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Linear Local Distance Coding MethodIn this section, we present a new coding method based on local distance vector which is a modification of the distance vector. Several methods taking advantage of locality, including LLC [3], LSC [4] and local NBNN [5], achieve improvements over their nonlocal versions [1, 6, 7]. Our proposed coding method maintains superior discriminative capability and effectiveness of the aforementioned coding methods. It provides better generalization capability by employing the distances between local descriptor and classes to estimate the image membership. Meanwhile, it preserves more discriminative information by avoiding coding process while obtaining image toclass distance. Furthermore, the LLDC method avoids poor estimates from isolated classes by eliminating the need to calculate distance vector for each class. Hence, the LLDC method can achieve superior image classification performance compared with the other coding schemes. Distance VectorDistance vector is an alternative discriminative pattern of local feature in the class specific manifold coordinate system. Let X = {xi, x_{2},x_{N}} e R^{DxN} be N D dimensional local features extracted from an image. It is assumed that the local feature is sampled from a classspecific manifold M^{c} = [mj, m2,..., ], which is constructed by clustering local features of the training images from the corresponding class c. Then the distance vector which denotes the distance between a local feature xi and class c is computed by where x_{i}^{c} denotes the mapped point of xi in class c. It can be computed by linearly combining its neighboring points in the manifold M^{c}. The LDC method calculates xc as follow:
where uc = [u^{c}n, u^{c}i2,..., u^{c}inc] is the linear coefficients of x_{;} on the manifold M^{c} and .N_{i}^{k} denotes the set of k nearest neighbors of x, on M^{c}. Then, (5.1) can be rewritten as Each local feature of an image is transformed to its distance vector d; = [d^{1}, df,..., d^{C}], where C is the class number. By generating image representation based on distance vector, the LDC method captures discriminative information and avoids the case where the discriminative features are dominated by outlier or noisy features. Therefore, using the linear SVM, the LDC method shows impressive image classification performance. However, distance vector is obtained by utilizing the approximate fast solution of the LLC coding method which inherently induces information loss. Meanwhile, distance vector treats every class equally because it is produced through calculating the distance from local feature to each class. Such operation easily brings in the uncorrelated information of classes which are far from query local feature, and consequently arouses unnecessary interference. Therefore, distance vector can be improved further to perform better in image classification tasks. 
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