# Gray-scale image--discrete Fourier transform (DFT) in frequency domain filter Fourier transform

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The opening crap.As always the opening nonsense, today introduced the discrete Fourier transform (DFT), learning to this, do not dare to say to Fourier have much understanding, but at least his family genealogy is smoothed Shun, this family does not matter quite complex, let how many people are confused for a long time, estimated that the university learned not many people, Work will not be able to go to study, but it is fortunate to feel that they have learned a bit of things, but also to solve the previous mind of a doubt, the latter hope to be able to learn the Fourier mathematical knowledge to solve practical problems in the algorithm. Discrete Fourier transform

first of all, the existence of DFT, we must make it clear that the computer can only handle discrete finite sequences, whether in the time domain, or the frequency domain if the calculation of the continuous formula, then the computer can not be implemented, so throughout the previous few Fourier, no one satisfies this condition, Although the Fourier series is discrete but the time domain and the frequency domain are infinite, the Fourier transform time domain and the frequency domain are infinite, moreover is continuous, the DTFT time domain is the discrete limited, but the frequency domain is the continuous infinite, therefore all does not satisfy the computer to the data request, but through the observation we can discover, to dtft slightly transforms, Or the Fourier series of discrete periodic sequences can be cropped to meet our needs of the data type, that is, limited and discrete data.

According to the sampling method of time domain, we make the equal interval sampling in frequency domain, and get the discrete sequence of frequency domain in the way of Dtft.

Or we can be the original signal, the real-time domain discrete finite signal, periodic replication, the period is greater than the length of the sequence to ensure that the signal shape is not changed, this discrete period sequence exists Fourier series, but the series is periodic discrete, that is, infinite, we intercept one of the complete period, we get the frequency domain discrete limited signal.

The discrete Fourier transform formula is very similar to the discrete-period Fourier series, for the sequence {X[n]} where 0=<n<n:

The x in the hat is the frequency domain sequence, and the corresponding conversion from the frequency domain to the time domain:

The above two equations are discrete Fourier transforms. As can be seen from the formula, both time and frequency domain signals are discrete sequences of length n.

Mathematical derivation under a brief derivation of the DfT, suppose X (t), where T is on the [0,l] interval, now samples the time domain, the sampling period is T, which is the new sequence x[n]=x (k*t), k=0,1,2,3,... N-1, a total of n sample points, where n=l/t, is a discrete sequence:
its Fourier transform is:
This equation is actually the discrete-time Fourier transform that was mentioned earlier. Following the sampling of its frequency domain, we know that according to the law of sampling, the sampling frequency must be greater than the maximum frequency of the original signal twice times, so the sampling period must be less than the original signal minimum period of one-second, so, a discrete finite sequence, its frequency maximum does not exceed its sampling period t twice times the reciprocal, That is, the frequency domain Maximum frequency w< (2*PI)/(2*t), the corresponding negative frequency should be-w>-(2*PI)/(2*t), so the discrete sequence corresponds to a frequency width of (2*PI)/T. Because the length of the original sequence is L, its minimum frequency interval is 2*pi/l. Then the discrete sequence length of the frequency domain is (2*pi/t)/(2*pi/l) =l/t=N, corresponding to the original sequence, the sampling interval of the GU frequency domain is 2*pi/n. the GU sampling sequence is::
the above formula is normalized, so that t=1 can get the DfT equation. Therefore, the complete DFT process is: for the continuous non-periodic signal, it is discretized, get Dtft, and then the frequency domain is discretized again, to obtain a DFT. or the finite discrete sequence is copied, so that the original signal into a discrete periodic signal, the Fourier series, to find the main value, that is, a continuous point in a cycle, is the corresponding DFT. Properties
SummarizeIn Summary, DFT is introduced roughly, because its calculation and discrete periodic signal Fourier series calculation is similar, and the formula is very easy to accept, so only a simple description, before a summary of all the Fourier basic knowledge, and then introduce two-dimensional Fourier, A circle around the outside will return to the image.

Gray-scale image--discrete Fourier transform (DFT) in frequency domain filter Fourier transform

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