Development of integer cosine transforms by the principle of dyadic symmetry

Abstract

The paper shows how to convert the order-8 cosine transforms into a family of integer cosine transforms (ICTs) using the theory of dyadic symmetry. The new transforms can be implemented using simple integer arithmetic. It was found that performance close to that of the DCT can be achieved with an ICT that requires only 4 bits for representation of its kernel component magnitude. Better performance can be achieved by some ICTs whose kernel components require longer bit lengths for representation. ICTs that require 3 bits or less for representation of their component magnitude are available but with degraded performance. The availability of many ICTs provides an engineer the freedom to tradeoff performance for simple implementation in designing a transform codec. IntroductionTransform coding can achieve a high data compression rate for image data. A transform coder comprises mainly two parts : the first part transforms highly correlated image data into weakly correlated coefficients using an orthogonal transform and the second part performs adaptive quantisation on coefficients to reduce the bit transmission rate. It has been widely accepted that among the many suboptimal orthogonal transforms the discrete cosine transform (DCT) has the best performance in both data compression and filtering for image data [I, 21. The DCT has also been shown to be asymptotically optimal for a first-order Markov source [3, 41, which is regarded as a good stochastic representation of image data. As block sizes of 8 and 16 are most appropriate for the transform coding of image data, therefore techniques for implementing order-8 and -16 DCTs economically and with fast computational time are very important for the realisation of a transform codec. The DCT can be implemented using either a programmable processor or dedicated hardware. For an order-n DCT, the magnitudes of the components of the 0th and (n/2)th basis vectors can be reduced to unity by suitably scaling components of these basis vectors. However, components of other basis vectors are irrational numbers which cannot be reduced to integers by simple scaling. Being real numbers, the components cause diffiPaper 67911 (E16), first received 24th October 1988 and in revised form 9th December 1988 The author is with the Department of Electronics, the Chinese University of Hong Kong, Shatin, N.T., Kong Kong 276 culties in both implementation methods. Dedicated hardware that computes the DCT using floating point arithmetic is very complex and expensive. On the other hand, a programmable processor which computes the DCT by executing a sequence of instructions does not usually have instructions that handle real numbers. Although floating point co-processors which have been specially tailored for processing floating point arithmetic can be found on the commercial market, they are also very expensive.A simple method to eliminate floating point arithmetic is to approximate the real magnitudes of the DCT components by M-bit integers, so that the DCT can be computed using integer ...

Integer lapped transforms and their applications to image coding

“…Section II reviews the theory and design of integer sinusoidal transforms, which was originally proposed by Cham [1]. Section III is devoted to the theory, design and implementation of the Integer Lapped Orthogonal Transform (ILOT).…”

“…The subscript and the superscript of and denote, respectively, the length and the type of the transformation. The type-II integer cosine transform (ICT), , proposed in [1] has the following matrix representation:…”

“…For example, means type-II discrete cosine transform and so on. Using the concept of dyadic symmetry, Cham [1] proposed the following construction for order-8 ICT (2.6) where , , , , , , and are integer parameters. It can be verified that the rows of are orthogonal to each other if…”

“…In [1], the constants , , , , , , and are chosen to be 8-b integers and the th entry of the normalization matrix , , is used to normalize the th row of so that is orthogonal. If is the -norm of the th row of , then is given by (2.9)…”

“…The remaining problem is to find similar integer transforms for the four-point DCT-II and the four-point DST-IV. Using the same concept as in [1], we propose the order-4 type-II integer cosine transform as follows: where , , and are integers to be determined. Like , and have to satisfy (3.3) Similarly, the order-4 type-IV integer sine transform is defined as follows:…”