Wold Features for Unsupervised Texture Segmentation Chun-Shien Lu
[email protected] Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC. Pau-Choo Chung
[email protected] Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC.
Abstract
[17][1] are either redundant ot insufficient. In contrast, Francos et al. [4][5] presented a computational texture model called Wold decomposition model to confirm the three relevant dimensionalities of texture to be the same as Rao et al. [14][15]. Therefore, the most representative texture features can be said to appear from the close agreement of Rao et al.’s psychological findings and Francos et al.’s computational model. The remaining issue is then how to obtain the desired texture features. Francos et al. [4][5] estimated the deterministic component from the DFT by the inverse transform of the frequencies identified from the 1-D and 2-D delta function in the periodogram. After removing the extracted deterministic component, the remaining part is regarded as the indeterministic component and a 2-D Levinson-type algorithm is applied to estimate the parameters of 2D causal AR model from the residual. Later Francos et al. [5] presented a more accurate but expensive parameter estimation approach by Maximum likelihood. In addition, Liu and Picard [7] also presented a new approach for Wold features without explicit decomposition as in Francos et al. [4][5]. They performed harmonicity test followed by detecting peak features for harmonic component and using MRSAR features [11] to replace the causal AR model in [4][5] for the indeterministic component. Though the concept of Wold decomposition is new but we realize that the corresponding Wold features may be not new. In fact, many methods about characterizing either the deterministic or the indeterministic components of textures exist but they are rarely integrated. Applications of Wold decomposition on Coding textures [16] and image retrieval [7] had been shown, but did not found on texture segmentation. In this paper, we suggest a new Wold features for texture segmentation by exploiting the wavelet transform to detect the information about scales and orientations of deterministic component, and the MRSAR parameters for indeterministic
An efficient texture representation for unsupervised segmentation is addressed based on the concept of Wold decomposition. Textures are described by the wavelet tuned to various scales and rotations to describe its deterministic component, and by the autogressive model to describe its indeterministic component. The wavelet features and the AR parameters capturing the perceptual properties, "periodicity", "directionality", and "randomness", respectively, have been proved to consistent with human texture perception. The performance of our approach is demonstrated on Brodatz textures and natural textured images. Keywords: Wold features Wavelet transform Autogressive model Feature selection Fusion Unsupervised Segmentation
1 Introduction Extraction of "good" features for texture analysis is still a challenging work until now. In the past, many texture features had been investigated for texture-related applications. However, what properties are perceptually meaningful is seldom considered. Tamura et al. [17] and Amadasun et al. [1] had earlierly proposed many visually relevant texture features. Afterwards, based on intuition and computational considerations, Rao [13] provided a texture taxonomy as strongly ordered, weakly ordered, and disordered textures. Followed, Rao et al. [14][15] identified relevant dimensions of textures and found that the most perceptual texture features are “periodicality”, “directionality”, and “complexity” from the psychological studies. Rao et al.’s exploratory research also indicated that some of the identified features in 1
component. The wavelet transform [8] and MRSAR [11] had been used in texture segmentation and some comparisons had been reported [7]. But no integration of them are found in texture segmentation. This motivates us to adopt the perceptually significant (Wold) features for texture segmentation, since chosing a “best” texture representation is very important. (a)
2 Wold Features 2.1
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Figure 1. Segmentation of four nondisordered textures: (a) original image, (b) result using WT features with four texture classes, (c) result using MRSAR features with three texture classes.
Wold Decomposition
In this subsection, we provide the major theorem of the Wold decomposition theory of homogeneous random fields. The proofs of the theorem and extensive presentations can be found in [4]. Let fy (m; n)g, (m; n) 2 Z 2 be a real valued, regular, and homogeneous random field. THEOREM (2D Wold decomposition [7]) A homogeneous regular random field fy (m; n)g can be represented uniquely by the following decomposition: 2D
y(m; n) = p(m; n) + g(m; n) + w(m; n):
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Field fp(m,n)g is half-plane deterministic and field fg(m,n)g is generalized evanescent. Both of them are deterministic. Field fw(m,n)g is purely-indeterministic and has a moving average (MA) representation. Fields fp(m; n)g, fg(m; n)g, and fw(m; n)g are mutually orthogonal.
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Figure 2. Segmentation of four disordered textures: (a) original image, (b) result using WT features with three texture classes, (c) result using MRSAR features with four texture classes.
2.2 Feature Extraction In the feature extraction stage, we explain why the proposed new Wold features are composed of wavelet features (WF) and simultaneous autogressive (SAR) model parameters. The individual feature about WF and SAR is not new but integration of them is perceptual meaningful. We still adopt multiresolution SAR (MRSAR) to model the indeterministic component of a texture as Liu and Picard did [7]. As for capturing the deterministic component, Tan [18] proposed a multi-channel filtering algorithm for edge detection of strongly ordered textures. To solve the feature selection problem, the peak frequencies with orientations in the spatial frequency domain of Gabor transform are first detected. Next, the major deterministic component is recovered from the selected peaks by inverse FFT. Hence, the required number of Gabor filters are reduced and the locations of the required filters are known. Tan’s method [18] implicitly made the assumptions about the textured images whose distinct spectral peaks correspond to the "periodicity" and "directionality" properties of textures. Similar to Gabor filters, the wavelet transform containing rich information about the scales and orientations had been used in texture segmentation [8]. Since textures are natural in a broad
range of scales, a multiscale representation is necessary. For texture segmentation problem, we adopt the multiscale representation/localization properties of wavelet transform to describe the deterministic component of a texture based on the following two causes: (1) The localization property of wavelets satisfy the need of local feature extraction for the texture segmentation problem. (2) The pre-defined threshold for detection of peak frequencies in [4, 7] can be avoided by using the feature reduction technique mentioned in [8]. In what follows, we examine two examples to demonstrate the success and failure of texture segmentation by the wavelet features (WF) and the MRSAR parameters. Fig. 1(a) is the synthetic image containing four non-disordered Brodatz textures. The upper left texture of Fig. 1(a) is weakly ordered, whereas the others are strongly ordered. The segmentation results are shown in Fig. 1(b) using WF with four textures correctly extracted and shown in Fig. 1(c) using MRSAR with only three textures detected. Another example shown in Fig. 2(a) is the synthetic image containing four disordered Brodatz textures. The segmentation results are shown in Fig. 2(b) with only three textures extracted 2
by WF and shown in Fig. 2(c) with four textures correctly extracted by MRSAR. It can be seen from the above two examples that the WF and the MRSAR are inefficient for disordered and non-disordered textures, respectively. In order to improve the segmentation capability, we suggest a new Wold feature set to be composed of wavelet features and MRSAR parameters. (a)
3 Feature Selection and Coarse Segmentation
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Figure 3. Segmentation of various ordered textures: (a) original image; (b) segmention result using Wold features, (c) boundaries outlined on (a).
From the extracted texture features, many of them containing useless information will degrade the segmentation performance. A feature selection process is required to choose useful information while preserving discriminable capability. Previous work for feature selection included transformation methods, user intervention and so on [2](see the references therein). Segmentation process is then performed based on the survived features. Usually the feature selection technique is time wasting and is isolated from the segmentation processes. In this paper, we propose a simple and fast feature selection technique. The results after feature selection are also the partial results of coarse segmentation. Our idea is to divide a feature vector with dimensionality M into M feature values. It is advantageous for clustering and feature reduction. To segment each feature image is then equivalent to a thresholding problem. The main problem is thus how to determine the number of clusters in each feature image. This is also the most difficult problem of unsupervised segmentation. In thresholding the feature image, identifying the peaks is crucial because the number of peaks is the number of clusters. We employ a histogram based thresholding method to help us find the peaks [19]. However, the number of peaks should be assigned a prior in their method. To conquer this problem, a commonly used within-between class criterion is proposed to determine the number of peaks in an iterative manner. Previous work to determine the number of clusters usually decide a number, Kmax , as the maximal number of texture classes. The true number of clusters is then determined by segmenting the image into k classes for 2 k Kmax and find the measurement optimizing some criteria [6][11]. This is, of course, a very time-wasting process. Though we still adopt a similar segmentation process to obtain the true number of classes, but our computational time is cheap. Owing to the number of peaks should be assigned in [19], if the histogram based thresholding technique can not find the desired number of peaks, this process is jumped out earlierly since higher number of peaks is impossible to exist. Further, if there is no more than one peak existing in a feature image, then the feature image is said to be indiscriminating and regarded as useless for segmentation. It will be discarded among the feature images.
4 Fusion: Integration of Coarsely Segmented Results After the feature selection stage, useful information are preserved and the relative segmentation results are yielded. Next, we will integrate these results to finish the whole segmentation work. Our previous fusion technique [8][9] is employed to generate new clusters by combining the features and to delete fragments if a region’s area is small enough. The fusion step is a re-labeling process by updating the labels of pixels until all individual results are combined. The only parameter needed is the size of a cluster considered to be not a “true” cluster. In this paper, the fusion is occurred in the WF and MRSAR features individually. Then, the individual fused results from WF and MRSAR are combined together to obtain the final result. After the fusion stage, the number of texture classes is automatically determined.
5 Experimental Results The number of scales used in wavelet transform depends on the image size and model order two with two resolutions are adopted for MRSAR. First, to demonstrate the efficiency of the suggested Wold features, the image containing various ordered textures is used, as shown in Fig. 3(a). In Fig. 3(a), two of them (strongly ordered) can not be discriminated by MRSAR features coming from Fig. 1(a) and the other two of them (disordered) can not be separated by WF coming from Fig. 2(a). By using the suggested Wold features, four texture classes are correctly extracted. In addition, we demonstrate the capability of our method in segmenting image with large number of textures. Fig. 4(a) consists of eight textures (boundaries has been outlined by our method) ever tested by Mao and Jain [11]. As they stated that it is difficult to decide the true number of textures. However, using our algorithm, eight texture classes are correctly extracted, as shown in Fig. 4(b). The other 3
References
image shown in Fig. 5(a) (with boundaries outlined) [12] is very difficult due to the number of texture classes (sixteen) is very large and the texture boundaries are not horizontal or vertical. By our method fourteen textures can be identified as illustrated in Fig. 5(b). The individually separated regions are shown in Fig. 5(c) for visual purpose. More examples can be found in [10].
[1] M. Amadasun and R. King, “Texture features corresponding to textural properties”, IEEE Trans. SMC, Vol. 19, 1989, pp. 1264-1274. [2] J. Bigu n, “Unsupervised feature reduction in image segmentation by local transforms”, Pattern Recognition Letters, Vol. 14, 1993, pp. 573-583. [3] P. Brodatz, "Textures: A Photographic Album for Artists and Designers", NY: Dover, 1966. [4] J. M. Francos, A. Z. Meiri, and B. Porat, "A unified texture model based on a 2-D Wold-like decomposition", IEEE Trans. Signal Processing, Vol. 41, 1993, pp. 2665-2678. [5] J. M. Francos et al., "Maximum likelihood parameter estimation of textures using a Wold-decomposition based model", IEEE Trans. Image Processing, Vol. 4, 1995, pp. 1655-1666. [6] A. K. Jain and F. Farrokhnia, "Unsupervised Texture Segmentation using Gabor Filters", Pattern Recognition, Vol. 24, pp. 1167-1182, 1991. [7] F. Liu and W. Picard, "Periodicity, Directionality, and Randomness: Wold features for image modeling and retrieval”, IEEE Trans. Pattern Anal. Machine Intell., Vol. 18, 1996, pp. 722-733.
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[8] C. S. Lu, P. C. Chung and C. F. Chen, “Unsupervised Texture Segmentation Via Wavelet Transform”, Pattern Recognition, Vol. 5, pp. 729-742, 1997. [9] C. S. Lu, W. L. Hwang, and P. C. Chung, “Segmentation of 3D Textured Image Using Continuous Wavelet Transform”, IEEE Conf. on Image Processing, Vol. I, pp. 235-238, 1997. [10] C. S. Lu, Ph. D. Thesis, Dept. of Electrical Engineering, National Cheng Kung University, Taiwan, ROC., 1998/6. [11] J. Mao and A. K. Jain, "Texture classification and segmentation using multiresolution simultaneous autogressive models", Pattern Recognition, Vol. 25, pp. 173-188, 1992.
(b) Figure 4. Segmentation of a 256 512 pixels 8texture image [11]: (a). Original image with boundaries superimposed; (b). Segmentation result illustrated with eight gray levels.
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6 Conclusions We have proposed a framework of unsupervised texture segmentation based on the new suggested Wold feature set. It is found that the accuracy of determining the “true” number of textures is improved with Wold features when the number of textures in an image is large. Unlike much existing methods, our feature selection and clustering techniques is coupled together. This has the advantage of lower time cost. The only parameter needed is the size of a region which can not be considered as a texture class. This value may be application dependent. Our texture segmentation system can be operated in two modes. In the human-made environment, our system can be switched to use wavelet features only since the perceived textures are mostly regular. Otherwise, our system uses MRSAR features in the natural environment.
[16] R. Sriram, J. M. Francos, and W. A. Pearlman, "Texture coding using a Wold decomposition model”, IEEE Trans. Image Processing, Vol. 5, 1996, pp. 1382-1386. [17] H. Tamura, S. Mori, and T. Yamawaki, “Texture features corresponding to visual perception”, IEEE Trans. SMC, Vol. 8, 1978, pp. 460473. [18] T. N. Tan, "Texture edge detection by modeling visual cortical channels", Pattern Recognition, Vol. 28, 1995, pp. 1283-1298. [19] D. M. Tsai and Y. H. Chen, "A fast histogram-clustering approach for multi-level thresholding", Pattern Recognition Letters, Vol. 13, 1992, pp. 245-252.
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(c) Figure 5. Segmentation of a 512 512 pixels 16-texture image [12]: (a) original image with boundaries superimposed, (b) segmentation result of (a) illustrated with 14 gray levels: two pair of textures are merged, (c) the figures show the separated regions for visual purpose.
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