# Illuminant Estimation Using Adaptive Neuro-Fuzzy Inference System
## 2 Related Works  
In this section, we give a brief introduction to statistics-based and combination methods, since our proposed method is mainly with respect to both of them.  
Statistics based methods. These algorithms directly compute some statistical measures of the input image to estimate illuminant color without complicated computational effort, although sometimes the accuracy cannot be assured. Most of these algorithms can be unified into a GE framework developed by [PERSON] et al. [23], which includes higher-order derivatives and the Minkowski family norm given as:  
\[\left(f\left|\frac{\partial^{n}\mathbf{f}^{\prime}(\mathbf{x})}{\partial \mathbf{x}^{n}}\right|^{p}d\mathbf{x}\right)^{\frac{1}{p}}=k\mathbf{e}^{n,p, \sigma}, \tag{1}\]  
Here \(\mathbf{f}^{\prime}=\mathbf{f}\otimes\mathbf{G}^{\sigma}\) denotes convolution of the image with a norm value, \(k\) is a scaling, and \(e^{n,p,\sigma}\) is the resulting illuminant estimate. The methods defined by different choices of the parameters \(n\), \(p\) and \(\sigma\) are denoted as \(GE^{n,p,\sigma}\), where the optimal values of these parameters may vary with different datasets. Some assumptions based algorithms are special instantiations of this framework [18; 20; 23; 24], WP and GW also being included: (1) \(GW\) (\(GE^{0,1,0}\)); (2) \(WP\) (\(GE^{0,\infty,0}\)); (3) \(SoG\) (\(GE^{0,6,0}\)); (4) 1-order gray edge (\(GE1\)) (\(GE^{1,1,6}\)); (5) 2-order gray edge (\(GE2\)) (\(GE^{2,1,5}\)); (6) General gray world (\(GGW\)) (\(GE^{0,13,2}\)).  
The other examples of statistics based methods include bright-and-dark color PCA (_PCA-based_) and local surface reflection (\(LSR\)) methods. The PCA-based method proposed by [PERSON] and [PERSON] [25] selects the color points with largest and smallest projections on the mean chromaticity, and then calculates its first PCA vector as the illuminant estimation. This method is fairly efficient and easy to compute. [PERSON] et al. [26] found that the ratio of the global summation of true surface reflectance to the global summation of locally normalized reflectance estimate in a scene is approximately achromatic for both indoor and outdoor scenes. Based on this substantial observation, LSR illuminant estimation model is developed which has only one free parameter and requires no explicit training process.  
Combination methods. Let \(E=\{e_{1},e_{2},\cdots\}\) be the set of the illumination estimates obtained from some statistics or learning based unitary algorithms, then combination methods are to combine the estimates \(e_{1}\),\(e_{2}\), \(\cdots\) into a single, final estimate by corresponding weights \(w_{1}\),\(w_{2}\), \(\cdots\). According to whether \(w_{1}\),\(w_{2}\), \(\cdots\) are constant or not, combination methods can be divided into two basic groups, i.e., static weight combination and dynamic weight combination. Static weights commonly are not suitable for the diversity of image characteristics; the related methods [18; 27], like Simple Averaging, Nearest-N%, and Median, etc., can not obtain robust illuminant estimation. On the other hand, dynamic weights vary with image features to guide the selection or combination of unitary estimates [10; 11; 18]. Various scene characteristics, including low-level properties (e.g., visual properties [10], 3D geometry [28; 29]), mid-level initial illumination estimates [30], high-level semantic content (e.g., semantic likelihood [31], indoor/outdoor classification [9]), can be used to find the best combination. Furthermore, the weights might be obtained by different algorithms, such as machine learning [19; 32; 33; 34], Fuzzy model [24], multi-objective optimization [20], graph-based semi-supervised [30], or one without prior training [9; 21]. In most combination methods, weight determination is usually accompanied by image classification and training regression models for different classes in order to deal with a wide range of image features [19]. It is a significant obstacle to reach better performances for combination algorithms.