## 5 Experimental Study and Result Analysis
### Analysis of Embedding Capacity through Modified Elias Gamma Encoding Process  
Let us assume that we have a run-length encoding sequence obtained as follows:  
\[R=(B,R_{0},R_{1},R_{2},\ldots,R_{N-1}) \tag{9}\]  
where \(B\) indicates the starting bit in the plane, and \(R_{X}\) indicates the run-length counts in which \(X=0,1,2,\ldots N-1\). Let us denote the number of bits required to represent \(R_{X}\) by using \(L_{X}\).  
Note that during the modified Elias gamma encoding process, the run-length sequence values will be incremented by one. Let us denote the run-length sequence obtained after incrementing the original sequence by \(\hat{R}\).  
\[\hat{R}=(B,\hat{R_{0}},\hat{R_{1}},\hat{R_{2}},\ldots,\hat{R_{N-1}}) \tag{10}\]  
Let us denote the number of bits required to represent \(\hat{R_{X}}\) by using \(\hat{L_{X}}\).  
While incrementing the actual run-length sequence by one, there are two possibilities:  
* Case 1 (best case): \(L_{X}=\hat{L_{X}}\;\;\forall X\in\{0,1,\ldots,N-1\}\) In this case, we will obtain the maximum embedding capacity through the modified Elias gamma encoding process. Note that in this case, we shall embed \(N\) additional bits through the encoding process without any change in the length of the binary sequence obtained after encoding. This is the best possible embedding capacity that we can additionally achieve through a modified Elias gamma encoding scheme. Note that in addition to this, we will use the empty space created through compression also to keep additional messages.
* Case 2 (worst case): \(L_{X}+1=\hat{L_{X}}\;\;\forall X\in\{0,1,\ldots,N-1\}\) In this case, every run-length sequence needs one extra bit for representation. Therefore, the overall overhead will become \(2N\) during the modified Elias gamma encoding process. However, we will have an opportunity to embed one-bit additional data on all the \(N\) sequences. Hence, the extra overhead will become \(N\) bits. However, note that the probability of happening this situation is very low. This situation will happen only when all the run-length sequences \(R_{X}\) are from \(\{2^{0}-1,2^{2}-1,2^{3}-1,\ldots,2^{K}-1\}\), where \(K\) can be any positive integer. If we consider an image of \(512\times 512\), there will be 262,144 pixels. The probability that one \(R_{X}\) from the set \(\{2^{0}-1,2^{2}-1,2^{3}-1,\ldots,2^{K}-1\}\) (where \(2^{K}-1\leq 262\),144) is \(\frac{18}{262\cdot 144}=6.86\times 10^{-5}\). This shows that the probability to obtain the \(R_{X}\) value, which will lead to the worst-case embedding capacity, is very close to 0.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Image Name** & **RLE + EG** &
\begin{tabular}{c} **Proposed Scheme** \\ **(RLE + MEG)** \\ \end{tabular} \\ \hline \multicolumn{3}{c}{Natural images} \\ \hline Airplane & 0.7758 & 0.7895 \\ \hline Peppers & 0.7869 & 0.7941 \\ \hline Boat & 0.6633 & 0.6857 \\ \hline Baboon & 0.3571 & 0.3534 \\ \hline \multicolumn{3}{c}{Medical images} \\ \hline Medical image-1 & 3.1844 & 3.2154 \\ \hline Medical image-2 & 3.1653 & 3.2313 \\ \hline Medical image-3 & 3.4800 & 3.5282 \\ \hline Medical image-4 & 3.2548 & 3.3199 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of embedding rate (in bpp) with the RLE + EG approach.