## 5 Analysis  
The autoencoder architectures presented in Section 4.1 can be further analyzed in terms of errors and semantic understanding of related visual features. The results from Sections 4.1 and 4.2 can be used to compare the overall quality of selected solutions and analyze the dependency between the autoencoder errors and particular image classes. This could be related to the level of adjustment of the set of selected features to the data set, for example, manually selected directions of lines are particularly irrelevant in the case of digits 0 and 5. Another important aspect is the inner complexity of the digits. Digit 1, which usually consists of one or two segments, is especially easy to model. The exact directions of the segments, however, do not fit any of the manually selected filters. Thus, only an automatically computed feature extractor was able to take full advantage of the simplicity of the shape of this digit, producing a remarkably low error rate for this class. The errors for the other nine classes differ only slightly in the case of the automatically chosen features, which means that this solution actually reflects the properties of the data set. Manually selected features were not digit-specific, which resulted in generally higher error rates and greater variance of errors among different digits in MF4 experiment. In the case of the manually designed features, the convolution filters were created arbitrarily, as illustrated in Table 3. However, another way of visual presentation of the features can be achieved by using the decoder part of the autoencoder architecture (see Figure 2). The results of decoding \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Original** & **Large** & **Increase** \\ \hline MF4 & 11.220 & 12.220 & 8.91\% \\ \hline AF4 & 8.304 & 8.342 & 0.46\% \\ \hline AF5 & 7.792 & 7.902 & 1.41\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The autoencoders trained in Section 4.1 were used with larger images, as shown in Figure 7. An average per-object error was calculated and compared to the original results, related to the objects with a single centered object. The information on the relative error increase is included in the table as well. Figure 7: In order to demonstrate that the proposed autoencoder architecture is size-independent, \(80\times 80\) images containing multiple digits from the MNIST data set [7] were generated. The extended images contain up to four objects. single points related to each of the manually chosen features are presented in Table 8--the relation to the filters shown in Table 3 is apparent. This technique of visualization can also be applied to the models with the automatically constructed features. The results of this approach are reported in Tables 9 and 10--apparently, this time the features are implicit and difficult to categorize semantically. As the shape of each visual feature is complex, and possibly context-dependent, we need an analysis that goes beyond simple visualization. Tables 11-13 provide information on the intensity of each feature in the data set, including both averaged results and those obtained for separate subsets consisting of different digits. The intensity is calculated as a sum of the elements of the respective encoding matrix (output of the encoder, as illustrated in Figure 2). It must be emphasized that because of different weights in neural network models related to each decoder/classifier, it is possible to compare only the values within the same table. However, the results presented still enable a thorough analysis that otherwise would be difficult to perform. The MF4 results (see Table 9) are especially easy to understand, as digits are rather taller than wide, vertical lines (feature #2) are the most visible among the data set. Remarkably, digit 1 contains almost no other features. The least intense feature is related to the backslash segments (#4), which occur mostly as a part of arc (digits 8, 3, and 0). As it was expected, digit 8 is especially rich in all kinds of features.