Question

Define power spectrum of 2d fourier transform

Answer 1

The Discrete Fourier Transform in 2D The Fourier transform is defined not only for one-dimensional signals but for functions of arbitrary dimension. Thus, two-dimensional images are nothing special from a mathematical point of view. For a two-dimensional, periodic function (e. g., an intensity image) g(u, v) of size M ×N, the discrete Fourier transform (2D DFT) is defined as G(m, n) = 1 √ MN M −1 u=0 N −1 v=0 g(u, v) · e−i2π mu M · e−i2π nv N = 1 √ MN M −1 u=0 N −1 v=0 g(u, v) · e−i2π( mu M + nv N ) (14.1) for the spectral coordinates m = 0 ...M −1 and n = 0 ...N −1. As we see, the resulting Fourier transform is again a two-dimensional function of the same size (M × N) as the original signal. Similarly, the inverse 2D DFT is defined as g(u, v) = 1 √ MN M −1 m=0 N −1 n=0 G(m, n) · ei2π mu M · ei2π nv N = 1 √ MN M −1 m=0 N −1 n=0 G(m, n) · ei2π( mu M + nv N ) (14.2)
 for the image coordinates u = 0 ...M −1 and v = 0 ...N −1. 343 14.1 Definition of the 2D DFT 14 The Discrete Fourier Transform in 2D 14.1.1 2D Basis Functions Equation (14.2) shows that a discrete two-dimensional, periodic function g(u, v) can be represented as a linear combination (i. e., as a weighted sum) of 2D sinusoids of the form ei2π( mu M + nv N ) = ei(ωmu+ωnv) (14.3) = cos - 2π mu M + nv N . CM,N m,n (u, v) + i·sin - 2π mu M + nv N . SM,N m,n (u, v) . (14.4) CM,N m,n (u, v) and SM,N m,n (u, v) are discrete, two-dimensional cosine and sine functions  with horizontal and vertical wave numbers n and m, respectively, and  the corresponding angular frequencies ωm, ωn: CM,N m,n (u, v) = cos - 2π mu M + nv N .= cos(ωmu + ωnv), (14.5) SM,N m,n (u, v) = sin - 2π mu M + nv N .= sin (ωmu + ωnv). (14.6) Each of these basis functions is periodic with M units in the horizontal direction and N units in the vertical direction. Examples Figures 14.1 and 14.2 show a set of 2D cosine functions CM,N m,n of size M × N = 16 × 16 for various combinations of wave numbers m, n = 0 ... 3. As we can clearly see, these functions correspond to a directed, cosine-shaped waveform whose orientation is determined by the wave numbers m and n. For example, the wave numbers m = n = 2 specify a cosine function CM,N 2,2 (u, v) that performs two full cycles in both the horizontal and vertical directions, thus creating a diagonally oriented, two-dimensional wave. Of course, the same holds for the corresponding sine functions. 14.1.2 Implementing the Two-Dimensional DFT As in the one-dimensional case, we could directly use the definition in Eqn. (14.1) to write a program or procedure that implements the 2D DFT. However, this is not even necessary. A minor rearrangement of Eqn. (14.1) into G(m, n) = 1 √ N N −1 v=0 ' 1 √ M M −1 u=0 g(u, v) · e−i2π mu M 1-dim. DFT of row g(·, v) ( · e−i2π nv N (14.7)