intvert.get_coeff_classes_2D

get_coeff_classes_2D(M, N, include_conjugates=True)

Returns a dictionary of classes of DFT coefficient frequencies for a 2D integer matrix.

Constructs a dictionary describing the equivalence classes of DFT coefficient frequencies for a 2D integer signal. The dictionary is structured as follows. It maps pairs \((D_1, D_2)\) of divisors of M and N to sets of coefficient classes. Each coefficient class is a FrozenSet of frequency pairs \((k, l)\). A coefficient class containing \((k, l)\) is in the set mapped to by \((D_1, D_2)\) whenever \(\gcd(k, M) = D_1\) and \(\gcd(l, N) = D_2\). If include_conjugates is False, frequencies made redundant by the signal being real are excluded. Otherwise, all frequences are included.

Parameters:

• M (int) – Height of the matrix.

• N (int) – Width of the matrix.

• include_conjugates (bool, optional) – Whether to include coefficients made redundant by the signal being real, by default True

Returns:

Dictionary mapping pairs of divisors of M and N to sets of coefficient classes.

Return type:

Dict[Tuple[int, int], Set[FrozenSet[Tuple[int, int]]]]

See also

select_coeffs_2D

selecting a partial set of DFT coefficients sufficient for inversion

get_coeff_classes_1D

analogous 1D function

Notes

If \({\bf X}\) is an integer matrix of size M, N, two DFT coefficients \(\tilde{X}_{kl}\) and \(\tilde{X}_{k'l'}\) are equivalent if there exists lambda relatively prime with M and N such that \(k = \lambda k' \pmod{M}\) and \(l = \lambda l' \pmod{N}\) [LV]. Assuming \({\bf X}\) is real implies \(\tilde{X}_{kl} = \tilde{X}_{M - k, N - l}^*\), so these DFT coefficients are trivially equivalent.

Examples

>>> intvert.get_coeff_classes_2D(4, 4)
{(4, 4): {frozenset({(0, 0)})}, (4, 1): {frozenset({(0, 1), (0, 3)})}, (4, 2): {frozenset({(0, 2)})}, (1, 4): {frozenset({(1, 0), (3, 0)})}, (1, 1): {frozenset({(3, 1), (1, 3)}), frozenset({(1, 1), (3, 3)})}, (1, 2): {frozenset({(3, 2), (1, 2)})}, (2, 4): {frozenset({(2, 0)})}, (2, 1): {frozenset({(2, 3), (2, 1)})}, (2, 2): {frozenset({(2, 2)})}}

>>> intvert.get_coeff_classes_2D(4, 4, False)
{(4, 4): {frozenset({(0, 0)})}, (4, 1): {frozenset({(0, 1)})}, (4, 2): {frozenset({(0, 2)})}, (1, 4): {frozenset({(1, 0)})}, (1, 1): {frozenset({(1, 1)}), frozenset({(1, 3)})}, (1, 2): {frozenset({(1, 2)})}, (2, 4): {frozenset({(2, 0)})}, (2, 1): {frozenset({(2, 1)})}, (2, 2): {frozenset({(2, 2)})}}