Image Processing

F(k) and G(k) are the Fourier transforms of the extended sequences fe(x) and ge(x), respectively.

The diagonal elements of D in eq. (4.30) are the eigenvalues of H. Following eq. (4.23),

$D\left(k,k\right)=\lambda \left(k\right)={\displaystyle \sum _{i=0}^{M-1}{h}_e\left(i\right)\exp \left(-j\frac{2\pi }{M}ki\right)=MH\left(k\right)k=0,1,\mathrm{...},M-1\left(4.33\right)}$

where H(k) is the Fourier transforms of the extended sequences he(x).

Combining eq. (4.31) to eq. (4.33) yields

$G\left(k\right)=M\times H\left(k\right)F\left(k\right)k=0,1,\mathrm{...},M-1\left(4.34\right)$

The right side of eq. (4.34) is the convolution of fe(x) and he(x) in the frequency domain. It can be calculated with the help of FFT.

Now, consider the 2-D cases (with noise). Taking eq. (4.28) into eq. (4.20), and multiplying both sides by W