# A Mixed Method with Effective Color Reduction
## 1 Introduction  
Color quantization is a process that attempts to reduce the colors of an image. Let us consider a color image including \(n\) pixels. When the image is represented using the RGB color space, each pixel \(p_{i}\) of this image, with \(1\leq i\leq n\) is represented by three integer values between 0 and 255 that indicate the amount of red, green and blue that define the pixel, \(p_{i}=(R_{i},G_{i},B_{i})\). Therefore, the palette used to represent this image can include \(256^{3}\) different colors. The color quantization problem consists of defining a new palette which includes fewer colors than the palette used to represent the original image. Once this palette containing \(q\) colors has been defined, it is used to represent the new image, called quantized image. The quantized palette should be defined in such a way that it allows a new image, as similar to the original as possible, to be obtained. Color quantization is not only a problem in itself, but it is also related to other operations applied to images, such as image segmentation [1; 2; 3; 4; 5], content-based image retrieval [6; 7; 8], texture analysis [9; 10; 11] and image watermarking [12; 13]. When the number of different colors of an image is reduced, the size of the file that contains the image is also reduced. This feature not only reduces the storage space required for the image but also accelerates the speed of transmission of the image. Therefore, although current devices can display images with many colors, reducing the colors of an image is still useful for solving other problems. Several practical scientific and industrial applications that involve color quantization are medical image analysis [14; 15; 16], nondestructive inspection of food quality [17; 18], biometric recognition [19; 20] or surveillance systems [21; 22]. The color quantization problem is complex, since the selection of the best colors to define the quantized palette is a NP-complete problem [23]. For this reason, several solution methods have been proposed to solve this problem. With respect to the strategy applied for finding a solution, the methods employed can be classified into two groups: splitting methods and clustering-based methods. The splitting methods operate on the color cube to define the quantized palette. This cube is progressively divided into smaller boxes until \(q\) boxes are obtained. At the end of this process each box defines one color in the quantized palette, and the quantized image is generated by representing thepixels of each box by the color of said box. This category includes methods such as the Median-cut [24], the Variance-based method [25], the Wu's method [26], the Binary splitting [27] and the Octree [28]. These methods differ in relation to the box that is selected to perform the next division as well as in the splitting axis and the splitting point used to divide the selected box. Clustering-based methods define clusters (groups) of similar pixels. Each cluster defines a color included within the new palette and that color is used to represent all the pixels of the group in the quantized image. Various clustering methods have been used to reduce the colors of an image. The K-means method is probably the most popular clustering method that has been used for this purpose [29; 30; 31; 32]. Neural networks have also been applied to this problem. In addition to the Neuquant method [33], other color quantization methods using Neural networks have been presented in [34; 35; 36]. Another group of algorithms based on this approach are those that use a swarm of individuals to solve a complex problem.