Condition on Word Length of Signals and Coefficients for DC Lossless Property of DWT

Abstract

A discrete wavelet transform (DWT) has been widely applied to various digital signal processing techniques. It has been designed under a certain condition such as perfect reconstruction, aliasing cancellation, regularity, vanishing moment, etc. This article introduces a new condition referred to “DC lossless”. It guarantees lossless reconstruction of a constant input signal (DC signal) instead of rounding of signal values and coefficient values inside a transform. The minimum word length of the values under the new condition is theoretically derived and experimentally verified. Since JPEG 2000 algorithm based on the discrete wavelet transform (DWT) was adopted as an international standard for digital cinema video coding [1], high speed and low power implementation of a DWT has been becoming an issue of great importance [2,3]. In designing a DWT, its coefficient values and signal values are assumed to be real numbers. However, in implementation, they are rounded to rational numbers so that they are expressed with finite word length representation in binary digit. Therefore it is inevitable to have rounding errors inside a DWT processing unit. In this article, we derive a condition on word length of coefficient values and that of signal values of a DWT such that the transform becomes lossless for a DC signal. Under this condition (DC lossless condition), it is theoretically guaranteed that an output signal contains no error in spite of rounding of coefficients and signals inside the DWT. We treat the irreversible 9-7 DWT adopted by the JPEG 2000 for lossy coding of image signals as an example. In case of the 5-3 DWT in JPEG 2000 for lossless coding, benefiting from its lifting structure [4-6], lossless reconstruction of any signal is guaranteed even though signals and coefficients are rounded. On the contrary, it does not hold for the 9-7 DWT because of scaling for adjusting DC gain of a low pass filter in a forward transform [7]. However, we have pointed out that it became possible to be lossless for a DC signal under a certain condition on word length of coefficients and signals [8]. This DC lossless condition is a necessary condition for the regularity which has been analyzed by numerous researchers to improve coding performance of a transform. When the regularity is not satisfied, the DWT has some problems such as a checker board artifact which is observed in a reconstructed signal as unnecessary high frequency noise in flat or