SUPERVISED TEXTURE CLASSIFICATION - SELECTION OF MOMENT LAGS Vladimiros Antoniades and Asoke K. Nandi Signal Processing Division, Department of Electronic and Electrical Engineering University of Strathclyde, 204 George Street, Glasgow G1 1XW, U.K. e-mail: vladimir@spd.eee.strath.ac.uk, asoke@eee.strath.ac.uk ABSTRACT This papers deals with supervised texture classification. The extracted features are the image second and third order moments. The number of possible moment lags for 2-D signals increases rapidly with the order of the moment even for small lag neighbourhoods. The paper focuses on the selection of moment lags that optimise classification performance. Lag selection also serves another purpose: it waives us from the trouble of calculating a large number of moments every time a new sample is to be classified. Lag selection is performed by a full stepwise feature selection method using four different feature evaluation measures. The selected moments are driven to four classifiers and comparative classification results are obtained. 1 INTRODUCTION Texture classification is an important task in various image processing problems, ranging from large scale satellite images to microscopic images used in medical applications. Classification is usually performed on a feature vector extracted from the images. Different kinds of features have been proposed in the literature for this task, including cooccurrence matrices, Gabor filtering, autoregressive models and fractals. In this work, we use the image second and third order moments as features for classification. A few relevant approaches are reported in the literature ([1],[2],[3],[4]). Second order statistics have been generally used for 'clean' signal classification whereas third order statistics perform favourably on signals corrupted by noise. In this paper, we do not examine images corrupted by noise. Instead, we focus our attention on both second and third order moment lag selection. The issue has been touched by [1] and [3]. Our approach is more thorough and gives favourable results. 2 FEATURE EXTRACTION We use as features two sets of the image moments. The second and third order moments of a 2-D stationary stochastic signal f x ( ) r are respectively defined as: m x f x f x x f 2 1 1 ( ) ( ) ( ) r r r r =< > m x x f x f x x f x x f 3 1 2 1 2 ( , ) ( ) ( ) ( ) r r r r r r r =< > where r x x x i i x i y = ( , ) , i = 12 , are the 2-D lags and < > denotes the ensemble average operator. Under the assumption of ergodicity, the above formulae hold for space averages too. For zero mean random fields, moments and cumulants up to the third order are identical. In the following analysis we will use both terms with the same meaning. With some similarity to [1], we choose as our first set of features all second order moments with lags r x 1 , so that - ≤ ≤ 4 4 1 1 x x x y , . Taking into account that permutation of lags does not change the value of the moments of any order, we reduce to 41 second order moments. It is well known that third order cumulants are insensitive to additive gaussian (or any other symmetrical) noise. However, noise insensitivity is not satisfied by moments with more than one pixels at the same position. In this work, we will not use any additive noise on our images, that is, we will only process 'clean' images, but we still impose the noise insensitivity restrictions so that our features can be used in the future when we deal with images corrupted by noise. We use as a second feature set all third order moments with lags - ≤ ≤ 3 3 1 1 2 2 x x x x x y x y , , , . If we remove redundancies arising from symmetries in the relative positions of the pixels and we also impose the noise insensitivity restrictions, we reduce to a set of 720 third order moments. Third order cumulants can be also estimated through the bispectrum. This approach reduces the computational time enormously but on the other hand needs a huge memory to store the resulting complete 4-D third order correlation function. There are ways to circumvent this difficulty on the price of worsening the estimation accuracy, but we will not present the analysis here due to space limitations. Within each estimation block (section 5), second order moments are normalised by the variance to get the autocorrelation coefficient. After normalisation, the moment with zero lags in both image directions becomes one so it does not offer any information and is discarded from the set. We therefore render to 40 second order moments. Third order moments are similarly normalised by the third order moment with zero lags, m f 3 00 00 (( , ),( , )).