Affine Invariant Texture Analysis Based on Structural Properties 1

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1 ACCV: The 5th Asian Conference on Coputer Vision, --5 January, Melbourne, Australia Affine Invariant Texture Analysis Based on tructural Properties Jianguo Zhang, Tieniu Tan National Laboratory of Pattern Recognition (NLPR), Institute of Autoation, Chinese Acadey of ciences, Beijing, 8, P.R.China {jgzhang, Abstract This paper presents a new texture analysis ethod based on structural properties. The texture features extracted using this algorith are invariant to affine transfor (including rotation, translation, scaling, and skewing). Affine invariant structural properties are derived based on texel areas. An area-ratio ap utilizing these properties is introduced to characterize texture iages. Histogra based on this ap is constructed for texture classification. Efficiency of this algorith for affine invariant texture classification is deonstrated using Brodatz textures.. Introduction Texture analysis is a fundaental issue in coputer vision and pattern recognition. Existing texture analysis ethods can be broadly split into three categories: statistical ethod, odel based ethods, and structural ethods []. In any practical applications, we cannot ensure that texture iages are acquired fro the sae viewpoint and usually these iages undergo affine or perspective distortion. Texture analysis ethods invariant to such distortion are highly desirable. More and ore efforts have been devoted to this iportant issue. However ost of the existing invariant texture analysis ethods only focus on rotation, translation, and scale transfor. urvey papers of the work on this iportant field ay be found in [][7]. Little work can be found on affine or perspective invariant texture analysis [][]. oe new algoriths on affine pattern analysis have been proposed recently [7][8][9][][]. Whether these ethods could be used for affine invariant texture analysis requires further investigation. tructural properties of texture eleents such as copactness area and perieter have been successfully used to character texture iage [][4]. oe invariant texture analysis ethods have been presented by eploying such properties [][4][5]. Unfortunately, these properties usually change as texture iages undergo affine transfors. Chang et al. [] deal with texture discriination by projective invariants. However their ethods are based on the use of cross ratios and are only suitable for textures with strong line properties. Furtherore, selection of feature points still reains a proble. In this paper, we focus on structural textures and investigate affine invariant texture classification by eploying structural properties. In ection, we give the atheatical details of the area ratio as affine invariants. In ection, an area-ratio ap utilizing these invariants is introduced to give a description of texture. Histogra of this ap is constructed for affine invariant texture classification. In ection 4, experients are described along with the presentation of the results. Conclusions are presented in ection 5.. Affine invariant structural properties Fro the structural point of view, texture is coposed of texture eleents called texels, which are arranged according to certain placeent rules. Area of a texel is the area of the region covered by the texel in an iage. They do not change with rotation, translation, but change under scaling or skewing transfor. Copactness of a texel is invariant to scale, but varies under skewing transfor. It has been pointed out that affine transfor can be splitted into This work is funded by research grants fro the NFC (Grant No and 69798) and the Chinese Acadey of ciences Page

2 three coponents based on ingular Value Decoposition, two of which are rotation and the other is skew [8]. Therefore the structural properties entioned above cannot be directly used for affine invariant texture analysis. In the following, we derive soe properties based on texel areas which are invariant to affine transfor. Theore : The ratio of the areas of two triangles does not change under affine transfor. Proof: For non-collinear points p ( x, y ) =,,, the area of the triangle fored by the three points is given by x x x P P P = y y y () Given an affine transfor represented by atrix a a a A = and the translation factor, let a a a point P ( x, y ) denote the affine transfored version of p x, y ) They have PP P x = y ( x y ax + ay + a = ax + ay + a x y = aa aa P P P () The ratio of the areas of triangle P P P and its transfored version P P P is thus given by P P P = aa aa () P P P uppose there is another triangle QQQ fored by Q ( x, y ), =,,. iilarly, we have: Q Q Q = a a a a (4) Q QQ By cobining Equation () and (4), we obtain: P P P P P P = (5) QQ Q Q Q Q This indicates that the ratio of the areas of two triangles is invariant to affine transfor. ince any closed planar shape can be approxiated by triangle eleents, the following conclusion can be easily drawn: Theore : The ratio of the areas of two closed planar shapes is invariant to affine transfor. These properties cannot be directly used for affine invariant texture analysis as they require the correspondences between texels and their transfored versions that are difficult to establish. In the following section we introduce an area ratio ap to deal with this proble. A histogra of this ap is established to present the siilarity easureent.. Affine invariant histogra based on area ratio ap For our algorith, the first step is to segent an iage into unifor intensity regions. This is iportant for classification, as it directly affects the classification accuracy. Texture iages often contain noise. Furtherore, presence of texture surfaces often shows non-unifor luinance and contrast. In such cases, a threshold that works well in one area of the iage ight work poorly in other areas. For this reason, adaptive thresholding [] is chosen (It ust be pointed that texel extraction is a very difficult and challenging proble, ore sophisticated segentation techniques ay be exploited but this is not the focus of this paper.). egentation often produces any sall holes in texel regions. This ay cause errors in classification. Morphological operations are perfored to eliinate these sall holes. After these operations, texels are extracted. The perforance of these operations is illustrated in Fig. The texels shown in Fig.(d) are directly extracted fro Fig.(b) (here we take white regions as texels). The texels shown in Fig.(e) are derived fro Fig.(c) that is obtained fro Fig.(b) after perforing orphological operation. We can see that Fig.(e) is less noisy than Fig.(d) and the perforance of texels extraction is greatly iproved. uppose that there are n texels in a texture iage. An area ratio ap atrix can be defined as follows (illustrated in Figure ): At the jth colun and ith row is the ratio of areas of texel i and texel j represented by area of texel i r ( i, j) = (6) area of texel j Thus a atrix r ( i, j) i, j =,... n is established. The size of the ratio atrix is n n. Page

( n )( n ) Notice that there are only independent eleents in this atrix. The probability of texel pairs having value r ( i, j) is defined as follows: 4.

Figure An exaple illustrating the process of texel extraction. (a): original texture iage (d5) selected fro Brodatz albu.

(d): texels extracted fro (b). (e): texels extracted fro (c) (less noisy than (d)). Texel index Texel index i r(,) r(,) r (,) r(,) Figure.

The second and the third row is the coresponding area-ratio histogras DH DH4 kr n( n + ) f r =, s = (7) s where Kr is the total nuber of texel pairs with area ratio r

It is obvious that these histogras are invariant to affine transfors and can be used as affine invariant texture features (it is iportant to note that the calculation

3 ( n )( n ) Notice that there are only independent eleents in this atrix. The probability of texel pairs having value r ( i, j) is defined as follows: 4. Experiental results To show that histogras based on area ratios are invariant to affine transfor and can be used for affine invariant texture classification, we have conducted a nuber of experients. The siilarity of the histogras is calculated as follows: Let C( k) ( k =,, n) be the cuulative distribution function derived fro histogra h (i), (a) (b) (c) (d) (e) Figure An exaple illustrating the process of texel extraction. (a): original texture iage (d5) selected fro Brodatz albu. (b): segented iage with the adaptive threshold ethod (c): binary iage after perforing orphological operations on (b) including dilation and erosion. (d): texels extracted fro (b). (e): texels extracted fro (c) (less noisy than (d)). Texel index Texel index i r(,) r(,) r (,) r(,) Figure. Illustration of area-ratio ap DH DH DH DH4 DH DH Figure. The first row is the four affine transfored versions of Brodatz texture D. The second and the third row is the coresponding area-ratio histogras DH DH4 kr n( n + ) f r =, s = (7) s where Kr is the total nuber of texel pairs with area ratio r and s is the total nuber of all possible texel pairs. An area ratio histogra h ( r) = f r is thus established based on the area-ratio atrix. It is obvious that these histogras are invariant to affine transfors and can be used as affine invariant texture features (it is iportant to note that the calculation of these features do not require the correspondence between texels).. k i.e., C( k) = h( i). Let the cuulative distribution i= functions of two histogras be C( K) and C ( K ). The distance between the two histogras is defined as D = C k) C ( ). We use the distance D ( k to represent the siilarity between two underlying textures. The saller the value of D, the ore siilar the two textures. 4.. Intra- and inter-class siilarities Figure and Figure 4 show four affine transfored Page

Table iilarity of the histogras shown in Figure and

88.8.65.6.89.95 DH.96.9..69.65.9.99 DH.88.9..78.74.

54.4.8.4 DH.89.9..8.4.8.7 DH4.95.99.8.88..4.7 Table

Reference texture D5 D D5 D D D5 D56 D74 D75 D4 8 9

5% D Matches with ( %) D5 D56 D74 D75 D4 versions

D4 DH Figure 5 Eight natural textures fro Brodatz

DH DH D5 DH4 Figure 4 The first row is the four

and the second row and the third row are the

transfored versions of texture D with the

Table tabulates the siilarity of these histogras.

distance D ) is very low within the sae textures,

This indicates that the area ratio histogras can be

4 Table iilarity of the histogras shown in Figure and Figure 4 Distance DH DH DH DH4 DH DH DH DH4 DH DH DH DH DH DH DH DH Table Confusion atrix for 8 textures shown in Figure 5 Reference texture D5 D D5 D D D5 D56 D74 D75 D4 8 9 Average classification accuracy=96.5% D Matches with ( %) D5 D56 D74 D75 D4 versions of texture D and four affine DH DH D D5 D56 D74 D75 D4 DH Figure 5 Eight natural textures fro Brodatz database used for texture classification DH D DH4 DH DH D5 DH4 Figure 4 The first row is the four affine transfored versions of Brodatz texture D, and the second row and the third row are the corresponding area-ratio histogras. transfored versions of texture D with the corresponding area ratio histogras. Table tabulates the siilarity of these histogras. It can be seen that the siilarity value (easured by distance D ) is very low within the sae textures, whereas it is uch higher between different textures. This indicates that the area ratio histogras can be used for affine invariant texture recognition. 4.. Texture classification In order to test the perforance of our Page 4

algorith for affine invariant texture classification, 8 natural textures fro the Brodatz texture albu are used as shown in Figure 5.

The area ratio histogra is used to characterize texture properties. The k-nearest neighbour algorith is eployed for texture classification. Table shows the results of texture classification.

This is because that there exist any sall texels in D5. When this iage undergoes affine transfor, these sall eleents do not strictly satisfy the affine atheatical odel due to quantization errors.

Therefore our ethod ay not be suitable for icro textures where texels are not easily identifiable. 4.

5 algorith for affine invariant texture classification, 8 natural textures fro the Brodatz texture albu are used as shown in Figure 5. Each texture is randoly affine-transfored into versions ( for training and for testing). Thus a total of 4 texture iages are constructed. The area ratio histogra is used to characterize texture properties. The k-nearest neighbour algorith is eployed for texture classification. Table shows the results of texture classification. Fro the confusion atrix, one can see that a classification accuracy of % is achieved for six texture classes (D5, D56, D75, D4, D and D). The lowest classification accuracy of 8% is shown for D5. This is because that there exist any sall texels in D5. When this iage undergoes affine transfor, these sall eleents do not strictly satisfy the affine atheatical odel due to quantization errors. This strongly affects the calculation of area ratios of these texels, which in turn results in the isclassification. Therefore our ethod ay not be suitable for icro textures where texels are not easily identifiable. 4.. Noise robustness This additional experient is perfored to investigate the noise robustness of the proposed affine invariant texture features. Three ost widely used noise odels (Gaussian white noise, alt&pepper noise and peckle noise) are siulated. Each texture iage is added with the three noises at different NR values. Exaples of texture iages corrupted by Gaussian white noise are shown in Figure 6. Here the definition of NR is described as: NR=- NR=-5 NR= NR=5 NR= NR=5 NR= NR=5 NR= NR= Figure 6. Texture iages fro Brodatz database corrupted by Gaussian white noise at different NR values Es NR = log (8) En where Es denotes the energy function of the original iage defined by s ( x, y) dxdy and En the energy function of noise signal defined by n ( x, y) dxdy. The classifier is trained with noise-free iages and tested with the noisy iages. No noisy iages are used for training. The Classification Accuracy Figure 7. Noise robustness of the affine invariant texture features perforance of our algorith under these noise odels at different noise level is shown in Figure 7. We can see that our algorith exhibits alost the sae perforance under different noise odels and produce proising correct recognition rates around 8% at very low signal to noise ratio (NR=5). This indicates that our ethod is quite robust to noise. 5. Conclusion ignal to noise ratio In this paper, we have presented a new ethod for texture classification that is invariant to affine transforation. We derive the affine invariant structural properties and introduce an area-ratio ap by utilizing these properties to characterize texture iages. Histogra based on this ap is constructed and successfully used for affine invariant texture classification. The results of the experients with natural textures for Brodatz albu have shown that the algorith perfors well. Furtherore, our algorith can also work well with very noisy texture iages. Notice that this ethod is based on texel extraction (although this is not the focus of this paper), so it is ore suitable for those textures where the texels can be easily defined. ince that the surface of each texel can also be considered as a sall texture iage, these surfaces describe the detailed texture inforation of the whole iage in a saller scale. Page 5

6 This feature should be taken into account in future research to give a ore robust and precise accuracy. 6. Reference [] M. Tuceryan and A. K. Jain, Texture analysis, in Handbook of Pattern Recognition and Coputer Vision (C. H. Chen, et. al., Eds), 99, pp [] T. N. Tan, Geoetric Transfor Invariant Texture Analysis, PIE, Vol. 488, (995) pp ,. [] R. K. Goyal, W. L. Goh, D. P. Mital and K. L. Chan, A Translation Rotation and cale Invariant Texture Analysis Technique Based on tructural Properties. Proceedings, Third International Conference on Autoation Technology (Autoation 994), Taipaei, July (994). [4] DP Mital, W. L. Goh, K. L. Chan and R K Goyal, A Translation Rotation and cale Invariant Texture Analysis Technique Based on Iage Granularity. Proceedings of Fifth International yposiu on Robotics and Manufacturing, Hawaii, August (994). [5] R K Goyal, W L Goh, D P Mital and K L Chan, Invariant Eleent Copactness for Texture Classification, The Third International Conference on Autoation, Robotics and Coputer Vision (ICARCV 94), pp.9-96, ingapore Noveber 9-, (994). [6] Jun ato and Roberto Cipolla, Extracting the Affine Transforation fro Texture Moents. Lecture notes in Coputer cience, Vol. 8, Coputer Vision-ECCV 94, pringer-verlag, Berlin Heidelberg. [7] J. G. Zhang and T. N. Tan, Brief Review of Invariant Texture Analysis Methods. Pattern Recognition (to appear). [8] J. Ben-Arie, Z. Wang, Pictorial Recognition of Objects Eploying Affine Invariance in Frequency Doain, IEEE trans. on pattern analysis and achine intelligence, Vol., No.6, June 998. [9] M. Pietikainen, T. Ojala, Z. Xu, Rotation-Invariant Texture Classification using Feature distributions. Pattern Recognition, pp. 4-5.(). [] Hyun-Ki hong, Yun-Chan Myung, Jong-oo Choi, -D Analysis of Projective Textures using tructural Approaches. Pattern Recognition, pp57-64, 999. [] Coloa Ballester and Manuel González, Affine Invariant Texture egentation and hape Fro Texture by Variational Methods. Journal of Matheatical Iaging and Vision 9, 4-7, 998. []. Chang, L.. Davis,.M. Dunn, A. Rosenfeld, and J.O. Eklundh, Texture Discriination by Projective Invariant. Pattern Recognition Letters, pp.7-4, May 987. [] R. J. Wall, The Gray Level Histogra for Threshold Boundary Deterination in Iage Processing to the cene egentation Proble in Huan Chroosoe Analysis, Ph.D. Thesis, University of California at Los Angeles, 974. [4]. Baheerathan, F. Albregtsen, H. E. Danislsen, New Texture features Based on the Coplexity Curve. Pattern Recognition pp (999). Page 6

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