# Information Fusion for Multi-Source Material Data: Progress and Challenges
## 3 Progress of Information Fusion for Multi-Source Material Data
### Algorithms for Material Data Fusion
#### 3.4.2 Feature-Level Fusion  
Feature-level fusion aims to perform data fusion based on feature information like categories, characteristics, primitives (edges or textures), or state vectors of the underlying output. Feature-level fusion comprehensively analyzes and processes feature information. Feature-level fusion algorithms can be mainly divided into three categories:  
1. Algorithms based upon probability-based methods, e.g., the Bayesian decision theory. The Bayesian decision theory is a classic pattern recognition method. The known model space \(\Omega\) includes \(C\) model classes, denoted as \(\Omega=(\omega_{1}\),..., \(\omega_{C})\), and the unknown samples are composed of \(N\)-dimensional real numbers, denoted as \(x=[x_{1},\)\(x_{2}\),..., \(x_{N}]^{N}\). According to the Bayesian decision theory with the minimum error rate, if the sample is within class \(j\), then this class is the model class with the largest posterior probability under the condition of the known sample \(x\). This decision process is expressed as \(x\rightarrow\omega_{j}\), such that \(\omega_{j}\) can be expressed by Equation (3).  
\[\omega_{j}=arg_{k=1,...,C}P(\omega_{k}|x) \tag{3}\]  
here, \(P(\omega_{k}|x)\) represents the posterior probability of the \(k^{\text{th}}\) class, \(k\in[1,\)\(C]\). The classical method for combining continuous conditional measures is to use the Bayes Rule, which (roughly) involves multiplying the measures together, and then normalizing via an integral operation. In the literature [41], the authors combined a class of data fusion rules to get the combination of two (or more) probability distributions in the presence of unknown correlations with the Bayesian data fusion for improving fusion performance.  
2. Algorithms based on the Sparse Representation Theory, which is mostly used for image fusion [42; 43; 44]. The Sparse Representation Theory was proposed for signal processing. Its basic idea is to use the prior knowledge of data sparseness to find a linear representation of the target signal from as few atoms as possible in an over-complete dictionary. The main steps in most sparse-representation-based image fusion methods include the following procedures:  
* Segment the source images into some overlapping patches, and rewrite each of these patches as a vector.
* Perform sparse representation on the source image patches using pre-defined or learned dictionaries.
* Combine the sparse representations by some fusion rules.
* Reconstruct the fused images from their sparse representations.  
3. Feature fusion algorithms based upon the Deep Learning Theory [45; 46]. In this kind of algorithm, fusion features are obtained by fusing features obtained from multiple neural networks. From the feed-forward neural network such as the perceptron, multi-layer perceptron to other feedback neural networks such as RBM (Restricted Boltzmann Machine), DBM (Deep Boltzmann Machine), and CNN (Convolutional Neural Network), depth learning has been demonstrated to be highly efficient for feature learning [45]. Deep learning can be used for feature learning in image fusion and object detection [47]. In machine learning, multi-modal data collected by various sensors are often used for fusion, while dynamic fusion of multi-modal data is often performed by the vectorization deep learning method [48; 49], and the model training method is used to achieve satisfactory results. When processing multi-modal data, it is necessary to use techniques such as image segmentation and the vector extraction of image sequences [50]. The defect of neural networks is its large computation complexity.