# A Novel Low-Complexity and Parallel Algorithm for DCT IV Transform and Its GPU Implementation
## 7 Conclusions  
The results presented in the previous sections show that the novel implemented DCT IV algorithm far exceeded our expectations. Due to the factorization method of the DCT transform, the reduced number of mathematical operations involved, and its parallel implementation, we obtained improvements in execution speed between 10.11- and 11.97-times for the implementation based on four sections (Tables 4 and 5) and between 11.81- and 16.04-times in the case of implementing the DCT algorithm based on eight independent sections (Tables 6 and 7). All these results were obtained based on the Ampere architecture. By using the numerical representation of the BFloat16 type, an existing representation that can be manipulated hardware only within the Ampere architecture, we will obtain an increase in performance between 1.21- and 1.88-times, in addition to the already existing one--Tables 10 and 11. The implementation of the new algorithm proposed in this study for the Volta architecture brought the following performance gains: (a) for the implementation of the algorithm based on the parallel execution of four sections, the increase in execution speed compared to the classical implementation of the DCT IV algorithm was between 5.41 and 12.1, (b) and in the case of the algorithm with eight sections, we obtained an increase in the execution speed between 8.08- and 14.41-times. Also, comparing the novel algorithm directly against recent algorithms [21; 40] under the same conditions, we notice that it surpasses them in all cases. A disadvantage of this algorithm is that it can only be parallelized on four or eight different units. This is due to the mathematical method of factorization, which is the basis of its implementation. This research aimed not to develop an algorithm that can efficiently use a GPU and fill out all its cores. But this is not a big problem for our algorithm. The DCT transform classical scenario usage is to be a component of the image compression algorithm. In such a case, it works on non-overlapping quadratic regions that, all together, make up the image. So, based on this approach, an ultra-HD image (3840 \(\times\) 2160) can be split into more than 32,000 quadratic regions, and the DCT IV will be computed simultaneously in parallel on all cores of the GPU on each area. In this mode, many, if not all, CUDA units will be used. \begin{table}
\begin{tabular}{c c c c c c} \hline
**DCT IV** & **Classical Implementation** & **Parallel Implementation** & \multicolumn{4}{c}{**Number of Writing/Reading from**} \\
**Algorithm** & **Execution Time—Mean** & **Execution Time—Mean** & \multirow{2}{*}{**Speedup**} & **Global Memory** & **Local Memory** \\
**Implementation** & & & & & \\ \hline Classical implementation & 0.276 & - & - & 2616 & 289 \\ \hline four sections & 0.276 & 0.051 & 5.41 & 152 & 94 \\ eight sections & 0.275 & 0.034 & 8.08 & 295 & 184 \\ \hline
1 st algorithm [21] & 0.188 & 0.047 & 4.00 & 480 & 196 \\ \hline
2 nd algorithm [40] & 0.187 & 0.044 & 4.25 & 162 & 72 \\ \hline \end{tabular}
\end{table} 
Table 13: Performances analysis of several DCT IV implementations (done on Jetson AGX Xavier development board) when GPU frequency was set to the maximum value. In conclusion, we can say that the new algorithm introduced is a very high-performance one capable of obtaining remarkable accelerations in the execution speed of the DCT IV transform through a parallel implementation on four sections and based on the reduced number of mathematical operations. If the requirements compel an additional acceleration in the execution speed, the implementation based on eight parallel sections of this algorithm can be used. Conceptualization, D.F.C. and D.M.D. ; methodology, D.F.C. and D.M.D. ; software, D.M.D. ; validation, D.F.C. and D.M.D. ; investigation, D.M.D. ; resources, D.M.D. ; writing--original draft preparation: D.F.C. and D.M.D. ; writing--review and editing: D.M.D. and D.F.C. ; supervision: D.F.C. All authors have read and agreed to the published version of the manuscript.