# Linear Local Distance Coding Method

In 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 non-local 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- to-class 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 Vector

Distance 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 class-specific 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 x*i* and class *c* is computed by

where x* _{i}^{c}* denotes the mapped point of x

*i*in class

*c*. It can be computed by linearly combining its neighboring points in the manifold

*M*The LDC method calculates

^{c}.*xc*as follow:

where uc = *[u ^{c}n, u^{c}i2,..., u^{c}inc]* is the linear coefficients of x

*on the manifold*

_{;}*M*and

^{c}*.N*denotes the set of

_{i}^{k}*k*nearest neighbors of x

*,*on

*M*Then, (5.1) can be rewritten as

^{c}.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.