## 1 Introduction  
With the rapid development of video services in recent years, how to efficiently compress video sequences has been considered as a very challenging task for transmitting video data over heterogeneous networks. To meet this demand, scalable video coding (SVC) has become a simple and flexible solution to enable seamless delivery by offering three different kinds of scalabilities, namely, temporal scalability, spatial scalability, and quality scalability [1; 2; 3; 4]. Generally speaking, there are two main categories on SVC: discrete cosine transform (DCT)-based hybrid video coding SVC and wavelet-based SVC. Owing to the intrinsic localization and multiresolution features of the discrete wavelet transform (DWT), video codecs based on motion-compensated three-dimensional (3-D) DWT have been studied extensively for use in SVC [5; 6; 7; 8]. 3-D wavelet-based SVC provides a natural way in producing embedded bitstreams with full scalability and fine granularity for in-network adaptation [9; 10]. In the 3-D wavelet-based SVC, temporal redundancy across frames is exploited by adopting the motion-compensated temporal filtering (MCTF) framework [11; 12; 13; 14], and spatial redundancy inside a frame is utilized by 2-D spatial transform. Such codecs do not suffer from the drift problem often exhibited by the DCT-based SVC codecs with their incorporated feedback loops. Consequently, over the last few years, much research has been devoted to the 3-D wavelet video coding, and it is meaningful to further explore its properties for developing a best coding approach for 3-D wavelet-based SVC. As in hybrid video coding-based SVC, rate-distortion optimization (RDO) plays a significant role in SVC in choosing the most suitable coding parameters under communication bandwidth constraints. A critical problem residing in RDO is the selection of the Lagrange multiplier that controls the RD trade-off [15; 16; 17; 18; 19; 20; 21; 22; 23]. [PERSON] and [PERSON] first derived the relationship among the Lagrange multiplier, distortion and bitrate. This simple and effective formula has been extensively adopted in hybrid video coding [24; 25]. Although the conventional Lagrange multiplier selection techniques have been developed and widely adopted in hybrid video coding, the existing formulations are not applicable to 3-D wavelet-based SVC. This is principally due to the fact that the Lagrange multiplier is determined by the quantization step. Different from hybrid video coding-based SVC where the quantization steps are implied in the base layer, quantization for wavelet-based SVC is usually approached in an embedded manner rather than using explicit quantization steps. Accordingly, in absence of the quantization step, the conventional Lagrange multiplier cannot be directly applied to 3-D wavelet-based SVC. The Lagrange multiplier selection technique used to solve the RDO problem is an important feature that contributes to the success of current video coding. However, solving the optimal Lagrange optimization problem in 3-D wavelet coding is more complicated because the energy difference between the pixel and wavelet domains is not conserved after applying the bi-orthogonal wavelet transform and MCTF technique. The latter leads to the intrinsic subband coupling across various temporal subbands during motion aligned temporal filtering. Moreover, the open-loop prediction structure employed in 3-D wavelet-based SVC further complicates the issue of Lagrange multiplier selection. This issue has been solved by assigning different weighting factors to the subbands, resulting in degrading unpleasant quality fluctuations [18; 19; 26; 27; 28]. For the Haar filter used in temporal filtering structure, [PERSON] proposed a method to derive various quantization weights associated with low-frequency and high-frequency subbands [29]. [PERSON] et al. have obtained weighting factors for other longer temporal filters, such as 5/3 and 9/7 filters [30; 31]. [PERSON] et al.