Low Complexity Implementation of Daubechies Wavelets for Medical Imaging Applications

Abstract

The Discrete Wavelet Transform (DWT) has extensively been used in a wide range of applications, including numerical analysis, image and video coding, pattern recognition, medical and telemetric imaging, etc. The invention of DWT decomposition by Mallat (Mallat, 1998) shows that the DWT can be viewed as a multiresolution decomposition of signal. This means it decomposes the signal into its components in different frequency bands. The Inverse DWT does the opposite, i.e. it reconstructs the signal from its octave band components. After its inclusion in JPEG2000 compression standard (Seo & Kim, 2007), significant research has been done to optimize the DWT implementation to reduce the computational complexity. Among a wide range of wavelets, the Daubechies wavelets include members ranging from highly localized to highly smooth and can provide excellent performance in image compression (Daubechies, 1992). Among the family members, the first two – Daubechies 4-tap (DAUB4) and Daubechies 6-tap (DAUB6) – are popular choices in medical imaging applications. While compressing medical images, the key here is to preserve as much critical information as possible in the reconstructed image so that accurate diagnosis is possible. There have been several efficient implementations of wavelet filters proposed for applications in image processing (Lee & Lim, 2006; Martina & Masera, 2007; Acharyya et al., 2009; Shi et al., 2009; Lai et al., 2009). But, the use of conventional fixed-point (FP) binary (or any other weighted) representation for implementing discrete wavelet coefficients (that are irrational in nature) introduces round-off or approximation errors at the very beginning of the process. The error is due to the lack of exact representation of the irrational numbers that form the coefficient basis. These errors tend to expand as the calculations progress through the architecture, degrading the quality of image reconstruction (Wahid et al., 2003). A lossless mapping technique, known as Algebraic Integer Quantization (AIQ), can be used to minimize the approximation error and efficiently compute the DAUB4 and DAUB6 coefficients (Wahid et al., 2004). The AIQ scheme is divided into two parts: the first stage is based on factorization and decomposition of transform matrices exploiting the symmetric structure. After the decomposition, we map the irrational transform basis coefficients using multidimensional algebraic integers that results in exact representation and simpler implementation. As a result, less error is introduced in the computation process that yields significantly better