# Color-Based Image Retrieval Using Proximity Space Theory
## 2 Proposed Method
### Similarity Measure Based on Dominance Perceptual Information
The ordinal relation between objects in terms of the attribute \(a_{x}\) or \(a_{y}\) is denoted by \(\succeq\), and \(l_{1}\succeq_{a_{x}}l_{2}\) or \(l_{1}\succeq_{a_{y}}l_{2}\) means that \(l_{1}\) is at least as good as \(l_{2}\) with respect to the attribute \(a_{x}\) or \(a_{y}\)._  
**Example 2**. : _To prove the feasibility of this method, we give an applied instance. Suppose \(m=3\), \(\Phi^{3}(1)=(7,9,1)\) (description of query image 1), where 7, 9 and 1 refer to the color features by quantization and \(\phi_{7}(1)>\phi_{9}(1)>\phi_{1}(1)\). Let \(\Phi^{3}(I_{1})\) (description of \(l_{1}\)), \(\Phi^{3}(I_{2})\) (description of \(l_{2}\)), \(\Phi^{3}(I_{3})\) (description of \(l_{3}\)), etc represent the set of Top-\(3\) color features of each image in image database, respectively. Thus, the similarity can be obtained by Equation (9) and results of the similarity are shown by Equation(10). \(I_{1}\), \(I_{2}\), \(\cdots\), and \(I_{13}\) represent images in the image database._  
_According to Equation (9), if 7 \(\succeq_{a_{x}}9\succeq_{a_{x}}1\), 7 \(\succeq_{a_{y}}1\succeq_{a_{y}}9\),_  
\[ \begin{array}{l}[7]_{\succeq_{a_{x}}}=\{7\}\\ [9]_{\succeq_{a_{x}}}=\{7,9\}\\ [1]_{\succeq_{a_{x}}}=\{7,9,1\},\\ \end{array}\begin{array}{l}[7]_{\succeq_{a_{y}}}=\{7\}\\ [1]_{\succeq_{a_{y}}}=\{7,1\}\\ [9]_{\succeq_{a_{y}}}=\{7,1,9\},\\ \end{array} \]  
\[ \begin{array}{l}\Phi^{3}(I)/\succeq_{a_{x}}=\{\{7\},\{7,9\},\{7,9,1\}\},\Phi ^{3}(I_{2})/\succeq_{a_{y}}=\{\{7\},\{7,1\},\{7,1,9\}\},\\ \end{array} \]  
\[ \begin{array}{l}S(\[PERSON]^{m}(I)/\succeq_{a_{x}},\[PERSON]^{m}(I_{2})/\succeq_{a_{y}} )=1-\frac{1}{3}\sum_{i=1}^{3}\frac{\min_{j}\big{|}[l_{i}^{x}]_{\succeq_{a_{x}}} \bigtriangleup[l_{j}^{y}]_{\succeq_{a_{y}}}\big{|}}{3}\\ =1-\frac{1}{3}\times(\frac{0}{3}+\frac{1}{3}+\frac{0}{3})\\ =1-\frac{1}{3}\times\frac{1}{3}\\ =\frac{8}{9}\approx 0.8,\\ \end{array} \]\[Sim=\left\{ \begin{array}{llll}1,&if&\[PERSON]^{3}(I_{1})=(7,9,1)\\ 0.8,&if&\[PERSON]^{3}(I_{2})=(7,1,9)\\ 0.8,&if&\[PERSON]^{3}(I_{3})=(9,7,1)\\ 0.8,&if&\[PERSON]^{3}(I_{4})=(1,7,9)\\ 0.8,&if&\[PERSON]^{3}(I_{5})=\{(7,9,I_{3})|I_{3}\
otin\[PERSON]^{3}(I)\}\\ 0.7,&if[PERSON]^{3}(I_{6})=\{(7,1,I_{3})|I_{3}\
otin\Phi^{3}(I)\}\\ 0.6,&if&\[PERSON]^{3}(I_{7})=(9,1,7)\\ 0.6,&if&\[PERSON]^{3}(I_{8})=(1,9,7)\\ 0.6,&if&\[PERSON]^{3}(I_{9})=(7,I_{2},I_{3})|I_{2},I_{3}\
otin\Phi^{3}(I)\}\\ 0.5,&if&\[PERSON]^{3}(I_{10})=\{(9,1,I_{3})|I_{3}\
otin\[PERSON]^{3}(I)\}\\ 0.4,&if&\[PERSON]^{3}(I_{11})=\{(9,I_{2},I_{3})|I_{2},I_{3}\
otin\[PERSON]^{3}(I)\}\\ 0.2,&if&\[PERSON]^{3}(I_{12})=\{(1,I_{2},I_{3})|I_{2},I_{3}\
otin\[PERSON]^{3}(I)\}\\ \vdots\\ 0,&if&\[PERSON]^{3}(I_{13})=\{(1,I_{2},I_{3})|I_{1},I_{2},I_{3}\
otin\[PERSON]^{3}(I)\}. \end{array} \right. \tag{10}\]  
As shown in Equation (10), we can see that the color feature's order of \(I_{1}\) is the same as the color feature's order of 1, so the similarity between \(I_{1}\) and 1 is 1. Image \(I_{5}\) strongly resembles (is strongly near) image 1 better than image \(I_{6}\), since the color feature's order of \(I_{5}\) is near to the color feature's order of 1. The image \(I_{13}\) does not resemble image 1, since both color features are different, so the similarity between \(I_{13}\) and 1 is 0. Therefore, it makes it easier to match images if there exist more identical elements between color features. Simultaneously, it is also critical whether corresponding elements between features vector are equal, such as \(I_{6}\) and image \(I_{7}\). The similarity measure proposed in this paper can improve the subjective and objective consistency of retrieval results.