Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 5)

Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 5)

5 Texture Characterization Using Autoregressive Models with Application

to Medical Imaging

Sarah Lee and Tania Stathaki

CONTENTS

5.1 Introduction5.1.1 One-Dimensional Autoregressive Modeling for BiomedicalSignals

5.1.2 Two-Dimensional Autoregressive Modeling for BiomedicalSignals

5.2 Two-Dimensional Autoregressive Model5.3 Yule-Walker System of Equations5.4 Extended Yule-Walker System of Equations in the Third-OrderStatistical Domain

5.5 Constrained-Optimization Formulation with Equality Constraints5.5.1 Simulation Results

5.6 Constrained Optimization with Inequality Constraints5.6.1 Constrained-Optimization Formulation with InequalityConstraints 1

5.6.2 Constrained-Optimization Formulation with InequalityConstraints 2

5.6.3 Simulation Results5.7 AR Modeling with the Application of Clustering Techniques5.7.1 Hierarchical Clustering Scheme for AR Modeling5.7.2 k-Means Algorithm for AR Modeling

5.7.3 Selection Scheme5.7.4 Simulation Results5.8 Applying AR Modeling to Mammography5.8.1 Mammograms with a Malignant Mass5.8.1.1 Case 1: mdb023

5.8.1.2 Case 2: mdb0285.8.1.3 Case 3: mdb058

2089_book.fm Page 185 Tuesday, May 10, 2005 3:38 PM

186 Medical Image Analysis

5.8.2 Mammograms with a Benign Mass5.8.2.1 Case 1: mdb069

5.8.2.2 Case 2: mdb0915.8.2.3 Case 3: mdb1425.9 Summary and ConclusionReferences

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-dimen-sional AR modeling technique has been used extensively for one-dimensional bio-medical signals, and some examples are given in Section 5.1.1 For two-dimensionalbiomedical signals, the idea of applying the two-dimensional AR modeling techniquehas 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 modelingtechnique whose results can be used to describe textured surfaces in images underthe assumption that every distinct texture can be represented by a different set oftwo-dimensional AR model coefficients The conventional Yule-Walker system ofequations 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 ofrealizations, i.e., simulations using the output of a same set of AR model coefficientsbut randomly generated driving input, are sufficiently low However, estimations failwhen large external noise is added onto the system; if the noise is Gaussian, we aretempted to work in the third-order statistical domain, where the third-order momentsare employed, and therefore the external Gaussian noise can be eliminated [1, 2].This method leads to higher variances from the estimated AR model coefficientsobtained from a number of realizations We propose three methods for estimation

of two-dimensional AR model coefficients The first method relates the extendedYule-Walker system of equations in the third-order statistical domain to the Yule-Walker system of equations in the second-order statistical domain through a con-strained-optimization formulation with equality constraints The second and thirdmethods use inequality constraints instead The textured areas of the images are thuscharacterized by sets of the estimated AR model coefficients instead of the originalintensities 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 finalestimation The proposed AR modeling method is also applied to mammography tocompare the AR model coefficients of the block of problematic area with thecoefficients 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 extendedYule-Walker system of equations in the third-order statistical domain, is explained

Texture Characterization Using Autoregressive Models 187

in Section 5.4 The proposed methods — constrained-optimization formulation withequality constraints and constrained-optimization formulations with inequality con-straints — are covered in Sections 5.5 and 5.6, respectively In Section 5.7, twoclustering techniques — minimum hierarchical clustering scheme and k-means algo-rithm — 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 modelingtechnique 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

FOR B IOMEDICAL S IGNALS

The output x[m] of the one-dimensional autoregressive (AR) can be written matically [3] as

to Thonet [6], a time-varying AR (TVAR) model is assumed for HR analysis: “thecomparison of the TVAR coefficients significance rate has suggested an increasinglinearity of HR signals from control subjects to patients suffering from a ventriculartachyarrhythmia.”

The AR modeling technique has also been applied to code and decode theelectrocardiogram (ECG) signals over the transmission between an ambulance and

a hospital [7] The AR model coefficients estimated in the higher-order statisticaldomain are transmitted instead of the real ECG signals The transmission results

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 ECGsignals, but this time the work was concentrated on estimating the AR model ordersfrom some conventional methods for two different mental tasks: math task andgeometric figure rotation Spectral density functions are derived after the order ofthe AR model is obtained, and a neural-network technique is applied to assign thetasks into their respective categories [8]

FOR B IOMEDICAL S IGNALS

The two-dimensional AR modeling technique has been applied to mammography[2, 9–11] Stathaki [2] concentrated on the directionalities of the tissue shown inmammograms, because healthy tissue has specific properties with respect to thedirectionalities “There exist decided directions in the observed X-ray images thatshow the underlying tissue structure as having distinct correlations in some specificdirection of the image plane” [2] Thus, by applying the two-dimensional AR mod-eling technique to these two-dimensional signals, the variations in parameters arecrucial in directionality characterization The AR model coefficients are obtainedwith the use of blocks of size between 2 × 2 and 40 × 40 and different “slices”(vertical, horizontal, or diagonal) (see Section 5.4 for details of slices) The prelim-inary study of a comparative nature on the subject of selecting cumulant slices inthe 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 greaterthe calcification, the greater will be the deviation of the texture parameters of thelesions from the norm [2]

5.2 TWO-DIMENSIONAL AUTOREGRESSIVE MODEL

The two-dimensional autoregressive (AR) model is defined [12] as

(5.2)

where p1×p2 is the AR model order, a ij is the AR model coefficient, and u[m,n] isthe driving input, which is assumed to have the following properties [2, 13]:

1 u[m,n] is non-Gaussian

2 Zero mean, i.e., E{u[m,n]} = 0, where E{⋅⋅⋅⋅} is the expectation operation

3 Second-order white, i.e., the input autocorrelation function isσu2δ[m,n]and σu2= E{u2[m,n]}

4 At least second-order stationary

x m n a x m ij i n j u m n

j p

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 ofstable 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 vectorthat 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 denotestransposition 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 theregion of support of AR model parameters [12]:

Two examples of TNSHP are shown in Figure 5.1 The shape of the dotted linesindicates the region of support when p1 = 1 and p2 = 3, and the shape of the solidlines is for p1 = p2 = 2

FIGURE 5.1 Examples of the truncated nonsymmetric half-plane region of support (TNSHP) for AR model parameters.

S TNSHP=   ={ i j, :i 1 2, ,
,p j1; = −p2,

, ,0 ,p2}∪ ii j i{  =, : 0;j=0 1, ,
,p2}

p1 = 1, p2 = 3

p1 = p2 = 2

i j

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

(5.3)

In Equation 5.3, the second-order moment, which is also regarded as

“autocor-relation,” is defined as Equation 5.4

(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

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 = 2 and p2 = 3, and the shape

filled with horizontal lines is the region of support of QP when p1 = p2 = 1

The Yule-Walker system of equations for a QP model can be written [12] as

Generalizing Equation 5.7 leads to the equations

where M xx is a matrix of size [(p1 + 1)(p2 + 1)] × [(p1 + 1)(p2 + 1)], and a l and h

are both vectors of size [(p1 + 1)(p2 + 1)] × 1

More explicitly, Equation 5.8 can be written as

i

p

2 0 0

h

0 0



a i= [ , , ,a i0 a i1… a ip2]T

These equations can be further simplified because the variance,σu2, is unknown,

and the AR model coefficient a00 is assumed to be 1 in general The Yule-Walkersystem of equations can be rewritten as

(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

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] × 1 vector of unknown AR model coefficients

r is a [(p1 + 1)(p2 + 1) − 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 coefficientswhen the power of the external noise is small compared with that of the signal

00 01 10 11

σσu

2

000

0 0

x

a a a

x

x

x

2 2 2

0 1

1 0

1 1

,,,

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 thatthe autocorrelation function (ACF) samples of a signal are sensitive to additiveGaussian noise because the ACF samples of Gaussian noise are nonzero [1, 15].Estimation of the AR model coefficients using the Yule-Walker system of equationsfor a signal with large external Gaussian noise is poor, therefore we are forced towork 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 two- dimensional signal, y[m,n], 1 ≤ m ≤ M, 1 ≤ n ≤ N, is estimated [1] by

(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 thesecond-order and third-order statistical domain is that in the latter version, wemultiply the output of the AR model by two shifted versions instead of just one inthe earlier version [1] The extended Yule-Walker system of equations in the third-order statistical domain can be written as shown in Equation 5.14 [11]

(5.14)

where γu = E{u3[m,n]} is the skewness of the input driving noise, and a00 = 1.From the derivation of the above relationship, it is evident that using Equation5.14 implies that it is unnecessary to know the statistical properties of the externalGaussian noise, because they are eliminated from the equations following the theorythat the third-order cumulants of Gaussian signals are zero [16] For a two-dimen-

sional AR model with order p1 × p2, we need at least a total of (p1 + 1)(p2 + 1)equations from Equation 5.14, where

where k1 + l1 + k2 + l2 ≠ 0 and k1,l1,k2,l2 ≥ 0 In this form, [(p1 + 1)(p2 + 1) − 1]

equations are required to determine the a ij parameters (for details, see the literature[17–21])

When the third-order cumulants are used, an implicit and additional degree offreedom 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, asshown 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 C 3y (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]

(5.16)

FIGURE 5.3 Different third-order cumulant slices for a one-dimensional signal.

Vertical Slice Horizontal Slice

2 1

C3y(i−k j1, −l1 −,i k2,j−l2)=C3y(i−k j1, −ll1 − −,i k1 c j1, − −l1 c2)

By applying the symmetry properties of cumulants we obtain

(5.17)

Let k = c1 + k1 and l = c1 + l1 Hence, the equations above take the form [2, 10, 11]

(5.18)

The extended Yule-Walker system of equations in the third-order statistical domain

is formed from Equation 5.18, with

2 1

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, we obtain

Let us write the system of equations for the model order p1 × p2 by

where

C is a [(p1 + 1)(p2 + 1) − 1] × [(p1 + 1)(p2 + 1) − 1] matrix of third-order cumulants

a is a [(p1 + 1)(p2 + 1) − 1] × 1 vector of unknown AR model coefficients

c is a [(p1 + 1)(p2 + 1) − 1] × 1 vector of third-order cumulants

In theory, everything seems to work properly However, in practice, one of themain problems we face when we work in the third-order statistical domain is thelarge variances that arise from the cumulant estimation [2]

5.5 CONSTRAINED-OPTIMIZATION FORMULATION

WITH EQUALITY CONSTRAINTS

A method for estimating two-dimensional AR model coefficients is proposed in thissection The extended Yule-Walker system of equations in the third-order statisticaldomain is related to the conventional Yule-Walker system of equations through aconstrained-optimization formulation with equality constraints [23] The Yule-Walker system of equations is used in the objective function, and we consider most

of the extended Yule-Walker system of equations in the third-order statistical domain

as the set of constraints In this work only, the last row of the extended Yule-Walkersystem of equations in the third-order statistical domain is eliminated The last row

is chosen after some statistical tests were carried out Eliminating any other rows

in this case did not lead to robust estimations It can be written mathematically [23] as

subject to C l a = −c l (5.22)where

y y

w = number of rows in matrix R in Equation 5.12

R i = ith row of the matrix R in Equation 5.12

r i = ith element of the vector r in Equation 5.12

and where C l is defined as matrix C in Equation 5.21 without the last row, c l is

defined as matrix c in Equation 5.21 without the last row, and a is a [(p1 + 1)(p2 +1) − 1] × 1 vector of unknown AR model coefficients We use sequential quadraticprogramming [24] to solve Equation 5.22

Two types of synthetic images of size 256 × 256 are generated for simulation purpose.The first one is a 2 × 2 AR symmetric model, which can be expressed as follows

Another type of synthetic image is created using a set of 2 × 2 nonsymmetric

AR model coefficients and is expressed as

The input driving noise to both systems is zero-mean, exponential-distributedwith variance σw2 = 0.5 The final image, y[m,n], is contaminated with external Gaussian noise, v[m,n], where y[m,n] = x[m,n] + v[m,n] The noise has zero mean

and unity variance The signal-to-noise ratio (SNR) of the system is calculated usingthe following equation

(5.23)

where σx2is the variance of the signal and σv2is the variance of the noise

The estimation results are evaluated using a relative error measurement defined

in the following equation [24]

a a a

ij j p

where is the estimated AR model coefficient, a ij is the original AR model

coef-ficient, and p1 × p2 is the AR model order

The simulation results obtained from 100 realizations can be found in Table 5.1for the symmetric model and in Table 5.2 for the nonsymmetric model In Table 5.1,the simulation results show that the proposed method is able to estimate symmetric

AR model coefficients in both low- and high-SNR systems The variances for the

100 realizations are small, particularly in the case of high-SNR system Similarperformance is obtained when the method is applied to the nonsymmetric AR model

Estimated Value

Variance (10 −−−−3 )

Estimated Value

Variance (10 −−−−3 )

Estimated Value

Variance (10 −−−−3 )

Estimated Value

Variance (10 −−−−3 )

5.6 CONSTRAINED OPTIMIZATION WITH

INEQUALITY CONSTRAINTS

Based on the constrained optimization with equality constraints method, two ods that use both the Yule-Walker system of equations and the extended Yule-Walkersystem of equations in the third-order statistical domain are proposed through con-strained-optimization formulations with inequality constraints Mathematically, itcan be written as

meth-subject to

−εεεε ≤ Ca + c ≤ εεεε (5.25)where

w = number of rows in matrix R in Equation 5.12

R i = ith row of the matrix R in Equation 5.12

r i = ith element of the vector r in Equation 5.12

a = a [(p1 + 1)(p2 + 1) − 1] × 1 vector of unknown AR model coefficients

and where C and c are as derived in Equation 5.21 and εεεε is defined as shown below.

Inequality constraints are introduced with an additional vector, εεεε Two methodsfor estimating εεεε are proposed here, and both are related to the average differencebetween the estimated AR model coefficients of each block and the average ARmodel coefficients of all the blocks We use sequential quadratic programming [24]

1 Divide the image into a number of blocks with a fixed size, z1 × z2, so

that B1 × B2 blocks can be obtained

2 Estimate the AR model coefficients of each block using the extended Walker system of equations in the third-order statistical domain in Equa-tion 5.21

Yule-3 From all of the AR model coefficient sets obtained, calculate the average

AR model coefficients, a A , [i, j] ≠ [0,0].

4 The ε value is calculated using the following equation

b B

b

B

2 2

2

1 1

1 1

where B1 × B2 is the number of blocks available, (b1, b2) is the block index,

is the matrix C in Equation 5.21 for the block (b1, b2), is the

vector c in Equation 5.21 for the block (b1, b2), and sum indicates the

sum-mation of all the items in a vector The vector, εεεε, is defined as εεεε = [ε,…,ε]T,

which is a [(p1 + 1)(p2 + 1) − 1] × 1 vector

5 Apply Equation 5.25 to obtain the AR model coefficient estimation

where is the (i × p1 + j)-th value of the vector

[(p1 + 1)(p2 + 1) − 1] × 1 vector

As shown in Section 5.1, the constrained-optimization formulations with inequality

constraints are applied to the output — y[m,n], 1 ≤ m ≤ 256, 1 ≤ n ≤ 256 — of both

the two-dimensional symmetric and nonsymmetric AR models shown below, tively

1

1

1 1

The output y[m,n] = x[m,n] + v[m,n], where v[m,n] is the additive Gaussian noise

with zero mean and unity variance

The results obtained using two different types of ε values are shown in thefollowing tables For the symmetric model, the results obtained from 100 realizationsfor the constrained-optimization formulation with inequality constraints 1 can befound in Table 5.3, and the results from the same formulation with inequalityconstraints 2 can be found in Table 5.4 and Table 5.5 for SNR equal to 5 and 30

dB, respectively For the nonsymmetric model, the results can be found in Table 5.6,Table 5.7, and Table 5.8 in the same order as for the symmetric model The ε values

of the constrained-optimization formulation with inequality constraints 1 is 9.0759

× 10−4 for the case of SNR equal to 5 dB and 6.8434 × 10−5 for the case of SNRequal to 30 dB for the symmetric model For the nonsymmetric model, the equivalentvalues are 8.2731 × 10−4 and 5.9125 × 10−5 The average ε values for each coefficientare also shown in the tables for both methods with constraint optimization withinequality constraints 2 (Table 5.4 and Table 5.5 for the symmetric model and Table5.7 and Table 5.8 for the nonsymmetric model)

From Table 5.3 and Table 5.6, the AR model coefficients — estimated forsymmetric and nonsymmetric models, respectively, using the constrained-optimiza-tion formulation with inequality constraints 1 — show high accuracy, as evidenced

Estimated Value

Variance (10 −−−−4 )

Estimated Value

Variance (10 −−−−4 )

by the small relative error in both low- and high-SNR systems In Table 5.4 andTable 5.7, the estimated results for the constrained-optimization formulation (withinequality constraints 2 and a 5-dB SNR for both the symmetric and nonsymmetric

AR models) are very close to the original AR model coefficient values except for

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

Estimated Value

Variance (10 −−−−3 )

Parameter

Real Value

Estimated Value

Variance (10 −−−−3 )

the coefficient a22 (whose variance for the 100 realizations of this coefficient is alsogreater than other coefficients) In the high-SNR system, as shown in Table 5.5 and

Table 5.8 for the symmetric and nonsymmetric AR models, respectively, the relativeerrors obtained are even smaller than in the low-SNR system, and the average εvalue for each coefficient is smaller than in the low-SNR system

Estimated Value

Variance (10 −−−−3 )

Estimated Value

Variance (10 −−−−3 )

Parameter

Real Value

Estimated Value

Variance (10 −−−−3 )