2d Fft Image Analysis Essay

You have a function of the spatial coordinates (x, y), the coordinates of the original image. Suppose, for clarity, that we are talking about a value from 0 to 255 for each (x, y) point in your original image. The transform is a function, again from 0 to 255, of the momentum coordinates (k1, k2) . The point (0, 0) - the sun - corresponds to the intensity of the constant part of the original function. Don't think, for a moment, to the fact that it represents an image, think of it like... a 2d bar chart or something like that. The constant is the average over the (periodically arranged) image. As you progress from the center you are sampling at higher frequencies (with sinusoidal and cosinusoidal function of increasing frequency). Given the spatial resolution of the details of your original image, you can see that the corners (high k1 frequency, high k2 frequency) are black (that is, the intensity of the transfor is low), and the central zone, lighter, correesponds to the "typical" spatial lenght of the details of your image. If you had took a picture of a more regular object (a grid?) you would have found a "typical" k corresponding to your "typycal" lenght (for example, this is the process that is used in physics to reconstructs the features of cristals).

The central line corresponds to the average values along the y direction for the various sampling frequencies along the x direction. It is roughly constant: this means that the average value of the image along the short side, independently of the frequency of sampling along the long side, is the same. This should be because the image exhibits a symmetry (the horizon) with a single feature (the girl) in a very concentrated region of space. It is relatively bright because the average value is influenced by the sky, which is mostly uniform and bright.

As an exercise, you could try to take a picture of a single/a few light object against a dark background and compare the results.

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