  As an extension of the 2D fractional Fourier transform (FRFT) and a special
case of the 2D linear canonical transform (LCT), the gyrator transform was
introduced to produce rotations in twisted space/spatial-frequency planes. It
is a useful tool in optics, signal processing and image processing. In this
paper, we develop discrete gyrator transforms (DGTs) based on the 2D LCT.
Taking the advantage of the additivity property of the 2D LCT, we propose three
kinds of DGTs, each of which is a cascade of low-complexity operators. These
DGTs have different constraints, characteristics, and properties, and are
realized by different computational algorithms. Besides, we propose a kind of
DGT based on the eigenfunctions of the gyrator transform. This DGT is an
orthonormal transform, and thus its comprehensive properties, especially the
additivity property, make it more useful in many applications. We also develop
an efficient computational algorithm to significantly reduce the complexity of
this DGT. At the end, a brief review of some important applications of the DGTs
is presented, including mode conversion, sampling and reconstruction,
watermarking, and image encryption.
