Signal Processing 87 (2007) 2850–2858 Fast communication A filter design strategy for binary field wavelet transform using the perpendicular constraint N.F. Law, W.C. Siu à Centre for Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong Received 15 June 2002; received in revised form 15 January 2004; accepted 29 May 2007 Available online 3 June 2007 Abstract Wavelet decomposition has recently been generalized to binary field in which the arithmetic is performed wholly in GF(2). In order to maintain an invertible binary wavelet transform with multiresolution properties, three constraints are placed on the filters, namely the bandwidth, the perfect reconstruction and the vanishing moment constraints. While these constraints guarantee the existence of the inverse filters, their form is unconstrained and could be signal length dependent. In this paper, we propose to use the perpendicular constraint to relate the forward and inverse filters. With this constraint, it is shown that the form of the inverse filters remains unchanged after the up-sampling operation associated with the wavelet transform. We also explore an efficient implementation structure in the binary filters so as to save memory space and reduce the computational complexity. A detailed comparison with the lifting implementation in the real field wavelet transform is carried out. It is found that the computational complexity of the binary filter is significantly less than that of the real field wavelet kernel. r 2007 Elsevier B.V. All rights reserved. Keywords: Binary wavelet transform; Filter design; Binary image processing; In-place implementation 1. Introduction Images in most applications are represented by a finite number of quantization levels such that image ranges are finite. For example, a gray scale image is commonly represented using 8 bits and ranges from 0 to 255. However, wavelet development is mostly concentrated on real-valued functions in which the data to be analyzed are real and the arithmetic performed is in the real field [1–3]. There have been several attempts to generalize wavelet decomposition to finite fields to take into account image character- istics [4–8]. For example, a formulation of the wavelet decomposition of a vector space over the finite field has been derived in [14]. Swanson and Tewfik [7] also proposed a binary wavelet transform (BWT) for binary images in which the arithmetic is performed wholly in GF(2). The binary field is particularly interesting because of its widespread use in error control coding. The design issue of the paraunitary filter banks over the binary field has been discussed in [15], while [16] has brought together the finite-field wavelet transforms and error correcting codes. Besides, [20] outlined a promising application of the BWT and its variant ARTICLE IN PRESS www.elsevier.com/locate/sigpro 0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.05.022 à Corresponding author. Fax: +852 23628439. E-mail address: enwcsiu@polyu.edu.hk (W.C. Siu).