# An Integrated Dictionary-Learning Entropy-Based Medical Image Fusion Framework
## 2 Related Works
### Sparse Representation and Dictionary Learning  
In signal processing, sparse representation is an extremely powerful way to acquire, represent and compress high-dimensional signals. According to the sparse representation theory, an image can be sufficiently reconstructed as a linear combination of the fewest possible atoms or transform bases in an over-complete dictionary [23]. The number of atoms in an over-complete dictionary is more than the dimension of signal, and can be used to represent more flexible and meaningful information. For instance, an image can be represented using fixed bases (i.e., Fourier, Wavelet), or concatenations of such bases. Sparse representation provides efficient compression and denoising estimators with simple diagonal operators to address the signal's natural sparsity [39]. The discrimination power of the discriminative methods with the reconstruction property and the sparsity of the sparse representation is combined [40]. A fixed dictionary can be easily obtained, but it depends on the data type. However, it is difficult to select a good adaptive over-complete dictionary for sparse representation-based image reconstruction and fusion techniques. DCT basis and wavelet basis are often used to form an over-complete dictionary that does not rely on input images. For dictionary-based image restoration, it is important to build a bridge between sparse representation and dictionary learning. [PERSON] [16] proposed an adaptive dictionary learning method called K-SVD that is related to the alternate minimization strategy MOD [41]. The dictionary is learned with the extracted patches either from globally or adaptively trained dictionary by applying SVD operations. The non-zero coefficients of the matrix are updated in dictionary update step at the same time. Comparing with the fixed dictionary based methods, K-SVD usually shows a better performance in image reconstruction. However, it takes high computational costs to do the iterative K-SVD learning processing, especially with the large number of training images. So the dimension of dictionary is constrained. For a certain type of source images, image fusion can also have good performance based on the fixed dictionary. [PERSON] [42] proposed a sparse K-SVD method to learn a sparse dictionary as a type of dictionary between fixed and trained dictionary. Each atom of the leaned sparse dictionary can be represented over a prespecified base dictionary. Based on the signal sparse representation theory, [PERSON] [43] proposed a novel image fusion scheme that conducted the sparse representation on overlapping patches instead of the whole image by using a small size of dictionary. As the key technique of the proposed solution, the simultaneous orthogonal matching pursuit technique sparsely decomposed different source images into the same subset of dictionary bases. Most of multi-focus image fusion solutions are subject to one or more of image fusion quality degradations, such as blocking artifacts, contrast decrease, sharpness reduction and so on. [PERSON] [44] utilized a dictionary learned from local patches of source images to do multi-focus image fusion in spatial domain. The corresponding pooled features were extracted by pooling sparse representations of relative sharpness pixels together, according to the learned dictionary. The pooled features are correlated with sparse representations of source images to get the corresponding pixel level score for the decision map of fusion. Markov Random Field (MRF) optimization regularized the decision map. Clustering of nonlocal patches is introduced to dictionary learning method. A centralized sparse representation (CSR) framework is proposed by [PERSON] that integrates local dictionary learning and nonlocal structure clustering [45]. PCA is used to learn an orthogonal dictionary for each cluster of input patches. All patches are sparsely encoded, and the decomposition coefficients of each cluster are close to a per-cluster mean representation.