Condition on Word Length of Signals and Coefficients for DC Lossless Property of DWT
Masahiro Iwahashi, Hitoshi Kiy · InTech eBooks · 2011
Discrete Wavelet Transforms: Algorithms and Applications 232 smooth region of a signal [9].It also brings about DC leakage which decreases the coding gain of a transform [10].The regularity has been structurally guaranteed for a two channel quadrature mirror filter bank (QMF) [9] and the DCT [10] respectively.However, since these previous methods were based on the lattice structure, these are not directly applicable to the lifting structure of the 9-7 DWT.Beside these relations to the regularity, the DC lossless condition itself is also considered to be important for white balancing of a video system in which the DC signal is used as a reference input for calibration [11].This article aims at deriving the DC lossless condition theoretically and clarifying the minimum word length of signals and coefficients.In conventional analysis, errors due to shortening of word length of signals (signal errors) were described as 'additive' to a signal [7,12].They were treated as independent and uniformly distributed white noise.On the other hand, errors due to rounding of coefficients (coefficient errors) were described as 'multiplicative' to a signal and evaluated with the sensitivity [13][14][15].It should be noted that the signal error and the coefficient error have been treated independently.Unlike those conventional approaches, we utilize mutual effect between rounding of signals and that of coefficients.Introducing a new model which unifies the coefficient error and the signal error, we define tolerance for those errors as a parameter to simultaneously control both of word length of signals and that of coefficients.As a result of our theoretical analysis, the minimum word length of signals and that of coefficients inside the lifting 9-7 DWT are derived under the DC lossless condition.We confirm that the minimum word length derived by our analysis is shorter than that determined by a conventional approach.We also confirm that the DWT under the condition does not have the checker board for a DC signal.This article is organized as follows.Chapter 2 defines a rounding operation and a rounding error, describes their basic properties in algebraic approach, and derives 'addition' formula and 'multiplication' formula of the rounding (modulo) operation.Application of these formulas to scaling of a signal value is introduced in chapter 3. Chapter 4 introduces the DC lossless DWT.Its usefulness is also described.Derivation process of conditions on word length of signals and coefficients is described in chapter 5.The new condition derived from the basic properties in chapter 2 is summarized in chapter 6.Other related condition derived from a conventional approach is also summarized.Theoretical results are verified and the minimum word length of the DC lossless DWT is clarified in chapter 7.This article is concluded in chapter 8. Rounding operation and its basic formulasThis chapter introduces basic properties of the rounding operation focusing on 'quotient', rather than 'remainder', in modulo operation.So far, 'remainder' had been attracted numerous mathematicians' attention and various basic properties were found such as the Chinese remainder theorem in the commutative algebra (commutative ring theory).On the contrary, 'quotient' plays an important role as a 'practical' value in finite word length implementation in modern computer systems.This chapter introduces an algebraic approach of expressing 'quotient' as a practical value, and 'remainder' as a rounding error, so that it can be applied to analyzing exact behavior of rounding errors in a complex calculation procedure.www.intechopen.