TITLE: Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?
QUESTION [13 upvotes]: If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$)
As Marek already stated, this is even not uniquely possible for $f(x)=c\in\mathbb C$, since the global phase cannot be re-determined. So please let me extend the question to

Under what circumstances is the phase-retrieval (up to a global phase) uniquely possible, and what ambiguities could arise otherwise?

REPLY [5 votes]: The right way to ask the question is: given a function $f\in L²(\mathbb{R})$, can $f$ be determined from $|f|$ and $|\widehat{f}|$ up to a multiplicative constant $c$ of modulus $|c|=1$.
This question dates back to Pauli and the answer is no. One can construct counter examples of the form $a\gamma(x-x_0)+b\gamma(x)+c\gamma(x+x_0)$ with $a,b,c$ properly chose ($\gamma(x)=e^{-\pi x²}$ the standard gaussian so that it is ots own Fourier transform). An other construction is as follows:
take $\chi=\mathbf{1}_{[0,1/2]}$ $(a_j)_{j\in\mathbb{Z}}$ a sequence with finite support (to simplify) and $f(x)=\sum_j a_j\chi(x-j)$ so that
$\hat f(\xi)=\sum_j a_je^{2i\pi j\xi}\hat\chi(\xi)$.
Now we want to construct a sequence $(b_j)$ such that $|a_j|=|b_j|$ and
$\left|\sum_j a_je^{2i\pi j\xi}\right|=\left|\sum_j b_je^{2i\pi j\xi}\right|$.
This can be done via a Riesz product: take $\alpha_1,\ldots,\alpha_N$ a finite real sequence, $\varepsilon_1,\ldots,\varepsilon_N$ a finite sequence of $\pm1$ and consider
$$
\prod_{k=1}^N (1+i\alpha_j\varepsilon_j\sin 2\pi 3^j\xi)=\sum a_j^{(\varepsilon)}e^{2i\pi j\xi}.
$$
Changing a $\varepsilon_j$ from $+1$ to $-1$ conjugates one of the factors on the left hand side, so it does not change the modulus. Now the same happens for the $a_j^{(\varepsilon)}$: each of them is either $0$ or a product of $i\alpha_j\varepsilon_j$ (up to a constant) -- the point is that it is not the sum of products of $i\alpha_j\varepsilon_j$'s, this is why we took the $3^j$ in the sine!.