The cause of Gibbs phenomenon
When we learn Fourier transform in a simple way, we know that there is a great debate in the field of mathematics about whether sine curves can be combined into an angular signal, and the protagonists of this debate are naturally Fourier and Lagrange. Of course, there is no problem with the two leading actors, and the plot has come to an end.
Until 1898, Albert Michelson made a harmonic analyzer. When he tested the square wave, he was surprised to find that the XNt of the square wave fluctuated around the discontinuous point, and the peak value of this fluctuation did not seem to decrease with the increase of n! So he wrote to Gibbs, a famous mathematical physicist at that time. Gibbs checked this result and expressed his opinion at will: with the increase of n, some fluctuations are compressed to discontinuous points, but for any finite value of n, the peak value of fluctuations remains unchanged, which is Gibbs phenomenon.
gibbs phenomenon
Because there are many mature fast algorithms, FFT algorithm, and its efficiency is close to the best, Fourier transform of images has been studied earlier and more widely. Its shortcoming lies in the so-called Gibbs phenomenon caused by the discontinuity of adjacent sub-image data at each boundary. This is because the two-dimensional Fourier transform of image data is essentially the Fourier expansion of two-dimensional images. Of course, this two-dimensional image should be considered periodic. Because the transform coefficients of the sub-image are discontinuous at the boundary, the restored sub-image will be discontinuous at its boundary. Therefore, the whole restored image composed of restored sub-images will present a block structure with the size of the sub-images faintly visible, which will affect the quality of the whole image. When the sub-image size is small, the situation is more serious.
Explanation of Gibbs phenomenon
Gibbs phenomenon means that the truncated approximation XNt of Fourier series of discontinuous signal Xt generally shows high-frequency fluctuation and excess near discontinuous points. If such approximation is used in practical situations, n should be chosen large enough to ensure that the total energy possessed by these fluctuations can be ignored. Of course, in the limit case, the energy of approximation error is zero, and the Fourier series of discontinuous signals such as square waves indicates convergence.
Gibbs phenomenon actually occurs because Fourier transform itself has many mature fast algorithms, such as FFT, and the efficiency is close to the best. However, because the two-dimensional Fourier transform of image data is actually the Fourier expansion of a two-dimensional image, of course, this two-dimensional image is considered periodic. Because the transform coefficients of the sub-image are discontinuous on the boundary, the restored sub-image will also be discontinuous on its boundary. Therefore, the whole restored image composed of restored sub-images will present a block structure with the size of the sub-images faintly visible, which will affect the quality of the whole image. This is the reason why Gibbs phenomenon appears at the breakpoint when Fourier transform analyzes square waves.