The Fourier transform of a lattice of vertical slits is composed of
equidistant peaks displayed along the horizontal direction going
through the Fourier space origin:
Object
"lattice" formed by the recurrence of equidistant and vertical slits & its Fourier transform 
The Fourier transform is then spread along the direction perpendicular to the slits, as evidences a lattice of horizontal slits:
Object "lattice" formed by the recurrence of equidistant and horizontal slits & its Fourier transform 
Then, one can obtain a two dimensional lattice simply by multiplying
these two one dimensional vertical & horizontal lattices:
2D lattice formed by two crossed 1D lattices & its Fourier transform 
The obtained Fourier transform
is a set of peaks displayed on a recantgular lattice, but contratry to
the direct object the vertical equidistance is larger than the
horizontal one in the Fourier space. The Fourier transform of a lattice
is then a lattice.
If, starting from this last Fourier transform one only keeps the
horizontal row going trough the origin (using a masking tool for
example), the image obtained by inverse Fourier transform will be a
lattice of vertical slits:
Using a
masking tool to hide all peaks excepted those of the horizontal
direction one creates a new Fourier space object, which corresponds to
a lattice of vertical slits in direct space 
Similarly, if using a masking tool one only keeps the vertical column
going through the origin, the image obtained by inverse Fourier
transform will be a lattice of horizontal slits:
Using a
masking tool to hide all peaks excepted those of the vertical direction
one creates a new Fourier space object, which corresponds to a lattice
of horizontal slits in direct space 
However, it should not be thought that one can select one of the
original 1D vertical or horizontal lattice using the corresponding
masking tool. Indeed, if the masking tool select only now a diagonal
row of the FT, the reconstructed image by inverse FT is a lattice of
slits perpendicular to that particular direction, and this lattice does
not exists as such in the original 2D lattice! Then, applying a mask on
a Fourier transform alters the corresponding image, and this is very
important for example in microscopy where one wishes to form an
enlarged but faithful image of a too tiny object using a apparatus
(microscope) which may modify the FT of the object and thus the
observed image.
Using a
masking tool to hide all peaks excepted those along a diagonal row of
the FT one creates a new Fourier space object, which corresponds to a
lattice of slits perpendicular to this particular direction 
