# Adaptive CoDT Feature Space

To combine the strengths of generative and discriminative approaches for image classification, we characterize our proposed CoDT features of a HEp-2 cell image by a gradient vector derived from a generative model of the training data to model the generation process of the features, and then we feed the output image representations into a discriminative classifier for the identification of HEp-2 cells.

Let *X = {x _{n}, n =* 1,2,

*N*} be a set of samples from the CoDT feature space of one HEp-2 cell image. The probability density distribution of the CoDT feature, which is used to model the generative process of elements in the feature space, is described by a GMM. Let

*X = {w*1,2,...,

_{t}, p_{t}, %t, t =*T}*denote the parameters of the GMM, where

*w*,

_{t}*/u*and

_{t}*X*respectively denotes the mixture weight, mean vector and covariance matrix of Gaussian

_{t}*t*. Then we can formulate

and *p _{t} (x_{n}* |X) is the Gaussian

*t*defined as

where *D* is the dimension of the CoDT feature. Actually, the GMM, which models the generation process of the CoDT features, can be regarded as a probabilistic codebook/vocabulary [21]. The parameters of GMM can be adaptively estimated by the Expectation Maximization (EM) algorithm [22] based on the training CoDT feature space. Briefly, EM algorithm can be implemented by the following two steps: Expectation step (E step): the posteriori probability for each training data *x _{n}* is given by

Maximization step (M step): the means, variances and mixture weights are updated by

)

)

)

Each Gaussian component can be treated as a micro-texton word of the micro- texton vocabulary while *w _{t}* corresponds to the relative frequency of word

*t*, represents the mean of the word and

*E*is the variation around the mean. In the followings, the covariance matrices are assumed to be diagonal and denoted by

_{t}*o*

_{t}= diag(E_{t}).