrix F is symmetric. Therefore

F−1 = F

The extensions are periodic. The extension of the DFT and unitary DFT of a sequence and their inverse transforms are periodic with period N. That is, in (3.10), v(k) = v(k + N) for every k.

The DFT is the sampled spectrum of the finite sequence u(n) extended by zeros outside the interval [0, N − 1]. If we define a zero-extended sequence

3.1.3.3The Fast Fourier Transform

The fast Fourier transform (FFT) was invented by Cooley and Tukey in 1965 [1]. Its purpose was to reduce the computation time of the DFT and the unitary DFT. It is a class of algorithms that requires O(N log2 N) operations, where one operation is a real multiplication and a real addition. The exact operation count depends on N as well as the particular choice of the algorithm in that class. Most common FFT algorithms require N = 2P, where P is a positive integer.

The DFT or unitary DFT of a real sequence {x(n), n = 0, . . . , N − 1} is conjugate symmetric about N/2.

From (3.10), we have

The DFT or unitary DFT of an N × 1 real sequence has N degrees of freedom and requires the same storage capacity as the sequence itself.

3.1.3.4 Circular Convolution Theorem

The DFT of the circular convolution of two sequences is equal to the product of their DFTs, that is, if

in which DFT{x(n)}N denotes the DFT of the sequence x(n) of size N. Hence one can calculate the circular convolution by first calculating the DFT of x2(n) using (3.21) and then taking its inverse DFT. Using FFT this will take O(N log2 N) operations compared to the N2 operations required for the direct computation of (3.19).

3.1.3.5Extension from One-Dimensional Signal Processing to Two-Dimensional Image Processing

Just as a one-dimensional signal can be represented by an orthogonal set of basis functions, an image can also be expanded in terms of a discrete set of basic arrays called basis images. These basis images can be generated by unitary matrices. An image transform provides a set of coordinates or basis vectors for the vector space.

3.1.3.6 Two-Dimensional Orthogonal and Unitary Transforms

By extending the one-dimensional orthogonal series expansion to a two-dimensional expansion, one has

k,l=0

where ak,l (m, n) denotes the image transform. Equations (3.22) and (3.23) form a set of complete orthogonal discrete basis functions satisfying the following properties:

k,l=0

The elements v(k, l) are called the transform coefficients and V = {v(k.l))}is called the transformed image. The orthonormality property assures that any truncated series expansion of the form

m,n=0

when the coefficients v(k, l) are given by (3.22).

3.1.3.7 Basis Images

Let ak denote the kth column of A T , the complex conjugate of the unitary matrix AT .Define the matrices

Equation (3.30) expresses any image U as a linear combination of the N2 matrices Ak,l , k, l = 0, . . . , N – 1, which are called the basis images.

3.1.3.8The Two-Dimensional DFT

The two-dimensional DFT of an N × N image {u(m, n)} is a separable transform defined as

22 Acoustical Imaging: Techniques and Applications for Engineers

3.1.4The Z-Transform

The Z-transform is a generalization of the Fourier transform to the two-dimensional complex regime. The Z-transform of the complex sequence x(m, n) is defined as

m,n=∞

where z1 and z2 are complex variables.

The Z-transform of the impulse response of a linear shift invariant discrete system is called its transfer function.

Applying the Z-transform to the convolution theorem and to (3.38), we have

Y (z1, z2 ) = H (z1, z2 ) X (z1, z2 )

or

This shows the transfer function of the systems, the ratio of the Z-transforms of the output, and the input sequences.

The inverse Z-transform is given by the double contour integral

where the contours of integration are counterclockwise and lie in the region of convergence. The region of convergence is defined as the set of values of z1, z2 for which the series converges uniformly.