# Color-Based Image Retrieval Using Proximity Space Theory
## 1 Introduction  
Image retrieval is the procedure of retrieving relevant images from a big database of images, which usually occurs in one of two ways: text-based image retrieval or content-based image retrieval. Text-based image retrieval describes an image by using one or more keywords, but manual annotation has a huge cost burden. To address this defect, content-based image retrieval is introduced in [1]. Content-based image retrieval differs from traditional retrieval methods, as it uses visual contents such as key points, colors and textures to retrieve similar images from large scale image databases. Content-based image retrieval has been widely employed in several research fields, such as machine vision, artificial intelligence, signal processing, etc. Any content-based image retrieval system has two steps that can be explained as follows. The first step is feature extraction; the key point in this step is to find a method of feature extraction to precisely describe the content of image. Color features, which are the basic characteristic of the image content, are widely used in image retrieval [2; 3; 4; 5]. For example, in [6], the color features was extracted in HSV color space by using color moment method. [PERSON] et.al. considered that only using grayscale information is imprecise in the image matching [7], because some images have same grayscale but have different color information, therefore they put color information into SURF descriptor making the precision better than only used SURF [8]. The second step is similarity measurement; in this step, the query image is selected, and the similarities between it and images of database are computed. In recent years, many different similarity computation methods have been proposed. Similarity is obtained by calculating the distance between two feature vectors, such as Euclidean, Manhattan and Chi square distance [9]. [PERSON] et al. calculated similarities between the query image and images in database by comparing the coding residual [10]. [PERSON] et al. proposed a similarity measure that is a sparse representation issue via matching the feature dictionaries between the query image and images of database [11]. With the rapid development of modern computer technology and soft computing theory, many modern algorithms and theories are applied to the image segmentation and image retrieval, which include fuzzy sets [12], rough set theory [13; 14; 15; 16; 17; 18; 19; 20], near sets [21], mathematical morphology, neural network, etc. Computational Proximity (CP) [21] is a novel method to deal with image processing, which can identify nonempty sets of points that are either close to each other or far apart by comparing descriptions of pixels points or region in digital images or videos. [PERSON] and [PERSON] [22] were the first to combine nearness in digital images with proximity spaces. In CP, pixel is an object, and the various features of pixel, such as grayscale, color, texture and geometry, are regarded as its attributes. For a point \(x\), \(\Phi(x)\) denotes the feature vector of \(x\), then  
\[\Phi(x)=(\phi_{1}(x),\cdots,\phi_{i}(x),\cdots,\phi_{n}(x)),i\in\{1,\cdots,n\}. \tag{1}\]  
Similarity between either points or regions in CP is defined in terms of the feature vectors that describe either the points or the regions. Feature vectors are expressed by introducing probe function. A probe \(\phi\) maps a geometric object to a feature value that is a real number [21]. The similarity of a pair of nonempty sets is determined by the number of the same element. Put another way, nonempty sets are strongly near, provided sets have at least one point in common (introduced in [23]). **Example 1**. : _[_21_]_ _Let \(A,B,C\) be a nonempty sets of points in a picture, \(\Phi=\{r,g,b\}\), where \(r(A)\) equals the intensity of the redness of \(A\), \(g(A)\) equals the intensity of the greenness of \(A\) and \(b(A)\) equals the intensity of the blueness of \(A\).