Fractional, canonical, and simplified fractional cosine transforms

The Fourier transform can be generalized into the fractional Fourier transform (FRFT), linear canonical transform (LCT), and simplified fractional Fourier transform (SFRFT). They extend the utilities of original Fourier transform, and can solve many problems that can not be solved well by original Fourier transform. We generalize the cosine transform. We derive the fractional cosine transform (FRCT), canonical cosine transform (CCT), and simplified fractional cosine transform (SFRCT). We show that they are very similar to the FRFT, LCT, and SFRFT, but they are much more efficient for dealing with the even, real even functions. For digital implementation, FRCT and CCT can save 1/2 of the real number multiplications, and SFRCT can save 3/4. We also discuss their applications, such as optical system analysis and space-variant pattern recognition.