# Intra-Mode Decision Based on Lagrange Optimization Regarding Chroma Coding
## 1 Introduction  
With the rapid development of multimedia technologies and communication networks, video services have undergone tremendous development in terms of range, from mobile HD video to ultra-HD video, experiencing explosive increases in applications such as those intended for entertainment, industry, medicine, sport, agriculture, and transportation [1]. While benefiting people's lives, some outstanding problems are involved in the storage and transmission of these huge amounts of video data. Lossy video coding is thus put forward to lessen space redundancy and transmission expenses that affect the quality of reconstructed images [2]. Consequently, many researchers investigate video compression schemes, and a series of standards have been successively established, such as Advanced Video Coding (AVC) [3] and High-Efficiency Video Coding (HEVC) [4], where hybrid coding structures are employed, including the basic codec modules of intra/inter-prediction, transform, quantization, entropy coding, rate control, and rate distortion optimization (RDO). Currently, the Joint Video Experts Term (JVET), founded by the ITU-T Video Coding Experts Group (VCEG) and the ISO/IEC Moving Picture Experts Group (MPEG), is devoted to formulating a new-generation video coding standard of versatile video coding (VVC) [5]. Novel compression technologies are being added to VVC with the aim of reducing the average bitrate to 50% with the same subjective video quality when compared to high-efficiency video coding (HEVC). In order to adapt to natural video characteristics, a quadtree with a nested multi-type tree (QT-MTT) coding structure is used to produce coding tree units (CTUs) for more flexible partitions, and extended multiple directional intra prediction modes are incorporated to accurately capture arbitrary edge directions [6]. In addition, multiple transform selection (MTS) is used for subsequent Context-based Adaptive Binary Arithmetic Coding (CABAC) to achieve an extraordinary performance improvement [7]. The hybrid encoders mentioned above employ prediction and residual coding to generate a video bitstream based on the redundant information of past-coded blocks [8]. Here, the prediction module refers to two aspects of intra and inter prediction, where intra prediction sufficiently explores spatial neighboring blocks to remove the potential correlation, and inter prediction corresponds to lessen inter-block redundancy as much as possible. Considering that a local scene would frequently change during a video sequence, intra prediction significantly contributes to coded video quality and efficiency through the use of a deep scheme to shrink the residual energy and subsequent bitstream size. Many studies focus on newly built or improved schemes for higher compression in prediction, and many directional prediction modes are continuously being investigated to obtain lower residual power, from the 9 modes previously available in AVC and the 35 modes in HEVC to the 67 modes currently available in VVC by fully utilizing contextual information [9]. The rich set of prediction patterns in intra coding can potentially achieve a relatively lower bitrate when representing blocks; however, it may simultaneously produce a burden on optimal selection. Usually, an encoder should compress a video by exhaustively trying all intra modes; however, the complexity and computation load increase drastically during operation. To address this issue, the RDO mode decision scheme is employed to determine the optimal mode to meet the best rate and distortion trade-off. Although research on intra-mode decisions using RDO has resulted in considerably positive benefits, most schemes are designed on the premise of ignoring the intra coding influence between the luma and chroma. Actually, prediction-based reconstructed luma samples could cause potential intra-distortion propagation to the chroma prediction block, which could impact the cost of rate and distortion (RD) in the final mode decision.