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2089_book.fm Page 185 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models with Application to Medical Imaging Sarah Lee and Tania Stathaki CONTENTS 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Introduction 5.1.1 One-Dimensional Autoregressive Modeling for Biomedical Signals 5.1.2 Two-Dimensional Autoregressive Modeling for Biomedical Signals Two-Dimensional Autoregressive Model Yule-Walker System of Equations Extended Yule-Walker System of Equations in the Third-Order Statistical Domain Constrained-Optimization Formulation with Equality Constraints 5.5.1 Simulation Results Constrained Optimization with Inequality Constraints 5.6.1 Constrained-Optimization Formulation with Inequality Constraints 5.6.2 Constrained-Optimization Formulation with Inequality Constraints 5.6.3 Simulation Results AR Modeling with the Application of Clustering Techniques 5.7.1 Hierarchical Clustering Scheme for AR Modeling 5.7.2 k-Means Algorithm for AR Modeling 5.7.3 Selection Scheme 5.7.4 Simulation Results Applying AR Modeling to Mammography 5.8.1 Mammograms with a Malignant Mass 5.8.1.1 Case 1: mdb023 5.8.1.2 Case 2: mdb028 5.8.1.3 Case 3: mdb058 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 186 Tuesday, May 10, 2005 3:38 PM 186 Medical Image Analysis 5.8.2 Mammograms with a Benign Mass 5.8.2.1 Case 1: mdb069 5.8.2.2 Case 2: mdb091 5.8.2.3 Case 3: mdb142 5.9 Summary and Conclusion References 5.1 INTRODUCTION In this chapter, we introduce texture characterization using autoregressive (AR) models and demonstrate its potential use in medical-image analysis The one-dimensional AR modeling technique has been used extensively for one-dimensional biomedical signals, and some examples are given in Section 5.1.1 For two-dimensional biomedical signals, the idea of applying the two-dimensional AR modeling technique has not been explored, as only a couple of examples can be found in the literature, as shown in Section 5.1.2 In the following sections, we concentrate on a two-dimensional AR modeling technique whose results can be used to describe textured surfaces in images under the assumption that every distinct texture can be represented by a different set of two-dimensional AR model coefficients The conventional Yule-Walker system of equations is one of the most widely used methods for solving AR model coefficients, and the variances of the estimated coefficients obtained from a large number of realizations, i.e., simulations using the output of a same set of AR model coefficients but randomly generated driving input, are sufficiently low However, estimations fail when large external noise is added onto the system; if the noise is Gaussian, we are tempted to work in the third-order statistical domain, where the third-order moments are employed, and therefore the external Gaussian noise can be eliminated [1, 2] This method leads to higher variances from the estimated AR model coefficients obtained from a number of realizations We propose three methods for estimation of two-dimensional AR model coefficients The first method relates the extended Yule-Walker system of equations in the third-order statistical domain to the YuleWalker system of equations in the second-order statistical domain through a constrained-optimization formulation with equality constraints The second and third methods use inequality constraints instead The textured areas of the images are thus characterized by sets of the estimated AR model coefficients instead of the original intensities Areas with a distinct texture can be divided into a number of blocks, and a set of AR model coefficients is estimated for each block A clustering technique is then applied to these sets, and a weighting scheme is used to obtain the final estimation The proposed AR modeling method is also applied to mammography to compare the AR model coefficients of the block of problematic area with the coefficients of its neighborhood blocks The structure of this chapter is as follows In Section 5.2 the two-dimensional AR model is revisited, and Section 5.3 describes one of the conventional methods, the Yule-Walker system of equations Another conventional method, the extended Yule-Walker system of equations in the third-order statistical domain, is explained Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 187 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 187 in Section 5.4 The proposed methods — constrained-optimization formulation with equality constraints and constrained-optimization formulations with inequality constraints — are covered in Sections 5.5 and 5.6, respectively In Section 5.7, two clustering techniques — minimum hierarchical clustering scheme and k-means algorithm — are applied to a number of sets of AR model coefficients estimated from an image with a single texture In Section 5.8, the two-dimensional AR modeling technique is applied to the texture characterization of mammography A relationship is established between the AR model coefficients obtained from the block containing a tumor and its neighborhood blocks The summary and conclusion can be found in Section 5.9 5.1.1 ONE-DIMENSIONAL AUTOREGRESSIVE MODELING FOR BIOMEDICAL SIGNALS The output x[m] of the one-dimensional autoregressive (AR) can be written mathematically [3] as p x m = − ∑ a i x m − i + u m (5.1) i =1 where a[i] is the AR model coefficient, p is the order of the model, and u[m] is the driving input AR modeling is among a number of signal-processing techniques that have been applied to biomedical signals, including the fast Fourier transform (FFT) used for frequency analysis; linear, adaptive, and morphological filters; and others [3] Some examples are given here According to Bloem and Arzbaecher [4], the one-dimensional AR modeling technique is applied to discriminate atrial arrhythmias based on the fact that AR modeling of organized cardiac rhythm produces residuals that are dominated by the impulse On the other hand, atrial fibrillation shows a residual containing decorrelated noise Apart from the cardiac rhythms, the AR modeling technique has been applied to apnea detection and to estimation of respiration rate [5] Respiration signals are assumed to be one-dimensional second-order AR signals, i.e., p = in Equation 5.1 Effective classification of different respiratory states and accurate detection of apnea are obtained from the functions of estimated AR model coefficients [5] In addition, the AR modeling method is applied to heart rate (HR) variability analysis [6], whose purpose is to study the interaction between the autonomic nervous system and the heart sinus pacemakers The long-term HR is said to be nonstationary because it has shown strong circadian variations According to Thonet [6], a time-varying AR (TVAR) model is assumed for HR analysis: “the comparison of the TVAR coefficients significance rate has suggested an increasing linearity of HR signals from control subjects to patients suffering from a ventricular tachyarrhythmia.” The AR modeling technique has also been applied to code and decode the electrocardiogram (ECG) signals over the transmission between an ambulance and a hospital [7] The AR model coefficients estimated in the higher-order statistical domain are transmitted instead of the real ECG signals The transmission results Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 188 Tuesday, May 10, 2005 3:38 PM 188 Medical Image Analysis were said to be safe and efficient, even in the presence of high noise (17 dB) [7] According to Palianappan et al [8], the AR modeling method is also applied to ECG signals, but this time the work was concentrated on estimating the AR model orders from some conventional methods for two different mental tasks: math task and geometric figure rotation Spectral density functions are derived after the order of the AR model is obtained, and a neural-network technique is applied to assign the tasks into their respective categories [8] 5.1.2 TWO-DIMENSIONAL AUTOREGRESSIVE MODELING FOR BIOMEDICAL SIGNALS The two-dimensional AR modeling technique has been applied to mammography [2, 9–11] Stathaki [2] concentrated on the directionalities of the tissue shown in mammograms, because healthy tissue has specific properties with respect to the directionalities “There exist decided directions in the observed X-ray images that show the underlying tissue structure as having distinct correlations in some specific direction of the image plane” [2] Thus, by applying the two-dimensional AR modeling technique to these two-dimensional signals, the variations in parameters are crucial in directionality characterization The AR model coefficients are obtained with the use of blocks of size between × and 40 × 40 and different “slices” (vertical, horizontal, or diagonal) (see Section 5.4 for details of slices) The preliminary study of a comparative nature on the subject of selecting cumulant slices in the area of mammography by Stathaki [2] shows that the directionality is destroyed in the area of tumor The three types of slices used give similar performance, except in the case of [c1,c2] = [1,0] The estimated AR model parameters tend to converge to a specific value as the size of the window increases [10] In addition, the greater the calcification, the greater will be the deviation of the texture parameters of the lesions from the norm [2] 5.2 TWO-DIMENSIONAL AUTOREGRESSIVE MODEL The two-dimensional autoregressive (AR) model is defined [12] as x m, n = − p1 p2 i=0 j =0 ∑ ∑ a x m − i, n − j + u m, n ij i, j ≠ 0, (5.2) where p1 × p2 is the AR model order, aij is the AR model coefficient, and u[m,n] is the driving input, which is assumed to have the following properties [2, 13]: u[m,n] is non-Gaussian Zero mean, i.e., E{u[m,n]} = 0, where E{⋅} is the expectation operation Second-order white, i.e., the input autocorrelation function is σu2δ[m,n] and σu2 = E{u2[m,n]} At least second-order stationary Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 189 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 189 The first condition is imposed to enable the use of third-order statistics A set of stable two-dimensional AR model coefficients can be obtained from two sets of stable one-dimensional AR model coefficients Let a1 be a row vector that represents a set of stable one-dimensional AR model coefficients and a2 be another row vector that represents a set of stable one-dimensional AR model coefficients, a, where a = a1T × a2 is a set of stable two-dimensional AR model coefficients and T denotes transposition When a1 is equal to a2, the two-dimensional AR model coefficients, a, are symmetric [14] 5.3 YULE-WALKER SYSTEM OF EQUATIONS The Yule-Walker system of equations is revisited for the two-dimensional AR model in this section The truncated nonsymmetric half-plane (TNSHP) is taken to be the region of support of AR model parameters [12]: { STNSHP = i, j : i = 1, 2, , p1; j = − p2, , 0, } { , p2 ∪ i, j : i = 0; j = 0,1, , p2 } Two examples of TNSHP are shown in Figure 5.1 The shape of the dotted lines indicates the region of support when p1 = and p2 = 3, and the shape of the solid lines is for p1 = p2 = j p1 = 1, p2 = i p1 = p2 = FIGURE 5.1 Examples of the truncated nonsymmetric half-plane region of support (TNSHP) for AR model parameters Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 190 Tuesday, May 10, 2005 3:38 PM 190 Medical Image Analysis The two-dimensional signal x[m,n] given in Equation 5.2 is multiplied by its shifted version, x[m − k,n − l], and under the assumption that all fields are wide sense stationary, the expectation of this multiplication gives us ∑∑am ij [ i , j ]≠ STNSHP 2x { k − i, l − j = E x m − k , n − l u m, n { = E x − k , −l u 0, } } (5.3) In Equation 5.3, the second-order moment, which is also regarded as “autocorrelation,” is defined as Equation 5.4 { m2 x k, l = E x m, n x m + k, n + l } (5.4) Because the region of support of the impulse response is the entire nonsymmetric half plane, by applying the causal and stable filter assumptions we obtain { } ∑ ∑ h i, j E {u −k − i, −l − j u 0, } E x − k − l u 0, = i , j ∈ S NSHP = h − k , −l σ (5.5) u Because h[k,l] is the impulse response of a causal filter, Equation 5.5 becomes E x − k, −l u 0, = h 0, σ u { } { for for ' k, l ∈ S NSHP k, l = 0, } ' where S NSHP = S NSHP ∪ 0, Because h[0,0] is assumed to be unity, the two-dimensional Yule-Walker equations [12] become E x − k, −l u 0, = σ u { } for for ' k, l ∈ S NSHP k, l = 0, (5.6) For simplicity in our AR model coefficient estimation methods, the region of support is assumed to be a quarter plane (QP), which is a special case of the NSHP Examples of QP models can be found in Figure 5.2 The shape filled with vertical lines indicates the region of support of QP when p1 = and p2 = 3, and the shape filled with horizontal lines is the region of support of QP when p1 = p2 = The Yule-Walker system of equations for a QP model can be written [12] as Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 191 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 191 j p1 = 2, p2 = p1 = p2 = i FIGURE 5.2 Examples of two quarter-plane region of supports for the AR parameters p1 p2 i=0 j =0 ∑∑ aij m2 x k − i, l − j = σ u for for k, l ∈ SQ' P k, l = 0, (5.7) Generalizing Equation 5.7 leads to the equations Mxxal = h (5.8) where Mxx is a matrix of size [(p1 + 1)(p2 + 1)] × [(p1 + 1)(p2 + 1)], and al and h are both vectors of size [(p1 + 1)(p2 + 1)] × More explicitly, Equation 5.8 can be written as M xx M xx 1 M xx p1 M xx −1 M xx M xx p1 − 1 a0 σ u h1 M xx − p1 − a1 = ap M xx M xx − p1 ( where = [ , ai1, …, aip2 ]T is a vector of size (p2 + 1) × h1 = [1,0,…,0]T is a vector of size (p2 + 1) × = [0,0,…,0]T is a vector of size (p2 + 1) × Copyright 2005 by Taylor & Francis Group, LLC ) (5.9) 2089_book.fm Page 192 Tuesday, May 10, 2005 3:38 PM 192 Medical Image Analysis m2 x i, m2 x i, −1 m i,1 m2 x i, M xx i = x m2 x i, p2 m2 x i, p2 − 1 of size (p2 + 1) × (p2 + 1) m2 x i, − p2 − is a matrix m2 x i, m2 x i, − p2 ( ) An example of the Yule-Walker system of equations for a × AR model is given below m2 x 0, m2 x 0, 1 m2 x 1, m2 x 1, 1 m2 x 0, −1 m2 x 0, m2 x −1, m2 x −1, 1 m2 x 1, −1 m2 x 1, m2 x 0, m2 x 0, 1 m2 x −1, −1 a σ u 00 m2 x −1, a01 = m2 x 0, −1 a10 m2 x 0, a11 (5.10) These equations can be further simplified because the variance, σu2, is unknown, and the AR model coefficient a00 is assumed to be in general The Yule-Walker system of equations can be rewritten as m2 x 0, m2 x 1, −1 m2 x 1, m2 x −1, 1 m2 x 0, m2 x 0, 1 m2 x 0, 1 m2 x −1, a01 m2 x 0, −1 a10 = − m2 x 1, m2 x 0, a11 m2 x 1, 1 (5.11) Let the Yule-Walker system of equations for an AR model with model order p1 × p2 be represented in the matrix form as Ra = −r (5.12) where R is a [(p1 + 1)(p2 + 1) − 1] × [(p1 + 1)(p2 + 1) − 1] matrix of autocorrelation samples a is a [(p1 + 1)(p2 + 1) − 1] × vector of unknown AR model coefficients r is a [(p1 + 1)(p2 + 1) − 1] × vector of autocorrelation samples 5.4 EXTENDED YULE-WALKER SYSTEM OF EQUATIONS IN THE THIRD-ORDER STATISTICAL DOMAIN The Yule-Walker system of equations is able to estimate the AR model coefficients when the power of the external noise is small compared with that of the signal Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 193 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 193 However, when the external noise becomes larger, the estimated values are influenced by the external noise statistics These results correspond to the well-known fact that the autocorrelation function (ACF) samples of a signal are sensitive to additive Gaussian noise because the ACF samples of Gaussian noise are nonzero [1, 15] Estimation of the AR model coefficients using the Yule-Walker system of equations for a signal with large external Gaussian noise is poor, therefore we are forced to work in the third-order statistical domain, where third-order cumulants are employed [2] Consider the system y[m,n] that is contaminated with external Gaussian noise v[m,n]: y[m,n] = x[m,n] + v[m,n] The third-order cumulant of a zero-mean twodimensional signal, y[m,n], ≤ m ≤ M, ≤ n ≤ N, is estimated [1] by Number of terms available ∑ ∑ y m, n y m + k , n + l y m + k , n + l 1 2 (5.13) The number of terms available is not necessarily the same as the size of the image because of the values k1, l1, k2, and l2 All the pixels outside the range are assumed to be zero The difference in formulating the Yule-Walker system of equations between the second-order and third-order statistical domain is that in the latter version, we multiply the output of the AR model by two shifted versions instead of just one in the earlier version [1] The extended Yule-Walker system of equations in the thirdorder statistical domain can be written as shown in Equation 5.14 [11] p1 p2 γ ∑ ∑ a C ( i − k , j − l , i − k , j − l ) = u ij i=0 3y j =0 2 k1 = k2 = l1 = l2 = (5.14) otherwise where γu = E{u3[m,n]} is the skewness of the input driving noise, and a00 = From the derivation of the above relationship, it is evident that using Equation 5.14 implies that it is unnecessary to know the statistical properties of the external Gaussian noise, because they are eliminated from the equations following the theory that the third-order cumulants of Gaussian signals are zero [16] For a two-dimensional AR model with order p1 × p2, we need at least a total of (p1 + 1)(p2 + 1) equations from Equation 5.14, where k1 k2 l1 l2 = = = = 0,…, p1 k2 0,…, p2 l1 in order to estimate the [(p1 + 1)(p2 + 1) − 1] unknown AR parameters and the skewness of the driving noise, γu Because we are only interested in estimating the AR model coefficients, we can rewrite Equation 5.13 as follows [2] Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 194 Tuesday, May 10, 2005 3:38 PM 194 Medical Image Analysis l Diagonal Slice Vertical Slice Horizontal Slice k FIGURE 5.3 Different third-order cumulant slices for a one-dimensional signal p1 p2 i=0 j =0 ∑ ∑ a C ( i − k , j − l , i − k , j − l ) = ij 3y 1 (5.15) where k1 + l1 + k2 + l2 ≠ and k1,l1,k2,l2 ≥ In this form, [(p1 + 1)(p2 + 1) − 1] equations are required to determine the aij parameters (for details, see the literature [17–21]) When the third-order cumulants are used, an implicit and additional degree of freedom is connected with the specific direction chosen for these to be used in the AR model [2] Such a direction is referred to as a slice in the cumulant plane, as shown on the graph for third-order cumulants for one-dimensional signals in Figure 5.3 [2, 22] Consider the third-order cumulant slice of a one-dimensional process, y, which can be estimated using C3y(k,l) = E{y(m) y(m+k) y(m+l)} [16] The diagonal slice indicates that the value of k is the same as the value of l, whereas the vertical slices have a constant k value, and the horizontal slices have a constant l value The idea can be extended into the third-order cumulants for two-dimensional signals In Equation 5.13, if k1 = l1 and k2 = l2, the slice is diagonal; if k1 and l1 remain constant, the slice is vertical; if k2 and l2 are constant, the slice is horizontal Let us assume that (k2,l2) = (k1+c1, l1+c2), where c1 and c2 are both constants Then [2] ( ) ( ) C y i − k1, j − l1 , i − k2 , j − l2 = C y i − k1, j − l1 , i − k1 − c1, j − l1 − c2 (5.16) Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 209 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 209 TABLE 5.12 AR Modeling Results of the Nonsymmetric Model with Application of Clustering Schemes (three clusters, SNR = dB) AR Model Coefficient a01 a02 a10 a11 a12 a20 a21 a22 Real Value Estimated Value (all) Estimated Value (MHC) Estimated Value (k-means) 0.5 0.4 0.4 0.2 0.16 0.3 0.15 0.12 0.4936 0.3879 0.3927 0.2036 0.1592 0.2884 0.1486 0.1163 0.4959 0.3926 0.3953 0.2061 0.1630 0.2926 0.1521 0.1205 0.4145 0.3259 0.3302 0.1710 0.1341 0.2425 0.1251 0.09807 0.06817 0.02801 0.04677 Relative error B1 B2 B3 BP B4 B6 B7 B5 Centre of the circle enclosing the abnormality B8 r – the given radius 2r FIGURE 5.5 Example of the mass and its × neighborhood in a mammogram For simplicity, we take the square block with the length equal to the given radius as the block of interest We form a × neighborhood around the block of interest and then estimate the AR model coefficients of each block, as shown in Figure 5.5 The order of the AR model is assumed to be × Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 210 Tuesday, May 10, 2005 3:38 PM 210 Medical Image Analysis FIGURE 5.6 The mammogram with the mass marked: mdb023 5.8.1 MAMMOGRAMS WITH A MALIGNANT MASS We take three examples of mammograms with a malignant mass The mammograms are named by their original index numbers in the database The origin of the system coordinates is the bottom-left corner 5.8.1.1 Case 1: mdb023 A well-defined mass with fatty-glandular background tissue is found in the square centered at (538, 681) with 59 pixels as its length [33], and the mammogram is shown in Figure 5.6 The AR model coefficients estimated from the block of size 59 × 59 centered at pixel (538, 681) and eight blocks in its × neighborhood are shown in Table 5.13 From the results, we find that the AR model coefficients of the block of tumor are almost symmetrical The degree of symmetry is calculated using Equation 5.31 [23], with smaller values indicating greater symmetry of the set of AR model coefficients, a11 − a01 × a10 (5.31) 5.8.1.2 Case 2: mdb028 In mammogram mdb028, which is shown in Figure 5.7, the malignant mass is found within the square centered (338, 314) with length as 113 pixels The background tissue is fatty [33] The AR model coefficients estimated from the block of size 113 × 113 centered at pixel (338, 314) and eight blocks in its neighborhood are shown in Table 5.14 The block, Bp, has the smallest value for the degree of symmetry, i.e., the AR model coefficients are more symmetric in this block than others Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 BP B5 B6 B7 B8 −0.9104 −0.9643 0.8759 0.0020 7.0822 −0.3474 −7.7154 5.2553 2.4479 −1.3119 −2.3145 −1.0768 −1.3301 −0.9151 1.2453 −0.0282 −1.0403 −1.0291 1.0696 0.0010 −1.1647 −0.7296 0.8944 −0.0446 −0.1890 −0.8346 0.0255 0.1322 −0.8935 −1.1864 1.0800 −0.0199 0.6717 −1.0080 −0.6707 −0.016 2089_book.fm Page 211 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.13 AR Model Coefficients for Blocks of Pixels in Mammogram mdb023 211 2089_book.fm Page 212 Tuesday, May 10, 2005 3:38 PM 212 Medical Image Analysis FIGURE 5.7 The mammogram with the mass marked: mdb028 5.8.1.3 Case 3: mdb058 The mammogram mdb058 is shown in Figure 5.8 A malignant mass is found in the square centered at (318, 359) with length equal to 55 pixels [33] The AR model coefficients estimated from the block of size 55 × 55 centered at pixel (318, 359) and eight blocks in its neighborhood are shown in Table 5.15 As in the previous cases, the AR model coefficients in block Bp are more symmetric than the other blocks 5.8.2 MAMMOGRAMS WITH A BENIGN MASS Apart from mammograms with a malignant mass, we also apply the same method to estimate the AR model coefficients of mammograms with a benign mass Three examples taken are mdb069, 091, and 142 from the database [33] 5.8.2.1 Case 1: mdb069 The mammogram mdb069 is shown in Figure 5.9 with its benign mass marked The background tissue is fatty, and the mass is situated in the square centered (462, 402) with 89 pixels as its length The AR model coefficients estimated from the block of size 89 × 89 centered at pixel (462, 402) and eight blocks in its neighborhood are shown in Table 5.16 The results obtained are similar to the results from the mammograms with a malignant mass, i.e., the block containing the benign mass can also be represented by a set of AR model coefficients that is more symmetric than the other blocks Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 BP B5 B6 B7 B8 −0.0939 −3.1346 2.2266 −1.9321 −0.9536 −3.3854 3.3406 −0.1121 −0.8448 0.4720 −0.6276 0.2288 −1.2181 −0.7490 0.9675 −0.0551 −1.0197 −1.0253 1.0450 0.0005 −1.0970 −0.9208 1.0176 −0.0075 −1.1433 −2.6646 2.8055 0.2410 1.4875 −0.1951 −2.2943 2.0041 −0.7789 −1.6102 1.3893 −0.1351 2089_book.fm Page 213 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.14 AR Model Coefficients for Blocks of Pixels in Mammogram mdb028 213 2089_book.fm Page 214 Tuesday, May 10, 2005 3:38 PM 214 Medical Image Analysis FIGURE 5.8 The mammogram with the mass marked: mdb058 5.8.2.2 Case 2: mdb091 Figure 5.10 shows the mammogram mdb091, whose background tissue is fatty The benign mass is situated in the square centered at (680, 494) with the length equal to 41 pixels The AR model coefficients estimated from the block of size 41 × 41 centered at pixel (680, 494) and eight blocks in its neighborhood are shown in Table 5.17 By comparing the degree of symmetry calculated for each block, the AR model coefficients from the block Bp are more symmetric 5.8.2.3 Case 3: mdb142 The mammogram mdb142 is shown in Figure 5.11, with its benign mass highlighted The background tissue is again fatty, and the mass is within the square centered at (347, 636), with length equal to 53 Table 5.18 shows the AR model coefficients estimated from the block of size 53 × 53 centered at pixel (347, 636) and eight blocks in its neighborhood The degree of symmetry is small for all the blocks, and block Bp has the smallest degree of symmetry 5.9 SUMMARY AND CONCLUSION In this chapter, we investigated the possibility of applying the two-dimensional autoregressive (AR) modeling technique to characterize textures in mammograms The two-dimensional AR model, the Yule-Walker system of equations, and the extended Yule-Walker system of equations in the third-order statistical domain were revisited Three methods for estimating AR model coefficients using both the Yule-Walker system Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 BP B5 B6 B7 B8 −0.6095 −1.6463 1.2561 −0.2528 −0.8822 −1.2162 1.0985 −0.0256 −0.9133 −0.6749 0.5886 0.0278 −0.5295 −5.0211 4.5482 −1.8893 −1.0697 −1.0366 1.1062 0.0026 12.5210 −1.6406 −11.8717 −8.6708 −0.4072 −0.6327 0.0399 0.2177 −0.9271 −0.8503 0.7775 0.0108 −1.1034 −0.8627 0.9662 −0.0142 2089_book.fm Page 215 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.15 AR Model Coefficients for Blocks of Pixels in Mammogram mdb058 215 2089_book.fm Page 216 Tuesday, May 10, 2005 3:38 PM 216 Medical Image Analysis FIGURE 5.9 The mammogram with the mass marked: mdb069 of equations and the extended Yule-Walker system of equations in the third-order statistical domain were proposed Their simulation results showed that these methods are able to estimate two-dimensional AR model coefficients in both low- and highSNR (signal-to-noise ratio) systems, and the variances generated from 100 realizations were sufficiently small The AR modeling results were further improved for images with a single texture by clustering methods Finally, one of the proposed methods was applied to characterize the texture of mammograms Preliminary observations concerned the fact that the × AR model coefficients representing the tumor area seemed to be more symmetric compared with the AR model coefficients of its neighbor blocks Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 Bp B5 B6 B7 B8 −18.1696 −19.0165 36.1743 309.4 −1.5331 −0.7074 1.2413 −0.1568 −0.8151 −1.2883 1.1037 −0.0536 −0.6781 −0.6861 0.3653 0.1000 −1.0925 −1.1285 1.2211 0.0118 −1.5952 −1.6451 2.2409 0.3832 −0.3248 −1.2173 0.5417 −0.1463 −0.9324 −0.6381 0.5708 0.0241 1.7081 −14.3620 11.5490 −35.9096 2089_book.fm Page 217 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.16 AR Model Coefficients for Blocks of Pixels in Mammogram mdb069 217 2089_book.fm Page 218 Tuesday, May 10, 2005 3:38 PM 218 Medical Image Analysis FIGURE 5.10 The mammogram with the mass marked: mdb091 Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 BP B5 B6 B7 B8 −1.0586 −0.7826 0.8416 −0.0132 −0.8702 −0.9344 0.8048 0.0083 −1.1097 −1.0844 1.1943 0.0091 −1.0722 0.7697 −0.6972 −0.1282 −1.0645 −1.0395 1.1042 0.0023 −0.8432 −1.1193 0.9626 −0.0188 −0.9022 −1.0801 0.9626 −0.0081 −0.3007 −1.0465 0.3465 −0.0318 −0.7324 −1.3942 1.1268 −0.1056 2089_book.fm Page 219 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.17 AR Model Coefficients for Blocks of Pixels in Mammogram mdb091 219 2089_book.fm Page 220 Tuesday, May 10, 2005 3:38 PM 220 Medical Image Analysis FIGURE 5.11 The mammogram with the mass marked: mdb142 Copyright 2005 by Taylor & Francis Group, LLC a01 a10 a11 Degree of symmetry Blocks B1 B2 B3 B4 BP B5 B6 B7 B8 −1.0429 1.0748 −1.0319 −0.0889 −0.8479 −1.1384 0.9858 −0.0213 −0.9107 −0.8749 0.7859 0.0110 −0.8579 −0.8168 0.6767 0.0255 −1.0396 −1.0595 1.0992 0.0022 −1.0608 −0.8636 0.9243 −0.0083 −0.7939 −2.2192 2.0136 −0.2517 −0.8885 −0.7471 0.6357 0.0280 −0.6772 −0.6094 0.2868 0.1259 2089_book.fm Page 221 Tuesday, May 10, 2005 3:38 PM AR Model Coefficient Texture Characterization Using Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.18 AR Model Coefficients for Blocks of Pixels in Mammogram mdb142 221 2089_book.fm Page 222 Tuesday, May 10, 2005 3:38 PM 222 Medical Image Analysis REFERENCES Giannakis, G.B., Mendel, J.M., and Wang, W., ARMA modeling using cumulants and autocorrelation statistics, Proc Int Conf Acoustics, Speech Signal Process (ICASSP), 1, 61, 1987 Stathaki, P.T., Cumulant-Based and Algebraic Techniques for Signal Modelling, Ph.D Thesis, Imperial College, London, 1994 Ifeachor, E.C., Medical Applications of DSP, presented at IEEE Younger Members Tutorial Seminar on DPS: Theory, Applications and Implementation, IEEE, Washington, DC, 1996 Bloem, D and Arzbaecher, R., Discrimination of atrial arrhythmias using autoregressive modelling, Proc Comput Cardiol., Durham, NC, USA 235–238, 1992 Nepal, K., Biegeleisen, E., and Ning, T., Apnea detection and respiration rate estimation through parametric modelling, Proc IEEE 28th Ann Northeast Bioeng Conf., Philadelphia, USA 277–278, 2002 Thonet, G et al., Assessment of stationarity horizon of the heart rate, Proc 18th Ann Int Conf IEEE Eng Medicine Biol Soc., Bridging Disciplines Biomedicine, 4, 1600, 1996 Economopoulos, S.A et al., Robust ECG coding using wavelet analysis and higherorder statistics, IEE Colloq Intelligent Methods Healthcare Medical Appl., 15/1-15/6, Digest number 1998/514, York, UK 1998 Palaniappan, R et al., Autoregressive spectral analysis and model order selection criteria for EEG signals, Proc TENCON 2000, 2, 126 Kuala Lumpur, Malaysia Stathaki, P.T and Constantinides, A.G., Noisy texture analysis based on higher order statistics and neural network classifiers, Proc IEEE Int Conf Neural Network Application to DSP, 324–329, 1993 10 Stathaki, P.T and Constantinides, A.G., Robust autoregressive modelling through higher order spectral estimation techniques with application to mammography, Proc 27th Ann Asilomar Conf Signals, Systems Comput., 1, 189, 1993 11 Stathaki, T and Constantinides, A.G., Neural networks and higher order spectra for breast cancer detection, Proc IEEE Workshop Neural Network for Signal Processing, 473–481, 1994 12 Kay, S.M., Modern Spectral Estimation: Theory and Application, Signal Processing Series, Prentice-Hall, Englewood Cliffs, NJ, 1987 13 Bhattacharya, S., Ray, N.C., and Sinha, S., 2-D signal modelling and reconstruction using third-order cumulants, Signal Process., 62, 61, 1997 14 Lee, S., Novel Methods on 2-D AR Modelling, M.Phil to Ph.D Transfer Report, Dept Electrical Electronic Engineering, Imperial College, London, 2003 15 Nikias, C.L and Raghuveer, M., Bispectrum estimation: a digital signal processing framework, Proc IEEE, 75, 869, 1987 16 Mendel, J.M., Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications, Proc IEEE, 79, 278, 1991 17 Giannakis, G.B., Cumulants: a powerful tool in signal processing, Proc IEEE, 75, 1333, 1987 18 Giannakis, G.B., Identification of nonminimum-phase systems using higher order statistics, IEEE Trans ASSP, 37, 360, 1989 19 Giannakis, G.B., On the identifiability of non-Gaussian ARMA models using cumulants, IEEE Trans Automatic Control, 35, 18, 1990 20 Giannakis, G.B., Cumulant-based order determination of non-Gaussian ARMA models, IEEE Trans ASSP, 38, 1411, 1990 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 223 Tuesday, May 10, 2005 3:38 PM Texture Characterization Using Autoregressive Models 223 21 Giannakis, G.B and Swami, A., On estimating noncausal nonminimum phase ARMA models of non-Gaussian processes, IEEE Trans ASSP, 38, 478, 1990 22 Dickie, J.R and Nandi, A.K., AR modelling of skewed signals using third-order cumulants, IEEE Proc Vision, Image Signal Process., 142, 78, 1995 23 Lee, S and Stathaki, T., Texture Characterisation Using Constrained Optimisation Techniques with Application to Mammography, presented at 5th Int Workshop on Image Analysis for Multimedia Interactive Services (WIAMIS), on CD, Lisbon, Portugal, 2004 24 Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization, Academic Press, New York, 1981 25 Lee, S., Stathaki, T., and Harris, F., Texture characterisation using a novel optimisation formulation for two-dimensional autoregressive modelling and k-means algorithm, 37th Asilomar Conf Signals, Systems Comput., 2, 1605, 2003 26 Lee, S., Stathaki, T., and Harris, F., A two-dimensional autoregressive modelling technique using a constrained optimisation formulation and the minimum hierarchical clustering scheme, 38th Asilomar Conf Signals, Systems Comput., 2, 1690, 2004 27 Johnson, S.C., Hierarchical clustering schemes, Psychometrika, 32, 241, 1967 28 Borgatti, S.P., How to explain hierarchical clustering, Connections, 17, 78, 1994 29 Anderberg, M.R., Clustering Analysis for Applications, Academic Press, New York, 1973; Statistics, John Wiley & Sons, New York, 1975 30 Hartigan, J.A., Clustering Algorithms, Wiley Series in Probability and Mathematical 31 Jain, A.K., Murty, M.N., and Flynn, P.J., Data clustering: a review, ACM Computing Surveys, 31, 264–323, 1999 32 Jain, A et al., Artificial Intelligence Techniques in Breast Cancer Diagnosis and Prognosis, Series in Machine Perception and Artificial Intelligence, 39, World Scientific, Singapore, London, 1–15, 2000 33 Mammographic Image Analysis Society (MIAS), MiniMammography Database; available on-line at http://www.wiau.man.ac.uk/services/MIAS/MIASmini.html, last accessed 5/25/2004 Copyright 2005 by Taylor & Francis Group, LLC [...]... error 0.008605 5.7 AR MODELING WITH THE APPLICATION OF CLUSTERING TECHNIQUES In Sections 5.3 to 5.6, the AR modeling methods are applied to the entire image In this section, we divide images into a number of blocks under the assumption that the texture remains the same throughout the entire image After applying an AR modeling method to each of these blocks, a number of sets of AR model coefficients are... texture analysis Masses and calcifications are two major abnormalities that radiologists look for in mammograms [32] We concentrate on the texture characterization of the mammogram with a mass under the assumption that the texture of the problematic area is different from the Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 208 Tuesday, May 10, 2005 3:38 PM 208 Medical Image Analysis. .. Pixels in Mammogram mdb058 215 2089_book.fm Page 216 Tuesday, May 10, 2005 3:38 PM 216 Medical Image Analysis FIGURE 5.9 The mammogram with the mass marked: mdb069 of equations and the extended Yule-Walker system of equations in the third-order statistical domain were proposed Their simulation results showed that these methods are able to estimate two-dimensional AR model coefficients in both low- and highSNR... constrained-optimization formulation with inequality constraints 1 — show high accuracy, as evidenced Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 202 Tuesday, May 10, 2005 3:38 PM 202 Medical Image Analysis TABLE 5.4 Results from Constrained-Optimization Formulation with Inequality Constraints 2 for Estimation of Two-Dimensional Symmetric AR Model Coefficients, SNR = 5 dB Parameter Real Value... low-SNR system, and the average ε value for each coefficient is smaller than in the low-SNR system Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 204 Tuesday, May 10, 2005 3:38 PM 204 Medical Image Analysis TABLE 5.8 Results from Constrained-Optimization Formulation with Inequality Constraints 2 for Estimation of Two-Dimensional Nonsymmetric AR Model Coefficients, SNR = 30 dB Parameter Real... , c2 C3 y i, 0 , c1 , c2 ( 2 C3 y i, p2 − 1 , c1 , c2 ( C3 y i, − p2 , c1 , c2 ( ( ) ) ) 2089_book.fm Page 196 Tuesday, May 10, 2005 3:38 PM 196 Medical Image Analysis The system in Equation 5.20 can be further simplified, as shown in Section 5.3 Let us take a 1 × 1 AR model as an example We apply a diagonal slice, i.e., [c1, c2] = [k−i, l−j]; therefore,... clusters be k 2 Randomly choose k sets of AR model coefficients and assign one set to one cluster Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 206 Tuesday, May 10, 2005 3:38 PM 206 Medical Image Analysis 3 For each of the rest of the sets of data, calculate the distance between the set and the mean of each cluster using Equation 5.29 Assign the set of AR model coefficients to the cluster... [24] ( )( p1 p2 i=0 j =0 ) ∑∑ 1 p1 + 1 p2 + 1 − 1 Copyright 2005 by Taylor & Francis Group, LLC aˆij − aij aij i, j ≠ 0, 0 (5.24) 2089_book.fm Page 198 Tuesday, May 10, 2005 3:38 PM 198 Medical Image Analysis TABLE 5.1 Results from Constrained-Optimization Formulation with Equality Constraints for Estimation of Two-Dimensional Symmetric AR Model Coefficients SNR = 5 dB SNR = 30 db Parameter... coefficients of each block, as shown in Figure 5.5 The order of the AR model is assumed to be 1 × 1 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 210 Tuesday, May 10, 2005 3:38 PM 210 Medical Image Analysis FIGURE 5.6 The mammogram with the mass marked: mdb023 5.8.1 MAMMOGRAMS WITH A MALIGNANT MASS We take three examples of mammograms with a malignant mass The mammograms are named by their... Autoregressive Models Copyright 2005 by Taylor & Francis Group, LLC TABLE 5.13 AR Model Coefficients for Blocks of Pixels in Mammogram mdb023 211 2089_book.fm Page 212 Tuesday, May 10, 2005 3:38 PM 212 Medical Image Analysis FIGURE 5.7 The mammogram with the mass marked: mdb028 5.8.1.3 Case 3: mdb058 The mammogram mdb058 is shown in Figure 5.8 A malignant mass is found in the square centered at (318, 359) with