Matrix factorization is a useful technique in process of image fusion. There are numerous applications of image fusion such as robot/machine vision, aerial military affairs, satellite imaging as well as medical imaging in magnetic resonance imaging (MRI) and positron emission tomography (PET). One of the techniques used for image processing is the Non-negative Matrix Factorization (NMF), which is argued to be more effective and efficient than the standard method of discrete wavelet transform. In a journal published in the Internal Conference on Image Processing concludes that a better performance of image fusion is acquired through the NMR method.

The Wavelet Transform method, which is commonly used for image fusion, uses wavelet functions that (unlike Fourier) analyze time and frequency. The wavelet coefficients are used as a filter to evaluate the information in a set of data and are placed in a matrix. After the wavelet transform method has been applied to two source image a fusion decision map (created from a set of defined rules) is applied to acquire the final wavelet coefficient map. Another transformation is applied and the result is the fusion of the two source image. In the article concerning NMF transformation argues that the Wavelet Transform method loses edge information since wavelets are often periodic and not continuously smooth. The authors advocate the use of the Non-negative Matrix Factorization that through experiment performed better than the Wavelet Transform method.

The NMF method transforms a set of data using non-negative constraints. Given a nxm matrix V, two matrix W and H are use to approximate the original matrix so that V ≈ WxH. The dimension of W_nr and H_rm are chosen such that (n+m)r < nm. The values in the matrix W represents a basis vector and the values in H are the weights of the columns in W. An objective function is defined that is the difference between the values in V and the values obtained from WxH.

There are numerous algorithms to solve for W and H such as the multiplicative update method to reach the minimum of this error squares function. The journal article on NMF evaluates images created from the NMF, wavlet transform and the Laplacian fusion method. The NMF appears to perform well when the original image is blurred, thus causing problems for the wavelet transform method to define borders. However, it is clear that further analysis of NMF must be performed in order it to determine its effectiveness in image fusion.

Sources:

Image Fusion Based on Non-negative Matrix Factorization

http://www.ece.lehigh.edu/SPCRL/IF/image_fusion.htm

http://en.wikipedia.org/wiki/Image_fusion#Standard_Image_Fusion__Methods

http://www.geosage.com/highview/imagefusion.html