Fourier domain - did I get my theory / terminology correctly?

Question

In terms of images.

The Fourier transform converts the spatial into the frequency (Fourier) domain. DC value = average value of sine waves (sine waves), F (0,0) and average brightness / gradilva of the image. Fourier has an output with a complex number .... you can get [amplitude and phase] or [real and imaginary] outputs.

What I don't get is what the volume number really is when you do FFT? I know that the image in the fourier area is the sum of the weighted sine waves, but that means the actual output.

What is spatial frequency? When I look at definitions, it defines it as the rate at which pixel values ​​change. What does it mean?

Answer 1

Thus, you can visualize it better if you parameterize the complex number information by the magnitude and phase value, it reflects the magnitude of the frequency component and phase, which gives you the position of the component in the image.

Spatial frequency usually refers to a change in the intensity value when moving through pixels. Thus, the edge of your image will have a high spatial frequency due to a sharp change in pixel values

Answer 2

The complex number outputs give you the magnitude and phase of the various vectors / signals of the base vector.

In the case of a one-dimensional signal, for example. audio, these basis vectors are complex sinusoids rotating around a unit circle with different (spatial) frequencies (which are integer multiples of the fundamental frequency) over time.

In the case of an image, the base vectors are two-dimensional (complex) tablets. Now components can also have different directions. Thus, each pixel in the Fourier space is equal to a specific combination of direction and frequency. Simply put, the direction and frequency of a planar wave in a spatial region starting in the upper left corner ends with one full period by the time it reaches the position of the corresponding pixel of the Fourier domain (in the most direct way).

To model a real sine or planetary wave, two complex basis vectors with frequencies of the same magnitude, but the negative frequency are superimposed so that their imaginary parts are canceled.

Answer 3

The result is complex because sine waves can be a combination of orthogonal sine waves and cosine waves, and a complex number is a mathematically good way to represent this sine wave.

Think of a picket fence of evenly spaced pickets throughout the image. The spatial frequency is the number of pickets in the image (horizontally or vertically). Move the picket fence in the image by half the picket (one quarter of the frequency), and the fence can switch between a sine wave, more like a cosine wave. Complex numeric representation will rotate.

What does it mean? The amazing Fourier theory is that each image (with the exception of some perverse mathematical constructions) can be represented or deconstructed into a bundle of orthogonal spatial frequency sinusoids. (Each image. Not only images of picket fences!) After converting the images to frequencies, you can make all kinds of DSPs on them, similar to sound processing or filtering, and then convert back.