Issue No. 04 - April (1979 vol. 1)
Anil K. Jain , MEMBER, IEEE, Department of Electrical Engineering, University of California, Davis, CA 95616.
A new family of unitary transforms is introduced. It is shown that the well-known discrete Fourier, cosine, sine, and the Karhunen-Loeve (KL) (for first-order stationary Markov processes) transforms are members of this family. All the member transforms of this family are sinusoidal sequences that are asymptotically equivalent. For finite-length data, these transforms provide different approximations to the KL transform of the said data. From the theory of these transforms some well-known facts about orthogonal transforms are easily explained and some widely misunderstood concepts are brought to light. For example, the near-optimal behavior of the even discrete cosine transform to the KL transform of first-order Markov processes is explained and, at the same time, it is shown that this transform is not always such a good (or near-optimal) approximation to the above-mentioned KL transform. It is also shown that each member of the sinusoidal family is the KL transform of a unique, first-order, non-stationary (in general), Markov process. Asymptotic equivalence and other interesting properties of these transforms can be studied by analyzing the underlying Markov processes.
Anil K. Jain, "A Sinusoidal Family of Unitary Transforms", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 1, no. , pp. 356-365, April 1979, doi:10.1109/TPAMI.1979.4766944