# An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System

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## Abstract

**:**

## 1. Introduction

## 2. Architecture of the Proposed Technique

## 3. Image Representation

#### 3.1. Tree Representation

#### 3.2. Feature Extraction

## 4. Similarity Measurement

#### 4.1. Global Similarity Measurement

#### 4.2. Local Similarity Measurement

#### 4.2.1. Node Similarity Measurement

- Fuzzification:The first step is to transform the input crisp values such as either ‘0’ or ‘1’ into grades of membership for the linguistic terms of fuzzy sets. The membership function is used to associate a grade with each linguistic term. Selecting a proper membership function is an application dependent problem. Some of most commonly used prototype membership functions are cone, exponential, and triangular functions. Two factors are considered when selecting the membership function for our system: the retrieval accuracy and the computational burden for evaluating a membership function. We chose the triangular function as the membership function since it has good expressiveness and high computational efficiency in the literature [30,33], as shown in Figure 5. The input and output linguistic terms of this paper are $\tilde{A}$ = {‘not similar’, ‘similar’} and $\tilde{B}$ = {‘not similar’, ‘similar’, ‘very similar’}, respectively. To satisfy the requirement of the membership function in the FIS, the input crisp values must be transformed into similarity values, ${x}_{f}$. The designed formula based on the Manhattan distance is defined as:$${x}_{f}=1-\frac{{D}_{f}}{\mathrm{max}({F}_{f}^{q},{F}_{f}^{r})}$$Using Equation (17), the five features of a node described in Section 3, namely, (1) convexity, (2) eccentricity, (3) compactness, (4) circle variance, and (5) elliptic variance are transformed to the similarity crisp values, ${x}_{f}$. The similarity crisp values are further converted into grades of membership ${\tilde{x}}_{f}$ for the linguistic terms of fuzzy sets.
- Fuzzy inference engine:The fuzzy inference engine employs fuzzy IF-THEN rules to express input-output relationships and models the qualitative inputs and reasoning process for creating the output. The law to design or build a set of fuzzy rules is based on a human being’s knowledge or experience, which depends on each different actual application. The IF part is mainly used to capture knowledge using the elastic conditions, and the THEN part can be utilized to give the conclusion in linguistic variable form. This IF-THEN rule is widely used by the fuzzy inference system to compute the degree to which the input data matches the condition of a rule. In this paper, there are five input variables and two linguistic terms, so we have ${2}^{5}=32$ possible rules. One of the fuzzy IF-THEN rules is represented by:$${R}^{L}:\text{\hspace{0.17em}}IF\text{\hspace{0.17em}}{\tilde{x}}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}{\tilde{A}}_{1}^{L},\dots ,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{5}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}{\tilde{A}}_{5}^{L},\text{\hspace{0.17em}}THEN\text{\hspace{0.17em}}\tilde{y}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}{\tilde{B}}^{L}$$$$\begin{array}{ll}{R}^{1}:& IF\text{\hspace{0.17em}}{\tilde{x}}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{2}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{3}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}\\ & and\text{\hspace{0.17em}}{\tilde{x}}_{4}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{5}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}THEN\text{\hspace{0.17em}}\tilde{y}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}very\text{\hspace{0.17em}}similar\\ \vdots & \\ {R}^{7}:& IF\text{\hspace{0.17em}}{\tilde{x}}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{2}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{3}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}\\ & and\text{\hspace{0.17em}}{\tilde{x}}_{4}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{5}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}THEN\text{\hspace{0.17em}}\tilde{y}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar\\ \vdots & \\ {R}^{17}:& IF\text{\hspace{0.17em}}{\tilde{x}}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{2}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{3}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}\\ & and\text{\hspace{0.17em}}{\tilde{x}}_{4}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\tilde{x}}_{5}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}similar,\text{\hspace{0.17em}}THEN\text{\hspace{0.17em}}\tilde{y}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}not\text{\hspace{0.17em}}similar\end{array}$$The output results are then aggregated using the Mamdani-type inference [33], a MAX-MIN compositional operator. There are two steps in the MAX-MIN compositional operator. In the first step, we use the minimum inference engine to integrate the fuzzy sets in the rule ${R}^{L}$, such that:$${\tilde{B}}^{L}(y)={\tilde{A}}_{1}^{L}(x)\wedge \dots \wedge {\tilde{A}}_{5}^{L}(x),\text{\hspace{1em}}L\in \left\{1,2,\dots ,{N}_{L}\right\}$$$$\tilde{B}(y)={\displaystyle \underset{L=1}{\overset{{N}_{L}}{{\displaystyle \cup}}}{\tilde{B}}^{L}}={\tilde{B}}^{1}(y)\oplus {\tilde{B}}^{2}(y)\oplus \dots \oplus {\tilde{B}}^{{N}_{L}}(y)$$
- Defuzzification:After the reasoning results, the fuzzy output is still a linguistic variable, and this linguistic variable needs to be converted into a crisp variable via the defuzzification process. Two commonly used methods of defuzzification are the center of area (COA) method and middle of maximum method (MOM). In this paper, the COA was used based on its better results; the formula of the COA is expressed as:$$\_nodefunc({O}_{j,i}^{q},{O}_{j,i}^{r})={y}^{*}=\frac{{\displaystyle \sum _{i=0}^{{N}_{ql}}\tilde{B}\left({y}_{i}\right){y}_{i}}}{{\displaystyle \sum _{i=0}^{{N}_{ql}}\tilde{B}\left({y}_{i}\right)}}$$

#### 4.2.2. Weighting Subtree Similarity Measurement

## 5. Experimental Results and Discussion

#### 5.1. Experiment Setup

_{2}norm) and the Chi-square distance, respectively.

#### 5.2. Analysis for Parameters Setting

#### 5.3. Analysis for the Effect of Fuzzy Inference System

#### 5.4. Performance of the Precision-Recall Rates

#### 5.5. Performance of Bull’s Eye Score and the Retrieval Ranking

#### 5.6. Performance of Efficiency

#### 5.7. Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Local geometric feature descriptors: (

**a**) convexity; (

**b**) eccentricity; (

**c**) compactness; (

**d**) circle variance; and (

**e**) elliptic variance.

**Figure 5.**The input and output membership functions. (

**a**) Input: ${x}_{1},{x}_{2},\dots ,{x}_{5}$; (

**b**) Output: $y$.

**Figure 6.**An example illustrating the weighting subtree generation. (

**a**) A tree; (

**b**) weighting subtree 1; (

**c**) weighting subtree 2; (

**d**) weighting subtree 3.

**Figure 7.**An unmatched problem between two subtrees. (

**a**) child nodes, n = 3; (

**b**) child nodes, m = 4; (

**c**) unmatched problem.

**Figure 9.**The analysis of the weighing combinations for parameters ${w}_{H}$ and ${w}_{T}$ using (

**a**) the BES and (

**b**) the average normalized modified retrieval rank (ANMRR) scores.

**Figure 10.**The comparison between the fuzzy inference system (FIS) and weighting assignment using precision and recall (P-R).

**Figure 12.**Comparison of the Bull’s eye and ANMRR scores of all classifications in the database. (

**a**) The Bull’s eye scores. (

**b**) The ANMRR scores.

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**MDPI and ACS Style**

Chen, C.-S.; Weng, C.-M. An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System. *Appl. Sci.* **2017**, *7*, 849.
https://doi.org/10.3390/app7080849

**AMA Style**

Chen C-S, Weng C-M. An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System. *Applied Sciences*. 2017; 7(8):849.
https://doi.org/10.3390/app7080849

**Chicago/Turabian Style**

Chen, Chin-Sheng, and Chi-Min Weng. 2017. "An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System" *Applied Sciences* 7, no. 8: 849.
https://doi.org/10.3390/app7080849