The mathematical principle in image processing 17--convolution theorem and its proof

Welcome to my Blog column "A detailed explanation of mathematical principles in image Processing"

Full-text catalogs see mathematical Principles in Image Processing (master)

http://blog.csdn.net/baimafujinji/article/details/48467225

A detailed explanation of the mathematical Principles in image processing (part of the link has been posted)

http://blog.csdn.net/baimafujinji/article/details/48751037

1.4.5 convolution theorem and its proof

Convolution theorem is an important property of Fourier transform. Convolution theorem indicates that the Fourier transform of function convolution is the product of Fourier transform of function. In other words, the convolution in one domain corresponds to the product in another domain, for example, the convolution in the time domain corresponds to the product in the frequency domain.

This theorem is also set up for variations of Fourier transforms, such as Laplace transform and Z-transform. It is important to note that the above notation is correct only for a particular form, because the transformation may be regularized in other ways, thus causing other constant factors to appear in the above relational formula.
Let us prove the time domain convolution theorem, the proof of the frequency domain convolution theorem is similar, the reader can prove it by itself.
Proof: the definition of convolution

Fourier transform the function of the signal in the frequency domain analysis, we can take the time domain signal as a linear superposition of a number of sine waves, the role of Fourier transform is to obtain the amplitude and phase of these signals. Since the fixed time domain signal is a number of fixed sine signal superposition, in the case of no change in amplitude, moving the signal on the time axis, it is equivalent to moving a number of sinusoidal signals at the same time, the phase change of these sinusoidal signals, but the amplitude is unchanged, reflected in the frequency domain is the Fourier transform results of the mode, and phase change. Therefore, the time-lapse property actually indicates that when a signal is shifted along the time axis, the size of each frequency component does not change, but the phase changes.
Since the nature of the Fourier transform is mentioned here, we will also add some relevant content about the Parseval's theorem. The theorem was first deduced by the French mathematician Parseval's (Marc-antoine Parseval) in 1799 of the theory of progression, which was then applied to the Fourier series. The expression of the Parseval's theorem is this:

To sum up, the original conclusion is proof.
We also introduced the Fourier series in the plural form, and we deduced the Parseval's equation corresponding to the complex form Fourier transform. Here again to give the Fourier series of the plural form of expression, the specific derivation process please refer to the previous article

The Parseval's theorem links the calculation of the energy or power of a signal with the spectral function or spectrum, indicating that the energy (power) contained in a signal is identical to the sum of the energy (power) of each component of the signal in the complete orthogonal function set. In other words, the total energy of the energy signal is equal to the continuous sum of the energy that each frequency component contributes separately, while the average power of the periodic power signal equals the sum of the power separately contributed by each frequency component.

The mathematical principle in image processing 17--convolution theorem and its proof