# Intra-Mode Decision Based on Lagrange Optimization Regarding Chroma Coding
## 3 Chroma Intra-Mode Decision in VVC  
### Chroma Intra Prediction  
In the new generation of video coding standard VVC, the intra prediction technique is still retained to remove the redundancy of adjacent pixels in the spatial domain, where chroma signals exhibit the visual correlation among the neighboring chroma units. However, the inter-correlation between the luma and chroma is also considered and applied in chroma intra prediction. Figure 1 depicts the components of the luma (Y) and chroma (Cband Cr) in one frame, which, to some extent, indicates the component relevancy in texture. Here, the test sequence of ParkScene, provided by JVET, has a resolution of 1280 \(\times\) 720.  
With the 4:2:0 YCbCr format, eight chroma modes are used for intra prediction, related to VER, HOR, PLANAR, DC, DM, LM, LM_T, and LM_L. Their serial numbers and corresponding operations are listed in Figure 2. Here, the luma and chroma blocks are shown in blue and yellow rectangles, respectively. The reconstructed blocks in gray act as references for intra prediction, and the pixels marked with circles are located in the coding blocks. The serial numbers are defined in [7] to distinguish the intra modes described below. The angular prediction of VER and HOR uses the nearest reference line to project each estimated pixel along the vertical and horizontal directions, respectively. For the DC mode, the predictor in the chroma block is the average of available neighboring reconstructed chroma pixels. Bi-linear interpolation is employed in PLANAR mode. In DM mode, the chroma block exploits the same intra prediction mode as the corresponding luma block. In VVC, the CCLM is related to the LM, LM_T, and LM_L, which operate based on different luma reference blocks.  
The modes of LM, LM_T, and LM_L take advantage of the reconstructed luma samples to predict the chroma block using a linear mode. The reconstructed luma CUs of 2N \(\times\) 2N  
Figure 1: Luma of Y and chroma of Cb and Cr in ParkScene sequence.  
Figure 2: Chroma prediction modes for intra coding.  
are first down-sampled to the same resolution as the chroma blocks of N \(\times\) N. Then, the intra chroma prediction is performed as follows:  
\[y_{p}(i,j)=\alpha x_{r}(i,j)+\beta, \tag{1}\]  
where \(y_{p}\) is the predicted pixel of the chroma block, and \(x_{r}\) denotes the reconstructed value in the collocated down-sampled luma block. The parameters of \(\alpha\) and \(\beta\) can be derived by the following:  
\[\alpha=\frac{C_{max}-C_{min}}{L_{max}-L_{min}}, \tag{2}\]  
\[\beta=C_{max}-\alpha C_{min}. \tag{3}\]  
Here, \(L_{max}\) and \(L_{min}\) represent the maximum and minimum down-sampled neighboring luma pixels, respectively. \(C_{max}\) and \(C_{min}\) are collocated chroma pixels. The process of LM is provided in Figure 3, and the parameters of \(\alpha\) and \(\beta\) can be obtained by linear regression using the blue and yellow samples. Then, the luma reference blocks derived from down-sampling are integrated into (1) for the chroma prediction block. In the same way, the prediction of LM_T and LM_L is similar to that of LM, with potential differences in the selection of luma reference blocks.