# An Analytic Transform Kernel Derivation Method for Video Codecs
## 3 Experimental Results and Analysis  
To verify the proposed method for the transform kernel derivation for VVC, the similarities between the proposed kernels and the original kernels in VTM-3.0 [27] were investigated by plotting the kernels in a graphical representation. Figure 19 shows the comparison of the transform kernels. It can be seen that the derived transform kernels are almost perfectly matched with the original transform kernels. As described in Section 3, only a few integer elements in the 32-point DST-7 kernel are tuned from the original kernels to obtain full transform kernels. Thus, it can be inferred that the proposed transform kernels are more likely to have identical coding gains. Experiment for verification of the proposed method were performed on the VVC reference software, VTM-3.0 [27], using CTC conditions [29]. The PC for simulation was Centos 7 OS with an Intel(r) Xeon(r) Silver 4114 CPU @ 2.20 GHz processor with 200 cores and 156 GB of RAM. Parallel processing is turned off to precisely measure the encoding and decoding complexity in the execution time. The coding performance was measured in BDBR (%) [30]. Table 2 shows the experimental results of the proposed method. As shown in Table 2, the coding performances of the proposed method are almost identical to the original VTM3.0 for all configurations. This is because the similarity of the proposed transform kernel with the original transform kernel is almost perfect. Figure 18: Application of the proposed 16\(-\)point grouped DST\(-\)7 transform to a fast algorithm in VVC. Figure 19: Comparison between the proposed and original transform kernels. Tables 3-5 show a comparison of the results of the proposed method for the AI, RA, and LDB configurations [31], respectively. Although COT [21] provides better coding gain, it fails to provide a 64-point DCT-2 transform kernel, as some rows of 64-point DCT-2 transform kernels are removed and replaced with selected row values of DST-7/DCT-8 transform kernels [21]. In addition, [21] used five transform kernels, that is, DCT-2, DST-7, DCT-8, DST-4, and DCT-4. Similarly, the unified matrix [23] presented excellent coding gain; however, the higher resolution class sequences, Classes A1 and A2 have higher coding loss. This results from the deviation of the proposed mathematically derived transform kernels with the original transform kernels. In [24, 25], owing to the mismatch in the proposed transform kernels with original transform kernels, coding losses can be seen. In our proposed method, no coding loss is observed in higher-resolution class sequences. The overall result showed a minor gain of 0.02% in the U-chroma. This is because the proposed transform kernels were in good agreement with the original transform kernels, as shown in Figure 19.  
Table 4 shows the overall experimental results of RA using the VTM-3.0 reference software under the CTC conditions. Coding loss can be seen in [23, 24, 25] but [21] showed less significant luma gain. Similarly, the proposed method showed no significant loss compared to the other methods. Table 5 shows the overall experimental results of LDB using the VTM-3.0 reference software under the CTC conditions. [21, 23, 25] showed higher coding loss, whereas the proposed method showed a 0.27% coding gain in V-chroma and negligible loss in U-chroma.