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Unsupervised segmentation of ultrasonic liver images by multiresolution fractal feature vector
Information Sciences 175 (2005) 177–199 www.elsevier.com/locate/ins
Unsupervised segmentation of ultrasonic liver images by multiresolution fractal feature vector Wen-Li Lee a, Yung-Chang Chen b,*, Ying-Cheng Chen b, Kai-Sheng Hsieh c a b
Department of Information Management, Kang Ning Junior College, Taipei, Taiwan 114, ROC Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan 300, ROC c Veterans General Hospital, Kaohsiung, Taiwan 813, ROC
Abstract The feasibility of selecting fractal feature vector based on multiresolution analysis to segment suspicious abnormal regions of ultrasonic liver images is described in this paper. The proposed feature extraction algorithm is based on the spatial-frequency decomposition and fractal geometry. Segmentation of various liver diseases reveals that the fractal feature vector based on multiresolution analysis is trustworthy. A quantitative characterization based on the proposed unsupervised segmentation algorithm can be utilized to establish an automatic computer-aided diagnostic system. As well, to increase the visual interpretation capability of ultrasonic liver image for junior physicians, off-line learning software is developed to investigate the visual criteria. 2005 Elsevier Inc. All rights reserved. Keywords: Multiresolution analysis; Fractal; Unsupervised segmentation
*
Corresponding author. Tel.: +886 3 5731153; fax: +886 3 5715971. E-mail address: [email protected] (Y.-C. Chen).
0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.01.007
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1. Introduction Among many imaging modalities, ultrasonic imaging has acquired widespread acceptance as an effective diagnostic tool to visualize human organs and soft tissues. It enables the operator to rapidly locate the desired image plane to display normal or pathological anatomy precisely. It is also safe in examination and economical in medical expenditure. Ultrasonic echoes from human tissues, as displayed in a B-scan image, form texture patterns that show unique characteristics of the imaging system as well as the tissues that are being imaged. These image textures represent the acoustic properties of human tissues and provide very useful tissue signatures regarding their soft structures. One application of diagnostic ultrasound is liver imaging, from which the most useful tissue differentiation techniques can be obtained based on the investigation of B-scan images [1–10]. Via visual interpretation of B-scan images, a clinician can diagnose the tissue by observing the brightness and texture compared to surrounding areas [11]. Hence, liver tissue characterization using ultrasound apparatus mainly depends on the ability of the clinician to observe certain textural characteristics. However, a visual criterion of diagnosing liver diseases principally depends on the clinical experience of physicians and it is extremely subjective. The characterization accuracy using only visual interpretation for diffused liver diseases is estimated to be around 72% [12,13]. Thus, further examination with other invasive methods is usually required, typically the liver needle biopsies. Liver biopsy is the standard clinical routine to diagnose liver disease. Nevertheless, there are associated morbidity and mortality. Therefore, a reliable non-invasive and quantitative method is desired in diagnosing liver diseases. Since ultrasound B-scan images present various granular structures as texture, the analysis of ultrasound image is analogous to the problem in texture analysis. The most difficult aspect for texture analysis is to define a set of meaningful features. Numerous approaches to the texture analysis problem have been proposed. The most successful ones are the spatial gray-level dependence matrices [14], the Fourier power spectrum [15], the gray-level difference statistics [16], LawsÕ texture energy measures [17], and filtering approaches [18–25]. Although they yield promising results to general texture analysis, they are unable to classify ultrasonic liver images adequately. Therefore, many researchers have fused several of these features to obtain an improved performance [10,26– 29]. Furthermore, Kadah et al. [28] and Oosterveld et al. [29] used the raw radio-frequency (RF) signal and conventional features to characterize liver diseases. While the use of raw RF data is attractive and has numerous advantages, it also has a few drawbacks. First, the sampling rate of the RF data must be very high. The amount of data stored and processed is thus much larger than that of image data. Furthermore, nonstandard equipment configuration may be required. Hence, if the distortions caused by B-scanned video image can
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be rectified, the gray-level distribution of a B-scan image characterizes liver tissues adequately. In the previous applications of these texture analysis algorithms that have been applied to characterize liver diseases, the supervised classification is adopted and the region of interest is given by experienced clinicians [18,22– 28]. Therefore, developing an unsupervised algorithm to decide the region of interest is essential in computer assisted diagnosis. Though the accuracy of those previous schemes has been improved; the experimental results are not trustworthy enough to be applied to other liver diseases [18]. Moreover, they share a common weakness in that their calculation is time-consuming. Thus, determining a set of more efficient and meaningful features for characterizing liver ultrasonic images requires further exploration. Wu et al. [18] have proposed a texture feature based upon the concepts of multiple resolution imagery and fractal geometry to detect liver diseases quickly and accurately. Nevertheless, a precise and unifying framework for multiple resolution imagery is required to further improve their algorithm [18]. The wavelet transform provides such a framework for the analysis of a signal that can locate energy in both the time and scale domain close to the theoretical bound given by the HeisenbergÕs uncertainty principle. Thus, multiresolution analysis based on wavelet transform is an excellent tool in providing spatial-frequency decomposition [12,30–38]. Fractal geometry as initially developed and explored by Mandelbrot [39] has had a major impact in modeling and analysis in natural and physical sciences. Fractal provides a proper mathematical framework to study the irregular and complex shapes that exist in nature. Hence, if the pixel intensity of ultrasound B-scan images is regarded as the height above a plane then the intensity surface can be viewed as a rough surface. An essential feature of fractal geometry is that it enables the characterization of irregularity that may not be treated generally in Euclidean geometry. Therefore, among fractal features, fractal dimension is one of the most significant features. Texture features based on fractal dimensions have been applied successfully to texture classification [18,23– 25,40,41]. It is well known that single fractal dimension is not sufficient to discriminate among most real-world textures since its dynamic range for an image is limited between two and three only. Wu et al. [18,23] have proposed a fractal feature vector based on multiple resolutions imagery or multi-threshold concepts. Alternately, it has been established previously that the fractal feature vector, based on standard pyramid wavelet transform and fractal geometry, is a proper feature-extraction method for texture classification [24, 25]. In this study, an unsupervised segmentation scheme by a fractal feature vector based on M-band wavelet transform for ultrasonic liver images is derived. This paper is organized as follows: Feature extraction approaches are presented in the following section. In Section 3, an unsupervised segmentation
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algorithm is introduced. The experimental results are given in Section 4. Finally, Section 5 summarizes the result and conclusions of our investigation.
2. Feature extraction approach 2.1. Estimation of fractal dimension Mandelbrot [39] defines fractal as a bounded set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension, where Hausdorff Besicovitch dimension or fractal dimension is a real number used to describe the shape and appearance of objects that have the property of self-similarity. The property of self-similarity or scaling, as exemplified by coastline, Von Koch curve, and the Mandelbrot set, is one of the central concepts of fractal geometry. The concept of self-similarity can be best utilized to estimate the fractal dimension. Consider a bounded set A in Euclidean n-space. The set A is said to be selfsimilar when A is the union of Nr non-overlapping copies of itself, each of which has been scaled down by a ratio r in all coordinates. The fractal dimension Df of A can be obtained from the relation [39], Df ¼ lim r!0
logðN r Þ logð1=rÞ
ð1Þ
While the definition of fractal dimension via self-similarity is straightforward, it can hardly be computed from the image data. However, a related measure of fractal dimension can be computed from a fractal set, A, in Rn . Several approaches exist to calculate fractal dimension in an image. However, some rely on a specific fractal model, such as fractional Brownian motions (fBm) [18,23,40–43]. Generally, computation using fBm is always over-simplified [18,23,40,43]. The computational errors may present no obvious advantage to texture analysis [43]. Hence, Liu and Chang [41] invoked an auxiliary tool, i.e. the increment of discrete-time fBm, to reduce computational error of the fractal dimension. Alternately, the most popular measure is box counting [44–48]. Among the varieties of box-counting approaches, the Differential Box Counting (DBC) method [44] that has a large dynamic range and computational efficiency is adopted herein. Hence, Nr is calculated as follows [44,45]: If an image of M · M pixels has been scaled down to L · L pixels where 1 < L 6 M/2 and L is an integer. Thus the scale ratio r is L/M. The image can be viewed as a three-dimensional (3D) space with (x, y) indicates the two-dimensional (2D) position and the third coordinate (z) denoting gray-level. The (x, y) space is partitioned into grids of size L · L. On each grid, there is a column of boxes of size L · L · L 0 . If the total number of gray-levels is G, then
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Fig. 1. Determination of nr.
L 0 = bL · G/Mc. Let the minimum and the maximum gray-level of the image in the (i, j)th grid both fall in the box number k and l, as illustrated in Fig. 1. Then nr ði; jÞ ¼ l k þ 1
ð2Þ
is the contribution of Nr in the (i, j)th grid. Taking contributions from all grids, the following is produced: X Nr ¼ nr ði; jÞ ð3Þ i;j
where Nr is counted for differing values of r (i.e. differing values of L). Then using (1), the fractal dimension Df can be estimated from the least-squares linear fitting of log(Nr) versus log(1/r). 2.2. General multiresolution analysis based on M-band wavelet transforms Via multiresolution analysis, a signal can be decomposed into numerous details at various resolutions where each resolution characterizes diverse physical structures in the signal. This coincides with processing visual information in the early stages of the human visual system. Campbell and Robson [49] first proposed the theory to support that the visual system decomposes the retinal image into a number of filtered images. Each filtered image contains intensity
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variations over a narrow range of frequency and orientation. The psychophysical experiments that suggested such decomposition used various grating patterns as stimuli and were based on adaptation techniques. Subsequent psychophysiological experiments provided additional evidence upholding the theory [50]. Therefore, the theory gives an impetus to describe a signal by decomposing it into subbands, and each subband can then be treated alone, based on its feature. Mallat [30] has illustrated that this multiresolution representation of an image can be interpreted as its decomposition by a wavelet basis set. Multiresolution analysis in the two-band case is referenced in [30–38]. Using M-band wavelets [32,34,36,37,51,52], the general multiresolution analysis provides a more flexible tiling of the time–scale plane than that resulting from the two-band multiresolution analysis, which is important for the analysis of middle-frequency or high-frequency signals as it reveals useful features from the subbands. Given a one-dimensional signal, the full discrete wavelet expansion can be represented as f ðtÞ ¼
1 X
cJ ðkÞM J =2 /ðM J t kÞ þ
1 X M 1 X 1 X j¼J
k¼1
d ‘;j ðkÞM j=2 w‘ ðM j t kÞ;
‘¼1 k¼1
ð4Þ where the expansion coefficients are determined by cJ ðkÞ ¼ hf ðtÞ; M J =2 /ðM J t kÞi
ð5Þ
d ‘;j ðkÞ ¼ hf ðtÞ; M j=2 w‘ ðM j t kÞi;
ð6Þ
and
where /(t) is called scaling function that satisfies the dilation equation: pffiffiffiffiffi X /ðtÞ ¼ M h0 ðkÞ/ðMt kÞ;
ð7Þ
k
with h0(k) denoting scaling filter and satisfying certain constraints [34,36,37]. w‘(t), ‘ = 1, 2, . . ., M 1, are called wavelet functions that satisfy the wavelet equations: pffiffiffiffiffi X h‘ ðkÞ/ðMt kÞ for ‘ ¼ 1; 2; . . . ; M 1; ð8Þ w‘ ðtÞ ¼ M k
with h‘(k) denoting the ‘th wavelet filter and satisfying certain constraints [34,36,37]. {/(t), w‘(t), ‘ = 1, 2, . . ., M 1} are mutually orthogonal functions and h,iis the inner product operator. The filters h0(k) and h‘(k), ‘ = 1, 2, . . ., M 1, play a central role in a given wavelet transform. To achieve M-band discrete wavelet transform, the explicit forms of /(t) and w‘(t) are not required but only depend on h0(k) and h‘(k),
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c
h0
↓3
h1
↓3
183
d2,J−1 d1,J−1
J
h2
↓3
h0
↓3
h1
↓3
h2
↓3
cJ−1
d2,J−2 d1,J−2
cJ−2
Fig. 2. Filter bank structure for a three-band wavelet system in one dimension.
which must meet several conditions [32,34,36,37,51,52]. Hence, an M-band wavelet basis can be completely specified by the choice of the scaling filter (i.e. h0(k)) and M 1 wavelet filters (i.e. h‘(k), ‘ = 1, 2, . . ., M 1). A typical M-channel filter bank based on MallatÕs fast algorithm [30] is shown in Fig. 2. That is, the filter bank furnishes an easy way to relate the coefficients of M-band wavelet analysis at various levels of decomposition. The above wavelet model can be generalized to any dimension. There are diverse extensions of one-dimensional wavelet transform to two dimensions. The easiest way to extend one-dimensional wavelet transform to two dimensions is the introduction of separable 2D scaling and wavelet functions as the tensor products of their one-dimensional wavelet basis functions along the horizontal and vertical directions. 2.3. The multiresolution fractal feature vector based on M-band wavelet transform An important aspect of texture analysis is to develop a set of texture measures that can successfully discriminate textures. The feature extraction scheme by traditional multiresolution analysis based on standard wavelet transform decomposes subimages recursively in the low frequency channel. The multiresolution fractal feature vector that Wu et al. [18] proposed simply uses the LL-band (i.e. low-pass filtering in all directions and decimation) in each scale. However, the most significant information of a texture often appears in the middle frequency channels rather than the low frequency ones. Further decomposition in only the lower frequency channels may not provide satisfactory discriminative information to texture analysis [19]. The M-band wavelet transform is an excellent means to portray signals at various scales, and decomposes a signal by projecting it onto a family of functions that are generated from a wavelet basis through its dilations and
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translations. Thus, an image is transformed into M2 resolution cells. This filtering can attain the desired regularizations inherently. General multiresolution analysis based on pyramid wavelet transform provides more significant features [24,25]. Hence, the fractal feature vector based on general multiresolution analysis has been defined as [24,25] m;iþ1 MF ðDm;i ; . . . ; Dfmnþ1;i ; . . . ; Dfmnþ1;iþk Þ; f ; Df
ð9Þ
where Dm;i denotes the fractal dimension of the ith subimage at scale level m. f In general, an M-band wavelet transform can be constructed by two levels of filter bank [32]. The M value can be different at varying levels. The general multiresolution analysis in Chen and coworkers [24,25] is implemented by the combinative structure of two-channel and three-channel filter bank as shown in Fig. 3. In fact, from signal processing viewpoint, this combinative structure is equal to six-channel filter bank [32]. At the first level, the LL-band is adopted and the other subbands are discarded. However, the other bands may provide additional useful information. From the observation of the Fourier spectrum of liver images shown in Fig. 4, LH-band (i.e. low-pass filtering along the abscissa and high-pass filtering along the ordinate) presents apparent information. Accordingly, the fractal dimension of LH-band at the first level should be added to the feature vector. Since the fractal dimension of each chosen subimage should provide valuable information about individual roughness, if the fractal dimension is less than threshold value a (a = 2.05), it means that this subimage has very smooth surface. Based on this observation, the fractal
h1
cJ
' 2
↓2
h1
h0
↓2
h h0
↓2
h1
↓2
↓2
h0 Row direction
↓2
cJ−1
' 1
h
Column direction First Level
h0'
h2'
d8,J−2
h1'
d7,J−2
' 0
h
d6,J−2
h2'
d5,J−2
' 1
h
d4,J−2
h0'
d3,J−2
' 2
h
d2,J−2
h1'
d1,J−2
' 0
h Row direction
cJ−2
Column direction Second Level
Fig. 3. The general multiresolution analysis: the combinative structure of two-channel and threechannel filter bank.
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Fig. 4. Liver images and its respective spectrum.
h0'
h0
↓2
h0
↓2
h1'
h0'
h2' Row Direction
Column Direction
Row Direction Column Direction
h0' h1'
h1'
h2' h0'
h1
↓2
h1'
h0'
h2'
(a) 1,4 D1,1 f Df
D 2,1 f
D 3,0 f
1,5 D1,2 f Df 1,6 D1,3 f Df
D1,7 f D 2,2 f
D1,8 f D1,9 f
(b) Fig. 5. Wavelet system structure for ultrasonic liver image: (a) the mixed structure of two-channel and three-channel filter bank, (b) spatial-frequency tiling.
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dimension of this subimage is too low to be used as a feature. Therefore, the number of features can be reduced to 12 rather than the original 41. The spatial-frequency tilting of wavelet system structure for ultrasonic liver images is demonstrated in Fig. 5. Herein, two levels with Daubechies 20-tap filter bank for the first level and three-channel orthogonal filter bank [33] for the second level were employed.
3. Unsupervised segmentation algorithm Having obtained the feature vector, the main task is how to produce segmentation. Let us assume that K texture categories, C1,C2, . . . ,CK, are present in the image. If the proposed texture features are capable of discriminating these categories then the patterns belonging to each category will form a cluster in the feature space that is compact and isolated from clusters corresponding to other texture categories. Pattern clustering algorithms are ideal vehicles for forming such clusters in the feature space. Segmentation algorithm accepts a set of features as input and assigns a class for each pixel. Fundamentally, this can be considered as a multidimensional data-clustering problem. In this study, the feature extraction part is emphasized. Thus, a traditional K-means clustering algorithm is adopted. A brief overview of the unsupervised K-means clustering algorithm is described below [53,54]. K-means clustering algorithm: Begin initializes l1, l2, . . . , lK Do classify n samples according to nearest li Recomputed li Until no change in li Return l1, l2, . . . , lK End Furthermore, determining the number of texture categories present in an image is very difficult problem. Relative indices provide a means of comparing clustering with different number of clusters and deciding which clustering is the best. A large number of cluster validity indices have been proposed in the literature [55,56]. Unfortunately, there is no single validity index that provides a satisfactory solution for a variety of cluster structures. In this study, the number of category is predefined. Hence, the segmentation algorithm based on multiresolution fractal feature vector is illustrated in the block diagram, as shown in Fig. 6. This algorithm consists of the following steps:
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Digitized ultrasonic liver image
2-level filter bank
Estimate fractal dimension of filtered images
Select feature images
Estimate fractal dimension of subimage of 64*64 in selected feature image
K-means segmentation algorithm Fig. 6. Block diagram for the segmentation scheme.
Step 1: The fractal dimension of input image is first estimated. Step 2: The input image is decomposed into 2 · 2 subimages by two-band filter bank. The fractal dimension of each subimage is calculated. Check the fractal dimension of each subimage by the extraction rule for feature in Section 2.3. Then the selected feature subimages are further decomposed into 3 · 3 subimages by three-band filter bank. The fractal dimension of each subimage is evaluated. Step 3: According to the above estimated fractal dimensions, the corresponding feature subimage is decided. Step 4: Set the square window length equal to 64 and apply to the selected feature image for computing the fractal dimension. Step 5: Apply the K-means clustering algorithm.
4. Experimental segmentation results and discussion In this section, we utilize one natural texture image and six ultrasonic liver images to investigate the performance of the proposed feature extraction methods. All the experiments were programmed in C/C++ language using Borland C++ Builder with a Celeron CPU 1.7 GHz personal computer under MSWindows XP professional environment.
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4.1. Segmentation of natural texture image To demonstrate the performance of the proposed algorithm, we first applied the proposed unsupervised segmentation algorithm to natural texture image. The image of size 384 · 384, 8-bit/pixel, is created by collaged subimages extracted from various natural textures from album images [57], as shown in Fig. 7(a). The segmentation is given in Fig. 7(b). The segmentation accuracy for the mosaic is 97.60%. This result is highly encouraging.
Fig. 7. (a) Texture D17D55, (b) segmentation of texture D17D55 by fractal feature vector, (c) segmentation of texture D17D55 by Gabor filter bank.
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Since Gabor filter approach provides another most promising method for texture feature extraction [55,56,58,59]. Meanwhile, texture descriptor based on Gabor filter bank (using 30 channels) has been approved for the Final Committee Draft of the MPEG-7 standard [60,61]. Hence, we compare the performances of these two different feature extraction methods in this study. Fig. 7(c) shows the segmentation result. The segmentation accuracy is 96.74%. The performance of feature extraction method based on the Gabor filter bank is slightly less than that of the fractal feature vector based on M-channel wavelet transform. That is, the discrimination provided by using the Gabor filter bank is not as good as that by using M-band wavelet system does in this study. The reason may be that the Gabor filter bank isnÕt orthogonal, which results in redundant features at different channels or scales. Furthermore, too many parameters (include orientation h, radial frequency l0, and the space constant of the Gaussian envelope r) are required to set. The chosen parameters will affect the experimental result. Similar results were achieved elsewhere [62– 64]. However, the proposed feature extraction algorithm is free parameter. In addition to accuracy, computational complexity is another essential consideration in a practical application. The execution time of feature extraction method based on the Gabor filter bank is 726 s, whereas the execution time of the fractal feature vector based on M-channel wavelet transform is 717 s. The difference of computational complexity is not obvious since the number of filters for two feature extraction method is both 20. 4.2. Segmentation of ultrasonic liver image From the above experiment, the result is highly heartening. Consequently, we proceed with segmentation of ultrasonic liver image. Four sets of ultrasonic liver images (six images) were taken: cirrhosis (two images), hepatitis B (two images), hepatitis C (one image) and normal (one image), as shown in Fig. 8. All ultrasonic liver images were captured from a phased-array system (Aloka SSD256, Tokyo, Japan) with a 3.5 MHz transducer and were stored as positive ones. These were then scanned by AGFAÕs DUOSCAN scanner with 32-pixel/ cm and 8-bit/pixel resolution. A specialized physician verified all digitized images. All liver diseases were biopsied for pathological diagnosis; therefore there is a basis of truth in this study. The segmented region is bounded from subcutaneous 3 cm to 11 cm. The segmentations for ultrasonic liver images are illustrated in Fig. 9. The information contained in the ultrasound image is a result of the complex interaction between ultrasound waves and tissue components. In ultrasound liver images, various granular structures are illustrated as texture. Generally, texture can be evaluated as being fine, coarse, grained or smooth. Therefore, it should be possible to extract tissue-relevant quantitative parameters from the image. Herein, a fractal feature vector based on M-channel
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Fig. 8. Ultrasonic liver image: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C, (f) normal.
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Fig. 9. Segmented region of ultrasonic liver image for given two-class: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C, (f) normal.
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wavelet transform was defined to determine the most useful information for the characterization of ultrasonic liver images. Additionally, the appropriate feature vector was selected from the Fourier spectrum. Thus, the number of features can be reduced to 12 rather than the original 41. The segmented regions have been verified by a senior physician and prove to be reasonable, as shown in Fig. 9. As well, the comparison for the two feature extraction method is given. In the segmentation of ultrasound liver images of size 376 · 220, the execution time of feature extraction method based on the Gabor filter bank is about 622 s, whereas of the fractal feature vector based on M-channel wavelet transform is 347 s. The difference of computational complexity is mainly caused by the number of filters, the length of filter and critical decimation. Hence, the proposed algorithm is computationally efficient. The results of segmented areas described above meet our expectation. However, is it possible to find further information from suspected pathological change region? For example, the association between hepatocellular carcinoma (HCC) and cirrhosis is strong. And the trabecular pattern is the most common and characteristic form of hepatoma. In this form, the tumor cells are arranged in anastomosing plates that are separated by a sinusoidal network. The structure thus resembles that of a normal liver; however the trabeculate is typically wider and less regular [65]. Hence, the hepatoma is rougher than that of a normal liver. The major characteristic of cirrhosis is the hepatic fibrosis that is associated with beginning nodules or fully established nodules [66]. The nodules cause a block structure in the image such that cirrhosis image is rougher than a normal liver image but less rough than hepatoma image. These two liver diseases have rough images of varying degrees. And thus, is it possible to detect suspicious hepatoma area from cirrhosis regions? Since hepatitis B, hepatitis C or alcoholic is the main cause of cirrhosis. Therefore, it is desired to further segment suspected cirrhosis or HCC areas from suspicious abnormal regions. Total number of classes is two in the previous experiment. The experimental result provides suspicious abnormal regions. Accordingly, the three classes are specified for previous ultrasonic images. The segmented results are illustrated in Fig. 10. The principle cause of the further analysis is that we hope to segment more pathological change tissues from the abnormal regions. Thus, the proposed unsupervised segmentation algorithm plays an important role in the automatic computer aided diagnostic system, since the previous works about the diagnosis of ultrasonic liver image are all based on the supervised classification scheme [22–28]. The test sample is extracted from the region of interest that is given by experienced physician. Then, the sample is classified based on the database of ultrasonic liver images. Apparently, the supervised classification is insufficient for automatic diagnose. Hence, the sufficient ultrasonic liver images of various liver diseases and the proposed unsupervised segmentation algorithm can establish an automatic computer aided diagnostic system. The block diagram of the
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Fig. 10. Segmented region of ultrasonic liver image for given three-class: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C.
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Digitized ultrasonic liver image
Feature vector extraction
Unsupervised segmentation
Extracted test sample from suspicious abnormal regions
Supervised classification
Suggestive certain liver disease Fig. 11. The block diagram of automatic computer aided diagnostic system.
system is illustrated in Fig. 11. First, the segmentation of ultrasonic liver image is executed. The suspicious pathological changes areas from the abnormal regions are recognized with the established database. Moreover, an application based on the proposed unsupervised segmentation is to set up an off-line investigating system. Since visual criteria of diagnosing liver diseases principally depend on the clinical experience of physicians, it is extremely subjective. A quantitative characterization scheme of ultrasonic liver images is essential in diagnosis. Thus, a software system can be developed, based on the proposed algorithm. The block diagram of the software system is depicted in Fig. 12. Via this software, a junior physician can enhance visual assessment capability from off-line database.
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Digitized ultrasonic liver image
Feature vector extraction
Number of classes assignment
Clustering
Investigation of segmented regions
Reasonable?
No
Yes End Fig. 12. The block diagram of learning software system.
5. Conclusion We have presented an unsupervised segmentation algorithm based on Mband wavelet transform and fractal geometry. Since the spatial-frequency plane can combine logarithmic and uniform spacing by the M-band wavelet transform, a more flexible decomposition of the entire frequency band can be achieved. The proposed scheme uses M-band wavelet transform to decompose an original texture image into a number of subbands. The subband images provide textural feature in different orientations and scales. Fractal dimension is
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used to furnish measurement of textural image. We applied the proposed unsupervised segmentation algorithm to natural texture image and four sets of ultrasonic liver images. The experimental results reveal that the combination of these two approaches furnishes valuable information to analyze ultrasonic liver images under study. Based on the proposed unsupervised segmentation algorithm, an automatic computer-aided diagnostic system can be established to increase the clinical discrimination of ultrasonic imaging. As well, software of off-line investigating system can be developed to enhance the visual interpretation capability of ultrasonic liver image for junior physicians.
References [1] M.F. Insana, R.F. Wagner, B.S. Garra, D.G. Brown, T.H. Shawker, Analysis of ultrasound image texture via generalized Rician statistics, Opt. Eng. 25 (1986) 743–748. [2] U. Reath, D. Schlaps, B. Limberg, Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease, J. Clin. Ultrasound 13 (1985) 87–99. [3] N.M. Botros, A microprocessor-based pattern recognition algorithm for in-vivo tissue differentiation, J. Clin. Eng. 13 (1988) 115–120. [4] R. Momenan, M.H. Loew, M.F. Insana, R.F. Wagner, B.S. Garra, Application of pattern recognition techniques in ultrasound tissue characterization, in: 10th Int. Conf. Pattern Recognition, vol. 1, 1990, pp. 608–612. [5] M.F. Insana, R.F. Wagner, B.S. Garra, R. Momenan, T.H. Shawker, Pattern recognition methods for optimizing multivariate tissue signatures in diagnostic ultrasound, Ultrasound Imaging 8 (1986) 165–180. [6] B.S. Garra, M.F. Insana, T.H. Shawker, R.F. Wagner, M. Bradford, M. Russell, Quantitative ultrasonic detection and classification of diffuse liver disease comparison with human observer performance, Invest. Radiol. 24 (1989) 196–203. [7] R.F. Wagner, M.F. Insana, G. Brown, Unified approach to the detection and classification of speckle texture in diagnostic ultrasound, Opt. Eng. 25 (1986) 743–748. [8] R. Momenan, M.F. Insana, R.F. Wagner, B.S. Garra, M.H. Loew, Application of clutter analysis and unsupervised learning to multivariate tissue characterization, J. Clin. Eng. 13 (1988) 455–461. [9] R. Momenan, R.F. Wagner, B.S. Garra, M.H. Loew, M.F. Insana, Image staining and differential diagnosis of ultrasound scans based on the Mahalanobis distance, IEEE Trans. Med. Imaging 11 (June) (1994) 37–47. [10] K. Ogawa, M. Fukushima, K. Kubota, N. Hisa, Computer-aided diagnostic system for diffuse liver diseases with ultrasonography by neural network, IEEE Trans. Nucl. Sci. 45 (6) (1998) 3069–3074. [11] G.J. W Simon, E.E. Jane, B. Nigal, E.H. Margaret, E.B. Joe, W. David, An ultrasound scoring system for the diagnosis of liver disease in cystic fibrosis, J. Hepatol. 22 (1995) 513–521. [12] G.F. Vawter, H. Shwachman, Cystic fibrosis in adults: an autopsy study, Pathol. Annu. 14 (1979) 357–382. [13] N.I. Sandford, P. Walsh, C. Matis, H. Baddeley, L.W. Powell, Is ultrasonography useful in the development of diffuse parenchyma liver disease, Gastroenterology 9 (1985) 186–191. [14] R.M. Haralick, K. Shanmugam, I. Dinstein, Texture features for image classification, IEEE Trans. Syst. Man Cybern. 3 (6) (1973) 610–621.
W.-L. Lee et al. / Information Sciences 175 (2005) 177–199
197
[15] G.O. Lendaris, G.L. Stanley, Diffraction pattern sampling for automatic pattern recognition, Proc. IEEE 58 (1970) 198–216. [16] J.S. Weszka, C.R. Dryer, A. Rosenfeld, A comparative study of texture measures for terrain classification, IEEE Trans. Syst. Man Cybern. SMC-6 (1979) 269–285. [17] K.I. Laws, Texture image segmentation, Ph.D. dissertation, Image Processing Inst., Univ. of Southern California, 1980. [18] C.M. Wu, Y.C. Chen, K.S. Hsieh, Texture features for classification of ultrasonic liver images, IEEE Trans. Med. Imaging 11 (June) (1992) 141–152. [19] T. Chang, C. Kuo, Texture analysis and classification with tree-structured wavelet transform, IEEE Trans. Image Process. 2 (October) (1993) 429–441. [20] A. Laine, J. Fan, Texture classification by wavelet packet signatures, IEEE Trans. Pattern Anal. Mach. Intell. 15 (November) (1993) 1186–1191. [21] M. Unser, Texture classification and segmentation using wavelet frames, IEEE Trans. Image Process. 4 (11) (1995) 1549–1560. [22] A. Mojsilovic´, M. Popovic´, S. Markovic´, M. Krstic´, Characterization of visually similar diffuse diseases from B-scan liver images using nonseparable wavelet transform, IEEE Trans. Med. Imaging 17 (4) (1998) 541–549. [23] C.M. Wu, Y.C. Chen, Multi-threshold dimension vector for texture analysis and its application to liver tissue classification, Pattern Recogn. 26 (1) (1993) 137–144. [24] Y.C. Chen, W.L. Lee, Texture classification using multiresolution fractal feature vector, in: Proc. 4th Asian Conf. on Computer Vision, 2000, pp. 204–209. [25] W.L. Lee, Y.C. Chen, K.S. Hsieh, Ultrasonic liver tissues classification by fractal feature vector based on M-band wavelet transform, in: The 2001 IEEE Int. Symp. on Circuits and Systems, vol. 2, 2001, pp. 1–4. [26] H. Sujana, S. Swarnamani, S. Suresh, Application of artificial neural networks for the classification of liver lesions by image texture parameters, Ultrasound Med. Biol. 22 (9) (1996) 1177–1181. [27] S. Pavlopoulos, E. Kyriacou, D. Koutsouris, K. Blekas, A. Stafylopatis, P. Zoumpoulis, Fuzzy neural network-based texture analysis of ultrasonic images, IEEE Eng. Med. Biol. 19 (1) (2000) 39–47. [28] Y.M. Kadah, A.A. Farag, J.M. Zurada, A.M. Badawi, A.M. Youssef, Classification algorithms for quantitative tissue characterization of diffuse liver disease from ultrasound images, IEEE Trans. Med. Imaging 15 (4) (1996) 466–478. [29] B.J. Oosterveld, J.M. Thijssen, P.C. Hartman, G.J.E. Rosenbusch, Detection of diffuse liver disease by quantative echography: dependence on a priori choice of parameters, Ultrasound Med. Biol. 19 (1) (1993) 21–25. [30] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell. 11 (July) (1989) 674–693. [31] O. Rioul, M. Vetterli, Wavelets and signal processing, IEEE Signal Process. Mag. 8 (October) (1991) 11–38. [32] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge, MA, 1996. [33] P.P. Vaidaynathan, Multirate Systems and Filter Banks, Prentice-Hall, New Jersey, 1993. [34] C.S. Burrus, R.A. Goponath, H. Guo, Introduction to Wavelets and Wavelet Transform: a Primerm, Prentice-Hall, New Jersey, 1998. [35] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Information Theory 36 (September) (1990) 961–1005. [36] R.A. Goponath, E. Odegard, C.S. Burrus, Optimal wavelet representation of signals and the wavelet sampling theorem, IEEE Trans. Circ. Syst. II 41 (4) (1994) 262–277. [37] M.K. Tsatsanis, G.B. Giannakis, Principal component filter banks for optimal multiresolution analysis, IEEE Trans. Signal Process. 43 (8) (1995) 1766–1777. [38] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.
198
W.-L. Lee et al. / Information Sciences 175 (2005) 177–199
[39] B.B. Mandelbrot, Fractal Geometry of Nature, Freeman Press, San Francisco, 1982. [40] C.C. Chen, J.S. Daponte, M.D. Fox, Fractal feature analysis and classification in medical imaging, IEEE Trans. Med. Imaging 6 (June) (1989) 133–142. [41] S.C. Liu, S. Chang, Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification, IEEE Trans. Image Process. 6 (8) (1997). [42] T. Lundahl, W.J. Ohely, S.M. Kay, R. Siffert, Fractional brownian motion: a maximum likelihood estimator and its application to image texture, IEEE Trans. Med. Imaging 5 (3) (1986) 152–161. [43] J.T.M. Verhoeven, J.M. Thijssen, Potential of fractal analysis for lesion detection in echographic images, Ultrasonic Imaging 15 (1993) 304–323. [44] N. Sarkar, B.B. Chaudhuri, An efficient differential box-counting approach to compute fractal dimension of image, IEEE Trans. Syst. Man Cybern. 24 (January) (1994) 115–120. [45] S. Buczkowski, S. Kyriacos, F. Nekka, L. Cartilier, The modified box-counting method: analysis of some characteristic parameters, Pattern Recogn. 31 (4) (1998) 411–418. [46] J.M. Keller, S. Chen, R.M. Crowniver, Texture description and segmentation through fractal geometry, CVGIP 45 (1989) 150–166. [47] J. Feng, W.-C. Lin, C.-T. Chen, Fractional box counting approach to fractal dimension estimation, in: Proc. 13th ICPR, II, 1996, pp. 854–858. [48] S. Chen, J.M. Keller, R.M. Crowniver, On the calculation of fractal features from images, IEEE Trans. Pattern Anal. Mach. Intell. 15 (10) (1993) 1087–1090. [49] F.W. Campbell, J.G. Robson, Application of Fourier analysis to the visibility of gratings, J. Physil. 197 (1968) 551–566. [50] R.L. De Valois, D.G. Albrecht, L.G. Thorell, Spatial-frequency selectivity of cells in macaque visual cortex, Vision Res. 22 (1982) 545–559. [51] P. Steffen, P.N. Heller, R.A. Goponath, C.S. Burrus, Theory of regular M-band wavelet bases, IEEE Trans. Signal Process. 41 (12) (1993) 3497–3511. [52] Y. Chitre, A.P. Dhawan, M-band wavelet discrimination of nature textures, Pattern Recogn. 32 (1999) 773–789. [53] R.O. Duda, P.E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, 1973. [54] M. Nadler, E.P. Smith, Pattern Recognition Engineering, Wiley, New York, 1993. [55] A.K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, in: Proc. IEEE Int. Conf. on Systems, Man and Cybernetics, 1990, pp. 14–19. [56] A.K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, Pattern Recogn. 24 (12) (1991) 1167–1186. [57] P. Brodatz, Texture: a Photographic Album for Artists and Designers, Dover, New York, 1966. [58] C.C. Chen, D.C. Chen, Multi-resolution Gabor filter in the texture analysis, Pattern Recogn. Lett. 17 (1996) 1069–1076. [59] O. Pichler, A. Teuner, B.J. Hosticka, A comparison of texture feature extraction using adaptive Gabor filtering, pyramidal and tree structured wavelet transforms, Pattern Recogn. 29 (5) (1996) 733–742. [60] P. Wu, B.S. Manjunath, S. Newsam, H.D. Shin, A texture descriptor for browsing and similarity retrieval, Signal Process.: Image Commun. 16 (2000) 33–43. [61] B.S. Manjunath, J.R. Ohm, V.V. Vasudevan, A. Yamada, Color and texture descriptors, IEEE Trans. Circ. Syst. Video Technol. 11 (11) (2001) 703–715. [62] T. Randen, J.H. Husoy, Multichannel filtering for image texture segmentation, Opt. Eng. 33 (August) (1994) 2617–2625. [63] T. Randen, J.H. Husoy, Filtering for texture classification: a comparative study, IEEE Trans. Pattern Anal. Mach. Intell. 21 (4) (1999) 291–310. [64] M. Acharyya, M.K. Kundu, An adaptive approach to unsupervised texture segmentation using M-band wavelet transform, Signal Process. 81 (2001) 1337–1356.
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199
[65] P.J. Scheuer, Pathologic types of hepatic tumors, in: P. Bannasch, D. Keppler, G. Weber (Eds.), Liver Cell Carcinoma, Kluwer-Academic, New York, 1989, p. 18. [66] J.H. Lefkowitch, Pathologic diagnosis of liver disease, in: D. Zakim, T.D. Boyer (Eds.), Hepatology: a Textbook of Liver Disease, W.B. Saunders, London, England, 1990, p. 719.
Unsupervised segmentation of ultrasonic liver images by multiresolution fractal feature vector Wen-Li Lee a, Yung-Chang Chen b,*, Ying-Cheng Chen b, Kai-Sheng Hsieh c a b
Department of Information Management, Kang Ning Junior College, Taipei, Taiwan 114, ROC Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan 300, ROC c Veterans General Hospital, Kaohsiung, Taiwan 813, ROC
Abstract The feasibility of selecting fractal feature vector based on multiresolution analysis to segment suspicious abnormal regions of ultrasonic liver images is described in this paper. The proposed feature extraction algorithm is based on the spatial-frequency decomposition and fractal geometry. Segmentation of various liver diseases reveals that the fractal feature vector based on multiresolution analysis is trustworthy. A quantitative characterization based on the proposed unsupervised segmentation algorithm can be utilized to establish an automatic computer-aided diagnostic system. As well, to increase the visual interpretation capability of ultrasonic liver image for junior physicians, off-line learning software is developed to investigate the visual criteria. 2005 Elsevier Inc. All rights reserved. Keywords: Multiresolution analysis; Fractal; Unsupervised segmentation
*
Corresponding author. Tel.: +886 3 5731153; fax: +886 3 5715971. E-mail address: [email protected] (Y.-C. Chen).
0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.01.007
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1. Introduction Among many imaging modalities, ultrasonic imaging has acquired widespread acceptance as an effective diagnostic tool to visualize human organs and soft tissues. It enables the operator to rapidly locate the desired image plane to display normal or pathological anatomy precisely. It is also safe in examination and economical in medical expenditure. Ultrasonic echoes from human tissues, as displayed in a B-scan image, form texture patterns that show unique characteristics of the imaging system as well as the tissues that are being imaged. These image textures represent the acoustic properties of human tissues and provide very useful tissue signatures regarding their soft structures. One application of diagnostic ultrasound is liver imaging, from which the most useful tissue differentiation techniques can be obtained based on the investigation of B-scan images [1–10]. Via visual interpretation of B-scan images, a clinician can diagnose the tissue by observing the brightness and texture compared to surrounding areas [11]. Hence, liver tissue characterization using ultrasound apparatus mainly depends on the ability of the clinician to observe certain textural characteristics. However, a visual criterion of diagnosing liver diseases principally depends on the clinical experience of physicians and it is extremely subjective. The characterization accuracy using only visual interpretation for diffused liver diseases is estimated to be around 72% [12,13]. Thus, further examination with other invasive methods is usually required, typically the liver needle biopsies. Liver biopsy is the standard clinical routine to diagnose liver disease. Nevertheless, there are associated morbidity and mortality. Therefore, a reliable non-invasive and quantitative method is desired in diagnosing liver diseases. Since ultrasound B-scan images present various granular structures as texture, the analysis of ultrasound image is analogous to the problem in texture analysis. The most difficult aspect for texture analysis is to define a set of meaningful features. Numerous approaches to the texture analysis problem have been proposed. The most successful ones are the spatial gray-level dependence matrices [14], the Fourier power spectrum [15], the gray-level difference statistics [16], LawsÕ texture energy measures [17], and filtering approaches [18–25]. Although they yield promising results to general texture analysis, they are unable to classify ultrasonic liver images adequately. Therefore, many researchers have fused several of these features to obtain an improved performance [10,26– 29]. Furthermore, Kadah et al. [28] and Oosterveld et al. [29] used the raw radio-frequency (RF) signal and conventional features to characterize liver diseases. While the use of raw RF data is attractive and has numerous advantages, it also has a few drawbacks. First, the sampling rate of the RF data must be very high. The amount of data stored and processed is thus much larger than that of image data. Furthermore, nonstandard equipment configuration may be required. Hence, if the distortions caused by B-scanned video image can
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be rectified, the gray-level distribution of a B-scan image characterizes liver tissues adequately. In the previous applications of these texture analysis algorithms that have been applied to characterize liver diseases, the supervised classification is adopted and the region of interest is given by experienced clinicians [18,22– 28]. Therefore, developing an unsupervised algorithm to decide the region of interest is essential in computer assisted diagnosis. Though the accuracy of those previous schemes has been improved; the experimental results are not trustworthy enough to be applied to other liver diseases [18]. Moreover, they share a common weakness in that their calculation is time-consuming. Thus, determining a set of more efficient and meaningful features for characterizing liver ultrasonic images requires further exploration. Wu et al. [18] have proposed a texture feature based upon the concepts of multiple resolution imagery and fractal geometry to detect liver diseases quickly and accurately. Nevertheless, a precise and unifying framework for multiple resolution imagery is required to further improve their algorithm [18]. The wavelet transform provides such a framework for the analysis of a signal that can locate energy in both the time and scale domain close to the theoretical bound given by the HeisenbergÕs uncertainty principle. Thus, multiresolution analysis based on wavelet transform is an excellent tool in providing spatial-frequency decomposition [12,30–38]. Fractal geometry as initially developed and explored by Mandelbrot [39] has had a major impact in modeling and analysis in natural and physical sciences. Fractal provides a proper mathematical framework to study the irregular and complex shapes that exist in nature. Hence, if the pixel intensity of ultrasound B-scan images is regarded as the height above a plane then the intensity surface can be viewed as a rough surface. An essential feature of fractal geometry is that it enables the characterization of irregularity that may not be treated generally in Euclidean geometry. Therefore, among fractal features, fractal dimension is one of the most significant features. Texture features based on fractal dimensions have been applied successfully to texture classification [18,23– 25,40,41]. It is well known that single fractal dimension is not sufficient to discriminate among most real-world textures since its dynamic range for an image is limited between two and three only. Wu et al. [18,23] have proposed a fractal feature vector based on multiple resolutions imagery or multi-threshold concepts. Alternately, it has been established previously that the fractal feature vector, based on standard pyramid wavelet transform and fractal geometry, is a proper feature-extraction method for texture classification [24, 25]. In this study, an unsupervised segmentation scheme by a fractal feature vector based on M-band wavelet transform for ultrasonic liver images is derived. This paper is organized as follows: Feature extraction approaches are presented in the following section. In Section 3, an unsupervised segmentation
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algorithm is introduced. The experimental results are given in Section 4. Finally, Section 5 summarizes the result and conclusions of our investigation.
2. Feature extraction approach 2.1. Estimation of fractal dimension Mandelbrot [39] defines fractal as a bounded set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension, where Hausdorff Besicovitch dimension or fractal dimension is a real number used to describe the shape and appearance of objects that have the property of self-similarity. The property of self-similarity or scaling, as exemplified by coastline, Von Koch curve, and the Mandelbrot set, is one of the central concepts of fractal geometry. The concept of self-similarity can be best utilized to estimate the fractal dimension. Consider a bounded set A in Euclidean n-space. The set A is said to be selfsimilar when A is the union of Nr non-overlapping copies of itself, each of which has been scaled down by a ratio r in all coordinates. The fractal dimension Df of A can be obtained from the relation [39], Df ¼ lim r!0
logðN r Þ logð1=rÞ
ð1Þ
While the definition of fractal dimension via self-similarity is straightforward, it can hardly be computed from the image data. However, a related measure of fractal dimension can be computed from a fractal set, A, in Rn . Several approaches exist to calculate fractal dimension in an image. However, some rely on a specific fractal model, such as fractional Brownian motions (fBm) [18,23,40–43]. Generally, computation using fBm is always over-simplified [18,23,40,43]. The computational errors may present no obvious advantage to texture analysis [43]. Hence, Liu and Chang [41] invoked an auxiliary tool, i.e. the increment of discrete-time fBm, to reduce computational error of the fractal dimension. Alternately, the most popular measure is box counting [44–48]. Among the varieties of box-counting approaches, the Differential Box Counting (DBC) method [44] that has a large dynamic range and computational efficiency is adopted herein. Hence, Nr is calculated as follows [44,45]: If an image of M · M pixels has been scaled down to L · L pixels where 1 < L 6 M/2 and L is an integer. Thus the scale ratio r is L/M. The image can be viewed as a three-dimensional (3D) space with (x, y) indicates the two-dimensional (2D) position and the third coordinate (z) denoting gray-level. The (x, y) space is partitioned into grids of size L · L. On each grid, there is a column of boxes of size L · L · L 0 . If the total number of gray-levels is G, then
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Fig. 1. Determination of nr.
L 0 = bL · G/Mc. Let the minimum and the maximum gray-level of the image in the (i, j)th grid both fall in the box number k and l, as illustrated in Fig. 1. Then nr ði; jÞ ¼ l k þ 1
ð2Þ
is the contribution of Nr in the (i, j)th grid. Taking contributions from all grids, the following is produced: X Nr ¼ nr ði; jÞ ð3Þ i;j
where Nr is counted for differing values of r (i.e. differing values of L). Then using (1), the fractal dimension Df can be estimated from the least-squares linear fitting of log(Nr) versus log(1/r). 2.2. General multiresolution analysis based on M-band wavelet transforms Via multiresolution analysis, a signal can be decomposed into numerous details at various resolutions where each resolution characterizes diverse physical structures in the signal. This coincides with processing visual information in the early stages of the human visual system. Campbell and Robson [49] first proposed the theory to support that the visual system decomposes the retinal image into a number of filtered images. Each filtered image contains intensity
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variations over a narrow range of frequency and orientation. The psychophysical experiments that suggested such decomposition used various grating patterns as stimuli and were based on adaptation techniques. Subsequent psychophysiological experiments provided additional evidence upholding the theory [50]. Therefore, the theory gives an impetus to describe a signal by decomposing it into subbands, and each subband can then be treated alone, based on its feature. Mallat [30] has illustrated that this multiresolution representation of an image can be interpreted as its decomposition by a wavelet basis set. Multiresolution analysis in the two-band case is referenced in [30–38]. Using M-band wavelets [32,34,36,37,51,52], the general multiresolution analysis provides a more flexible tiling of the time–scale plane than that resulting from the two-band multiresolution analysis, which is important for the analysis of middle-frequency or high-frequency signals as it reveals useful features from the subbands. Given a one-dimensional signal, the full discrete wavelet expansion can be represented as f ðtÞ ¼
1 X
cJ ðkÞM J =2 /ðM J t kÞ þ
1 X M 1 X 1 X j¼J
k¼1
d ‘;j ðkÞM j=2 w‘ ðM j t kÞ;
‘¼1 k¼1
ð4Þ where the expansion coefficients are determined by cJ ðkÞ ¼ hf ðtÞ; M J =2 /ðM J t kÞi
ð5Þ
d ‘;j ðkÞ ¼ hf ðtÞ; M j=2 w‘ ðM j t kÞi;
ð6Þ
and
where /(t) is called scaling function that satisfies the dilation equation: pffiffiffiffiffi X /ðtÞ ¼ M h0 ðkÞ/ðMt kÞ;
ð7Þ
k
with h0(k) denoting scaling filter and satisfying certain constraints [34,36,37]. w‘(t), ‘ = 1, 2, . . ., M 1, are called wavelet functions that satisfy the wavelet equations: pffiffiffiffiffi X h‘ ðkÞ/ðMt kÞ for ‘ ¼ 1; 2; . . . ; M 1; ð8Þ w‘ ðtÞ ¼ M k
with h‘(k) denoting the ‘th wavelet filter and satisfying certain constraints [34,36,37]. {/(t), w‘(t), ‘ = 1, 2, . . ., M 1} are mutually orthogonal functions and h,iis the inner product operator. The filters h0(k) and h‘(k), ‘ = 1, 2, . . ., M 1, play a central role in a given wavelet transform. To achieve M-band discrete wavelet transform, the explicit forms of /(t) and w‘(t) are not required but only depend on h0(k) and h‘(k),
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c
h0
↓3
h1
↓3
183
d2,J−1 d1,J−1
J
h2
↓3
h0
↓3
h1
↓3
h2
↓3
cJ−1
d2,J−2 d1,J−2
cJ−2
Fig. 2. Filter bank structure for a three-band wavelet system in one dimension.
which must meet several conditions [32,34,36,37,51,52]. Hence, an M-band wavelet basis can be completely specified by the choice of the scaling filter (i.e. h0(k)) and M 1 wavelet filters (i.e. h‘(k), ‘ = 1, 2, . . ., M 1). A typical M-channel filter bank based on MallatÕs fast algorithm [30] is shown in Fig. 2. That is, the filter bank furnishes an easy way to relate the coefficients of M-band wavelet analysis at various levels of decomposition. The above wavelet model can be generalized to any dimension. There are diverse extensions of one-dimensional wavelet transform to two dimensions. The easiest way to extend one-dimensional wavelet transform to two dimensions is the introduction of separable 2D scaling and wavelet functions as the tensor products of their one-dimensional wavelet basis functions along the horizontal and vertical directions. 2.3. The multiresolution fractal feature vector based on M-band wavelet transform An important aspect of texture analysis is to develop a set of texture measures that can successfully discriminate textures. The feature extraction scheme by traditional multiresolution analysis based on standard wavelet transform decomposes subimages recursively in the low frequency channel. The multiresolution fractal feature vector that Wu et al. [18] proposed simply uses the LL-band (i.e. low-pass filtering in all directions and decimation) in each scale. However, the most significant information of a texture often appears in the middle frequency channels rather than the low frequency ones. Further decomposition in only the lower frequency channels may not provide satisfactory discriminative information to texture analysis [19]. The M-band wavelet transform is an excellent means to portray signals at various scales, and decomposes a signal by projecting it onto a family of functions that are generated from a wavelet basis through its dilations and
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translations. Thus, an image is transformed into M2 resolution cells. This filtering can attain the desired regularizations inherently. General multiresolution analysis based on pyramid wavelet transform provides more significant features [24,25]. Hence, the fractal feature vector based on general multiresolution analysis has been defined as [24,25] m;iþ1 MF ðDm;i ; . . . ; Dfmnþ1;i ; . . . ; Dfmnþ1;iþk Þ; f ; Df
ð9Þ
where Dm;i denotes the fractal dimension of the ith subimage at scale level m. f In general, an M-band wavelet transform can be constructed by two levels of filter bank [32]. The M value can be different at varying levels. The general multiresolution analysis in Chen and coworkers [24,25] is implemented by the combinative structure of two-channel and three-channel filter bank as shown in Fig. 3. In fact, from signal processing viewpoint, this combinative structure is equal to six-channel filter bank [32]. At the first level, the LL-band is adopted and the other subbands are discarded. However, the other bands may provide additional useful information. From the observation of the Fourier spectrum of liver images shown in Fig. 4, LH-band (i.e. low-pass filtering along the abscissa and high-pass filtering along the ordinate) presents apparent information. Accordingly, the fractal dimension of LH-band at the first level should be added to the feature vector. Since the fractal dimension of each chosen subimage should provide valuable information about individual roughness, if the fractal dimension is less than threshold value a (a = 2.05), it means that this subimage has very smooth surface. Based on this observation, the fractal
h1
cJ
' 2
↓2
h1
h0
↓2
h h0
↓2
h1
↓2
↓2
h0 Row direction
↓2
cJ−1
' 1
h
Column direction First Level
h0'
h2'
d8,J−2
h1'
d7,J−2
' 0
h
d6,J−2
h2'
d5,J−2
' 1
h
d4,J−2
h0'
d3,J−2
' 2
h
d2,J−2
h1'
d1,J−2
' 0
h Row direction
cJ−2
Column direction Second Level
Fig. 3. The general multiresolution analysis: the combinative structure of two-channel and threechannel filter bank.
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Fig. 4. Liver images and its respective spectrum.
h0'
h0
↓2
h0
↓2
h1'
h0'
h2' Row Direction
Column Direction
Row Direction Column Direction
h0' h1'
h1'
h2' h0'
h1
↓2
h1'
h0'
h2'
(a) 1,4 D1,1 f Df
D 2,1 f
D 3,0 f
1,5 D1,2 f Df 1,6 D1,3 f Df
D1,7 f D 2,2 f
D1,8 f D1,9 f
(b) Fig. 5. Wavelet system structure for ultrasonic liver image: (a) the mixed structure of two-channel and three-channel filter bank, (b) spatial-frequency tiling.
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dimension of this subimage is too low to be used as a feature. Therefore, the number of features can be reduced to 12 rather than the original 41. The spatial-frequency tilting of wavelet system structure for ultrasonic liver images is demonstrated in Fig. 5. Herein, two levels with Daubechies 20-tap filter bank for the first level and three-channel orthogonal filter bank [33] for the second level were employed.
3. Unsupervised segmentation algorithm Having obtained the feature vector, the main task is how to produce segmentation. Let us assume that K texture categories, C1,C2, . . . ,CK, are present in the image. If the proposed texture features are capable of discriminating these categories then the patterns belonging to each category will form a cluster in the feature space that is compact and isolated from clusters corresponding to other texture categories. Pattern clustering algorithms are ideal vehicles for forming such clusters in the feature space. Segmentation algorithm accepts a set of features as input and assigns a class for each pixel. Fundamentally, this can be considered as a multidimensional data-clustering problem. In this study, the feature extraction part is emphasized. Thus, a traditional K-means clustering algorithm is adopted. A brief overview of the unsupervised K-means clustering algorithm is described below [53,54]. K-means clustering algorithm: Begin initializes l1, l2, . . . , lK Do classify n samples according to nearest li Recomputed li Until no change in li Return l1, l2, . . . , lK End Furthermore, determining the number of texture categories present in an image is very difficult problem. Relative indices provide a means of comparing clustering with different number of clusters and deciding which clustering is the best. A large number of cluster validity indices have been proposed in the literature [55,56]. Unfortunately, there is no single validity index that provides a satisfactory solution for a variety of cluster structures. In this study, the number of category is predefined. Hence, the segmentation algorithm based on multiresolution fractal feature vector is illustrated in the block diagram, as shown in Fig. 6. This algorithm consists of the following steps:
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Digitized ultrasonic liver image
2-level filter bank
Estimate fractal dimension of filtered images
Select feature images
Estimate fractal dimension of subimage of 64*64 in selected feature image
K-means segmentation algorithm Fig. 6. Block diagram for the segmentation scheme.
Step 1: The fractal dimension of input image is first estimated. Step 2: The input image is decomposed into 2 · 2 subimages by two-band filter bank. The fractal dimension of each subimage is calculated. Check the fractal dimension of each subimage by the extraction rule for feature in Section 2.3. Then the selected feature subimages are further decomposed into 3 · 3 subimages by three-band filter bank. The fractal dimension of each subimage is evaluated. Step 3: According to the above estimated fractal dimensions, the corresponding feature subimage is decided. Step 4: Set the square window length equal to 64 and apply to the selected feature image for computing the fractal dimension. Step 5: Apply the K-means clustering algorithm.
4. Experimental segmentation results and discussion In this section, we utilize one natural texture image and six ultrasonic liver images to investigate the performance of the proposed feature extraction methods. All the experiments were programmed in C/C++ language using Borland C++ Builder with a Celeron CPU 1.7 GHz personal computer under MSWindows XP professional environment.
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4.1. Segmentation of natural texture image To demonstrate the performance of the proposed algorithm, we first applied the proposed unsupervised segmentation algorithm to natural texture image. The image of size 384 · 384, 8-bit/pixel, is created by collaged subimages extracted from various natural textures from album images [57], as shown in Fig. 7(a). The segmentation is given in Fig. 7(b). The segmentation accuracy for the mosaic is 97.60%. This result is highly encouraging.
Fig. 7. (a) Texture D17D55, (b) segmentation of texture D17D55 by fractal feature vector, (c) segmentation of texture D17D55 by Gabor filter bank.
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Since Gabor filter approach provides another most promising method for texture feature extraction [55,56,58,59]. Meanwhile, texture descriptor based on Gabor filter bank (using 30 channels) has been approved for the Final Committee Draft of the MPEG-7 standard [60,61]. Hence, we compare the performances of these two different feature extraction methods in this study. Fig. 7(c) shows the segmentation result. The segmentation accuracy is 96.74%. The performance of feature extraction method based on the Gabor filter bank is slightly less than that of the fractal feature vector based on M-channel wavelet transform. That is, the discrimination provided by using the Gabor filter bank is not as good as that by using M-band wavelet system does in this study. The reason may be that the Gabor filter bank isnÕt orthogonal, which results in redundant features at different channels or scales. Furthermore, too many parameters (include orientation h, radial frequency l0, and the space constant of the Gaussian envelope r) are required to set. The chosen parameters will affect the experimental result. Similar results were achieved elsewhere [62– 64]. However, the proposed feature extraction algorithm is free parameter. In addition to accuracy, computational complexity is another essential consideration in a practical application. The execution time of feature extraction method based on the Gabor filter bank is 726 s, whereas the execution time of the fractal feature vector based on M-channel wavelet transform is 717 s. The difference of computational complexity is not obvious since the number of filters for two feature extraction method is both 20. 4.2. Segmentation of ultrasonic liver image From the above experiment, the result is highly heartening. Consequently, we proceed with segmentation of ultrasonic liver image. Four sets of ultrasonic liver images (six images) were taken: cirrhosis (two images), hepatitis B (two images), hepatitis C (one image) and normal (one image), as shown in Fig. 8. All ultrasonic liver images were captured from a phased-array system (Aloka SSD256, Tokyo, Japan) with a 3.5 MHz transducer and were stored as positive ones. These were then scanned by AGFAÕs DUOSCAN scanner with 32-pixel/ cm and 8-bit/pixel resolution. A specialized physician verified all digitized images. All liver diseases were biopsied for pathological diagnosis; therefore there is a basis of truth in this study. The segmented region is bounded from subcutaneous 3 cm to 11 cm. The segmentations for ultrasonic liver images are illustrated in Fig. 9. The information contained in the ultrasound image is a result of the complex interaction between ultrasound waves and tissue components. In ultrasound liver images, various granular structures are illustrated as texture. Generally, texture can be evaluated as being fine, coarse, grained or smooth. Therefore, it should be possible to extract tissue-relevant quantitative parameters from the image. Herein, a fractal feature vector based on M-channel
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Fig. 8. Ultrasonic liver image: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C, (f) normal.
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Fig. 9. Segmented region of ultrasonic liver image for given two-class: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C, (f) normal.
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wavelet transform was defined to determine the most useful information for the characterization of ultrasonic liver images. Additionally, the appropriate feature vector was selected from the Fourier spectrum. Thus, the number of features can be reduced to 12 rather than the original 41. The segmented regions have been verified by a senior physician and prove to be reasonable, as shown in Fig. 9. As well, the comparison for the two feature extraction method is given. In the segmentation of ultrasound liver images of size 376 · 220, the execution time of feature extraction method based on the Gabor filter bank is about 622 s, whereas of the fractal feature vector based on M-channel wavelet transform is 347 s. The difference of computational complexity is mainly caused by the number of filters, the length of filter and critical decimation. Hence, the proposed algorithm is computationally efficient. The results of segmented areas described above meet our expectation. However, is it possible to find further information from suspected pathological change region? For example, the association between hepatocellular carcinoma (HCC) and cirrhosis is strong. And the trabecular pattern is the most common and characteristic form of hepatoma. In this form, the tumor cells are arranged in anastomosing plates that are separated by a sinusoidal network. The structure thus resembles that of a normal liver; however the trabeculate is typically wider and less regular [65]. Hence, the hepatoma is rougher than that of a normal liver. The major characteristic of cirrhosis is the hepatic fibrosis that is associated with beginning nodules or fully established nodules [66]. The nodules cause a block structure in the image such that cirrhosis image is rougher than a normal liver image but less rough than hepatoma image. These two liver diseases have rough images of varying degrees. And thus, is it possible to detect suspicious hepatoma area from cirrhosis regions? Since hepatitis B, hepatitis C or alcoholic is the main cause of cirrhosis. Therefore, it is desired to further segment suspected cirrhosis or HCC areas from suspicious abnormal regions. Total number of classes is two in the previous experiment. The experimental result provides suspicious abnormal regions. Accordingly, the three classes are specified for previous ultrasonic images. The segmented results are illustrated in Fig. 10. The principle cause of the further analysis is that we hope to segment more pathological change tissues from the abnormal regions. Thus, the proposed unsupervised segmentation algorithm plays an important role in the automatic computer aided diagnostic system, since the previous works about the diagnosis of ultrasonic liver image are all based on the supervised classification scheme [22–28]. The test sample is extracted from the region of interest that is given by experienced physician. Then, the sample is classified based on the database of ultrasonic liver images. Apparently, the supervised classification is insufficient for automatic diagnose. Hence, the sufficient ultrasonic liver images of various liver diseases and the proposed unsupervised segmentation algorithm can establish an automatic computer aided diagnostic system. The block diagram of the
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Fig. 10. Segmented region of ultrasonic liver image for given three-class: (a) cirrhosis, (b) cirrhosis, (c) hepatitis B, (d) hepatitis B, (e) hepatitis C.
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Digitized ultrasonic liver image
Feature vector extraction
Unsupervised segmentation
Extracted test sample from suspicious abnormal regions
Supervised classification
Suggestive certain liver disease Fig. 11. The block diagram of automatic computer aided diagnostic system.
system is illustrated in Fig. 11. First, the segmentation of ultrasonic liver image is executed. The suspicious pathological changes areas from the abnormal regions are recognized with the established database. Moreover, an application based on the proposed unsupervised segmentation is to set up an off-line investigating system. Since visual criteria of diagnosing liver diseases principally depend on the clinical experience of physicians, it is extremely subjective. A quantitative characterization scheme of ultrasonic liver images is essential in diagnosis. Thus, a software system can be developed, based on the proposed algorithm. The block diagram of the software system is depicted in Fig. 12. Via this software, a junior physician can enhance visual assessment capability from off-line database.
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Digitized ultrasonic liver image
Feature vector extraction
Number of classes assignment
Clustering
Investigation of segmented regions
Reasonable?
No
Yes End Fig. 12. The block diagram of learning software system.
5. Conclusion We have presented an unsupervised segmentation algorithm based on Mband wavelet transform and fractal geometry. Since the spatial-frequency plane can combine logarithmic and uniform spacing by the M-band wavelet transform, a more flexible decomposition of the entire frequency band can be achieved. The proposed scheme uses M-band wavelet transform to decompose an original texture image into a number of subbands. The subband images provide textural feature in different orientations and scales. Fractal dimension is
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used to furnish measurement of textural image. We applied the proposed unsupervised segmentation algorithm to natural texture image and four sets of ultrasonic liver images. The experimental results reveal that the combination of these two approaches furnishes valuable information to analyze ultrasonic liver images under study. Based on the proposed unsupervised segmentation algorithm, an automatic computer-aided diagnostic system can be established to increase the clinical discrimination of ultrasonic imaging. As well, software of off-line investigating system can be developed to enhance the visual interpretation capability of ultrasonic liver image for junior physicians.
References [1] M.F. Insana, R.F. Wagner, B.S. Garra, D.G. Brown, T.H. Shawker, Analysis of ultrasound image texture via generalized Rician statistics, Opt. Eng. 25 (1986) 743–748. [2] U. Reath, D. Schlaps, B. Limberg, Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease, J. Clin. Ultrasound 13 (1985) 87–99. [3] N.M. Botros, A microprocessor-based pattern recognition algorithm for in-vivo tissue differentiation, J. Clin. Eng. 13 (1988) 115–120. [4] R. Momenan, M.H. Loew, M.F. Insana, R.F. Wagner, B.S. Garra, Application of pattern recognition techniques in ultrasound tissue characterization, in: 10th Int. Conf. Pattern Recognition, vol. 1, 1990, pp. 608–612. [5] M.F. Insana, R.F. Wagner, B.S. Garra, R. Momenan, T.H. Shawker, Pattern recognition methods for optimizing multivariate tissue signatures in diagnostic ultrasound, Ultrasound Imaging 8 (1986) 165–180. [6] B.S. Garra, M.F. Insana, T.H. Shawker, R.F. Wagner, M. Bradford, M. Russell, Quantitative ultrasonic detection and classification of diffuse liver disease comparison with human observer performance, Invest. Radiol. 24 (1989) 196–203. [7] R.F. Wagner, M.F. Insana, G. Brown, Unified approach to the detection and classification of speckle texture in diagnostic ultrasound, Opt. Eng. 25 (1986) 743–748. [8] R. Momenan, M.F. Insana, R.F. Wagner, B.S. Garra, M.H. Loew, Application of clutter analysis and unsupervised learning to multivariate tissue characterization, J. Clin. Eng. 13 (1988) 455–461. [9] R. Momenan, R.F. Wagner, B.S. Garra, M.H. Loew, M.F. Insana, Image staining and differential diagnosis of ultrasound scans based on the Mahalanobis distance, IEEE Trans. Med. Imaging 11 (June) (1994) 37–47. [10] K. Ogawa, M. Fukushima, K. Kubota, N. Hisa, Computer-aided diagnostic system for diffuse liver diseases with ultrasonography by neural network, IEEE Trans. Nucl. Sci. 45 (6) (1998) 3069–3074. [11] G.J. W Simon, E.E. Jane, B. Nigal, E.H. Margaret, E.B. Joe, W. David, An ultrasound scoring system for the diagnosis of liver disease in cystic fibrosis, J. Hepatol. 22 (1995) 513–521. [12] G.F. Vawter, H. Shwachman, Cystic fibrosis in adults: an autopsy study, Pathol. Annu. 14 (1979) 357–382. [13] N.I. Sandford, P. Walsh, C. Matis, H. Baddeley, L.W. Powell, Is ultrasonography useful in the development of diffuse parenchyma liver disease, Gastroenterology 9 (1985) 186–191. [14] R.M. Haralick, K. Shanmugam, I. Dinstein, Texture features for image classification, IEEE Trans. Syst. Man Cybern. 3 (6) (1973) 610–621.
W.-L. Lee et al. / Information Sciences 175 (2005) 177–199
197
[15] G.O. Lendaris, G.L. Stanley, Diffraction pattern sampling for automatic pattern recognition, Proc. IEEE 58 (1970) 198–216. [16] J.S. Weszka, C.R. Dryer, A. Rosenfeld, A comparative study of texture measures for terrain classification, IEEE Trans. Syst. Man Cybern. SMC-6 (1979) 269–285. [17] K.I. Laws, Texture image segmentation, Ph.D. dissertation, Image Processing Inst., Univ. of Southern California, 1980. [18] C.M. Wu, Y.C. Chen, K.S. Hsieh, Texture features for classification of ultrasonic liver images, IEEE Trans. Med. Imaging 11 (June) (1992) 141–152. [19] T. Chang, C. Kuo, Texture analysis and classification with tree-structured wavelet transform, IEEE Trans. Image Process. 2 (October) (1993) 429–441. [20] A. Laine, J. Fan, Texture classification by wavelet packet signatures, IEEE Trans. Pattern Anal. Mach. Intell. 15 (November) (1993) 1186–1191. [21] M. Unser, Texture classification and segmentation using wavelet frames, IEEE Trans. Image Process. 4 (11) (1995) 1549–1560. [22] A. Mojsilovic´, M. Popovic´, S. Markovic´, M. Krstic´, Characterization of visually similar diffuse diseases from B-scan liver images using nonseparable wavelet transform, IEEE Trans. Med. Imaging 17 (4) (1998) 541–549. [23] C.M. Wu, Y.C. Chen, Multi-threshold dimension vector for texture analysis and its application to liver tissue classification, Pattern Recogn. 26 (1) (1993) 137–144. [24] Y.C. Chen, W.L. Lee, Texture classification using multiresolution fractal feature vector, in: Proc. 4th Asian Conf. on Computer Vision, 2000, pp. 204–209. [25] W.L. Lee, Y.C. Chen, K.S. Hsieh, Ultrasonic liver tissues classification by fractal feature vector based on M-band wavelet transform, in: The 2001 IEEE Int. Symp. on Circuits and Systems, vol. 2, 2001, pp. 1–4. [26] H. Sujana, S. Swarnamani, S. Suresh, Application of artificial neural networks for the classification of liver lesions by image texture parameters, Ultrasound Med. Biol. 22 (9) (1996) 1177–1181. [27] S. Pavlopoulos, E. Kyriacou, D. Koutsouris, K. Blekas, A. Stafylopatis, P. Zoumpoulis, Fuzzy neural network-based texture analysis of ultrasonic images, IEEE Eng. Med. Biol. 19 (1) (2000) 39–47. [28] Y.M. Kadah, A.A. Farag, J.M. Zurada, A.M. Badawi, A.M. Youssef, Classification algorithms for quantitative tissue characterization of diffuse liver disease from ultrasound images, IEEE Trans. Med. Imaging 15 (4) (1996) 466–478. [29] B.J. Oosterveld, J.M. Thijssen, P.C. Hartman, G.J.E. Rosenbusch, Detection of diffuse liver disease by quantative echography: dependence on a priori choice of parameters, Ultrasound Med. Biol. 19 (1) (1993) 21–25. [30] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell. 11 (July) (1989) 674–693. [31] O. Rioul, M. Vetterli, Wavelets and signal processing, IEEE Signal Process. Mag. 8 (October) (1991) 11–38. [32] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge, MA, 1996. [33] P.P. Vaidaynathan, Multirate Systems and Filter Banks, Prentice-Hall, New Jersey, 1993. [34] C.S. Burrus, R.A. Goponath, H. Guo, Introduction to Wavelets and Wavelet Transform: a Primerm, Prentice-Hall, New Jersey, 1998. [35] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Information Theory 36 (September) (1990) 961–1005. [36] R.A. Goponath, E. Odegard, C.S. Burrus, Optimal wavelet representation of signals and the wavelet sampling theorem, IEEE Trans. Circ. Syst. II 41 (4) (1994) 262–277. [37] M.K. Tsatsanis, G.B. Giannakis, Principal component filter banks for optimal multiresolution analysis, IEEE Trans. Signal Process. 43 (8) (1995) 1766–1777. [38] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.
198
W.-L. Lee et al. / Information Sciences 175 (2005) 177–199
[39] B.B. Mandelbrot, Fractal Geometry of Nature, Freeman Press, San Francisco, 1982. [40] C.C. Chen, J.S. Daponte, M.D. Fox, Fractal feature analysis and classification in medical imaging, IEEE Trans. Med. Imaging 6 (June) (1989) 133–142. [41] S.C. Liu, S. Chang, Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification, IEEE Trans. Image Process. 6 (8) (1997). [42] T. Lundahl, W.J. Ohely, S.M. Kay, R. Siffert, Fractional brownian motion: a maximum likelihood estimator and its application to image texture, IEEE Trans. Med. Imaging 5 (3) (1986) 152–161. [43] J.T.M. Verhoeven, J.M. Thijssen, Potential of fractal analysis for lesion detection in echographic images, Ultrasonic Imaging 15 (1993) 304–323. [44] N. Sarkar, B.B. Chaudhuri, An efficient differential box-counting approach to compute fractal dimension of image, IEEE Trans. Syst. Man Cybern. 24 (January) (1994) 115–120. [45] S. Buczkowski, S. Kyriacos, F. Nekka, L. Cartilier, The modified box-counting method: analysis of some characteristic parameters, Pattern Recogn. 31 (4) (1998) 411–418. [46] J.M. Keller, S. Chen, R.M. Crowniver, Texture description and segmentation through fractal geometry, CVGIP 45 (1989) 150–166. [47] J. Feng, W.-C. Lin, C.-T. Chen, Fractional box counting approach to fractal dimension estimation, in: Proc. 13th ICPR, II, 1996, pp. 854–858. [48] S. Chen, J.M. Keller, R.M. Crowniver, On the calculation of fractal features from images, IEEE Trans. Pattern Anal. Mach. Intell. 15 (10) (1993) 1087–1090. [49] F.W. Campbell, J.G. Robson, Application of Fourier analysis to the visibility of gratings, J. Physil. 197 (1968) 551–566. [50] R.L. De Valois, D.G. Albrecht, L.G. Thorell, Spatial-frequency selectivity of cells in macaque visual cortex, Vision Res. 22 (1982) 545–559. [51] P. Steffen, P.N. Heller, R.A. Goponath, C.S. Burrus, Theory of regular M-band wavelet bases, IEEE Trans. Signal Process. 41 (12) (1993) 3497–3511. [52] Y. Chitre, A.P. Dhawan, M-band wavelet discrimination of nature textures, Pattern Recogn. 32 (1999) 773–789. [53] R.O. Duda, P.E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, 1973. [54] M. Nadler, E.P. Smith, Pattern Recognition Engineering, Wiley, New York, 1993. [55] A.K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, in: Proc. IEEE Int. Conf. on Systems, Man and Cybernetics, 1990, pp. 14–19. [56] A.K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, Pattern Recogn. 24 (12) (1991) 1167–1186. [57] P. Brodatz, Texture: a Photographic Album for Artists and Designers, Dover, New York, 1966. [58] C.C. Chen, D.C. Chen, Multi-resolution Gabor filter in the texture analysis, Pattern Recogn. Lett. 17 (1996) 1069–1076. [59] O. Pichler, A. Teuner, B.J. Hosticka, A comparison of texture feature extraction using adaptive Gabor filtering, pyramidal and tree structured wavelet transforms, Pattern Recogn. 29 (5) (1996) 733–742. [60] P. Wu, B.S. Manjunath, S. Newsam, H.D. Shin, A texture descriptor for browsing and similarity retrieval, Signal Process.: Image Commun. 16 (2000) 33–43. [61] B.S. Manjunath, J.R. Ohm, V.V. Vasudevan, A. Yamada, Color and texture descriptors, IEEE Trans. Circ. Syst. Video Technol. 11 (11) (2001) 703–715. [62] T. Randen, J.H. Husoy, Multichannel filtering for image texture segmentation, Opt. Eng. 33 (August) (1994) 2617–2625. [63] T. Randen, J.H. Husoy, Filtering for texture classification: a comparative study, IEEE Trans. Pattern Anal. Mach. Intell. 21 (4) (1999) 291–310. [64] M. Acharyya, M.K. Kundu, An adaptive approach to unsupervised texture segmentation using M-band wavelet transform, Signal Process. 81 (2001) 1337–1356.
W.-L. Lee et al. / Information Sciences 175 (2005) 177–199
199
[65] P.J. Scheuer, Pathologic types of hepatic tumors, in: P. Bannasch, D. Keppler, G. Weber (Eds.), Liver Cell Carcinoma, Kluwer-Academic, New York, 1989, p. 18. [66] J.H. Lefkowitch, Pathologic diagnosis of liver disease, in: D. Zakim, T.D. Boyer (Eds.), Hepatology: a Textbook of Liver Disease, W.B. Saunders, London, England, 1990, p. 719.