Crystallinity — means periodicity, periodicity — means pattern. Fourier transform of your data can expand accessible information about the analyzed sample. Mainly, the Fourier transform is represented as an indefinite integral:. For understanding the logic behind the origin of this integral watch an awesome video:. However, indefinite integral presumes indefinite continuous data, which kinda do not exist in the digital world. For this purpose, the classical Fourier transform algorithm can be expressed as a Discrete Fourier transform DFT , which converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform:.
For more information on FFT with some code examples in Python, I highly recommend the blog post below:. Why sound?
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Just because we know that this is a bunch of sinusoidal waves stacked together, what can be more appropriate for our Fun with Fourier?! The raw waveform signal looks like this:. Now we can feed it to a 1D CNN, to see what we can get:.
Not bad! The most simple way is to use the built-in function numpy. So, in this case, it would be better to use the Lambda function inside a CNN model. Luckily Keras has its own FFT layers:.
Thus we can train CNN without any additional costs. As a result, after epochs using the same test-train split dataset, it was possible to reach Even though the above-mentioned approach gave us some improvement in accuracy, there are few moments to be addressed. Therefore, our CNN can recognize the classes, from the perspective of the frequencies, with almost similar accuracy when the net is fed with raw waveform data.
Other than this, the spectra of typical images have no discernable order, appearing random. Of course, images can be contrived to have any spectrum you desire. As shown in c , the polar form of an image spectrum is only slightly easier to understand.
The low-frequencies in the magnitude have large positive values the white corners , while the high-frequencies have small positive values the black center. Figure d shows an alternative way of displaying an image spectrum. Since the spatial domain contains a discrete signal, the frequency domain is periodic. In other words, the frequency domain arrays are duplicated an infinite number of times to the left, right, top and bottom. This brings the bright pixels at the four corners of c together in the center of d. Figure illustrates how the two-dimensional frequency domain is organized with the low-frequencies placed at the corners.
For the real part and the magnitude, the upper-right quadrant is a mirror image of the lower-left, while the upper-left is a mirror image of the lower-right. This symmetry also holds for the imaginary part and the phase, except that the values of the mirrored pixels are opposite in sign. One of the points is the positive frequency, while the other is the matching.
In equation form, this symmetry is expressed as:. These equations take into account that the frequency spectrum is periodic, repeating itself every N samples with indexes running from 0 to N This symmetry makes four points in the spectrum match with themselves. Each pair of points in the frequency domain corresponds to a sinusoid in the spatial domain. As shown in a , the value at corresponds to the zero frequency sinusoid in the spatial domain, i.
There is only one point shown in this figure, because this is one of the points that is its own match.
Discrete Fourier transform
As shown in b , c , and d , other pairs of points correspond to two-dimensional sinusoids that look like waves on the ocean. One-dimensional sinusoids have a frequency , phase , and amplitude. Two dimensional sinusoids also have a direction. The frequency and direction of each sinusoid is determined by the location of the pair of points in the frequency domain. As shown, draw a line from each point to the zero frequency location at the outside corner of the quadrant that the point is in, i.
The direction of this line determines the direction of the spatial sinusoid, while its length is proportional to the frequency of the wave. This results in the low frequencies being positioned near the corners, and the high frequencies near the center. When the spectrum is displayed with zero frequency at the center Fig.
How the FFT works
This organization is simpler to understand and work with, since all the lines are drawn to the same point. Another advantage of placing zero at the center is that it matches the frequency spectra of continuous images.
When the spatial domain is continuous, the frequency domain is aperiodic. Would my understanding of it be correct? If not, could somebody explain it to me in simple, plain english?
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Or, does anybody have anything to add to it? Last but not least, could somebody explain the "discrete fourier transform"? The fourier transform decomposes an image into its sine and cosine components. Put simply, sine and cosine are waves starting at a minimum and maximum respectively. In the real world, we can't tell whether a wave that we observe started at a maximum or minimum point, and therefore we can't really distinguish between the two.
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Therefore, sine and cosine are simply referred to as sinusoids. When applying the FT to an image, we transform it from its spatial domain into a "frequency domain", which in essence is the image represented in terms of its variation in colour and brightness over time well, not time, but space.
That is, over a number of pixels. And what are its benefits over other methods? For example, one application in literature is in shape recognition or noise elimination. In basic terms, how could one go about shape recognition using the FT? At a conceptual level, the Fourier Transform tells you what is happening in the image in terms the frequencies of those sinusoids. For example, if you have a picture of a plain wall, the values of the pixels change very little as you go from left to right or from top to bottom.
In the frequency domain that means that your image contains low frequencies, but no high frequencies. On the other hand, if you have a picture of a picket fence, then the values of the pixels change all the time as you go from left to right.
Related Fourier Transform in image processing. And a little Digital Signal Processing
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