Abstract Sparse representation based classification has led to interesting image recognition results, while the dictionary used for sparse coding plays a key role in it. This paper presents a novel dictionary learning (DL) method to improve the pattern classification performance. Based on the Fisher discrimination criterion, a structured dictionary, whose dictionary atoms have correspondence to the class labels, is learned so that the reconstruction error after sparse coding can be used for pattern classification. Meanwhile, the Fisher discrimination criterion is imposed on the coding coefficients so that they have small within-class scatter but big between-class scatter. A new classification scheme associated with the proposed Fisher discrimination DL (FDDL) method is then presented by The past several years have witnessed the rapid development of the theory and algorithms of sparse representation (or coding) [30] and its successful applications in image restoration [1] [2] [3] and compressed sensing [4] . Recently sparse representation techniques have also led to promising results in image classification, e.g. face recognition (FR) [5-7, 10, 31], digit and texture classification [8] [9] [11] [12] , etc. The success of sparse representation based classification owes to the fact that a high-dimensional image can be represented or coded by a few representative samples from the same class in a low-dimensional manifold, and the recent progress of l 0 -norm and l 1 -norm minimization techniques [28] . In sparse representation based classification, there are two phases: coding and classification. First, the query signal/image is collaboratively coded over a dictionary of atoms with some sparsity constraint, and then classification is performed based on the coding coefficients and the dictionary. The dictionary for sparse coding could be predefined. For example, Wright et al. [5] directly used the training samples of all classes as the dictionary to code the query face image, and classified the query face image by evaluating which class leads to the minimal reconstruction error. Although this so called sparse representation based classification (SRC) scheme shows interesting FR results, the dictionary used in it may not be effective enough to represent the query images due to the uncertain and noisy information in the original training images. The number of atoms of such a dictionary can also be very big, which increases the coding complexity. In addition, using the original training samples as the dictionary could not fully exploit the discriminative information hidden in the training samples. On the other hand, using analytically designed off-the-shelf bases as dictionary (e.g., [8] uses Haar wavelets and Gabor wavelets as the dictionary) might be universal to all types of images but will not be effective enough for specific type of images such as face, digit and texture images. In fact, all the above mentioned problems of predefined dictionary can be addressed, at least to some extent, by learning properly a non-parametric dictionary from the original training samples. Dictionary learning (DL) aims to learn from the training samples the space where the given signal could be well represented or coded for processing. Many DL methods have been proposed for image processing [1, 3, 17] and classification [9] [10] [11] [12] [13] [14] [15] [16] . One representative DL method for image processing is the KSVD algorithm [17], which learns an over-complete dictionary from a training dataset of natural image patches. However, KSVD is not suitable for classification tasks because it only requires that the learned dictionary could faithfully represent the training samples. Based on KSVD, Mairal et al. [14] added a discriminative reconstruction constraint in the DL model to gain discrimination ability, and used the learned dictionary for texture segmentation and scene analysis; however, this method is not convex and it does not explore the discrimination capability of sparse coding coefficients. Later, Mairal et al. [9] proposed a discriminative DL method by training a classifier of the coding coefficients, and verified their method for digit recognition and texture classification. used an incoherence promoting term to make the dictionaries associated with different classes as independent as possible. These methods use the reconstruction error associated with each class as the discriminative information for classification, but they do not enforce discriminative information into the sparse coding coefficients. In this paper we propose a new discriminative DL framework which employs the Fisher discrimination criterion to learn a structured dictionary (i.e. the dictionary atoms have correspondence to the class labels so that the reconstruction error associated with each class can be used for classification). Meanwhile, the Fisher discrimination criterion is imposed on the coding coefficients to make them discriminative. To this end, in the DL process we make the sparse coding coefficients have small within-class scatter but big between-class scatter, and at the same time we make each class-specific sub-dictionary in the whole structured dictionary have good representation ability to the training samples from the associated class but poor representation ability for other classes. With the proposed Fisher discrimination based DL (FDDL) method, both the reconstruction error and the coding coefficient will be discriminative, and hence a new classification scheme is proposed to exploit such information. The FDDL method is applied to face, digit and gender recognition to evaluate its performance. Compared with the SRC method [5] , not only higher classification accuracy is got by FDDL, but also a smaller dictionary can be learnt (e.g., on the Extended Yale B database, the learnt dictionary by FDDL with only 8 atoms per class could still achieve better FR performance than SRC with 20 atoms per class). Compared with other state-of-the-art methods, FDDL has competitive performance in various pattern recognition tasks. The rest of this paper is organized as follows. Section 2 briefly introduces the SRC scheme in [5] . Section 3 presents the proposed FDDL model. Section 4 describes the optimization procedure of FDDL. Section 5 presents the FDDL based classifier. Section 6 conducts experiments, and Section 7 concludes the paper. Wright et al. [5]  where γ is a scalar constant. ii.) Do classification via  To improve the performance of previous DL methods, we propose here a novel
Yang et al. REF considered Fisher's discriminative information , where is the within-class scatter of and is the between-class scatter of , and are the parameters.
Fisher Discrimination Dictionary Learning for sparse representation
2011 International Conference on Computer Vision
2011 International Conference on Computer Vision
Computer Science