intvert.sample_2D

sample_2D(signal, known_coeffs={})

Sample DFT coefficients of a 2D integer signal.

Samples a subset of the DFT coefficients of a 2D integer signal in frequency space. DFT coefficients besides the sampled ones are set to 0, and the inverse DFT is returned. If known_coeffs is given, the DFT frequencies in known_coeffs are sampled. Otherwise, a minimial set of DFT coefficients is sampled to ensure uniqueness of recovery. To make the returned signal real, conjugate pairs of DFT coefficients are both sampled, as described in Notes.

Parameters:

• signal (arraylike) – Signal to be sampled.

• known_coeffs (Dict, optional) – Dictionary of coefficients to sample, structured as in get_coeff_classes_2D, by default None

Returns:

out – Real space signal sampled along the last two axes.

Return type:

mpfr ndarray

See also

select_coeffs_2D

selecting a partial set of known DFT coefficients

invert_2D

inverting the sampled matrix to recover the original integer matrix

sample_1D

analogous 1D function

Notes

Let \((N_1, N_2)\) be the size of the last two axes of signal. If the frequency \((k, l)\) is to be sampled, the frequency \((N_1 - k, N_2 - l)\) is also sampled. This ensures that the sampled signal in real space is real-valued. If we write \(S\) as the set of known DFT frequencies, which satisfies \((k, l) \in S \implies (N_1 - k, N_2 - l) \in S\), the returned signal \(\overline{\bf X}\) is given entrywise by

\[\overline{X}_{mn} = \sum_{(k, l) \in S} \tilde{X}_{kl} e^{-2\pi{\rm i}(mk/N_1 + nl/N_2)}.\]