How I Came up with the Discrete Cosine Transform Nasir Ahmed Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, New Mexico 87131

Home , Digital signal, Nasir Ahmed (engineer)

mxT*L. BImL4L. PRocEsSlNG 1,4-5 (1991)

How I Came Up with the Discrete Cosine Transform Nasir Ahmed Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, New Mexico 87131

During the late sixties and early seventies, there to study a “cosine transform” using Chebyshev poly- was a great deal of research activity related to digital nomials of the form orthogonal transforms and their use for image data compression. As such, there were a large number of T,(m) = (l/N)‘/“, m = 1, 2, . . . , N transforms being introduced with claims of better performance relative to others transforms. Such compari- em- lh) h = 1 2 N T,(m) = (2/N)‘%os 2N t ,...) . sons were typically made on a qualitative basis, by viewing a set of “standard” images that had been sub- jected to data compression using transform coding The motivation for looking into such “cosine func- techniques. At the same time, a number of researchers tions” was that they closely resembled KLT basis were doing some excellent work on making compari- functions for a range of values of the correlation coef- sons on a quantitative basis. In particular, researchers ficient p (in the covariance matrix). Further, this at the University of Southern California’s Image Pro- range of values for p was relevant to image data per- cessing Institute (Bill Pratt, Harry Andrews, Ali Ha- taining to a variety of applications. bibi, and others) and the University of California at Much to my disappointment, NSF did not fund the Los Angeles (Judea Pearl) played a key role. In this proposal; I recall one reviewer’s comment to the effect regard, the variance criterion and the rate distortion that the whole idea seemed “too simple.” Hence I de- criterion were developed and used extensively as per- cided to work on this problem with my Ph.D. student formance measures for evaluating image data com- Mr. T. Natarajan and my friend Dr. Ram Mohan Rao pression. In addition, the Karhunen-Loeve trans- at the University of Texas at Arlington. In fact, I re- form (KLT) evolved as the optimal transform for com- member dedicating the whole summer of 1973 to work parison purposes. With this as background, I can now on this problem. The results that we got appeared too address the DCT issue. good to be true, and I therefore decided to consult What intrigued me was that the KLT was indeed Harry Andrews later that year at a conference in New the optimal transform on the basis of the mean- Orleans. We were both invited speakers at a session square-error criterion and the first-order Markov pro- on Walsh Functions. Harry suggested that I check out cess model, and yet there was no efficient algorithm the performance of this “cosine transform” using the available to compute it. As such, the focus of my re- rate distortion criterion. He then sent me the com- search was to determine whether it would be possible puter program to do so. The results again showed that to come up with a good approximation to the KLT this transform performed better than all the others, that could be computed efficiently. An approach that and its performance compared very closely with that I thought might be worth looking into was Chebyshev of the KLT. When I sent the results back to Harry interpolation, a neat discussion of which was available Andrews, he suggested that I publish them. As such, I in a text book (Computer Evaluation of Mathematical sent them to the IEEE Computer Transactions, and Functions, by C. T. Fike, Prentice-Hall, 1968, Sec- the paper was then published in the January 1974 tions 7.4 and 7.5). This was in early 1972, and I wrote issue. I recall that we decided to send it in as a corre- a proposal to the National Science Foundation (NSF) spondence item in order to get it published with mini-

1051.2004/91$1.50 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. mum delay. Little did we realize at that time that the Honeywell, Inc., St. Paul, Minnesota. He was with Kansas State resulting “DCT” would be widely used in the future! University, Manhattan, from 1968 to 1983. Since 1983 he has been a Professor in the Electrical and Computer Engineering Depart- It is indeed gratifying to see that the DCT is now ment at the University of New Mexico, Albuquerque. He became essentially a “standard” in the area of image data the Chairman of this department in July 1989. In August 1985 he compression via transform coding techniques. was awarded one of the twelve Presidential Professorships at the University of New Mexico. He is the leading author of Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, 1975), and Discrete-Time Signals and Systems (Reston, 1983), and coau- thor of Computer Science Fundamentals (Merrill, 1979). He is also NASIR AHMED was born in Bangalore, India, in 1940. He re- the author of a number of technical papers in the area of digital ceived the B.S. degree in electrical engineering from the University signal processing. Dr. Ahmed was an Associate Editor for the of Mysore, India, in 1961, and the MS. and Ph.D. degrees from the IEEE Transactions on Acoustics, Speech, and Signal Processing University of New Mexico in 1963 and 1966, respectively. From (1982-1984) and is currently an Associate Editor for the IEEE 1966 to 1968 he worked as Principal Research Engineer in the area Transactions on Electromagnetic Compatibility (Walsh Functions of information processing at the Systems and Research Center, Applications).