Quincunx Sampling:

In case of quincunx sampling, number of samples is reduced by a factor of two compared to square sampling (R=S). Pure horizontal and vertical sinusoids can be reconstructed to the same frequency as with square sampling (R=S).

For sinusoids of diagonal orientations the maximum allowable frequency is lower,

Also for

Hexagonal sampling results in the sampling conditions,

for

or

For arbitrary orientation of sinusoids, hexagonal sampling schemes guarantee equal conditions omnidirectionally. Number of samples is reduced by a factor of compared to square sampling approach

Shear sampling:

The shear sampling is a specific case which physically has the same sampling positions as a rectangular grid, but artificially defines a neighbourhood relationship between samples which may actually not be direct neighbours of each other. The boundary of the baseband relating to horizontal sinusoids is in parallel to the the same as in case of rectangular sampling. For the vertical frequency, two lines of slope and intercept establish the sampling conditions .

Sampling conditions:

for

or

The reconstruction of a continuous signal by interpolation from the discrete signal is typically performed by a low pass filter having a frequency response which retains the baseband. This means that the actual base-band layout and hence the conditions for alias free reconstruction depend on the pass-band shape of the lowpass filter used for reconstruction.

Shear sampling allows reconstructing signals which clearly would violate the rigid sampling conditions of the equivalent rectangular grid, this is penalised by forbidding other frequency components.

On the whole, the bandwidth range (i.e. area of baseband) of signal frequencies which can be reconstructed alias-free is always identical, independent of shear factor. For any sampling system, the bandwidth range is identical to the determinant of the frequency sampling matrix . The definition of baseband allows certain degrees of freedom, indeed.