Texture Classi cation by Higher Order Local Autocorrelation Features

Texture Classi cation by Higher Order Local Autocorrelation Features T. Kurita

and

N. Otsu

Electrotechnical Laboratory 1-1-4 Umezono, Tsukuba, Japan 305

Abstract This paper proposes a texture classi cation method which makes use of higher order local autocorrelation features. Higher order autocorrelation is an extension of autocorrelation and is shiftinvariant. For practical application, we restrict the order up to two (three points relations) and the range of consideration to within a local 3 2 3 region. To obtain from detail to rough local statistical information on texture, features are extracted from each of the multi-resolution images. Then a classi er which makes use of those primitive features is designed by learning from training examples. In experiments, discriminant analysis and multi-layer Perceptron are used as classi ers.

1 Introduction Texture analysis plays an important role in the interpretation and understanding real-world images such as photomicrographs, aerial photographs, or satellite images. Many features have been proposed to measure texture properties for statistical texture analysis[1]. Usually local features are computed at each pixel in a texture image and a set of statistics are derived from them. According to the number of pixels which de ne the local feature, statistics are classi ed into rst-order, secondorder, and higher-order. Co-occurrence matrix, Fourier power spectrum, autoregression model are well-known and are based on second-order statistics. Interval cover, run length, fractal dimension can be regarded as higher-order features. Recently importance of multi-resolution (multi-channel) processing has been pointed out and multi-channel texture analysis methods using the 2-D Gabor lters have been proposed[2, 3]. The authors have proposed a scheme for adaptive image recognition systems (Fig.1)[4]. It consists of two stages of feature extraction; Geometrical Feature Extraction (GFE) and Statistical Feature Extraction (SFE). At the rst stage, general and primitive features based on higher-order local

Adaptive Learning x

y

MVA Output Teacher Input Geometrical Feature Extraction

Statical Feature Extraction

Figure 1: The scheme for adaptive image recognition systems. autocorrelation, which are shift-invariant and additive, are extracted. Then those features are combined to provide new eective features for the desired recognition tasks. The way of combination of primitive features is learned from the given supervised training examples. To apply the scheme for face recognition, the authors have used higherorder local autocorrelation features extracted from each of pyramidal images [5]. In this paper the scheme is applied to texture classi cation. Since higher-order autocorrelation is an extension of autocorrelation, it is expected that they also include information which can not extracted by second-order statistics. To obtain from detail to rough local statistical information on texture, higher order local features are extracted from each of Pyramidal images. Then those primitive features are combined to classify textures by learning from training examples. Discriminant analysis and multi-layer Perceptron are used as the classi ers.

2 Primitive Features 2.1

Higher Order Local Autocorrelation

It is well known that the autocorrelation function is shift-invariant. An extension of the autocorrela-

tion function to higher orders has been presented in [6]. Let a region in an image be denoted by P . Then the N th-order autocorrelation functions with N displacements a1 ; . . . ; aN is de ne by xN f (a1 ; . . . ; aN )

=

Z

P

f (r)f (r +a1 )

1 1 1 f (r+aN )dr;

(1) where f (r) denotes gray-level at the position r. Since the number of these autocorrelation functions obtained by the combination of the displacements over the region P are enormous, we must reduce them for practical application. First, we restrict the order N up to two (N = 0; 1; 2). It is seen that the 0th-order autocorrelation is just the averaged gray-level of the region P . We also restrict the range of displacements to within a local 3 2 3 region, the center of which is the reference point (Fig.2 (a)). By eliminating the displacements which are equivalent in the shift in the scanning, the number of the patterns of the displacements are reduced to 25. Fig.2 (b) shows the patterns, where the symbol 3 represents \don't care".

(a) Local 3 2 3 region and the reference point

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(b) 25 local 3 2 3 mask patterns

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Figure 3: An example of Pyramid Image 2.2

Features on a Resolution Pyramid

The features computed from the image of the highest resolution includes only very local and detail information. Often more rough information is useful for texture classi cation. In multi-channel texture analysis [2, 3], features extracted from images with several resolutions are utilized. A pyramidal image data structure gives a set of images of dierent resolutions from the highest to lowest [7]. Fig.3 shows an example of pyramid image. A set of the higher-order local autocorrelation features extracted from each of the images in the pyramidal structure includes from detail to rough information on texture. They can be good primitive features on the whole.

3 Classi ers Each of higher-order local autocorrelation features extracted from the Pyramidal images is considered to be insucient for texture classi cation. However it is expected that they, in total, have enough information to classify textures. Thus they are combined to get new eective features for texture classi cation.

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Figure 2: Local mask patterns for higher-order local autocorrelation features Thus the higher-order local autocorrelation features are obtained by scanning the image over P with the 25 local 3 2 3 masks and by computing the sums of the products of the corresponding pixels. The features are obviously shift-invariant.

3.1

Linear Discriminant Analysis

Most simple way to combine features is to use weighted linear combination y = AT x;

(2)

where A = [aij ] is a weighting matrix and the symbol T denotes the transpose. For the purpose of classi cation, linear discriminant analysis (DA) is one of the best methods to determine the weights. Suppose that a set of feature vectors fxi g are extracted from K dierent texture images fCk gKk=1 .

Then the within-class and the between-class covariance matrices of the primitive features are computed for this training samples as

XK k6k k XK k(xk 0 xT )(xk 0 x T )T

6W =

!

(3)

;

=1

6B =

k=1

!

;

Multi-layer Perceptron

It is also possible to use multi-layer Perceptron (MLP)[8] as a classi er. Since a 2-layer MLP is capable of forming an arbitrarily close approximation to any continuous nonlinear mapping[9], it is expected that a good classi er can be constructed. Fig.4 shows an example of 2-layer MLP. A

B

z x y

Figure 4: An example of MLP Output vector z = (z1 ; . . . ; zK )T of 2-layer MLP is computed for an input vector x as zk = f (k ) k

=

XJ jk j + j=1

b y

b0k

Table 1: Recognition Rates

(4)

where !k , x k , x T , and 6k denote a priori probability of class Ck (usually set to be equally 1=K ), the mean vector of class Ck , the total mean vector, and the covariance matrix of class Ck , respectively. Then the optimal weighting matrix A is given by the solution of the following eigen-equation 6B A = 6W A3 (AT 6W A = I ); (5) where 3 is a diagonal matrix of eigenvalues and I denotes the unit matrix. The j -th column of A is the eigenvector corresponding to the j -th largest eigenvalue. Thus, the N new features yj are evaluated in its importance for discrimination in terms of the eigenvalues. The maximum number N is bounded by min(K 0 1; M ). To identify the class of the texture, we can use a simple classi er which checks the distances from an input y to class means fy k g and classi es the input to such class Ck that gives the least distance. 3.2

Set A Set B HLC+DA 99.6% 96.2% LC+DA 93.2% 80.6% HLC+MLP 99.7% 98.4%

yj

=

j

=

f (

)

XI j ij i + i=1

a x

a0j

(6)

where f (1) is the sigmoid nonlinearity. The weights and bjk are from the i-th input node to the j -th hidden node and from the j -th hidden node to the k -th output node. For classi cation of K classes, desired outputs are usually set to K -dimension binary vectors in which one element is unity corresponding to the correct class and all others are zero. In this case, outputs of the network are interpreted as estimates of Bayesian a posteriori probabilities[10]. Thus the input is classi ed by nding maximum elements of the output vector. The back-propagation algorithm and its variations[11, 12] are available to determine the weights on the network classi er from the training samples.

aij

4 Experiments To assess the ability of this method, we have performed experiments with 30 texture images taken from the book by P.Brodatz[13]. Images are shown in Fig.5. The size of the original texture images is 256 2 256 pixels. From each of the images, 144 dierent local regions with 64 2 64 pixels were extracted and used as samples for the classi cation experiment. Higher-order local autocorrelation features were extracted from each of the images in the pyramidal structure with 4 layers. Classi cation experiment was performed for two set of textures (Set A and B). Set A contains 22 dierent textures (D1, D3-D6, D8-D12, D14-D22, D24, D28, D29). Set B contains all of 30 dierent textures. In Table 1, the recognition rates were summarized, where HLC and LC denote higher order local autocorrelation features and local autocorrelation features (corresponding to the masks No.1 to No.5), respectively. For HLC+MLP, the number of hidden nodes was set to 30 for both data sets and the table shows the recognition rates after 3000 iterations of back-propagation learning. From the comparison between HLC+DA and LC+DA, it is noticed that the recognition rate was

improved by using higher order local autocorrelation features. The recognition rates of HLC+MLP were better than those of HLC+DA, but the improvements were not so much.

References D1

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Figure 5: Original texture images used in the classi cation experiments.

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