Andrey S. Krylov,
Faculty of Computational Mathematics and Cybernetics, Moscow State University,
Moscow, Russia
email:
Andrey V. Kutovoi,
Faculty of Computational Mathematics and Cybernetics, Moscow State University,
Moscow, Russia
email:
Wee Kheng Leow,
School of Computing, National University of Singapore,
Singapore
email:
Contents
The report considers the task of texture identification using Hermite expansion method. Three methods to analyze textured images in terms of onedimensional Hermite transform are described and compared. The methods were tested with Brodatz textures and real photo. Hierarchical coding without subtractions was found as the most effective method for the Hermite transform based texture identification. The method was also applied to the problem of texture based image segmentation.
Keywords: Hermite transform, texture identification, image segmentation.
1. INTRODUCTION
Texture nowadays is an important term in the world of computer vision, graphics applications and computer modeling. Although the term �texture� is wide known, it is difficult to construct a formal definition for it. Informal definitions such as surface property, etc. are intuitively clear for a human, but they do not reflect any approach to the textured images analysis. So it is better to say that texture is related to periodical luminosity fluctuations in the image, which let us interpret the surface as a homogenous structure. Textures can be characterized using properties such as regularity, coarseness, contrast and directionality. A system that is required to deliver meaningful judgments concerning texture, needs to be able to extract a description of the image data in a form that explicitly captures these properties.
The task of texture processing first came up with the task of making the automatic image recognition systems. Textures in the context of image recognition can be used in different ways: image segmentation; detecting the physical properties and materials of the surfaces represented in the image; detecting the shape and the position of the objects and surfaces in the image.
Our first aim will be the task of texture identification. During the past decades there have been presented different approaches [2]: statistical, geometrical, structural, modelbased and signal processing. The focus of this paper is to present a new approach in the concept of the signal processing methods in texture analysis. The approach can also be referred to as filtering approach.
The signal processing method is based on applying a set of bandpass filters to extract the socalled feature vectors from the image. Each filter responds most strongly to a selected spatialfrequency and orientation bands. The 2D Gabor filters h(x,y) are very popular to be applied to images[5],[1]:
These functions are often used because they are oriented, have the easy tunable carrier frequency f, and are good localized both in the spatial and frequency domains [1].
2. HERMITE FUNCTIONS DEFINITION
In this paper the Hermite functions were taken as an alternative to the Gabor functions. These functions are defined as:
,
They also can be evaluated recursively:
These functions vanish at infinity and satisfy the equation
,
and are the eigenfunctions of the Fourier transform.
The Hermite functions also satisfy an important feature in image processing, as they derivate a full orthonormal in system of functions.
Hermite functions of the first six orders are plotted in fig.1. Different applications of Hermite functions in image processing can be found in [610].
Fig.1
Now we describe the scheme of the feature vectors retrieval. We consider the 1d function f(x). It can be expanded into the series


where� a_{i} is the ith Fourier coefficient, and Y_{i}(x) �is �the ith Hermite function.
The 2d Hermite function can be introduced as� a separable product of the corresponding 1d functions:
but in this current work we use
In this paper we will describe and compare three methods to analyze textured images in terms of Hermite transform. They will be referred to as the ordinary transform, the hierarchical coding with subtractions and the hierarchical coding without subtractions.
For n=4 and n=15 the functions are plotted in fig.2
Fig.2
As can be seen in fig.2, the functions have a strongly expressed orientation. To perform an adequate analysis the functions should be turned at several different angles. In this work 8 orientation angles were used: 0, 22.5, 45, 67.5, 90, 112.5, 135, 157.5. Applying of this method to Gabor functions is described in [5].
In this approach to get the feature vectors we consider the functions y _{n}(x,y) where n1=0..64, and 6 energy coefficients are caclulated:
E_{1} = (a _{0})^{2} + (a _{1})^{2} ,
E_{2} = (a _{2})^{2}+(a _{3})^{2} + (a _{4})^{2},
E_{3}= (a _{5})^{2} +(a _{6})^{2}+(a _{7})^{2}+(a _{8})^{2},
�
E_{6} = (a _{33})^{2}+(a _{34})^{2} + � + (a _{63})^{2} +(a _{64})^{2} ,
f(x,y) is the source image.
Hence with the 8 orientations we get a 48dimensional feature vector for each texture example.
3.2 Hierarchical Coding
When decomposing a function by the standard scheme in context of the texture analysis some problem arise. The most significant of them is that the images from the real world are unstructured, i.e. luminosity, scale, size, perspective of objects vary, and can cause inaccuracy during computer processing. Discussing the task of texture retrieval the problem can be interpreted as follows: human can easily detect a bricked wall in an image while variations in scale, size, perspectives can make the task unsolvable for computer vision. To avoid the mentioned problem we took the Martens� idea of the Hierarchical coding[3].
This approach lets us use several sets of patterns, inspired by the different features that can arise in an image[3]. These sets distinguish in spatial scale. So the basis Hermite functions with changing of spatial scale repeatedly take part in forming the different coefficients of the feature vector. The scheme of calculating the feature vector components is as follows:
E_{1} = (a _{0}^{(1)})^{2} ^{}+ (a _{1}^{(1)})^{2} ,
E_{2} = (a _{0}^{(2)})^{2}+(a _{1}^{(2)})^{2} + (a _{2}^{(2)})^{2} +(a _{3}^{(2)})^{2} ,
�
E_{6} = (a _{0}^{(6)})^{2}+(a _{1}^{(6)})^{2} + � + (a _{62}^{(6)})^{2} +(a _{63}^{(6)})^{2} ,
f(x) is the source image,
f^{(}^{i1)}(x)= a _{0}^{(i1)} × y _{0}(x) + � + a _{k}^{(i1)} × y _{k}(x) is the source function�s approximation at step i1; k=2^{(i1)}1.
��������������� As the maximum order of the Hermite function in the current scheme is 63 we say that we have a 6level hierarchy. An important feature in the hierarchical coding is that at each level of the hierarchy we stretch the segment of the functions� approximation to the segment [A_{j}; A_{j}] where A_{j}is found from the condition
where k is the highest Hermite function order at the current

level. We can say that we rescale our source function to be localized at the same segment as the mentioned Hermite function. This approach makes the method of texture detection invariant to the sizes of the objects in the source textured images.
Similarly to the ordinary coding, combining with 8 orientations we get a 48dimensional feature vector.
3.3 Hierarchical Coding Without Subtractions
This approach is the same as the hierarchical coding, but the subtraction is not performed. So we get the following feature vector:
E_{1} = (a _{0}^{(1)})^{2} ^{}+ (a _{1}^{(1)})^{2} ,
E_{2} = (a _{0}^{(2)})^{2}+(a _{1}^{(2)})^{2} + (a _{2}^{(2)})^{2} +(a _{3}^{(2)})^{2} ,
�
E_{6} = (a _{0}^{(6)})^{2}+(a _{1}^{(6)})^{2} + � + (a _{62}^{(6)})^{2} +(a _{63}^{(6)})^{2} ,
f(x) is the source image.
Combining with 8 orientations, we get a 48dimensional feature vector.
The results of applying these methods to some textured images can be seen in fig.3

Texture sample 
Standard coding 
Hierarchical coding 
Hierarchical coding without subtractions 
A1 

A2 

A3 

A4 

B1 

B2 

B3 

B4 

C1 

C2 

C3 

C4 
Fig.3 Structures of different textures give different patterns in the 2D representations of the feature vectors. In the 2D representations, spatial frequency levels increase from bottom up along the vertical axis, and orientation changes from left to right along the horizontal axis and wraps around.
4. RESULTS
To compare the resulting feature vectors we define the difference between two texture examples as
a, b are the two textures, a _{i}, b _{i} _{}are the components of the feature vectors of a and b respectively.
In table1 some results are reflected:
a 
b 
Standard coding 
Hierarchical coding 
Hierarchical coding without subtractions 
a1 
a2 
0.004505 
0.005349 
0.003739 
b1 
b2 
0.009905 
0.008160 
0.007742 
c1 
c2 
0.017778 
0.011848 
0.009946 
a3 
a4 
0.008807 
0.006047 
0.002422 
b3 
b4 
0.020075 
0.010667 
0.006275 
c3 
c4 
0.128322 
0.101591 
0.080176 
a1 
b1 
0.488420 
0.492238 
0.491653 
b1 
c1 
0.315644 
0.304597 
0.305442 
Table1. The differences between the a and b texture samples in different coding schemes.
The results in the Table 1 show that the hierarchical coding without subtractions algorithm gives better results for the Brodatz texture samples.
5.� RESULTS OF APPLYING TO THE TEXTURE SEGMENTATION TASK
��������������� The proposed hierarchical coding without subtractions algorithm to obtain texture feature vectors has been tested with images of natural scenes to perform the texturebased image segmentation. All images used were 8bit grayscale. To perform the analysis the images were split into 50x50(fig.4) and 20x20(fig.5) square blocks. For each square block the texture feature vector was calculated and by comparing them we found the regions of the similar textures. Some of the results are shown in figs.45 (fig.4 presents an artificial textured image with blocks of Brodatz texture samples and fig.5 is a natural scene image). Both the fig.4 and fig.5 contain the source image and two segmented images. The threshold parameter of the texture similarity was chosen manually. The �b� images were segmented with a big threshold parameter and the �c� images were segmented with approximately 10 times smaller threshold parameter. We must also notice that depending on the current image set (i.e. air photography) the threshold parameter can be obtained automatically. As is well seen from the presented illustrations our method gives adequate results for the images of natural scenes.
6.� CONCLUSIONS
The Hermite coding schemes can be effectively used to analyze textured images. Further work is going on to a) design the feature vectors which are obtained using the 2d Hermite functions and b) apply and fasten the algorithm for the pattern recognition and image segmentation tasks.
A 
b 
c 



Fig.4 The result of applying the texture segmentation algorithm to the artificial textured image. With the small threshold parameter all the different textures are separated correctly (image �c�).
Fig.5 The result of applying the texture segmentation algorithm to the �bricked wall� image. The �strongly bricked� part of the wall is found as a homogenous surface fragment in both the �b� and �c� images, and the �c� image is separated more strictly due to the more precise threshold parameter selection.