Phương pháp chẩn đoán hình ảnh (Phần 4)

Phương pháp chẩn đoán hình ảnh (Phần 4)

2089_book.fm Page 87 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images of the Carotid Plaque for the Assessment of Stroke Christodoulos I Christodoulou, Constantinos S Pattichis, Efthyvoulos Kyriacou, Marios S Pattichis, Marios Pantziaris, and Andrew Nicolaides CONTENTS 3.1 3.2 3.3 Introduction 3.1.1 Ultrasound Vascular Imaging 3.1.2 Previous Work on the Characterization of Carotid Plaque Materials The Carotid Plaque Multifeature, Multiclassifier System 3.3.1 Image Acquisition and Standardization 3.3.2 Plaque Identification and Segmentation 3.3.3 Feature Extraction 3.3.3.1 Statistical Features (SF) 3.3.3.2 Spatial Gray-Level-Dependence Matrices (SGLDM) 3.3.3.3 Gray-Level Difference Statistics (GLDS) 3.3.3.4 Neighborhood Gray-Tone-Difference Matrix (NGTDM) 3.3.3.5 Statistical-Feature Matrix (SFM) 3.3.3.6 Laws’s Texture Energy Measures (TEM) 3.3.3.7 Fractal Dimension Texture Analysis (FDTA) 3.3.3.8 Fourier Power Spectrum (FPS) 3.3.3.9 Shape Parameters 3.3.3.10 Morphological Features Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 88 Tuesday, May 10, 2005 3:38 PM 88 Medical Image Analysis 3.3.4 3.3.5 3.3.6 Feature Selection Plaque Classification 3.3.5.1 Classification with the SOM Classifier 3.3.5.2 Classification with the KNN Classifier Classifier Combiner 3.3.6.1 Majority Voting 3.3.6.2 Weighted Averaging Based on a Confidence Measure 3.4 Results 3.4.1 Feature Extraction and Selection 3.4.2 Classification Results of the SOM Classifiers 3.4.3 Classification Results of the KNN Classifiers 3.4.4 Results of the Classifier Combiner 3.4.5 The Proposed System 3.4.5.1 Training of the System 3.4.5.2 Classification of a New Plaque 3.5 Discussion 3.5.1 Feature Extraction and Selection 3.5.2 Plaque Classification 3.5.3 Classifier Combiner 3.6 Conclusions and Future Work Appendix 3.1 Texture-Feature-Extraction Algorithms Acknowledgment References 3.1 INTRODUCTION There is evidence that carotid endarterectomy in patients with asymptomatic carotid stenosis will reduce the incidence of stroke [1] The current practice is to operate on patients based on the degree of internal carotid artery stenosis of 70 to 99% as shown in X-ray angiography [2] However, a large number of patients may be operated on unnecessarily Therefore, it is necessary to identify patients at high risk, who will be considered for carotid endarterectomy, and patients at low risk, who will be spared from an unnecessary, expensive, and often dangerous operation There are indications that the morphology of atherosclerotic carotid plaques, obtained by high-resolution ultrasound imaging, has prognostic implications Smooth surface, echogenicity, and a homogeneous texture are characteristics of stable plaques, whereas irregular surface, echolucency, and a heterogeneous texture are characteristics of potentially unstable plaques [3–6] The objective of the work described in this chapter was to develop a computeraided system based on a neural network and statistical pattern recognition techniques that will facilitate the automated characterization of atherosclerotic carotid plaques, recorded from high-resolution ultrasound images (duplex scanning and color flow imaging), using texture and morphological features extracted from the plaque images The developed system should be able to automatically classify a plaque into (a) symptomatic (because it is associated with ipsilateral hemispheric symptoms) Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 89 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 89 and (b) asymptomatic (because it is not associated with ipsilateral hemispheric events) As shown in this chapter, it is possible to identify a group of patients at risk of stroke based on texture features extracted from high-resolution ultrasound images of carotid plaques The computer-aided classification of carotid plaques will contribute toward a more standardized and accurate methodology for the assessment of carotid plaques This will greatly enhance the significance of noninvasive cerebrovascular tests in the identification of patients at risk of stroke It is anticipated that the system will also contribute toward the advancement of the quality of life and efficiency of health care An introduction to ultrasound vascular imaging is presented in Subsection 3.1.1, followed by a brief survey of previous work on the characterization of carotid plaque In Section 3.2, the materials used to train and evaluate the system are described In Section 3.3, the modules of the multifeature, multiclassifier carotid-plaque classification system are presented Image acquisition and standardization are covered in Subsection 3.3.1, and the plaque identification and segmentation module is described in Subsection 3.3.2 Subsections 3.3.3 and 3.3.4 outline, respectively, the feature extraction and feature selection The plaque-classification module with its associated calculations of confidence measures is presented in Subsection 3.3.5, and the classifier combiner is described in Subsection 3.3.6 In the following Sections 3.4 and 3.5 the results are presented and discussed, and the conclusions are given in Section 3.6 Finally, in the appendix at the end of the chapter, the implementation details are given for the algorithms used to extract texture features 3.1.1 ULTRASOUND VASCULAR IMAGING The use of ultrasound in vascular imaging became very popular because of its ability to visualize body tissue and vessels in a noninvasive and harmless way and to visualize in real time the arterial lumen and wall, something that is not possible with any other imaging technique B-mode ultrasound imaging can be used to visualize arteries repeatedly from the same subject to monitor the development of atherosclerosis Monitoring of the arterial characteristics like the vessel lumen diameter, the intima media thickness (IMT) of the near and far wall, and the morphology of atherosclerotic plaque are very important in assessing the severity of atherosclerosis and evaluating its progression [7] The arterial wall changes that can be easily detected with ultrasound are the end result of all risk factors (exogenous, endogenous, and genetic), known and unknown, and are better predictors of risk than any combination of conventional risk factors Extracranial atherosclerotic disease, known also as atherosclerotic disease of the carotid bifurcation, has two main clinical manifestations: (a) asymptomatic bruits and (b) cerebrovascular syndromes such as amaurosis fugax, transient ischemic attacks (TIA), or stroke, which are often the result of plaque erosion or rupture, with subsequent thrombosis producing occlusion or embolization [8, 9] Carotid plaque is defined as a localized thickening involving the intima and media in the bulb, internal carotid, external carotid, or common femoral arteries (Figure 3.1) Recent studies involving angiography, high-resolution ultrasound, Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 90 Tuesday, May 10, 2005 3:38 PM 90 Medical Image Analysis 10:30:00 RT PROX ICA (a) 7R.R 28.8 cm/s 0.29:23 RT PROX ICA (b) FIGURE 3.1 (Color figure follows p 274.) (a) An ultrasound B-scan image of the carotid artery bifurcation with the atherosclerotic plaque outlined; (b) the corresponding color image of blood flow through the carotid artery, which physicians use to identify the exact plaque region Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 91 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 91 thrombolytic therapy, plaque pathology, coagulation studies, and more recently, molecular biology have implicated atherosclerotic plaque rupture as a key mechanism responsible for the development of cerebrovascular events [10–12] Atherosclerotic plaque rapture is strongly related to the morphology of the plaque [13] The development and continuing technical improvement of noninvasive, high-resolution vascular ultrasound enables the study of the presence of plaques, their rate of progression or regression, and most importantly, their consistency The ultrasonic characteristics of unstable (vulnerable) plaques have been determined [14, 15], and populations or individuals at increased risk for cardiovascular events can now be identified [16] In addition, high-resolution ultrasound facilitates the identification of the different ultrasonic characteristics of unstable carotid plaques associated with amaurosis fugax, TIAs, stroke, and different patterns of computed tomography (CT) brain infarction [14, 15] This information has provided new insight into the pathophysiology of the different clinical manifestations of extracranial atherosclerotic cerebrovascular disease using noninvasive methods Different classifications have been proposed in the literature for the characterization of atherosclerotic plaque morphology, resulting in considerable confusion For example, plaques containing medium- to high-level uniform echoes were classified as homogeneous by Reilly [17] and correspond closely to Johnson’s [18] dense and calcified plaques, to Gray-Weale’s [19] type and 4, and to Widder’s [20] type I and II plaques (i.e., echogenic or hyperechoic) A recent consensus on carotid plaque characterization has suggested that echodensity should reflect the overall brightness of the plaque, with the term “hypoechoic” referring to echolucent plaques [21] The reference structure to which plaque echodensity should be compared with is blood for hypoechoic plaques, the sternomastoid muscle for the isoechoic, and the bone of the adjacent cervical vertebrae for the hyperechoic ones 3.1.2 PREVIOUS WORK ON THE CHARACTERIZATION OF CAROTID PLAQUE There are a number of studies trying to associate the morphological characteristics of the carotid plaques as shown in the ultrasound images with cerebrovascular symptoms A brief survey of these studies is given below Salonen and Salonen [3], in an observational study of atherosclerotic progression, investigated the predictive value of ultrasound imaging They associated ultrasound observations with clinical endpoints, risk factors for common carotid and femoral atherosclerosis, and predictors of progression of common carotid atherosclerosis On the basis of their findings, the assessment of common carotid atherosclerosis using B-mode ultrasound imaging appears to be a feasible, reliable, valid, and cost-effective method Geroulakos et al [2] tested the hypothesis that the ultrasonic characteristics of carotid artery plaques are closely related to symptoms and that the plaque structure may be an important factor in producing stroke, perhaps more than the degree of stenosis In their work, they manually characterized carotid plaques into four ultrasonic types: echolucent, predominantly echolucent, predominantly echogenic, and echogenic An association was found of echolucent plaques with symptoms and cerebral infarctions, which provided further evidence that echolucent plaques are unstable and tend to form embolisms Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 92 Tuesday, May 10, 2005 3:38 PM 92 Medical Image Analysis El-Barghouty et al [4], in a study with 94 plaques, reported an association between carotid plaque echolucency and the incidence of cerebral computed tomography (CT) brain infarctions The gray-scale median (GSM) of the ultrasound plaque image was used for the characterization of plaques as echolucent (GSM ≤ 32) and echogenic (GSM > 32) Iannuzzi et al [22] analyzed 242 stroke and 336 transient ischemic attack (TIA) patients and identified significant relationships between carotid artery ultrasound plaque characteristics and ischemic cerebrovascular events The results suggested that the features more strongly associated with stroke were either the occlusion of the ipsilateral carotid artery or wider lesions and smaller minimum residual lumen diameter The features that were more consistently associated with TIAs included low echogenicity of carotid plaques, thicker plaques, and the presence of longitudinal motion Wilhjelm et al [23], in a study with 52 patients scheduled for endarterectomy, presented a quantitative comparison between subjective classification of the ultrasound images, first- and second-order statistical features, and a histological analysis of the surgically removed plaque Some correlation was found between the three types of information, where the best-performing feature was found to be the contrast Polak et al [5] studied 4886 individuals who were followed up for an average of 3.3 years They found that hypoechoic carotid plaques, as seen on ultrasound images of the carotid arteries, were associated with increased risk of stroke The plaques were manually categorized as hypoechoic, isoechoic, or hyperechoic by independent readers Polak et al also suggested that the subjective grading of the plaque characteristics might be improved by the use of quantitative methods Elatrozy et al [24] examined 96 plaques (25 symptomatic and 71 asymptomatic) with more than 50% internal carotid artery stenosis They reported that plaques with GSM < 40, or with a percentage of echolucent pixels greater than 50%, were good predictors of ipsilateral hemispheric symptoms related to carotid plaques Echolucent pixels were defined as pixels with gray-level values below 40 Furthermore, Tegos et al [25], in a study with 80 plaques, reported a relationship between microemboli detection and carotid plaques having dark morphological characteristics on ultrasound images (echolucent plaques) Plaques were characterized using first-order statistics and the gray-scale median of the ultrasound plaque image AbuRahma et al [6], in a study with 2460 carotid arteries, correlated ultrasonic carotid plaque morphology with the degree of carotid stenosis As reported, the higher the degree of carotid stenosis, the more likely it is to be associated with ultrasonic heterogeneous plaque and cerebrovascular symptoms Heterogeneity of the plaque was more positively correlated with symptoms than with any degree of stenosis These findings suggest that plaque heterogeneity should be considered in selecting patients for carotid endarterectomy Asvestas et al [26], in a pilot study with 19 carotid plaques, indicated a significant difference of the fractal dimension between the symptomatic and asymptomatic groups Moreover, the phase of the cardiac cycle (systole/diastole) during which the fractal dimension was estimated had no systematic effect on the calculations This study suggests that the fractal dimension, estimated by the proposed method, could be used as a single determinant for the discrimination of symptomatic and asymptomatic subjects Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 93 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 93 In most of these studies, the characteristics of the plaques were usually subjectively defined or defined using simple statistical measures, and the association with symptoms was established through simple statistical analysis In the work we are about to describe in this chapter, a large number of texture and morphological features were extracted from the plaque ultrasound image and were analyzed using multifeature, multiclassifier methodology 3.2 MATERIALS A database of digital ultrasound images of carotid arteries was created such that for each gray-tone image, there was also a color image indicating the blood flow The color images were necessary for the correct identification of the plaques as well as their outlines The carotid plaques were labeled as symptomatic after one of the following three symptoms was identified: stroke, transient ischemic attack, or amaurosis fugax Two independent studies were conducted In the first study with Data Set 1, a total of 230 cases (115 symptomatic and 115 asymptomatic) were selected Two sets of data were formed at random: one for training the system and another for evaluating its performance For training the system, 80 symptomatic and 80 asymptomatic plaques were used, whereas for evaluation of the system, the remaining 35 symptomatic and 35 asymptomatic plaques were used A bootstrapping procedure was used to verify the correctness of the classification results The system was trained and evaluated using five different bootstrap sets, with each training set consisting of 160 randomly selected plaques and the remaining 70 plaques used for evaluation In the second study, where the morphology features were investigated, a new Data Set of 330 carotid plaque ultrasound images (194 asymptomatic and 136 symptomatic) were analyzed For training the system, 90 asymptomatic and 90 symptomatic plaques were used; for evaluation of the system, the remaining 104 asymptomatic and 46 symptomatic plaques were used 3.3 THE CAROTID PLAQUE MULTIFEATURE, MULTICLASSIFIER SYSTEM The carotid plaque classification system was developed following a multifeature, multiclassifier pattern-recognition approach The modules of the system are described in the following subsections and are illustrated in Figure 3.2 In the first module, the carotid plaque ultrasound image was acquired using duplex scanning, and the gray level of the image was manually standardized using blood and adventitia as reference In the second module, the plaque region was identified and manually outlined by the expert physician In the feature-extraction module, ten different texture and shape feature sets (a total of 61 features) were extracted from the segmented plaque images of Data Set using the following algorithms: statistical features (SF), spatial gray-level-dependence matrices (SGLDM), gray-level difference statistics (GLDS), neighborhood gray-tone-difference matrix (NGTDM), statistical-feature matrix (SFM), Laws’s texture energy measures (TEM), fractal dimension texture analysis (FDTA), Fourier power spectrum (FPS), and shape parameters Copyright 2005 by Taylor & Francis Group, LLC Patient Image Acquisition and Standardization Carotid plaque ultrasound image Plaque Identification & Segmentation Carotid plaque segmented image Feature Extraction and Selection Feature set Classifier Feature set Classifier Feature set n • Identification and outline of the plaque region manually by the human expert Classifier n • Modular multiclassifier system using the neural SOM classifier • Modular multiclassifier system using the statistical KNN classifier Weighting Factors • Combining using majority voting • Combining by averaging the confidence measures FIGURE 3.2 Flowchart of the carotid plaque multifeature, multiclassifier classification system (From Christodoulou, C.I et al., IEEE Trans Medical Imaging, 22, 902–912, 2003 With permission.) Medical Image Analysis Texture features using: • Statistical Features • Spatial Gray Level Dependence Matrices • Gray Level Diff Statistics • Neighborhood Gray Tone Difference Matrix • Statistical Feature Matrix • Laws Text Energy Meas • Fractal Dimension Texture • Fourier Power Spectrum • Shape Parameters Diagnosis: • Symptomatic • Asymptomatic • • • • • • • Carotid plaque ultrasound image acquisition using duplex scanning • Image standardization Classifier Combiner 2089_book.fm Page 94 Tuesday, May 10, 2005 3:38 PM 94 Copyright 2005 by Taylor & Francis Group, LLC THE CAROTID PLAQUE MULTIFEATURE MULTICLASSIFIER SYSTEM 2089_book.fm Page 95 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 95 Following the feature extraction, several feature-selection techniques were used to select the features with the greatest discriminatory power For the classification, a modular neural network using the unsupervised self-organizing feature map (SOM) classifier was implemented The plaques were classified into two types: symptomatic or asymptomatic For each feature set, an SOM classifier was trained, and ten different classification results were obtained Finally, in the system combiner, the ten classification results were combined using: (a) majority voting and (b) weighted averaging of the ten classification results based on a confidence measure derived from the SOM For the sake of comparison, the above-described modular system was also implemented using the KNN statistical classifier instead of the SOM 3.3.1 IMAGE ACQUISITION AND STANDARDIZATION The protocols suggested by the ACSRS (asymptomatic carotid stenosis at risk of stroke) project [1] were followed for the acquisition and quantification of the imaging data The ultrasound images were collected at the Irvine Laboratory for Cardiovascular Investigation and Research, Saint Mary’s Hospital, U.K., by two ultrasonographers using an ATL (model HDI 3000, Advanced Technology Laboratories, Leichworth, U.K.) duplex scanner with a 4- to 7-MHz multifrequency probe Longitudinal scans were performed using duplex scanning and color flow imaging [27] B-mode scan settings were adjusted so that the maximum dynamic range was used with a linear postprocessing curve The position of the probe was adjusted so that the ultrasonic beam was vertical to the artery wall The time gain compensation (TGC) curve was adjusted (gently sloping) to produce uniform intensity of echoes on the screen, but it was vertical in the lumen of the artery, where attenuation in blood was minimal, so that echogenicity of the far wall was the same as that of the near wall The overall gain was set so that the appearance of the plaque was assessed to be optimal and noise appeared within the lumen It was then decreased so that at least some areas in the lumen appeared to be free of noise (black) The resolution of the images was on the order of 700 × 500 pixels, and the average size and standard deviation of the segmented images was on the order of 350 ± 100 × 100 ± 30 pixels The scale of the gray level of the images was in the range from to 255 The images were standardized manually by adjusting the image so that the median graylevel value of the blood was between 15 and 20 and the median gray-level value of the adventitia (artery wall) was between 180 and 200 [27] The image was linearly adjusted between the two reference points, blood and adventitia This standardization using blood and adventitia as reference points was necessary to extract comparable results when processing images obtained by different operators and equipment and vascular imaging laboratories 3.3.2 PLAQUE IDENTIFICATION AND SEGMENTATION The plaque identification and segmentation tasks are quite difficult and were carried out manually by the expert physician The main difficulties are due to the fact that the plaque cannot be distinguished from the adventitia based on brightness level difference, or using only texture features, or other measures Also, calcification and Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 96 Tuesday, May 10, 2005 3:38 PM 96 Medical Image Analysis Symptomatic Plaques Median = 19.60, Entropy = 5.51, Coars = 8.56 Median = 1.43, Entropy = 3.65, Coarseness = 4.96 Median = 6.05, Entropy = 4.35, Coars = 5.55 Median = 5.32, Entropy = 4.10, Coarseness = 5.45 Asymptomatic Plaques Median = 40.13, Entropy = 6.86, Coars = 27.16 Median = 36.45, Entropy = 7.17, Coarseness = 59.83 Median = 58.92, Entropy = 7.93, Coars = 20.76 Median = 50.79, Entropy = 7.65, Coarseness = 44.30 FIGURE 3.3 Examples of segmented symptomatic and asymptomatic plaques Selected texture values are given for the following features: median (2), entropy (14), and coarseness (36) (The numbers in parentheses denote the serial feature number as listed in Table 3.1.) acoustic shadows make the problem more complex The identification and outlining of the plaque were facilitated using a color image indicating the blood flow (see Figure 3.1) All plaque images used in this study were outlined using their corresponding color blood flow images This guaranteed that the plaque was correctly outlined, which was essential for extracting texture features characterizing the plaque correctly The procedure for carrying out the segmentation process was established by a team of experts and was documented in the ACSRS project protocol [1] The correctness of the work carried out by the single expert was monitored and verified by at least one other expert However, the extracted texture features depend on the whole of the plaque area and are not significantly affected if a small portion of the plaque area is not included in the region of interest Figure 3.1 illustrates an ultrasound image with the outline of the carotid plaque and the corresponding color blood flow image Figure 3.3 illustrates a number of examples of symptomatic and asymptomatic plaques that were segmented by an expert physician Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 121 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images A3.1.2.2 121 Texture Measures A.3.1.2.2.1 Angular Second Moment The angular second moment is a measure for homogeneity of the image { ( )} f1 = ∑ ∑ p i, j i j (3.13) A.3.1.2.2.2 Contrast The contrast is a measure of the amount of local variations present in the image f2 = N g −1  Ng  Ng ∑ n  ∑ ∑ p ( i, j ) i =1  i =1 j =1 i= j = n (3.14)  A.3.1.2.2.3 Correlation Correlation is a measure of gray-tone linear dependencies ∑ ∑ ( i, j ) p ( i, j ) − µ µ x f3 = i y j (3.15) σ xσ y where µx, µy, and σx, σy, are the mean and standard deviation values of px and py A.3.1.2.2.4 Sum of Squares: Variance f4 = ∑ ∑ ( i − µ ) p ( i, j ) i A.3.1.2.2.5 Inverse Difference Moment f5 = ∑ ∑ + (i1− j ) p (i, j ) i A.3.1.2.2.6 (3.16) j (3.17) j Sum Average Ng f6 = ∑ ip (i ) x+ y i=2 Copyright 2005 by Taylor & Francis Group, LLC (3.18) 2089_book.fm Page 122 Tuesday, May 10, 2005 3:38 PM 122 A.3.1.2.2.7 Medical Image Analysis Sum Variance Ng f7 = ∑ (i − f ) () px + y i i=2 A.3.1.2.2.8 (3.19) Sum Entropy Ng ∑ p (i ) log { p (i )} (3.20) ∑ ∑ p (i, j ) log ( p (i, j )) (3.21) f8 = x+ y x+ y i=2 A.3.1.2.2.9 Entropy f9 = i A.3.1.2.2.10 j Difference Variance f10 = variance of px−y A.3.1.2.2.11 (3.22) Difference Entropy N g −1 f11 = ∑ p (i ) log { p (i )} x− y x− y (3.23) i=0 A.3.1.2.2.12/13 Information Measures of Correlation f12 = HXY − HXY max HX , HY { } f13 = (1 − exp[−2.0(HXY2 − HXY)])1/2 HXY = − ∑ ∑ p (i, j ) log ( p (i, j )) i j where HX and HY are entropies of px and py , and Copyright 2005 by Taylor & Francis Group, LLC (3.24) (3.25) (3.26) 2089_book.fm Page 123 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images ∑ ∑ p (i, j ) log { p (i ) p ( j )} (3.27) ∑ ∑ p (i )p ( j ) log { p (i ) p ( j )} (3.28) HXY = − x i HXY = − y j x i A3.1.2.3 123 y x y j Extracted SGLDM Features For a chosen distance d (in this work d = was used), we have four angular graylevel-dependence matrices, i.e., we obtain four values for each of the above 13 texture measures The mean and the range of the four values for each of the 13 texture measures compose a set of 26 texture features that can be used for classification Some of the 26 features are strongly correlated with each other, and a featureselection procedure can be applied to select a subset or linear combinations of them In this work, the mean values and the range of values were computed for each feature for d = 1, and they were used as two different feature sets A3.1.3 GRAY-LEVEL-DIFFERENCE STATISTICS (GLDS) The gray-level-difference-statistics algorithm [31] uses first-order statistics of local property values based on absolute differences between pairs of gray levels or of average gray levels to extract texture measures Let I(x,y) be the image intensity function, and for any given displacement δ ≡ (∆x,∆y), let Iδ(x,y) = |I(x,y) − I(x+∆x, y+∆y)| Let pδ be the probability density of Iδ(x,y) If there are m gray levels, this has the form of an m-dimensional vector whose ith component is the probability that Iδ(x,y) will have value i The probability density pδ can be easily computed by counting the number of times each value of Iδ(x,y) occurs, where ∆x and ∆y are integers In a coarse texture, if δ is small, Iδ(x,y) will be small, i.e., the values of pδ should be concentrated near i = Conversely, in a fine texture, the values of pδ should be more spread out Thus, a good way to analyze texture coarseness would be to compute, for various magnitudes of δ, some measure of the spread of values in pδ away from the origin Four such measures are as follows A3.1.3.1 Contrast Contrast is the second moment of pδ, i.e., its moment of inertia about the origin CON = ∑ i p (i ) i A3.1.3.2 Angular Second Moment ASM is defined as Copyright 2005 by Taylor & Francis Group, LLC δ (3.29) 2089_book.fm Page 124 Tuesday, May 10, 2005 3:38 PM 124 Medical Image Analysis ASM = ∑ p (i ) (3.30) δ i ASM is small when the pδ(i) values are very close and large when some values are high and others low A3.1.3.3 Entropy Entropy is defined as ENT = − ∑ p (i ) log ( p (i )) δ (3.31) δ i This is largest for equal pδ(i) values and small when they are very unequal A3.1.3.4 Mean Mean is defined as ( ) ∑ i p (i ) MEAN = m (3.32) δ i This is small when the pδ(i) are concentrated near the origin and large when they are far from the origin The above features were calculated for δ = (0, 1), (1, 1), (1, 0), (1, −1), and their mean values were taken A3.1.4 NEIGHBORHOOD GRAY-TONE-DIFFERENCE MATRIX (NGTDM) Amadasun and King [28] proposed the neighborhood gray-tone-difference matrix to extract textural features that correspond to visual properties of texture Let f(k,l) be the gray tone of a pixel at (k,l) having gray-tone value i Then we can find the average gray tone over a neighborhood centered at, but excluding, (k,l) ( ) Ai = A k , l = d d   f k + m, l + n W −  m=− d n=− d  ∑∑ (  ) (3.33)  where (m,n) ≠ (0,0), d specifies the neighborhood size, and W = (2d + 1)2 The neighborhood size d = was used in this work Then the ith entry in the NGTDM is () s i = ∑i− A i = if Ni = Copyright 2005 by Taylor & Francis Group, LLC for i ∈ Ni if Ni ≠ (3.34) 2089_book.fm Page 125 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 125 where {Ni} is the set of all pixels having gray tone i The textural features are defined in the following subsections A3.1.4.1 Coarseness Coarseness is defined as fcos  = ε +   Gb ∑ i=0  pi s(i)    −1 (3.35) where Gh is the highest gray-tone value present in the image, and ε is a small number to prevent fcos from becoming infinite For an N × N image, pi is the probability of occurrence of gray-tone value i, and is given by pi = Ni/n2 (3.36) where n = N − 2d Amadasun and King [28] define an image as coarse when the primitives composing the texture are large and texture tends to possess a high degree of local uniformity in intensity for fairly large areas Large values of fcos represent areas where gray-tone differences are small A3.1.4.2 Contrast Contrast is defined as fcon  =  Ng Ng −  Gh Gh i=0 j=0  pi p j i − j   n  ) ∑∑ ( ) ( Gh  i=0  ∑ s (i ) (3.37) where Ng is the total number of different gray levels present in the image High contrast means that the intensity difference between neighboring regions is large A3.1.4.3 Busyness Busyness is defined as Gh fbus = Gh Gh i=0 j=0 ∑ p s (i ) ∑ ∑ ip − jp , i i=0 i j pi ≠ 0, p j ≠ (3.38) A busy texture is one in which there are rapid changes of intensity from one pixel to its neighbor Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 126 Tuesday, May 10, 2005 3:38 PM 126 Medical Image Analysis A3.1.4.4 Complexity Complexity is defined as ∑ ∑ {( i − j ) ( n ( p + p ))}{ p s (i ) + p s ( j )} , p ≠ 0, p ≠ Gh fcom = Gh i i=0 j i j i j (3.39) j=0 A texture is considered complex when the information content is high, i.e., when there are many primitives in the texture, and more so when the primitives have different average intensities A3.1.4.5 Strength Strength is defined as  Gh fstr =   i=0  Gh ∑ ∑( j=0  pi + p j i − j   ε +   )( ) Gh  ∑ s (i ) , p ≠ 0, p ≠ i i=0 j (3.40)  A texture is generally referred to as strong when the primitives that compose it are easily definable and clearly visible A3.1.5 STATISTICAL-FEATURE MATRIX (SFM) The statistical-feature matrix [32] measures the statistical properties of pixel pairs at several distances within an image that are used for statistical analysis Let I(x,y) be the intensity at point (x,y), and let δ = (∆x,∆y) represent the intersample-spacing distance vector, where ∆x and ∆y are integers The δ contrast, δ covariance, and δ dissimilarity are defined as CON(δ) ≡ E{[I(x,y) − I(x + ∆x, y + ∆y)]2} (3.41) COV(δ) ≡ E{[I(x,y) − η] [I(x + ∆x, y + ∆y) − η]} (3.42) DSS(δ) ≡ E{[I(x,y) − I(x + ∆x, y + ∆y)]} (3.43) where E{ } denotes the expectation operation, and η is the average gray level of the image A statistical-feature matrix (SFM), Msf, is an (Lr + 1) × (2Lc + 1) matrix whose (i,j) element is the d statistical feature of the image, where d = (j − Lc, i) is an intersample spacing distance vector for i = 0,1,…,Lr, j = 0,1,…,Lc, and where Lr, Lc are the constants that determine the maximum intersample spacing distance In a similar way, the contrast matrix (Mcon), covariance matrix (Mcov), and dissimilarity matrix (Mdss) can be defined as the matrices whose (i,j) elements are the d contrast, Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 127 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 127 d covariance, and d dissimilarity, respectively Based on the SFM, the following texture features can be computed: coarseness, contrast, periodicity, and roughness A3.1.5.1 Coarseness Coarseness is defined as FCRC = c ∑ DSS (i, j ) n (3.44) (i , j )∈Nr where c is a normalizing factor, Nr is the set of displacement vectors defined as Nr = {(i,j) : |i|, |j| ≤r}, and n is the number of elements in the set A pattern is coarser than another when the two differ only in scale, with the magnified one being the coarser and having a larger FCRS value The definition of coarseness given here is different from the definition given for NGTDM in Equation 3.35 A3.1.5.2 Contrast Contrast is defined as 12 FCON   = CON i, j     (i , j )∈Nr  ∑ ( ) (3.45) Contrast measures the degree of sharpness of the edges in an image A3.1.5.3 Periodicity Periodicity is defined as FPER = ( M dss − M dss valley M dss ) (3.46) where M dss is the mean of all elements in Mdss, and Mdss(valley) is the deepest valley in the matrix Periodicity measures the appearance of periodically repeated patterns in the image A3.1.5.4 Roughness Roughness is defined as ( h v FRGH = D(f ) + D(f ) Copyright 2005 by Taylor & Francis Group, LLC )2 (3.47) 2089_book.fm Page 128 Tuesday, May 10, 2005 3:38 PM 128 Medical Image Analysis where Df is the fractal dimension in horizontal and vertical directions Df = − H, and E{|∆I|} = k(δ)H, where H can be estimated from the dissimilarity matrix because the (i, j + Lc) element of the matrix is E{|∆I|}, with δ = (j,i) The larger the Df, the rougher is the image In this study, an intersample spacing distance vector δ = (4,4) was used A3.1.6 LAWS’S TEXTURE ENERGY MEASURES (TEM) Laws’s texture energy measures [33, 34], are derived from three simple vectors of length 3: L3 = (1, 2, 1), E3 = (−1, 0, 1), and S3 = (−1, 2, −1) These three vectors represent, respectively, the one-dimensional operations of center-weighted local averaging, symmetric first differencing for edge detection, and second differencing for spot detection If these vectors are convolved with themselves, we obtain new vectors of length 5: L5 = (1, 4, 6, 4, 1), E5 = (−1, −2, 0, 2, 1), and S5 = (−1, 0, 2, 0, −1) By further self-convolution, we obtain new vectors of length 7: L7 = (1, 6, 15, 20, 15, 6, 1), E7 = (−1, −4, −5, 0, 5, 4, 1), and S7 = (−1, −2, 1, 4, 1, −2, −1), where L7 again performs local averaging, E7 acts as edge detector, and S7 acts as spot detector If we multiply the column vectors of length l by row vectors of the same length, we obtain Laws’s l × l masks In this work, the following combinations were used to obtain × masks: LL = L7t L7 LE = L7t E7 LS = L7t S7 EL = E7t L7 EE = E7t E7 ES = E7t S7 SL = S7t L7 SE = S7t E7 SS = S7t S7 In order to extract texture features from an image, these masks are convoluted with the image, and statistics (e.g., energy) of the resulting image are used to describe texture The following texture features were extracted: LL: EE: SS: LE: texture energy from LL kernel texture energy from EE kernel texture energy from SS kernel average texture energy from LE and EL kernels LE = (LE + EL)/2 ES: average texture energy from ES and SE kernels ES = (ES + SE)/2 LS: average texture energy from LS and SL kernels LS = (LS + SL)/2 The averaging of matched pairs of energy measures gives rotational invariance Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 129 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images A3.1.7 129 FRACTAL DIMENSION TEXTURE ANALYSIS (FDTA) Mandelbrot [35] developed the fractional Brownian motion model to describe the roughness of natural surfaces It considers naturally occurring surfaces as the end result of random walks Such random walks are basic physical processes in our universe [34] An important parameter to represent a fractal dimension is the fractal dimension Df, estimated theoretically by Equation 3.48 [34] E ( ∆I ) = c( ∆r ) ( 6− D f ) (3.48) where E( ) denotes the expectation operator, ∆I is the intensity difference between two pixels, c is a constant, and ∆r is the distance between two pixels A simpler method is to estimate the H parameter (Hurst coefficient) from Equation 3.49 E(|∆I|) = k(∆r)H (3.49) where k = E(|∆I|) ∆r=1 By applying the log function we obtain logE(|∆I|) = logk + H log(∆r) (3.50) From Equation 3.50, the H parameter can be estimated, and the fractal dimension Df can be computed from the relationship Df = − H (3.51) A smooth surface is described by a small value of the fractal dimension Df (large value of the parameter H), and the reverse applies for a rough surface Given an M × M image, the intensity difference vector is defined as IDV ≡ [id(1), id(2), … id(s)] (3.52) where s is the maximum possible scale, and id(k) is the average of the absolute intensity difference of all pixel pairs with vertical or horizontal distance k The value of the parameter H can be obtained by using least squares linear regression to estimate the slope of the curve of id(k) vs k in log–log scales If the image is seen under different resolutions, then the multiresolution fractal (MF) feature vector is defined as MF ≡ (H(m), H(m-1), …, H(m-n+1)) (3.53) where M = 2m is the size of the original image, H(k) is the H parameter estimated from image I(k), and n is the number of resolutions chosen The multiresolution fractal (MF) feature vector describes also the lacunarity of the image It can be used for the separation of textures with the same fractal dimension Df by considering all Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 130 Tuesday, May 10, 2005 3:38 PM 130 Medical Image Analysis but the first components of the MF vectors In this work, H was computed for four different resolutions A3.1.8 FOURIER POWER SPECTRUM (FPS) The discrete Fourier transform [31, 34] of an N × N picture is defined by F (u, v) = N n −1 ∑ f (i, j)e −2 π −1 ( iu + jv ) (3.54) i , j=0 where ≤ u and v ≤ N − The sample Fourier power spectrum is defined by Φ(u,v) ≡ F(u,v) F*(u,v) = |F(u,v)|2 (3.55) where Φ is the sample power spectrum, and * denotes the complex conjugate Coarse texture will have high values of |F|2 concentrated near the origin, whereas in fine texture, the values will be more spread out The standard set of texture features used are ring- and wedge-shaped samples of the discrete FPS A3.1.8.1 Radial Sum Radial sum is defined as Φr1 ,r2 ≡ ∑ F (u, v) (3.56) r12 ≤u + v r22 for various values of the inner and outer radii r1 and r2 A3.1.8.2 Angular Sum Angular sum is defined as Φθ1 ,θ2 ≡ ∑ F(u, v) θ1 ≤ tan −1 ( v / u )≤θ2 for various angles θ1 and θ2 A3.1.9 SHAPE PARAMETERS The following shape parameters were derived: Copyright 2005 by Taylor & Francis Group, LLC (3.57) 2089_book.fm Page 131 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 131 X-coord max length: the length of the X-coordinate of the rectangular window where the plaque segment is enclosed Y-coord max length: the length of the Y-coordinate of the rectangular window where the plaque segment is enclosed Area: the number of pixels of the plaque segment Perimeter: the number of pixels that define the outline of the plaque segment Perimeter2/Area: parameter calculated to characterize areas with irregular outline A3.1.10 MORPHOLOGICAL FEATURES Morphological image processing makes it possible to detect the presence of specified patterns at different scales We consider the detection of isotropic features that show no preference to particular directions The simplest structural element for nearisotropic detection is the cross ‘+’ consisting of five image pixels Thus, we considered pattern spectra based on a flat '+' structural element B Formally, the pattern spectrum is defined in terms of the discrete-size transform (DST) We define the DST using Equation 3.58 [36–38] f → (…, d − k ( f ; B),; d −1 ( f ; B), d ( f ; B),…, d1 ( f ; B),…, d k ( f ; B),…) (3.58)  fokB − fo ( k + 1) B, k ≥  d k ( f ; B) =   f • k B − f • ( k − 1) B, k ≤ (3.59) where where o denotes an open operation, and • denotes the close operation The gray-scale DST is a multiresolution image-decomposition scheme that decomposes an image f into residual images fokB − fo(k + 1) B, for k > 0, and f • |k | B − f • (|k | +1) B for k < The pattern spectrum of a gray-scale image f, in terms of a structuring element B, is given by  fokB − fo ( k + 1) B , k ≥  Pf ;B ( k ) = d k ( f ; B) =   f • k B − f • ( k − 1) B , k ≤  (3.60) where f = ∑ f (x, y), x ,y Copyright 2005 by Taylor & Francis Group, LLC f ( x , y) ≥ (3.61) 2089_book.fm Page 132 Tuesday, May 10, 2005 3:38 PM 132 Medical Image Analysis We note that in the limit, as k → ∞, we have that the resulting image ƒ•kB − ƒ•(k + 1)B converges to the zero image Also, we note that with increasing values of k, ƒ•kB is a subset of the original image For k ≥ 0, we can thus normalize the pattern spectrum by dividing by the norm of the original image ||ƒ|| Similarly, as k → ∞, ||ƒ•kB|| converges to NM max ƒ(x, y), where it is assumed that the image is of size N by M, and that images are extended using a constant extension by replicating boundary values Hence, for k < 0, we can normalize the pattern spectrum by dividing by NM max ƒ(x, y) − ||ƒ|| Thus, to eliminate undesired variations, all the pattern spectra were normalized ACKNOWLEDGMENT The material for this study was collected in the context of a European Union project (Biomed Program, PL 950629) carried out in centers all over Europe and coordinated by the St Mary’s Hospital, London, U.K The aim of the project was to evaluate the value of noninvasive investigations in the identification of individuals with asymptomatic carotid stenosis at risk of stroke (ACSRS) The texture and morphological analysis carried out in this work was partly funded through the project Integrated System for the Support of the Diagnosis for the Risk of Stroke (IASIS) (supported by the 5th Annual Program for the Financing of Research and by the Research Promotion Foundation of Cyprus) as well as through the project Integrated System for the Evaluation of Ultrasound Imaging of the Carotid Artery (TALOS) (supported by the Program for Research and Technological Development 2003–2005 and by the Research Promotion Foundation of Cyprus) Partial funding was also obtained from the Research Committee of the University of Cyprus for the research activity support of C S Pattichis for the years 2003 and 2004 REFERENCES Nicolaides, A., Asymptomatic carotid stenosis and the risk of stroke (the ACSRS study): identification of a high-risk group, chap 38 in Cerebrovascular Ischaemia Investigation and Management, Med-Orion Publishing, London, 1996, pp 435–441 Geroulakos, G., Domjan, J., Nicolaides, A., Stevens, J., Labropoulos, N., Ramaswami, G., Belcaro, G., and Belcaro, G., Ultrasonic carotid artery plaque structure and the risk of cerebral infarction on computed tomography, J Vasc Surg., 20, 263–266, 1994 Salonen, J.T and Salonen, R., Ultrasound B-mode imaging in observational studies of atherosclerotic progression, Suppl II Circ., 87, 56–65, 1993 El-Barghouty, N., Geroulakos, G., Nicolaides, A., Androulakis, A., 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images of carotid atherosclerotic plaque: correlation with visual classification and histological examination, IEEE Trans Medical Imaging, 17, 910–922, 1998 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 134 Tuesday, May 10, 2005 3:38 PM 134 Medical Image Analysis 24 Elatrozy, T., Nicolaides, A., Tegos, T., and Griffin, M., The objective characterisation of ultrasonic carotid plaque features, Eur J Vasc Endovasc Surg., 16, 223–230, 1998 25 Tegos, T., Kalodiki, E., Nicolaides, A., Sabetai, M., Dhanjil, S., El-Atrozy, T., Ramaswami, G., Daskalopoulos, M., Robless, P., Pare, G., Byrd, S., and Kalomiris, K., Correlation of microemboli detected in the middle cerebral artery on transcranial Doppler with the echomorphology of the carotid atherosclerotic plaque, in Proc VIII Mediterranean Conference on Medical and Biological Engineering and Computing, MEDICON ‘98, Lemesos, Cyprus, 1998 26 Asvestas, P., Golemati, S., Matsopoulos, G., Nikita, K., and Nicolaides, A., Fractal 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terrain classification, IEEE Transactions on Systems, Man., and Cybernetics, 6, 269–285, 1976 32 Wu, C.-M and Chen, Y.-C., Statistical feature matrix for texture analysis, CVGIP: Graphical Models Image Process., 54, 407–419, 1992 33 Laws, K.I., Rapid texture identification, SPIE, 238, 376–380, 1980 34 Wu, C.-M., Chen, Y.-C., and Hsieh, K.-S., Texture features for classification of ultrasonic liver images, IEEE Trans Medical Imaging, 11, 141–152, 1992 35 Mandelbrot, B.B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982 36 Dougherty, E.R., An Introduction to Morphological Image Processing, SPIE Optical Engineering Press, Bellingham, WA, 1992 37 Dougherty, E.R and Astola, J., An Introduction to Nonlinear Image Processing, SPIE Optical Engineering Press, Bellingham, WA, 1994 38 Maragos, P., Pattern spectrum and multiscale shape representation, IEEE Trans Pattern Anal Machine Intelligence, 11, 701–715, 1989 39 Christodoulou, C.I., Pattichis, C.S., Pantziaris, M., and Nicolaides, A., Texture-based classification of atherosclerotic carotid plaques, IEEE Trans Medical Imaging, 22, 902–912, 2003 40 Haykin, S., Neural Networks: a Comprehensive Foundation, Macmillan College Publishing, New York, 1994 41 Kohonen, T., The self-organizing map, Proc IEEE, 78, 1464–1480, 1990 42 Kittler, J., Hatef, M., Duin, R., and Matas, J., On combining classifiers, IEEE Trans Pattern Anal Machine Intelligence, 20, 226–239, 1998 43 Perrone, M.P., Averaging/modular techniques for neural networks, in The Handbook of Brain Theory and Neural Networks, Arbib, M.A., Ed., MIT Press, Cambridge, MA, 1995, pp 126–129 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 135 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 135 44 Christodoulou, C.I., Pattichis, C.S., Pantziaris, M., Tegos, T., Nicolaides, A., Elatrozy, T., Sabetai, M., and Dhanjil, S., Multifeature texture analysis for the classification of carotid plaques, in Proc Int Joint Conf Neural Networks, IJCNN '99, Washington, DC, 1999 45 Christodoulou, C.I., Kyriacou, E., Pattichis, M.S., Pattichis, C.S., and Nicolaides, A., A comparative study of morphological and other texture features for the characterization of atherosclerotic carotid plaques, Computer Analysis of Images and Patterns, 10th International Conference CAIP’2003, Springer-Verlag, Groningen, Netherlands, 2003, pp 503–511 46 Kovalev, V and Petrou, M., Texture analysis in three dimensions as a cue to medical diagnosis, in Handbook of Medical Imaging: Processing and Analysis, Bankman, I.N., Ed., Academic Press, New York, 2000, pp 231–247 47 Pattichis, C.S., Kyriacou, E., Christodoulou, C.I., Pattichis, M.S., Loizou, C., Pantziaris, M., and Nicolaides, A., Cardiovascular: ultrasonic imaging in vascular cases, in Wiley Encyclopaedia of Biomedical Engineering, Wiley, New York, will be published in 2005 Copyright 2005 by Taylor & Francis Group, LLC [...]... contrast, (3) busyness, (4) complexity, and (5) strength 3.3.3.5 Statistical-Feature Matrix (SFM) The statistical-feature matrix [32] measures the statistical properties of pixel pairs at several distances within an image, which are used for statistical analysis Based on the SFM, the following texture features were computed: (1) coarseness, (2) contrast, (3) periodicity, and (4) roughness The constants... have intensities of gray-level i and gray-level j Based on the probability density functions, the following texture measures [30] were computed: (1) angular second moment, (2) contrast, (3) correlation, (4) sum of squares: variance, (5) inverse difference moment, (6) sum average, (7) sum variance, (8) sum entropy, (9) entropy, (10) difference variance, (11) difference entropy, and (12, 13) information... property values based on absolute differences between pairs of gray levels or of average gray levels to extract the following texture measures: (1) contrast, (2) angular second moment, (3) entropy, and (4) mean These features were calculated for displacements δ = (0, 1), (1, 1), (1, 0), (1, −1), where δ ≡ (∆x, ∆y), and their mean values were taken Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm... can be found in Appendix 3.1 at the end of the chapter 3.3.3.1 Statistical Features (SF) The following statistical features were computed [29]: (1) mean value, (2) median value, (3) standard deviation, (4) skewness, and (5) kurtosis 3.3.3.2 Spatial Gray-Level-Dependence Matrices (SGLDM) The spatial gray-level-dependence matrices as proposed by Haralick et al [30] are based on the estimation of the second-order... image are used to describe texture The following texture features were extracted: (1) LL, texture energy from LL kernel, (2) EE, texture energy from EE kernel, (3) SS, texture energy from SS kernel, (4) LE, average texture energy from LE and EL kernels, (5) ES, average texture energy from ES and SE kernels, and (6) LS, average texture energy from LS and SL kernels 3.3.3.7 Fractal Dimension Texture... to describe texture 3.3.3.9 Shape Parameters The following shape parameters were calculated from the segmented plaque image: (1) X-coordinate maximum length, (2) Y-coordinate maximum length, (3) area, (4) perimeter, and (5) perimeter2/area Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 99 Tuesday, May 10, 2005 3:38 PM Texture and Morphological Analysis of Ultrasound Images 99 3.3.3.10... plaques retain their positive values Step 2: Calculate the average confidence Calculate the average of the n confidence measures that is the final output of the system combiner as 1 conf = n n ∑ conf j (3 .4) j =1 Step 3: Classify plaque If conf < 0, then the plaque is classified as symptomatic, else if conf > 0, then the plaque is classified as asymptomatic Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm... [45] Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 108 Tuesday, May 10, 2005 3:38 PM 108 Medical Image Analysis FIGURE 3.4 Box plots of the features gray-scale median (2), entropy ( 14), and coarseness (36) for the symptomatic and asymptomatic plaques (The numbers in parentheses denote the serial feature number as listed in Table 3.1.) The notched box shows the median, lower and upper... p i, j i j 2 (3.13) A.3.1.2.2.2 Contrast The contrast is a measure of the amount of local variations present in the image f2 = N g −1  Ng  Ng ∑ n  ∑ ∑ p ( i, j ) 2 i =1  i =1 j =1 i= j = n (3. 14)  A.3.1.2.2.3 Correlation Correlation is a measure of gray-tone linear dependencies ∑ ∑ ( i, j ) p ( i, j ) − µ µ x f3 = i y j (3.15) σ xσ y where µx, µy, and σx, σy, are the mean and standard deviation... HXY 1 max HX , HY { } f13 = (1 − exp[−2.0(HXY2 − HXY)])1/2 HXY = − ∑ ∑ p (i, j ) log ( p (i, j )) i j where HX and HY are entropies of px and py , and Copyright 2005 by Taylor & Francis Group, LLC (3. 24) (3.25) (3.26)