Pattern Recognition Letters 2 (1984) 319-327 September 1984 North-Holland Properties and some fast algorithms of the Haar transform in image processing and pattern recognition P.C. MALI, B.B. CHAUDHURI and D. DUTTA MAJUMDER Electronics and Communication Sciences Unit, Indian Statistical Institute, Calcutta 700 035, India Received 4 August 1983 Revised 24 May 1984 Abstract: Properties of the Haar transform in image processing and pattern recognition are investigated. A lower bound of the performance of the Haar transform relative to that of the Karhunen-Loeve transform for first-order Markov processes is found. It is proved that the Haar transform is inferior to the Walsh-Hadamard transform for such processes. A unique condition is presented which, if satisfied by the elements of a matrix, will make the Karhunen-Loeve transform of the matrix and the Haar transform equivalent. Some fast algorithms are given to realize the diagonal elements of a Haar transformed matrix. Key words'." Haar transform, image processing, pattern recognition, Karhunen-Loeve transform, fast algorithm. 1. Introduction The data compression, feature selection and diagonalisation properties of a sinusoidal family of unitary transforms have recently been studied and compared with those of Karhunen-Loeve transform (KLT) (Ahmed et al. (1974), Jain (1979)). A similar study on other fast unitary transforms such as the Walsh-Hadamard transform (WHT), Haar transform (HT), Slant transform (ST), etc. is also very useful. Although some practical results have been reported (Pratt et al. (1974)) a detailed analytical study of the properties appears to be lacking. To this end the present authors reported recently some properties and fast algorithms on WHT for image models (Mali et al. (1983)). This paper is an extension of this work for HT. The extension is useful and important because HT may be fruitfully applied for feature selection, data coding, edge detection and multiplexing problems (Shore (1973)). Also, HT is locally as well as globally sensitive and is computationally least expensive among the unitary transforms referred to above. At first, a relative measure of performance of any unitary transform with that of KLT will be defined. A lower bound of the relative measure is found for images modeled by a first-order Markov process in Section 2. In Section 3 it is shown that the relative measure for WHT is better than that for HT for any intermediate case. The image model is represented by a first-order Markov process and both numerical and analytical results are shown. In Section 4 a unique condition is presented, which, if satisfied by the elements of any matrix, will make the KLT of the matrix and the HT equivalent. For such a matrix, therefore, the computa- tionally less efficient KLT is unnecessary. In Section 5 a fast algorithm is proposed to find the diagonal elements of the two-dimensional HT of any matrix. If the matrix is symmetrical, as is the covariance matrix 0167-8655/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland) 319