# Intra-Mode Decision Based on Lagrange Optimization Regarding Chroma Coding
## 2 Related Works  
Considering the non-uniformity between the luma and chroma, the coding structures and the intra mode prediction of the chroma component are autonomous from the luma component in VVC [10]. In terms of chroma intra mode prediction, [PERSON] et al. [11] developed a cross-component linear prediction mode (CCLM) to remove visual redundancies, where the intra-chroma CU utilizes cross-component correlations with the luma-reconstructed CU to predict pixels on the basis of a linear model (LM). [PERSON] et al. [12] proposed that the LM parameters for the prediction of the original chroma are error-sensitive, and they proposed three novel LM-like modes to identify some potentially problematic conditions in parameter estimation. [PERSON] et al. [13] also explained that it is difficult to derive optimal LM parameters, especially when there are insufficient training samples for prediction blocks with small sizes, and proposed a multi-mode linear model based on up-sampling chroma samples. [PERSON] et al. [14] proposed to utilize the intra prediction mode of the luma block covering the center position as the chroma candidate mode. The aforementioned research on intra mode prediction primarily focuses on the use of local reference blocks. Nevertheless, it is crucial to acknowledge that blocks with similar textures may reside in more distant regions and can serve as viable references in the prediction process. The block matching approach employed in an intra block copy (IBC) method [15] was purposely designed for natural video data, where a coding block is predicted by a reconstructed block residing within the same frame. Here, the utilization of IBC achieves a good balance between high efficiency and low complexity. Furthermore, a combined intra-mode approach [16] was developed by utilizing both local and non-local correlations, which integrates block matching and angular prediction to effectively handle complex video content. The non-local blocks could be accurately predicted, thereby removing the spatial redundancy. Inspired by the decline in the accuracy of intra modes as the reference distance expanded, the position-dependent intra prediction combination (PDPC) [17] was introduced with more available reference samples, where the prediction pixels are determined based on the positions of samples and the intra mode. While some researchers held that the intra mode prediction block could be obtained using the neighboring samples, it was acknowledged that the reference line might exhibit significant discontinuity compared to the original samples. Therefore, multiple reference lines [18] were supported to improve the accuracy of prediction. All of the approaches above are beneficial to coding efficiency for intra coding, and some of them have been adopted into the VVC codec [19]. Regarding intra-mode decision, the technique of Lagrange optimization played a critical role in chroma coding, as it aimed to obtain the best prediction mode by calculating the minimum of the Lagrange cost, thereby achieving a balance between bitrate and distortion (RD). Several researchers devoted themselves to estimating the Lagrange multipliers, \(\lambda\), in order to discover improvements in coding efficiency. Initially, some heuristic schemes were proposed to determine the \(\lambda\) using empirical expressions or iterative processes [20; 21], but these methods were generally unsuitable for practical applications due to their high complexity and lack of theoretical foundation. Thus, the study of accurate RD models became appealing, aiming to solve the Lagrange multiplier based on the source probability distribution through analytical means. The popular logarithmic relationship between bitrate and distortion was presented based on the high-rate approximation curve for entropy-constrained scalar quantization. Consequently, the corresponding Lagrange multiplier could be accurately calculated in terms of the quantization parameter [22].