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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 4, AUGUST 1998 541

Characterization of Visually Similar Diffuse Diseasesfrom B-Scan Liver Images Using Nonseparable

Wavelet TransformAleksandra Mojsilovic,* Miodrag Popovic, Member, IEEE, Srdjan Markovic, and Miodrag Krstic

Abstract This paper describes a new approach for texturecharacterization, based on nonseparable wavelet decomposition,and its application for the discrimination of visually similardiffuse diseases of liver. The proposed feature-extraction algo-rithm applies nonseparable quincunx wavelet transform and usesenergies of the transformed regions to characterize textures.Classification experiments on a set of three different tissue typesshow that the scale/frequency approach, particularly one basedon the nonseparable wavelet transform, could be a reliablemethod for a texture characterization and analysis of B-scan liverimages. Comparison between the quincunx and the traditional

wavelet decomposition suggests that the quincunx transformis more appropriate for characterization of noisy data, andpractical applications, requiring description with lower rotationalsensitivity.

Index Terms Classification, quincunx sampling, texture,wavelet transform.

I. INTRODUCTION

DIAGNOSTIC ultrasound has been an useful clinical tool

for imaging organs and soft tissues in the human body,

for more than two decades [1]. Currently, one of its impor-

tant applications is imaging of the liver, and many methods

for differentiation between normal and abnormal tissues arebased on the examination of B-scan images. However, for

diagnosing diffuse diseases, such as cirrhosis, particularly in

its early phase, steatosis, or hepatitis, clinical ultrasound is

not reliable enough, and accurate identification is usually

performed by needle biopsy. Although these diffuse diseases

are quite different, the main obstacle for diagnosing them

is very subtle visual difference between their sonograms.

For example, echosonographic images of steatosis and early

cirrhosis are very similar and it is very difficult, even for

an experienced clinician, to perform the diagnosis about the

existence, type, and the level of a disease. Therefore, a reliable

noninvasive method for early detection and differentiation of

these two diseases is clearly desirable. One possible approachcan be found in texture analysis, because steatosis and cirrhosis

Manuscript received April 29, 1997; revised July 21, 1998. The AssociateEditor responsible for coordinating the review of this paper and recommendingits publication was A. Manduca. Asterisk indicates corresponding author.

*A. Mojsilovic is with Bell Laboratories, Lucent Technologies, 600 Moun-tain Avenue, Murray Hill, NJ 07972 USA (e-mail: [email protected]).

M. Popovic and S. Markovic are with the School of Electrical Engineering,University of Belgrade, 11011 Belgrade, Yugoslavia.

M. Kristic is with the Institute for Digestive Diseases, Clinical Center of Serbia, 11011 Belgrade, Yugoslavia.

Publisher Item Identifier S 0278-0062(98)08545-0.

Fig. 1. Liver tissue samples taken from different patients and from differentparts of liver. First row: normal tissue samples; second row: cirrhosis samples;third row: steatosis samples.

produce different changes in acoustical properties of the liver

tissue, which can be detected by ultrasound as a textural pattern

different from the normal one (see Fig. 1). In addition to visual

interpretation of B-scan images, texture data indicating the

possible presence and the extent of illness, could provide more

reliable diagnosis, and may eventually help to avoid the use of

biopsy for identifying diffuse liver diseases. Many researchers

have studied the problem of liver tissue classification [2]–[11].

Initial attempts to characterize diffuse diseases have utilized

different signal processing techniques in order to obtain useful

information from the raw radio-frequency signal [2], [3]. In

a series of papers, Momenan et al. showed that second-order statistical parameters from envelope-detected or intensity

echo signals have discriminatory power in human liver [4],

[5]. Some researchers have treated the task of liver tissue

quantification from the point of description and classification

with numerical texture measures. Nicholas et al. were among

the first who used textural features of the B-scan images,

showing their potential to discriminate between livers and

spleens of normal humans [6]. Wu et al. [7] have applied

fractal-based statistics and compared them with other texture

measures, for distinguishing between hepatoma, cirrhosis, and

0278–0062/98$10.00 © 1998 IEEE

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MOJSILOVIC et al.: CHARACTERIZATION OF VISUALLY SIMILAR DIFFUSE DISEASES 543

a signal can be computed via the following analysis and

synthesis formulas:

(4)

The mother wavelet can be constructed from the scalingfunction as

(5)

where . In the wavelet literature [16],

[17], many different sets of coefficients can be found,

corresponding to wavelet bases with different properties. In

the case of the discrete wavelet transform (DWT), coefficients

play an important role, since they can be used for the

DWT computation, instead of the explicit forms for and

. It is shown [18] that, starting from the original signal, discrete signals [the approximation

of at resolution ] and (information content lost

between higher resolution and lower resolution ) can

be computed as

(6)

where and . This decomposition

can be understood as passing of signal through a pairof low-pass and high-pass filters and , followed by

the subsampling with a factor two.

IV. WAVELET TRANSFORM IN TWO DIMENSIONS:

CONNECTION TO THE TEXTURE ANALYSIS

There are various extensions of one-dimensional (1-D)

wavelet transform to two dimensions. The simplest way to gen-

erate a 2-D wavelet transform is to apply two 1-D transforms

separately. Thus, image decomposition can be computed with

separable filtering along the abscissa and ordinate, by using

the same pyramidal algorithm as in the 1-D case [18]. This

corresponds to the case of separable sampling described by thesampling matrix in (1). As shown in Fig. 3, this separable

transform (ST) decomposes images with a multiresolution

scale factor of two, providing at each resolution level one

low-resolution subimage and three spatially oriented wavelet

coefficient subimages. Another solution for the application

of the wavelet transform to higher dimensions is to use

nonseparable sampling and nonseparable filters. The simplest

transform of that type, known as quincunx transform (QT),

uses nonseparable and nonoriented filters, followed by the

nonseparable sampling represented by the matrix in (1).

Hence, the scaling function and corresponding wavelet family

Fig. 3. The division of spectrum after two iterations of the traditional dyadicwavelet decomposition.

Fig. 4. Filter bank performing the quincunx wavelet transform.

are

(7a)

(7b)

Since this transform is performed with the two-

channel filter bank shown in Fig. 4. The Fourier expression

for the output of channel is

(8)

where

and

(9)

are coset and modulation vectors for the case of quincunx

sampling. For more general applications, such as texture

synthesis, to assure the cancellation of the aliasing terms at the

output of the analysis/synthesis filter bank [14], the high-pass

filter should be designed as

(10)

This decomposition results in one low-resolution subimage

and one nonoriented wavelet subimage. Fig. 5 illustrates the

idealized partition of the frequency domain after four iter-

ations of quincunx decomposition. At each level, the input

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544 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 4, AUGUST 1998

Fig. 5. The division of spectrum after four iterations of the quincunxtransform.

image is decomposed with the multiresolution scale factor .

This is very nice property for description of small textured

images, since the analysis is twice as fine as the separable

multiresolution decomposition. The spectral decompositions

shown in Figs. 3 and 5 suggest several advantages of thequincunx transform for tissue characterization from B-scan

images. First, the separable sampling provides only rectangular

divisions of spectrum, with increased sensitivity to horizontal

and vertical edges. This could be important for the analysis

of directional textures, but it yields a rotationally sensitive

description, which is not desirable in this application. Due to

the shape of the low-pass and high-pass filters, it should be

expected that the quincunx decomposition has lower orienta-

tion sensitivity than separable decomposition. Still, it is not

completely rotationally insensitivefollowing the rotation of

the QT Voronoi cell around the origin, rotational sensitivity

increases up to 45 (where it reaches maximum), and then

decreases again, reaching the complete invariance for 90 .

Second, the energy of natural textures is mainly concentrated

in the mid-frequencies, with the insignificant energy along

diagonals. Therefore, the quincunx low-pass filter will preserve

more of the original signal energy, and its implementation in

the iterated filter bank could provide more reliable description

of texture. Finally, the diamond shape of the low-pass filter

in the quincunx case, plays the crucial role for the extraction

of texture features in the presence of noise, since it cuts off

diagonal high frequencies, where the most significant portion

of noise is contained. Thus, when working with noisy samples

(as in our case) the spectral decomposition performed on the

quincunx lattice represents a better solution than traditionalapproach based on the separable sampling.

V. DESIGN OF 2-D DIAMOND-SHAPED FILTERS

When constructing a filter bank performing the DWT for

the texture characterization, a number of design requirements

have to be fulfilled. First, since images are mostly smooth, the

analysis should be performed with a smooth mother wavelet.

On the other hand, to achieve fast computation, the filters have

to be short, affecting the smoothness of the associated wavelet.

In more general applications, such as texture synthesis, it

would be nice to construct a perfect reconstruction filter bank,

leading to the selection of orthogonal bases. Furthermore, the

filters should be symmetric, so they can be easily cascaded

without any additional phase compensation. For a two-channel,

real, finite impulse response case, linear phase and orthogo-

nality are mutually exclusive, but by using 2-D biorthogonal

filters it is possible to relax the orthogonality requirement,

yet preserving other important characteristics [16]. Finally, in

order to achieve the computation of the continuous wavelet

transform by iterating the low-pass branch of the filter bank,

the low-pass filter with the sufficient number of zeros at the

points of replicated spectra has to be used [17]. Unfortunately,

due to the difficult design of nonseparable filters there are

only few solutions satisfying all these properties. Therefore,

we have decided to apply the McClellan transform [19]

(11)

and to map coefficients of the selected 1-D filter into a 2-D

filter defined on the quincunx lattice. The transform obtained

ensures that all properties of 1-D filters are also satisfied in the

2-D case. Derivation of the transform is given in the Appendix.

VI. TISSUE CLASSIFICATION

A. Data Acquisition

The ultrasound images used in this research were obtained

on Toshiba SSA-100 equipment, with 3.5-MHz transducer

frequency. A Series 151 Image processor, from Imaging

Technology Inc., on IBM-PC, was used to capture images with

512 512 pixels and 256 gray-level resolution. Three sets of

images have been taken: normal (37 images from ten subjects),

steatosis (65 images from ten patients), and cirrhosis (20

images from ten patients). Since this study addresses the issue

of diagnosing early cirrhosis and steatosis, we have constrainedthe selection of images to those from patients in early stage

of the disease. From each image, two blocks of 64 64 pixels

(approximately 2 cm 2 cm in actual dimensions) have been

selected. Blocks were chosen to include only liver tissue,

without blood vessels, acoustic shadowing, or any type of

distortion. In that way, the whole data set contained 244 tissue

samples for training and classification. The training and test

sets (each with 122 samples) were composed out of all blocks

from independent images.

In patients with steatosis and cirrhosis, the final diagno-

sis was confirmed by liver biopsy and histology, since this

presents the gold standard for diagnosing diffuse diseases. The

needle diameter was 1.6 mm, and the obtained tissue cylinderswere at least 2 cm.

The estimation of a texture quality was performed with the

four-level quincunx decomposition, yielding feature vectors

with maximal length of five. The size of the smallest subimage

in the quincunx pyramid is 16 16 pixels and further decom-

position would yield unreliable estimates of texture quality.

Fig. 6 shows the first three levels of the quincunx pyramid,

for one representative sample from each tissue class. The

decomposition was performed by using the three biorthogonal

families that provide short filters with the best tradeoff between

the filter lengths and the regularity of corresponding wavelet.

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(a)

(b)

(c)

Fig. 6. Quincunx decomposition of: (a) cirrhosis tissue, (b) normal tissue,and (c) steatosis tissue.

A brief description and 1-D filter coefficients for each family

are given in Table I. Table II presents coefficients of the

corresponding diamond-shaped filter. To illustrate the 1-D to2-D mapping, the frequency responses of the second family

and its diamond-shaped counterparts are shown in Fig. 7.

B. Texture Feature Extraction

Since the filter bank performing the QT represents one spe-

cial case of the local linear transform approach for the texture

characterization, iterations of the quincunx decomposition

can be seen as a -channel filter bank, whose outputs

serve for the estimation of texture quality

in the corresponding frequency subband. The texture is then

characterized by the set of first-order probability density

functions estimated at the output of each channel. More

compact representation can be achieved using the channel

variances var . The statistical justification for this approach

can be found in [20]. Another, psychophysical justification was

offered by Pratt et al. [21], who showed that natural textures

are visually indistinguishable if they possess the same first

and second-order statistics. Also, the reliability of channel

variances has been proved in numerous wavelet-based texture

characterization algorithms [11], [12], [22], [23]. Since the

high-pass conditions and imply

that , for , channel variances are

represented through energies calculated at

the output of each channel with

(12)

where represents the image field, # denotes the number

of pixels in , and stands for wavelet

coefficient subimages obtained at the th decomposition level.

Finally, due to the low-pass condition , to obtain

the variance in the last low-pass channel, the mean of

the image should be removed.

C. Classification

Assuming that feature vectors for normal, cirrhosis, and

steatosis tissue have approximately normal distribution (with

mean vector and covariance matrix ), the minimum error

discriminant function between the feature vector and class

is the Bayes distance

(13)

For each texture class, the mean and the covariance were

estimated using 122 texture samples with the leave-one-out

method [24]. The remaining samples were then classified.

VII. RESULTS

The classification results obtained using all features of

quincunx decomposition are presented in Table III. The overall

classification accuracy is 90%. Specificity of the method was

92%. Sensitivity in the detection of cirrhosis and steatosis

were: 92 and 97%, respectively. These results correspond todecomposition with the second pair of filters (see Table II,

set 2). Similar results were obtained with the other two

sets, indicating that the texture classification scheme in this

application is almost insensitive to the selection of the wavelet

base.

As already mentioned in the data-acquisition section, the

training and classification sets were formed from independent

images. Therefore, it is to be expected that in the classification

phase both blocks from one image should be diagnosed

identically. This is true for normal and steatosis classes, but

in the cirrhosis case, one block from an image was diagnosed

differently from the other. More detailed examination showed

that this one, as well as many other misclassified samples wasaffected by noise. Some of them had inhomogeneous structure

due to small blood vessels present in that part of liver. Hence,

it is obvious that the location of the region of interest (ROI)

within an image has dominant effect on the classification. This

issue is addressed in more detail in Section VIII.

A. Comparison Between Separable and

Nonseparable Transform

To compare the performance of the nonseparable quincunx

decomposition with the performance of the separable wavelet

transform, we have also performed two-level ST with the

same filter family (Table I, set 2), on the same data set. Thisdecomposition results in maximum of seven features and the

same size of the smallest subimage. Using the same method

for training and classification and using the complete feature

set resulted in an 88% of overall classification accuracy. By

decreasing the number of features from five to three (for

QT), and from seven to three (for ST), the classification

rate remained constant in the quincunx case, whereas in the

ST case the overall classification rate decreased to 82%. By

inspection of misclassified samples, we have found that almost

all samples that were misclassified with the QT, were also

misclassified using the ST.

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TABLE IFILTER COEFFICIENTS FOR THE 1-D BIORTHOGONAL WAVELET BASIS USED FOR THE DESIGN OF 2-DDIAMOND-SHAPED FILTERS. THE FIRST FAMILY BELONGS TO THE SO-CALLED “SPLINE FILTERS” OR

“BINOMIAL FILTERS” IN [15], BECAUSE THE UNDERLYING SCALING FUNCTION IS A B-SPLINE. THE SECOND

FAMILY IS ALSO A “SPLINE VARIANT” BUT WITH LESS DISSIMILAR LENGTHS (AS OPPOSED TO BINOMIAL

FILTERS). THE EXAMPLE WE HAVE USED HAS THE SHORTEST FILTERS WITHIN

THIS FAMILY. THE THIRD SET OF COEFFICIENTS BELONGS TO FILTERS CLOSE TO

ORTHONORMAL FILTERS WHERE SCALING FUNCTION AND WAVELETS ARE VERY SIMILAR

TABLE IIFILTER COEFFICIENTS FOR THE 2-D BIORTHOGONAL DIAMOND-SHAPED FILTERS

(a)

(b)

Fig. 7. Magnitudes of the frequency response for: (a) 1-D low-pass andhigh-pass prototype filters and (b) their 2-D counterparts. The filters corre-spond to biorthogonal spline families with low-pass and high-pass filters of similar length. The coefficients of 1-D prototypes correspond to the secondfamily in Table I.

TABLE IIICLASSIFICATION RESULTS OBTAINED WITH THE QUINCUNX DECOMPOSITION

normal cirrhosis steatosis overa ll

misclassification/total 4/37 1/20 6/65 11/22

classification rate 89% 95% 91% 90%

B. Sensitivity to Rotation

We were also interested in testing the ability of nonseparable

decomposition to classify textures with respect to rotation.In the second experiment, without changing already esti-

mated class prototypes, we have classified texture samples

rotated at 5 , 23 , 45 , and 90 . Samples for the experiment

were obtained in the following manner. First, the whole

image was rotated for the determined degree, and then two

tissue samples were taken from the same positions as in

the previous experiment. The rotation was performed using

a bicubic interpolation. The QT yielded overall accuracy

of 88%, 88%, 80%, and 90% (for 5 , 23 , 45 , and 90 ,

respectively), whereas ST gave 82%, 81%, 56%, and 81%.

These results follow from the analysis in Section IV. It is

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obvious that the quincunx decomposition performs better

than the separable decomposition, but the complete rotation

invariance is not achieved since the classification accuracy is

not maintained at different angles. One way to achieve the

complete invariance is to use any circularly shaped spectral

decomposition, such as hexagonal decomposition, steereable

pyramid, or Gabor transform [25], [26]. Each of these trans-

forms has very fine orientation selectivity and produces a

number of features that correspond to a certain orientation.

Hence, for the selected transform, by averaging all features

from the same resolution level, we will produce one rota-

tionally insensitive feature. Unfortunately, by achieving the

rotational insensitivity, we are loosing valuable features in

terms of general description, and the overall accuracy of

the method can decrease. Another solution would be the

construction of tissue prototypes for different orientations.

For example, instead of one cirrhosis class we would have

classes, for different degrees of orientation. In that case,

instead of classification into three groups, we would perform

classification into groups. Although promising, for reliable

estimation of class prototypes, this approach requires a hugeamount of data and even more sophisticated feature-extraction

techniques.

C. Comparison with Other Texture Description Methods

To compare the performance of different feature-extraction

techniques in diagnosing early diffuse diseases, we have

compared the wavelet-based approach with: 1) gray-level

concurrence (GLC) matrices, as defined in [7], and [10]; 2)

fractal texture measures [7]; and 3) Fourier measures [7].

In this experiment, using the same data set, the following

measures have been calculated.

1) Contrast, angular second moment, and correlation for theGLC method [7], [10]. The GLC matrices were formed

for all combination of displacement and

and angles and .

2) Ring and wedge energies, derived from

the Fourier power spectrum [6], for all combinations

and

.

3) Multithreshold fractal features proposed for liver tissue

classification as defined in [7].

The highest overall classification accuracy was: 87% for the

GLC method, 82% for Fourier measures, and 69% for fractal

measures, indicating the superiority of the wavelet approach.

VIII. DISCUSSION AND CONCLUSION

The goal of this research was the detection of an optimal

feature-extraction technique for description of different ultra-

sound textures. Therefore, we have investigated the utility

of wavelet decompositions as feature-extraction methods in

discrimination among diffuse liver diseases. We have applied

a nonseparable quincunx transform with a multiresolution

scale factor , traditional approach based on the separable

wavelet transform, and compared them with the previously

used approaches. The classification results obtained with both

wavelet transforms are very promising. Based on the ex-

perimental results, we conclude that the quincunx transform

is more appropriate for characterization of noisy data, and

for practical applications requiring description with lower

rotational sensitivity. The results obtained in the study, clearly,

illustrate the utility of sophisticated image processing algo-

rithms in the ultrasound diagnostics of liver. However, for

the general application of the method, many issues still have

to be developed. It is obvious from our results, as well

as from the previous work [10], [27], that the location of

the ROI within an image has the dominant effect on the

classification. This is due to the fact that ultrasound pattern

exhibits different behavior along the path of the acoustic pulse,

resulting in the prominently different distribution of horizontal

edges along axis of an image. Also, the lateral speckle pattern

is strongly depth dependent [27]. Finally, echocardiogram is

formed according to the sector, while images are sampled

along the rectangular grid. During the selection of tissue

samples we have done our best to choose the data set as

homogenous as possible. This was done by taking the samples

from an isolated region in the center of an image withoutgreat changes in the cursor position, but our selection was

severely constrained by the presence of major blood vessels

and acoustic shadowing. It should be also mentioned that the

images were recorded by different clinicians who were free

to adjust ultrasound settings (within the acceptable range) in

order to visually optimize images. We feel that this degree

of freedom is required, so the whole method could follow

natural processing of each individual physician, but it is to be

expected that a more rigorous selection of tissue samples could

increase classification accuracy. At this point, the advantages

of the wavelet approach over other methods become obvious.

First, since the mean values of all DWT channels are zero, the

changes in the gain setting could not affect the results. Also,it has been shown that the wavelet transform is powerful ana-

lytical tool for assessment of the singularities [16], [17]. Since

visual differences between the early diffuse diseases are hardly

distinguishable, due to minor changes in tissue structure, the

wavelet transform is appropriate for “zooming” into those dif-

ferences. Furthermore, the ultrasound noise is a high-frequency

phenomenon, mostly affecting a single image point. Hence, by

eliminating from the classification high-frequency channels at

the highest resolution level, the impact of ultrasound noise

can be significantly reduced. Finally, the proposed feature

method is independent of the classification technique. The

wavelet-based texture measures can be used with minimum-

distance-type classifiers, Bayesian-type classifiers, or as inputsto a neural network. Since previous investigations showed that

the classification technique has significant influence to the final

diagnosis [8], [10], further research should be performed in

that direction.

APPENDIX

DERIVATION OF THE MCCLELLAN TRANSFORM

FOR DESIGN OF DIAMOND-SHAPED FILTERS

The mapping of a 1-D zero-phase symmetric filter

into a 2-D filter starts with the Fourier

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transform of written as a function of

(A1)

(A2)

where , , and , and

is the length of the filter. The next step is to write

as a polynomial of . This can be achieved

by using the Chebyshev polynomials, since can be

written as , where is the th-order Chebyshev

polynomial. Hence

(A3)

The idea of the McClellan transform is to replace by a

zero-phase 2-D filter . This will result in an overall

zero-phase 2-D filter

(A4)

Typically is chosen as

(A5)

Therefore, the transform maps the frequency re-

sponse of a 1-D filter to the contours in plane. The

shape of the obtained contours is determined only by the

transformation parameters , which give the structure of

a function . In this section we will consider the problem

of choosing the transformation parameters so that the contours

produced by the transformation, for various const, have a

diamond shape in plane. In that case, the coefficients

for the low-order transformation can be determined as follows.

First, we have to impose the structure of the transformation.

The simplest form for , in accordance with the

lattice points of the quincunx sampling scheme, is

(A6)

Since our objective is to design a 2-D low-pass filter based on

the 1-D low-pass prototype, we must require that 1-D origin

maps to 2-D origin. This gives the first constraint to the filter

(A7)

In the context of wavelet-based filter banks, we would like to

transform a 1-D filter into a 2-D one, such that zeros at aliasing

frequencies , are preserved

(A8)

By substituting an aliasing frequency into (A8) we

obtain the additional constraint for the filter design as

(A9)

Since the shape of the low-pass filter produced with the

transform has to follow the shape of the Voronoi cell for the

quincunx lattice, we impose the band-edge-contour constraint

to minimize the following error function:

(A10)

where corresponds to the Voronoi cell of the quincunx

lattice. In this particular case we minimize under

the constraints (A7) and (A9). Consequently, the minimization

of the error function is the linear problem. Hence, we obtain

the simplest, first-order transform for a design of diamond-

shape filters

(A11)

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