1051-8215 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCSVT.2019.2948306, IEEE Transactions on Circuits and Systems for Video Technology IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, 2019 1 Efficient Rationalization of Triplet Halfband Filter Banks and Its Application to Image Compression Jayanand P. Gawande, Member, IEEE, Amol D. Rahulkar, Member, IEEE and Raghunath S. Holambe Abstract—This paper presents a novel approach for rationaliza- tion of irrational coefficients of filters designed using three-step lifting scheme. The existing three-step lifting schemes result in the irrational filter coefficients which require infinite precision for implementation. In this work, Euler-Frobenius halfband polynomials are designed to obtain the required kernels of three- step lifting structure. Next, the efficient rationalization of the designed filter banks (FBs) is proposed to reduce the arithmetic complexity. The proposed rational FBs preserve perfect recon- struction, near-orthogonality and regularity properties of wavelet FBs. These filters with rational coefficients are then used in image compression algorithms to compress the test images in well- known Classic, EPFL, RAISE, FiveK datasets and chromosome image datasets. The performance of the proposed rational FBs is compared with the existing rational FBs. It is observed that the performance of the proposed rational FBs is improved in terms of computational cost, encoding-decoding time and compression ratio as compared to existing rational FBs. Index Terms—Filter banks, halfband polynomial, image com- pression, rational coefficients, wavelets. I. I NTRODUCTION T HE discrete wavelet transform (DWT) has been applied in many signal and image processing applications such as data compression, feature extraction, communications, de- noising, etc. The wavelet transform has been widely used in image compression applications because of its excellent multi-resolution analysis (MRA) features and better time- frequency localization property. The DWT-based image coding algorithms (EZW [1], SPIHT [2], JPEG-2000 [3]) present superior compression performance than the discrete cosine transform (DCT) based image coding (JPEG standard) in terms of compression efficiency, speed and embedded bit-stream representation. JPEG-2000 has been accepted as a standard for image compression by International Standard Organisation (ISO). The highest compression performance for given com- pression ratios have been obtained for a wide variety of images using SPIHT. Hence, it is the most widely used wavelet-based algorithm for image compression. The newer image coding standard JPEG XR has half the complexity of JPEG-2000 while preserving image quality. A lifting-based four-channel J. P. Gawande is with Department of Instrumentation Engineering, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai 400706, INDIA. e-mail: jayanandpg@gmail.com A. D. Rahulkar is with Department of Electrical and Electronics En- gineering, National Institute of Technology, Goa 403401, INDIA. e-mail: amolrahulkar 000@yahoo.com R. S. Holambe is with Department of Instrumentation Engineering, SGGS Institute of Engineering and Technology, Nanded 431606, INDIA. e-mail: rsholambe@gmail.com hierarchical lapped transform (HLT) is used in JPEG XR [4]. Recently, the new wavelet-based coding algorithms have been reported to improve the image compression performance in literature [5], [6], [7], [8], [9]. However, the performance of a wavelet-based coding algorithm extensively depends on the choice of an appropriate wavelet. Hence, the correlation between the image characteristics and the properties of the designed wavelets needs to be investigated. During the last decades, custom designs have been signifi- cantly used to meet the computational demand of image and multimedia processing. In recent years, DWT emerged as a heart of image and video coding and as a valuable tool for wide variety of image processing applications and computer graphics. The performance of image processing applications based on DWT highly dependent on the choice of suitable wavelet filter banks (FBs) with its desirable properties. It is observed that most of wavelet FBs designed with irra- tional coefficients. It is necessary to represent these irrational coefficients with infinite precision. Due to high precision requirement, the hardware complexity and computation time increased a lot. Thus, it is necessary to design the wavelet FBs with rational coefficients in order to reduce hardware resources and processing time. The DWT is implemented using two-channel wavelet FBs and can be classified into orthogonal and biorthogonal. The biorthogonal wavelet FBs are preferred in image processing applications because of their linear phase (symmetry) and smoothness (regularity) characteristics. The linear phase prop- erty is useful in better bit allocation and less quantization noise in image compression.The more regular wavelet filters are important in order to minimize the effects of quantization noise and unwanted artifacts in reconstructed images. The perfect reconstruction (PR) condition is used to design the two-channel biorthognal wavelet FBs. The regularity can be achieved by imposing vanishing moments (VMs) (zeros at z = −1). Several design techniques of well-known biorthog- onal wavelet FBs have been described in [10], [11] and [12]. The design of these FBs and spline family of wavelets is based on the spectral factorization of a class of halfband polynomials. The well-known Cohen-Duabechies-Feauveau (CDF) 9/7 [10] filter bank (FB) has been adopted in JPEG-2000 image compression standard [3]. The CDF 9/7 FB is designed by factorization of Lagrange halfband polynomial (LHBP), which has maximum number of zeros at z = −1 and achieves more regularity. However, the filters in CDF 9/7 FB have irrational coefficients which require infinite precision in hardware implementation. The LHBP filters do not have any degree of freedom and thus there is no direct control 19 IEEE. Personal use of this material is permitted. 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