1 Introduction
Many real images are often corrupted by noise in their acquisition, such as biomedical images like Singlephoton emission computed tomography (SPECT) images, and the image itself often contains both smooth and textured regions. Wavelet decomposition is a heavily used tool in image processing and classification; it enables the decomposition of an image into varies frequency subbands, similar to the way the human visual system operates [1]. Because of this property, Wavelets has been widely adopted in image denoising and texture classification [2][3]. However, there has been limited work on image clustering and segmentation with Wavelets coefficients.
Most conventional image clustering or segmentation algorithms, such as Kmeans, Fuzzy Cmeans (FCM), Gaussian mixture model (GMM), and Active contour without edges (ACWE), are based only on image intensities. Some improvements in the clustering algorithms to incorporate information about spatial context were proposed
[4][5][6], these methods usually require some modifications of the objective functions, thus they complicate calculations and increases computational time. A Wavelet based image clustering scheme was introduced by Porter and Canagarajah [7], the authors proposed a Kmeans image segmentation algorithm using the optimal Wavelet features derived from the image. In their method, the algorithm assigns pixels into two categories, smooth regions and textured regions by a certain threshold in the Wavelet domain, then Kmeans algorithm was applied in the different categories. In this paper, we propose to apply clustering or segmentation algorithms directly on the Wavelet domain with a weighting parameter to the Wavelet coefficients. The weighting parameter provides the possibility to control the different information to be segmented, the schematic of the algorithm is shown in Fig. 1. Three algorithms, Kmeans_w, FCM_w, and ACWE_w are proposed, and they are the modified versions of the conventional Kmeans, FCM, and ACWE algorithms to incorporate Wavelet coefficients. By the nature of the Wavelet transform, the spatial information is automatically being incorporated in the Wavelet coefficients, so the penalty terms for incorporating the spatial context [4][5][6] are not necessary. The detailed methods and implementation will be described in section 2.2 Image Clustering and Segmentation in Wavelet Domain
2.1 Wavelet Decomposition
The wavelet transform represents the singularity content of an image at multiple scales [8]. In our wavelet decomposition scheme, as shown in Fig. 2
, the transform is applied on the entire image and the first Wavelet tree was extracted and then vectorized. The vectorized Wavelet tree represents the feature vector of the first pixel. We then shift the image by one pixel and apply the transform again, iteratively.
Fig. 3 shows the labeled subbands of a 3level transform, the subbands , , and are called the details, where , represents the maximum decomposition level; the subband is called the smooth approximation; the subbands, , , , and , are recognized as lowfrequency bands [7]. We then introduced a weighting parameter to the lowfrequency bands in the vectorized Wavelet feature vector, the vector is defined as:
where the represents the feature vector of pixel, is the vectorization operator, and weighting parameter biases the differences in lowfrequency bands when computing the squared differences in the clustering/segmentation algorithms, the effect of this parameter will be discussed in the next subsections.
2.2 Kmeans using Wavelet feature vectors
The Kmeans clustering algorithm aims to minimize the squared distances between all pixel intensity and the cluster center[9]. The algorithm iteratively minimizes the following objective function[10]:
(1) 
where is the image domain, is the number of classes, is the observation at pixel j, and is the centroid of class .
The objective function of proposed Kmeans_w is define as:
(2) 
where is the feature vector at pixel , the term is the centroid of feature vectors of class , and the hard membership function is defined as:
(3) 
and
(4) 
The algorithm computes the squared error between the feature vector at each pixel and the mean feature vector of each cluster; this operation spread the energy to each subbands evenly. The weighting parameter , which we introduced previously, controls the weight of the lowfrequency subbands. With , the squared error in lowfrequency subbands gets emphasized, on the other hand, the squared error in lowfrequency subbands gets suppressed when .
2.3 Fuzzy cmeans using Wavelet feature vectors
The FCM clustering algorithm was proposed by Bezdek, it is an improvement of the hard membership Kmeans clustering algorithm[11]. With the similar idea as Kmeans, The algorithm assigns soft membership of each class to a given pixel, depending on the similarity of the pixel intensity value to a particular class relative to all other classes. In the FCM_w, pixel intensities were replaced by the Wavelet feature vectors:
(5) 
the following constrains remains the same as the conventional FCM:
(6) 
where is the image domain, is the number of classes, represents the soft membership function, is the feature vector at pixel , is the centroid of feature vectors of class , and parameter is a weighting exponent that satisfies , it controls the amount of fuzzy overlap between clusters[11]. The larger values indicate a greater degree of overlap (and vice versa). By controlling in feature vectors , we control the importance of the lowfrequency bands.
Applying the Lagrange multiplier method to minimize the objective function , an iterative update expression of membership functions and cluster centroids can be obtained.
2.4 Active contour without edges using Wavelet feature vectors
ACWE was developed by Chan and Vese[12], the algorithm is based on the techniques of curve evolution, the MumfordShah functional for segmentation, and level sets[13][14][15]. The ChanVese function is defined as:
(7) 
where denotes the level set function. The parameters and are fixed to be (i.e. ), is a scale parameter for determining the weight of the length term. The term denotes the Heaviside function of defined as[12]:
it represents the foreground and background regions in image , and represents the length of the contour of the region. and denote the mean intensities of foreground and background regions defined by and . The segmentation of a region can be obtained by minimizing respect to , and . To apply ACWE in the wavelet domain, the energy function (i.e., ) was modified, and the objective function of the ACWE_w is:
(8) 
where is the Wavelet feature vector, and indicate the mean Wavelet feature vectors inside foreground and background regions, respectively. Again, by adjusting the weighting parameter, , we decide how much information in the lowfrequency bands we choose to incorporate.
3 Results
The proposed Kmeans and FCM algorithms were tested on binarized minefield image, as shown in Fig.
4;the proposed ACWEW algorithm was tested on the simulated Quantitative Bone SPECT image using the NURBSbased XCAT phantom [16], an example slice is shown in Fig. 5 (a). In Fig. 5 (b), the green curve indicates the rectangular region of interest that we cropped, Fig. 5 (c) represents the cropped image with the initial contour (red curve) of ACWE, and Fig. 5 (c) contains the ground truth of lesion regions (red curves) and bone regions (blue curves).
The effect of using different orthogonal and biorthogonal Wavelet filter pairs was also studied. The filter coefficients are shown in Table. 1.
8/8  Orthogonal pair (’’) 

6/10  Biorthogonal pair (’’) 
9/7  Biorthogonal pair (’’) 
The results obtained by the conventional 2class Kmeans and FCM algorithm are the same, as shown in Fig. 6. The clustering results were corrupted by noises because the algorithms depend only on local intensity values and spatial information was missing. Fig. 7 shows the results generated by the proposed Kmeans_w and FCM_w algorithms. The first column in Fig. 7 represents the results produced by Kmeans_w with the weight parameter, ; the second column shows the results produced by FCM_w; the first row were generated with orthogonal filter pairs, , shown in the Table 1; the second and the third row were generated with biorthogonal filter pairs, and , respectively. With , the importance of highfrequency information (i.e., noises) gets suppressed, and the algorithms depend more on lowfrequency information. Thus, clustering algorithms produce smoother results. The results also suggest that the use of biorthogonal filter pairs in Kmeans_w and FCM_w algorithms could lead to a better clustering result.
Fig. 8 shows the results produced by conventional ACWE algorithm. Comparing to the ground truth (Fig. 5 (d)), the algorithm failed to obtain a good segmentation for either bone or lesion. The results generated by the ACWE_w algorithm are shown in Fig. 9. When , the algorithm incorporates highfrequency patterns to segment the bone regions. In contrast, when , the algorithm focuses more on the lowfrequency patterns, which provides the ability to segment lesion regions. The results also suggest that for the ACWE_w algorithm, the differences in the resulting segmentation is negligible with different Wavelet filter pairs.
4 Conclusion
Image clustering and segmentation algorithms often concentrate only on local pixel intensities, although many algorithms have been proposed to incorporate spatial context [4][5][6], the modified clustering/segmentation algorithms could still be sensitive to noises. In this paper, we proposed to apply clustering/segmentation algorithms in the Wavelet domain, and we introduced a weighting parameter to control the importance of the lowfrequency subbands. The results provided that the proposed methods perform robust to noises, and the algorithms converge to different meaningful regions based on different frequency band.
5 Acknowledgment
The authors would like to thank Shuwen Wei who provided expertise that greatly assisted this work.
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