Research On Matrix Decomposition Based Image Representation Theory And Its Application

Data representation is a fundamental problem in image processing and pattern recog-nition tasks. A good representation can typically reveal the latent structure of data, and further facilitate these tasks in terms of learnability and computational complexity. How-ever, in many real applications, the input data matrix is generally of very high dimension, which brings the curse of dimensionality for further data processing. To solve this prob-lem, matrix factorization approaches are used to explore two or more lower dimensional matrices whose product provides a good approximation for the original data matrix. In addition, sparsity is an attribute characterizing a mass of natural and manmade signals, and has played a vital role in the success of many sparse decomposition based image rep-resentation techniques such as sparse coding, dictionary learning, sparse auto-encoders and independent component analysis. Decomposing data into a sparse and discrimina-tive linear combination of features is an important and well-studied problem. The major contributions of the paper are:1. We propose a topographic non-negative matrix factorization algorithm, called TN-MF. Specifically, TNMF incorporates a topographic constraint to intuitively pro-mote the sparseness of encoding factor. Meanwhile, this constraint forces features to be organized in a topographical map by pooling together structure-correlated features belonging to the same hidden topic, which is beneficial to learn complex invariances (such as scale and rotational invariance). Some experiments carried out on three standard datasets validate the effectiveness of our method in comparison to the state-of-the-art approaches.2. We propose a semi-supervised class-driven non-negative matrix factorization method to associate class label with each basis vector by introducing an inhomo-geneous representation cost constraint. This constraint forces the learned basis vectors to represent better for their own classes but worse for the others. Therefore, data samples in the same class will have similar representations, and thereby the discriminability in new representations could be boosted. Some experiments car-ried out on several standard databases validate the effectiveness of our method in comparison to the state-of-the-art approaches.3. To exploit the class information, we extend the unsupervised reconstruction inde- pendent component analysis method (RICA) to a supervised one, namely d-RICA, by introducing a class-driven discrimination constraint. This constraint minimizes the inhomogeneous representation energy and maximizes the homogeneous rep-resentation energy simultaneously, which will make a data sample uniquely repre-sented by the over-complete basis vectors from the corresponding class. In essence, it's equivalent to maximizing the between-class scatter and minimizing the within-class scatter in an implicit way.4. Since nonlinearly separable data structure generally exists in the original data s-pace, a linear discriminant might not be complex enough for most real-world data. To enhance the expressiveness of the discriminant, kernel trick can be used to map the nonlinearly separable data structure into a linearly separable case in a high di-mensional feature space. Therefore, we develop the kernel extensions of RICA and d-RICA respectively, i.e., kRICA and d-kRICA, to represent the data in the feature space. Experimental results validate the effectiveness of kRICA and d-kRICA for classification tasks.