# Color-Based Image Retrieval Using Proximity Space Theory
## 2 Proposed Method
### Similarity Measure Based on Dominance Perceptual Information  
Feature extraction and similarity measurement are two important steps in image retrieval. In this paper, color feature is selected as the retrieval feature, and we quantify color features into 72 numbers, \(\Phi=\{0,1,\cdots,71\}\). If an image \(X\) has \(T\) pixels, and there are \(Q\) pixels with the color feature \(c\) in an image, \(\phi_{c}(X)=\frac{Q}{T}\) and \(c\in\Phi\). For example, \(c=0\), \(\phi_{0}(X)\) represents the probability that the color feature equals 0 in the image \(X\), \(\phi_{1}(X)\) represents the probability that the color feature equals 1 in the image \(X\), and \(\phi_{71}(X)\) represents the probability that the color feature equals 71 in the image \(X\); then, we can obtain that \(\Phi^{\prime}(X)=\{\phi_{0}(X),\phi_{1}(X),\cdots,\phi_{71}(X)\}\), using which we can get a color histogram (see Figure 1) of a query image. The Top-m color features \(\Phi^{m}(X)\) is obtained by descending order of \(\Phi^{\prime}(X)\), \(\Phi^{m}(X)=\{l_{1}^{x},l_{2}^{x},\cdots,l_{m}^{x}\},l_{i}\in\Phi=\{0,1,2, \cdots,71\}\). In this paper, we take Top-m dominant color features in the HSV space to match images to improve the efficiency of retrieval; therefore, we use color features vector \(\Phi^{m}(X)\) as the distinguishing description of image \(X\). In addition, image similarity measurement is also important, such as Euclidean distance formula or cosine formula. However, if we use Euclidean distance formula or cosine formula to compute the similarity between the two images, in essence, it will lose significance. For example, the similarity between matrix \([1,2,3]\) and matrix \([2,4,6]\) is equal to 1 by cosine formula. Inspired by  
Figure 1: (**a**) The query image; and (**b**) the color histogram of image (**a**). dominance rough set theory, we propose a novel dominance granule structure similarity method as similarity measurements, defined as follows:  
**Definition 1**. : _Let \(\Phi^{m}(X)=\{l_{1}^{x},l_{2}^{x},\cdots,l_{m}^{x}\},\Phi^{m}(Y)=\{l_{1}^{y},l_{2} ^{y},\cdots,l_{m}^{y}\}\) and \(\Phi^{m}(X),\Phi^{m}(Y)\subseteq\Phi\), where \(X\) and \(Y\) represent the query image and the image of image database, respectively; and \(a_{x}\) and \(a_{y}\) represent the color attribute of \(X\) and \(Y\), respectively. A is a set of all pre-order relationships on \(\[PERSON]^{m}(X)\) and \(\[PERSON]^{m}(Y)\). \(\forall\succeq_{a_{x}}\succeq_{a_{y}}\in A\)[26], \(\[PERSON]^{m}(X)/\succeq_{a_{x}}=\{[l_{1}^{x}]_{\succeq_{a_{x}}},[l_{2}^{x}]_{ \succeq_{x}},\cdots,[l_{m}^{x}]_{\succeq_{x}}\}\) and \(\[PERSON]^{m}(Y)/\succeq_{a_{y}}=\{[l_{1}^{y}]_{\succeq_{a_{y}}},[l_{2}^{y}]_{ \succeq_{a_{y}}},\cdots,[l_{m}^{y}]_{\succeq_{a_{y}}}\}\) are the corresponding dominance granule structure, then the similarity between \(X\) and \(Y\) can be denoted by_  
\[S(\[PERSON]/\succeq_{a_{x}}\[PERSON]/\succeq_{a_{y}})=1-\frac{1}{|\[PERSON]|-1}\sum_{i=1}^{ |\[PERSON]|}\frac{\big{|}[l_{i}]_{\succeq_{x}}\bigtriangleup[l_{i}]_{\succeq_{y}} \big{|}}{|\Phi|}. \tag{8}\]  
_because Equation (8) needs to calculate the global color feature similarity, the efficiency is reduced due to the large amount of calculation. Therefore, the Top-m color features of the image are selected for image matching, and Equation (8) is improved as follows:_  
\[S(\Phi^{m}(X)/\succeq_{a_{x}},\Phi^{m}(Y)/\succeq_{a_{y}})=1-\frac{1}{m}\sum_{ i=1}^{m}\frac{\min_{j}\big{|}[l_{i}^{x}]_{\succeq_{x}}\bigtriangleup[l_{j}^{y}]_{ \succeq_{a_{y}}}\big{|}}{m},j\in\{1,2,\cdots,m\}, \tag{9}\]  
_where \(|\Phi|\) is the number of all the elements in set \(\Phi\), \(\big{|}[l_{i}^{x}]_{\succeq_{ax}}\bigtriangleup[l_{j}^{y}]_{\succeq_{a_{y}}} \big{|}=\big{|}[l_{i}^{x}]_{\succeq_{x}}\cup[l_{j}^{y}]_{\succeq_{a_{y}}} \big{|}-\big{|}[l_{i}^{x}]_{\succeq_{a_{x}}}\cap[l_{j}^{y}]_{\succeq_{a_{y}}} \big{|}\), and \([l_{i}]_{\succeq_{ax}}=\{l_{j}|l_{j}\succeq_{a_{x}}l_{i},1\leq j\leq m\}\),\([l_{i}]_{\succeq_{a_{y}}}=\{l_{j}|l_{j}\succeq_{a_{y}}l_{i},1\leq j\leq m\}\).