# An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System
## 5 Experimental Results and Discussion
### Performance of Efficiency  
The computationally most expensive parts of our technique are the local descriptor in image representation and the local similarity measurement. In the local descriptor, the tree structure is first used to organize \(k\) objects in an image; therefore, the order of the tree structure grows linearly with the number of objects and holes in the image. Next, the local features of each tree node are obtained by the five geometric features: convexity, eccentricity, compactness, circle variance, and elliptic variance. The computation of these five features is based on the \(N\) points of the boundary. In [8], the time complexity of these five features is \(O(N)\). Accordingly, the total time complexity of extracting the local descriptor is \(O(k\cdot 5\cdot N)\) in the worst case, i.e., \(O(k\cdot N)\).  
The local similarity measurement is mainly based on the weighting subtree and FIS. Therefore, the computational complexity of the local similarity algorithm is dependent on the number of subtrees, the number of children of a subtree node, and the operation time of FIS, \(\_nodefunc()\). For obtaining the maximum similarity between two subtree sets of \(st^{q}\) and \(st^{r}\) subtrees, the bipartite matching method using the Hungarian algorithm is used to assign the \(st^{q}\cdot st^{r}\) similarities. The Hungarian algorithm has a polynomial-time complexity of \(O(n_{1}\cdot n_{2}^{2})\)[21]. In addition, bipartite matching is also used in the similarity measurement of the child level between two subtrees. Here, we assume that the number of child nodes per subtree is the same; hence, the \(st^{q}\) and \(st^{r}\) subtrees have \(n\) and \(m\) child nodes, respectively. Let \(n\) and \(m\) be the maximum number of child nodes. Therefore, the local similarity measurement has a complexity of \(O(st^{q}\cdot(st^{r})^{2}+st^{q}\cdot st^{r}\cdot(nm^{2}))\) without the computational complexity of the node similarity. In this paper, Mamdani-type inference is used in our FIS. This inference has \(O(x_{f}\cdot N_{L})\) time complexity, with \(x_{f}\) and \(N_{L}\) being the number of input dimensions and the number of fuzzy rules [34], resepctively. Finally, the total time complexity of the worst case is \(O(st^{q}\cdot(st^{r})^{2}+st^{q}\cdot st^{r}\cdot(nm^{2}))+n\cdot m\cdot x_{f} \cdot N_{L}\).  
Figure 13: Comparison of the ranking performance for the query in class 1.  
This experiment also provides an analysis of the efficiency performance of the retrieval system. Twelve classes (416 trademark images) are used to assess the computational burden. The efficiency index is the average execution time for the twelve classes. Table 2 shows the computational burden for the proposed technique, ZMEG, and ZMLDP. As can be seen in terms of image representation, the computation of the global and local descriptors in the proposed technique is highly efficient. In particular, the proposed novel similarity measure based on the weighted subtree and the corresponding fuzzy sets is more efficient than that of ZMLDP. The computational advantage range compared to that of ZMEG and ZMLDP is between 151 and 193 times greater. Clearly, the results of the proposed technique are superior to the results of the other approaches. Based on the convincing experimental results, the proposed technique not only provides high accuracy but also retrieves the query image in 2 ms.