# Frequency-Domain and Spatial-Domain MLMVN-Based Convolutional Neural Networks
## 5 Simulation Results and Discussion
### MLMVN as a CNN in the Frequency Domain
It was established that the reason is similar to that for CNNMVN: the network's errors decrease during the training process, and the normalization factors in the error-backpropagation process drive these errors towards zero, resulting in minimal changes in neuron weights and a slowdown in learning. Like CNNMVN, this network demonstrates the capability to learn over many epochs without overfitting. Figures 8 and 9 illustrate the stable accuracy rate of the testing dataset during the training process. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Topology**} & \multicolumn{2}{c}{**Number of Frequencies**} & **Epoch When Maximum** & \multirow{2}{*}{**Accuracy**} \\  & **Used** & **Accuracy Reached** & & \\ \hline h2048-o10 & Images transformed by (1) & 91 & 87.48 \\ h3072-o10 & Images transformed by (1) & 51 & 87.94 \\ \hline h2048-o10 & 7 & 178 & 89.81 \\ h3072-o10 & 7 & 118 & 90 \\ \hline h2048-o10 & 8 & 180 & 89.85 \\ h3072-o10 & 8 & 165 & 89.89 \\ \hline h2048-o10 & 9 & 43 & 89.99 \\ h3072-o10 & 9 & 109 & 90 \\ \hline h2048-h3072-o10 & 7 & 24 & 88.87 \\ h3072-h2048-o10 & 7 & 23 & 88.63 \\ \hline h2048-h3072-o10 & 8 & 81 & 88.51 \\ h3072-h2048-o10 & 8 & 147 & 88.19 \\ \hline h2048-h3072-o10 & 9 & 30 & 88.5 \\ h3072-h2048-o10 & 9 & 60 & 88.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Classification results for MLMVN as a CNN in a frequency domain for Fashion MNIST dataset. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Topology**} & \multicolumn{2}{c}{**Number of Frequencies**} & **Epoch When Maximum** & \multirow{2}{*}{**Accuracy**} \\  & **Used** & **Accuracy Reached** & & \\ \hline h1024-o10 & Images transformed by (1) & 199 & 93.18 \\ h2048-o10 & Images transformed by (1) & 190 & 94.13 \\ \hline h1024-o10 & 5 & 145 & 97.86 \\ h2048-o10 & 5 & 193 & 98.11 \\ \hline h1024-o10 & 6 & 183 & 97.91 \\ h2048-o10 & 6 & 85 & 98.09 \\ \hline h1024-o10 & 7 & 200 & 97.78 \\ h2048-o10 & 7 & 157 & 97.86 \\ \hline h1024-h2048-o10 & 5 & 139 & 97.18 \\ h2048-h1024-o10 & 5 & 137 & 96.85 \\ h2048-h2048-o10 & 5 & 45 & 97.3 \\ h2048-h3072-o10 & 5 & 187 & 97.58 \\ \hline h1024-h2048-o10 & 6 & 40 & 97.08 \\ h2048-h1024-o10 & 6 & 124 & 96.9 \\ h2048-h2048-o10 & 6 & 113 & 97.41 \\ h2048-h3072-o10 & 6 & 95 & 97.44 \\ \hline h1024-h2048-o10 & 7 & 25 & 96.68 \\ h2048-h1024-o10 & 7 & 38 & 96.66 \\ h2048-h2048-o10 & 7 & 64 & 96.71 \\ h2048-h3072-o10 & 7 & 65 & 97.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification results for MLMVN as a CNN in a frequency domain for MNIST dataset. Figure 8: MLMVN as a Frequency domain CNN accuracy for MNIST dataset using 1–5, 1–6, 1–7 frequencies and the original image as input. Figure 9: Accuracy of MLMVN as a frequency-domain CNN for Fashion MNIST dataset using 1–7, 1–8, 1–9 frequencies and the original image as input. Comparative Analysis of Convolved Images Produced by CNNMVN and MLMVN as a CNN in the Frequency Domain  
It is interesting to analyze the convolutions performed by CNNMVN in the spatial domain and by MLMVN in the frequency domain. In both networks, these convolutions result from the learning process. By analyzing these convolutions, we can see which details they extract from images to recognize them. While both networks perform convolutions, these convolutions appear to be quite different from each other. This is illustrated in Figure 12 using one image from the MNIST dataset (class \"3\"--Figure 12a). CNNMVN creates convolutional kernels that perform high-pass (Figure 12c) and medium-pass filtering (Figure 12b). The latter can be compared to unsharp masking, well known in image processing as a family of filters that distinguish image details. This can clearly be seen from the feature maps resulting from the respective convolutions. We may also conclude that these convolutions are quite similar to the ones performed by kernels in the first convolutional layer of a traditional CNN. MLMVN performs convolution in the frequency domain. Figure 12d depicts an image resulting from frequency-domain downsampling.