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A Sinusoidal Family of Unitary Transforms
April 1979 (vol. 1 no. 4)
pp. 356-365
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.
Citation:
Anil K. Jain, "A Sinusoidal Family of Unitary Transforms," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 1, no. 4, pp. 356-365, April 1979, doi:10.1109/TPAMI.1979.4766944
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