2DFouriertransformation

The two-dimensional Fourier transformation is a mathematical operation that converts a function f(x, y) defined on the plane into its frequency components. In the continuous form, the 2D Fourier transform is F(u, v) = ∫∫ f(x, y) e^{-i 2π (u x + v y)} dx dy, and the inverse transform is f(x, y) = ∫∫ F(u, v) e^{i 2π (u x + v y)} du dv. Normalization conventions vary; some definitions place 1/(2π)^2 factors in the inverse. The transform assumes f is integrable (or square-integrable) and is used to analyze spatial frequency content.

For digital data, the discrete 2D Fourier transform applies to an M by N array f[m, n]. It is defined as F[k, l] = ∑_{m=0}^{M-1} ∑_{n=0}^{N-1} f[m, n] e^{-i 2π (k m / M + l n / N)}, with the inverse f[m, n] = (1/(MN)) ∑_{k=0}^{M-1} ∑_{l=0}^{N-1} F[k, l] e^{i 2π (k m / M + l n / N)}. The 2D transform is separable, meaning it can be computed efficiently by applying one-dimensional transforms along rows and then along columns. Fast algorithms, such as the fast Fourier transform (FFT), are widely used for this purpose.

Key properties include linearity, and the correspondence between spatial and frequency operations: translation in space results in a phase shift in the frequency domain; convolution in space corresponds to multiplication in frequency, and vice versa. The transform supports applications in image processing, including filtering (low-pass, high-pass), restoration, compression, and feature extraction. Practical use also involves considerations of boundary conditions, sampling, and the effects of padding to control spectral leakage.