## 3 Materials and Methods
### Ensemble of DIR Systems  
Within our ensemble of retrievals, we introduce two distinct techniques: \"Vote-Based DIR\" (VBDIR) and \"The Strong Decision-Based DIR\" (SDBDIR). In SDBDIR, we define \(\text{R}_{i}\) as an ordered list of K results, sorted by their similarity to a query image Q, for each of the weak DIR systems denoted as \(\text{DIR}_{i}\). We then apply an OR (logical \"or\") operation to combine these lists \(R_{i}\) for \(i=1\ldots N\), where \(N\) represents the number of weak DIR systems. This operation results in a global candidate set R, the union of all individual lists: \(R=R_{1}\cup R_{2}\cup\ldots\cup R_{1|N|}\). The global candidate set R is subsequently provided to the strong DIR system, the final decision-maker. The strong DIR system ranks the documents within the set R based on its criteria and computation. Ultimately, the ranked list produced by the strong DIR system is returned as the final result of the retrieval process. In the VBDIR approach, both each weak DIR system and the strong DIR system \(\text{DIR}_{i}\) are assigned a weight (\(w_{i}\)). This \(w_{i}\) reflects the significance of \(\text{DIR}_{i}\) in determining the final ranking of relevant documents. Each weak DIR system produces a list of the K most relevant document images denoted as \(R_{i}=\{d_{i1}\), \(d_{i2}\), \(\ldots\), \(d_{iK}\}\) for a given query \(Q\), along with their respective similarity scores \(sim_{i}(d_{ik})\) for \(k\in\{1\ldots k\}\). The union of the documents from all weak DIR systems is represented as \(R=R_{1}\cup R_{2}\cup\ldots R_{1|N|}\), and we determine the membership of each document (\(d_{i}\)) in the list \(R_{i}\) using the indicator function \(I\) (\(d_{j}\), \(R_{i}\)), as shown in Equation (1). In essence, VBDIR combines the outputs of multiple weak and strong DIR systems, taking into account their respective weights, to create a unified ranking of relevant documents based on the votes and preferences of each system. \[I\big{(}d_{j},R_{i}\big{)}= \begin{cases}1&\text{if }d_{j}\in R_{i}\\ 0&\text{if }d_{j}\
ot\in R_{i}\end{cases}  \tag{1}\]  
\[\pi_{j}=\sum_{i\in weak}I\big{(}d_{j},S_{i}\big{)}\cdot w_{i}\cdot\frac{sim_{ i}(d_{j})}{sim_{i}(d_{i}1)} \tag{2}\]  
\[\pi^{\prime}_{j}=\pi_{j}+\sum_{i\in strong}w_{i}\cdot\frac{sim_{i}(d_{j})}{sim_ {i}(d_{i}1)} \tag{3}\]  
The initial scores (\(\pi_{j}\)) for each document \(d_{j}\) are calculated as the sum of weights, which is determined based on the membership function \(I\)(\(d_{j}\), \(R_{i}\)), the \(w_{i}\) assigned to the DIR system, and the ratio of \(sim_{i}(d_{j})\) to \(sim_{i}(d_{i1})\). Equation (2) represents the formally expressed calculation. In this equation, \(\pi_{j}\) represents the initial score for document \(d_{j}\). The summation \(\Sigma\) is performed over all the weak DIR systems and the strong DIR system. For each system, \(i\), the \(w_{i}\), the membership functions \(I\)(\(d_{j}\), \(R_{i}\)), and the similarity ratio \(sim_{i}(d_{j})/sim_{i}(d_{i1})\) are considered in the computation. This scoring mechanism combines various factors to determine the initial relevance score for each document, considering the preferences of different DIR systems and their respective importance weights. Once the initial assignments of \(\pi_{j}\) for each document have been computed, the similarity of each document is updated based on the scores provided by the strong DIR systems. This update process involves considering the outputs and assessments of the strong DIR systems to refine the document similarity scores, potentially altering their rankings or relevance based on the additional insights and criteria provided by the strong DIR systems. In the final rank, the documents are sorted based on the updated scores \(\pi^{\prime}_{j}\), as shown in Equation (3). Ideally, strong DIR systems should carry higher weights than weak DIR systems in this ranking process. This approach is designed to mitigate errors that may occur within the strong DIR systems.