image: f(x,y)
fourier transform of f is F(u,v)
my Goal is to compute its power spectrum. [denoted by P(u,v)]
the first way to compute is by using the magnitude of fourier transform: $|F(u,v)|=\sqrt{R^2(u,v)+I^2(u,v)}$
R is the real part of F & I is the imaginary part. $P(u,v)=|F(u,v)|^2$
the second way is by using autocorrelation function and taking its fourier transform: $P(u,v)=\text{Fourier of}\{R_{ff}(n_1,n_2)\}$
$R_{ff}$ denotes autocorrelation of f with the asumption that f is wide sense stationary & ergodic stochastic process (or random field).
$R_{ff}=\underset{N \rightarrow \infty}{\text{lim}} \sum_{k_1=-N}^{N} \sum_{k_1=-N}^{N} f(k_1,k_2) f^*({k_1}-{n_1},{k_2}-{n_2})$
My question is: Are these two approach identical? If yes, How can one show these approaches lead to equal results?