i think i may have accidentally tuned out during this part of lecture, but what do the three dots correspond to? based on what we were talking about later in the lecture i assumed they had something to do with the frequency and/or the distance between the sine waves, but i just wanted clarification on that
motoole2
We will use the Fourier decomposition described on this slide to explain what's going on here.
Suppose that the 2D signal has the form $f(x,y) = 2 * \cos(f_x * x)+1$.
The signal can be represented through the sum of three different basis functions:
$$f(x,y) = 1 * f_{-1} + 1 * f_{0} + 1 * f_{1}$$
where
$$f_{1} = \cos(f_x * x) + j \sin(f_x * x)$$
$$f_{0} = \cos(0 * x) + j \sin(0 * x)$$
$$f_{-1} = \cos(-f_x * x) + j \sin(-f_x * x)$$
The three dots represents the amplitude of these basis functions.
Note that the frequency domain representation of a real valued signal is always conjugate symmetric about the origin, e.g., $f_{-1} = conj(f_{1})$. This results in the imaginary terms cancelling each other out.
i think i may have accidentally tuned out during this part of lecture, but what do the three dots correspond to? based on what we were talking about later in the lecture i assumed they had something to do with the frequency and/or the distance between the sine waves, but i just wanted clarification on that
We will use the Fourier decomposition described on this slide to explain what's going on here.
Suppose that the 2D signal has the form $f(x,y) = 2 * \cos(f_x * x)+1$. The signal can be represented through the sum of three different basis functions: $$f(x,y) = 1 * f_{-1} + 1 * f_{0} + 1 * f_{1}$$ where $$f_{1} = \cos(f_x * x) + j \sin(f_x * x)$$ $$f_{0} = \cos(0 * x) + j \sin(0 * x)$$ $$f_{-1} = \cos(-f_x * x) + j \sin(-f_x * x)$$
The three dots represents the amplitude of these basis functions.
Note that the frequency domain representation of a real valued signal is always conjugate symmetric about the origin, e.g., $f_{-1} = conj(f_{1})$. This results in the imaginary terms cancelling each other out.