Bản chất của hình ảnh y sinh học (Phần 7)

Although periodic texture may be modeled as repetitions of textons, not many methods have been developedfor the structural analysis of texture 444].. In this chapter, we shall explore th

Analysis of T extur e

Texture is one of the important characteristics of images, and texture analysis

is encountered in several areas
432, 433, 434, 435, 436, 437, 438, 439, 440, 441,442] We nd around us several examples of texture: on wooden furniture,cloth, brick walls, oors, and so on We may group texture into two generalcategories: (quasi-) periodic and random If there is a repetition of a textureelement at almost regular or (quasi-) periodic intervals, we may classify thetexture as being (quasi-) periodic or ordered the elements of such a texture arecalled textons
438] or textels Brick walls and oors with tiles are examples

of periodic texture On the other hand, if no texton can be identied, such

as in clouds and cement-wall surfaces, we can say that the texture is random.Rao
432] gives a more detailed classication, including weakly ordered ororiented texture that takes into account hair, wood grain, and brush strokes

in paintings Texture may also be related to visual and/or tactile sensationssuch as neness, coarseness, smoothness, granularity, periodicity, patchiness,being mottled, or having a preferred orientation
441]

A signicant amount of work has been done in texture characterization
441,

442, 439, 432, 438] and synthesis
443, 438] see Haralick
441] and Haralickand Shapiro
440] (Chapter 9) for detailed reviews According to Haralick

et al
442], texture relates to information about the spatial distribution ofgray-level variation however, this is a general observation It is important

to recognize that, due to the existence of a wide variety of texture, no singlemethod of analysis would be applicable to several dierent situations Sta-tistical measures such as gray-level co-occurrence matrices and entropy
442]characterize texture in a stochastic sense however, they do not convey aphysical or perceptual sense of the texture Although periodic texture may

be modeled as repetitions of textons, not many methods have been developedfor the structural analysis of texture
444]

In this chapter, we shall explore the nature of texture found in biomedicalimages, study methods to characterize and analyze such texture, and investi-gate approaches for the classication of biomedical images based upon texture

We shall concentrate on random texture in this chapter due to the extensiveoccurrence of oriented patterns and texture in biomedical images, we shalltreat this topic on its own, in Chapter 8

7.1 Texture in Biomedical Images

A wide variety of texture is encountered in biomedical images Oriented ture is common in medical images due to the brous nature of muscles andligaments, as well as the extensive presence of networks of blood vessels, veins,ducts, and nerves A preferred or dominant orientation is associated with thefunctional integrity and strength of such structures Although truly periodictexture is not commonly encountered in biomedical images, ordered texture

tex-is often found in images of the skins of reptiles, the retina, the cornea, thecompound eyes of insects, and honeycombs

Organs such as the liver are made up of clusters of parenchyma that are

of the order of 1;2 mm in size The pixels in CT images have a typicalresolution of 1
1mm, which is comparable to the size of the parenchymalunits With ultrasonic imaging, the wavelength of the probing radiation is ofthe order of 1;2mm, which is also comparable to the size of parenchymalclusters Under these conditions, the liver appears to have a speckled randomtexture

Several samples of biomedical images with various types of texture areshown in Figures 7.1, 7.2, and 7.3 see alsoFigures 1.5, 1.8, 9.18, and 9.20

It is evident from these illustrations that no single approach can succeed incharacterizing all types of texture

Several approaches have been proposed for the analysis of texture in ical images for various diagnostic applications For example, texture mea-sures have been derived from X-ray images for automatic identication ofpulmonary diseases
433], for the analysis of MR images
445], and processing

med-of mammograms
165, 275, 446] In this chapter, we shall investigate the ture of texture in a few biomedical images, and study some of the commonlyused methods for texture analysis

na-7.2 Models for the Generation of Texture

Martins et al
447], in their work on the auditory display of texture in ages (seeSection 7.8),outlined the following similarities between speech andtexture generation The sounds produced by the human vocal system may begrouped as voiced, unvoiced, and plosive sounds
31, 176] The rst two types

im-of speech signals may be modeled as the convolution im-of an input excitationsignal with a lter function The excitation signal is quasi-periodic when weuse the vocal cords to create voiced sounds, or random in the case of unvoicedsounds Figure 7.4 (a)illustrates the basic model for speech generation

Speech signal Unvoiced

Textured image Random

Ordered (a)

(b)

FIGURE 7.4

(a) Model for speech signal generation (b) Model for texture synthesis produced with permission from A.C.G Martins, R.M Rangayyan, and R.A.Ruschioni, \Audication and sonication of texture in images", Journal ofElectronic Imaging,10(3): 690 { 705, 2001 cSPIE and IS&T

Re-Texture may also be modeled as the convolution of an input impulse eldwith a spot or a texton that would act as a lter The \spot noise" model ofvan Wijk
443] for synthesizing random texture uses this model, in which theFourier spectrum of the spot acts as a lter that modies the spectrum of a2D random-noise eld Ordered texture may be generated by specifying thebasic pattern or texton to be used, and a placement rule The placement rulemay be expressed as a eld of impulses Texture is then given by the convolu-tion of the impulse eld with the texton, which could also be represented as a

lter A one-to-one correspondence may thus be established between speechsignals and texture in images Figure 7.4 (b) illustrates the model for tex-ture synthesis: the correspondence between the speech and image generationmodels in Figure 7.4 is straightforward

7.2.1 Random texture

According to the model in Figure 7.4, random texture may be modeled as a

ltered version of a eld of white noise, where the lter is represented by aspot of a certain shape and size (usually of small spatial extent compared tothe size of the image) The 2D spectrum of the noise eld, which is essentially

a constant, is shaped by the 2D spectrum of the spot Figure 7.5illustrates

a random-noise eld of size 256
256 pixels and its Fourier spectrum Parts(a) { (d) of Figure 7.6show two circular spots of diameter 12 and 20 pixelsand their spectra parts (e) { (h) of the gure show the random texturegenerated by convolving the noise eld in Figure 7.5 (a) with the circularspots, and their Fourier spectra It is readily seen that the spots have lteredthe noise, and that the spectra of the textured images are essentially those ofthe corresponding spots

Figures 7.7and7.8illustrate a square spot and a hash-shaped spot, as well

as the corresponding random texture generated by the spot-noise model andthe corresponding spectra the anisotropic nature of the images is clearly seen

a (quasi-) periodic eld of impulses, whereas the latter uses a random-noise

eld Once again, the spectral characteristics of the texton could be seen as

a lter that modies the spectrum of the impulse eld (which is essentially a2D eld of impulses as well)

(a) (b)

FIGURE 7.5

(a) Image of a random-noise eld (256
256 pixels) (b) Spectrum of the image

in (a) Reproduced with permission from A.C.G Martins, R.M Rangayyan,and R.A Ruschioni, \Audication and sonication of texture in images",Journal of Electronic Imaging,10(3): 690 { 705, 2001 cSPIE and IS&T

Figure 7.9 (a)illustrates a 256
256 eld of impulses with horizontal odicity px= 40 pixels and vertical periodicity py= 40 pixels Figure 7.9 (b)shows the corresponding periodic texture with a circle of diameter 20 pixels

peri-as the spot or texton Figure 7.9 (c) shows a periodic texture with the textonbeing a square of side 20 pixels, px= 40 pixels, and py = 40 pixels Figure7.9 (d) depicts a periodic-textured image with an isosceles triangle of sides

12 16 and 23 pixels as the spot, and periodicitypx= 40 pixels andpy = 40pixels See Section 7.6for illustrations of the Fourier spectra of images withordered texture

7.2.3 Oriented texture

Images with oriented texture may be generated using the spot-noise model

by providing line segments or oriented motifs as the spot Figure 7.10shows

a spot with a line segment oriented at 135o and the result of convolution ofthe spot with a random-noise eld the log-magnitude Fourier spectra of thespot and the textured image are also shown The preferred orientation of thetexture and the directional concentration of the energy in the Fourier domainare clearly seen in the gure SeeFigure 7.2for examples of oriented texture

in mammograms See Chapter 8 for detailed discussions on the analysis oforiented texture and several illustrations of oriented patterns

(a) (b)

FIGURE 7.6

(a) Circle of diameter 12 pixels (b) Circle of diameter 20 pixels (c) Fourierspectrum of the image in (a) (d) Fourier spectrum of the image in (b).(e) Random texture with the circle of diameter 12 pixels as the spot (f) Ran-dom texture with the circle of diameter 20 pixels as the spot (g) Fourierspectrum of the image in (e) (h) Fourier spectrum of the image in (f) Thesize of each image is 256
256 pixels Reproduced with permission fromA.C.G Martins, R.M Rangayyan, and R.A Ruschioni, \Audication andsonication of texture in images", Journal of Electronic Imaging, 10(3): 690{ 705, 2001 cSPIE and IS&T

FIGURE 7.7

(a) Square of side 20 pixels (b) Random texture with the square of side

20 pixels as the spot (c) Spectrum of the image in (a) (d) Spectrum ofthe image in (b) The size of each image is 256
256 pixels Reproducedwith permission from A.C.G Martins, R.M Rangayyan, and R.A Ruschioni,

\Audication and sonication of texture in images", Journal of ElectronicImaging,10(3): 690 { 705, 2001 cSPIE and IS&T

(a) (b)

FIGURE 7.8

(a) Hash of side 20 pixels (b) Random texture with the hash of side 20 pixels

as the spot (c) Spectrum of the image in (a) (d) Spectrum of the image in(b) The size of each image is 256
256 pixels Reproduced with permissionfrom A.C.G Martins, R.M Rangayyan, and R.A Ruschioni, \Audicationand sonication of texture in images", Journal of Electronic Imaging, 10(3):

690 { 705, 2001 cSPIE and IS&T

Or-of sides 12 16 and 23 pixels as the spot px = 40 pixels and py = 40 els The size of each image is 256
256 pixels Reproduced with permissionfrom A.C.G Martins, R.M Rangayyan, and R.A Ruschioni, \Audicationand sonication of texture in images", Journal of Electronic Imaging, 10(3):

pix-690 { 705, 2001 cSPIE and IS&T

(a) (b)

FIGURE 7.10

Example of oriented texture generated using the spot-noise model in

Fig-ure 7.4: (a)Spot with a line segment oriented at 135o (b) Oriented texturegenerated by convolving the spot in (a) with a random-noise eld (c) and(d) Log-magnitude Fourier spectra of the spot and the textured image, re-spectively The size of each image is 256
256 pixels

7.3 Statistical Analysis of Texture

Simple measures of texture may be derived based upon the moments of thegray-level PDF (or normalized histogram) of the given image Thekthcentralmoment of the PDFp(l) is dened as

mk=LX ; 1

l =0 (l;
f)k p(l) (7.1)where l= 0 1 2 ::: L;1 are the gray levels in the image f, and
f is themean gray level of the image given by

High-Byng et al
448] computed the skewness of the histograms of 24
24(3:12
3:12 mm) sections of mammograms An average skewness measurewas computed for each image by averaging over all the section-based skewnessmeasures of the image Mammograms of breasts with increased broglandu-lar density were observed to have histograms skewed toward higher density,resulting in negative skewness On the other hand, mammograms of fattybreasts tended to have positive skewness The skewness measure was found

to be useful in predicting the risk of development of breast cancer

7.3.1 The gray-level co-occurrence matrix

Given the general description of texture as a pattern of the occurrence of graylevels in space, the most commonly used measures of texture, in particular ofrandom texture, are the statistical measures proposed by Haralick et al
441,442] Haralick's measures are based upon the moments of a joint PDF that

is estimated as the joint occurrence or co-occurrence of gray levels, known asthe gray-level co-occurrence matrix (GCM) GCMs are also known as spatialgray-level dependence (SGLD) matrices, and may be computed for variousorientations and distances

The GCMP( d
 )(l1 l2) represents the probability of occurrence of the pair

of gray levels (l1 l2) separated by a given distanced at angle  GCMs areconstructed by mapping the gray-level co-occurrence counts or probabilitiesbased on the spatial relations of pixels at dierent angular directions (specied

by) while scanning the image from left-to-right and top-to-bottom

Table 7.1shows the GCM for the image inFigure 7.11with eight gray levels(3 b=pixel) by considering pairs of pixels with the second pixel immediatelybelow the rst For example, the pair of gray levels 12

occurs 10 times in theimage Observe that the table of counts of occurrence of pairs of pixels shown

in Table 11.2 and used to compute the rst-order entropy also represents aGCM, with the second pixel appearing immediately after the rst in the samerow Due to the fact that neighboring pixels in natural images tend to havenearly the same values, GCMs tend to have large values along and around themain diagonal, and low values away from the diagonal

Observe that, for an image with B b=pixel, there will be L = 2B graylevels the GCM is then of size L
L Thus, for an image quantized to

8 b=pixel, there will be 256 gray levels, and the GCM will be of size 256

256 Fine quantization to large numbers of gray levels, such as 212 = 4 096levels in high-resolution mammograms, will increase the size of the GCM tounmanageable levels, and also reduce the values of the entries in the GCM Itmay be advantageous to reduce the number of gray levels to a relatively smallnumber before computing GCMs A reduction in the number of gray levelswith smoothing can also reduce the eect of noise on the statistics computedfrom GCMs

GCMs are commonly formed for unit pixel distances and the four angles

of 0o 45o, 90o, and 135o (Strictly speaking, the distances to the diagonallyconnected neighboring pixels at 45o and 135o would be p

2 times the pixelsize.) For anM
N image, the number of pairs of pixels that can be formedwill be less than MN due to the fact that it may not be possible to pair thepixels in a few rows or columns at the borders of the image with another pixelaccording to the chosen parameters (d )

A 16
16 part of the image inFigure 2.1 (a) quantized to 3b=pixel, shown

as an image and as a 2D array of pixel values

7.3.2 Haralick's measures of texture

Based upon normalized GCMs, Haralick et al
441, 442] proposed severalquantities as measures of texture In order to dene these measures, let usnormalize the GCM as

The texture measures are then dened as follows

The energy featureF1, which is a measure of homogeneity, is dened as

The contrast featureF2is dened as

The correlation measure F3, which represents linear dependencies of graylevels, is dened as

The inverse dierence moment, a measure of local homogeneity, is denedas

F8=;

2( X L ; 1)

k =0 px + y(k) log2
px + y(k)]: (7.18)Entropy, a measure of nonuniformity in the image or the complexity of thetexture, is dened as

in a manner similar to that given by Equations 7.16 and 7.17 for its sumcounterpart

The dierence entropy measure is dened as

F11=;

L X ; 1

k =0 px ; y(k) log2
px ; y(k)]: (7.20)Two information-theoretic measures of correlation are dened as

F12= Hxy;Hxy1

maxfHx Hy g

(7.21)and

of the second largest eigenvalue ofQ, where

Q(l1 l2) =LX ; 1

k =0

p(l1 k)p(l2 k)

px(k)py(k) : (7.25)The subscriptsdandin the representation of the GCMP( 
d )(l1 l2) havebeen removed in the denitions above for the sake of notational simplicity.However, it should be noted that each of the measures dened above may bederived for each value of d and  of interest If the dependence of textureupon angle is not of interest, GCMs over all angles may be averaged into asingle GCM The distancedshould be chosen taking into account the samplinginterval (pixel size) and the size of the texture units of interest More details

on the derivation and signicance of the features dened above are provided

tex-along the diagonal corresponding to the gray levels present in the texture ements A measure of association is the2statistic, which may be expressedusing the notation above as

Parkkinen et al
449] discussed some limitations of the 2 statistic in theanalysis of periodic texture, and proposed a measure of agreement given by

Haralick's measures have been applied for the analysis of texture in severaltypes of images, including medical images Chan et al
450] found the threefeatures of correlation, dierence entropy, and entropy to perform better thanother combinations of one to eight features selected in a specic sequence.Sahiner et al
428, 451] dened a \rubber-band straightening transform"(RBST) to map ribbons around breast masses in mammograms into rect-angular arrays (see Figure 7.26), and then computed Haralick's measures oftexture Mudigonda et al
165, 275] computed Haralick's measures usingadaptive ribbons of pixels extracted around mammographic masses, and usedthe features to distinguish malignant tumors from benign masses details ofthis work are provided in Sections 7.9 and 8.8 See Section 12.12 for a dis-cussion on the application of texture measures for content-based retrieval andclassication of mammographic masses

7.4 Laws' Measures of Texture Energy

Laws
452] proposed a method for classifying each pixel in an image basedupon measures of local \texture energy" The texture energy features rep-

resent the amounts of variation within a sliding window applied to several

ltered versions of the given image The lters are specied as separable 1Darrays for convolution with the image being processed

The basic operators in Laws' method are the following:

L3 =
1 2 1 ]

E3 =
;1 0 1 ]

The operatorsL3 E3 andS3 perform center-weighted averaging, symmetric

rst dierencing (edge detection), and second dierencing (spot detection),respectively
453] Nine 3
3 masks may be generated by multiplying thetransposes of the three operators (represented as vectors) with their directversions The result ofL3TE3 gives one of the 3
3 Sobel masks

Operators of length ve pixels may be generated by convolving theL3 E3and S3 operators in various combinations Of the several lters designed byLaws, the following ve were said to provide good performance
452, 453]:

where represents 1D convolution

The operators listed above perform the detection of the following types offeatures: L5 { local averageE5 { edges S5 { spots R5 { ripples and W5{ waves
453] In the analysis of texture in 2D images, the 1D convolutionoperators given above are used in pairs to achieve various 2D convolutionoperators (for example,L5L5 =L5TL5 andL5E5 =L5TE5), each of whichmay be represented as a 5
5 array or matrix Following the application of theselected lters, texture energy measures are derived from each ltered image

by computing the sum of the absolute values in a 15
15 sliding window.All of the lters listed above, except L5, have zero mean, and hence thetexture energy measures derived from the ltered images represent measures

of local deviation or variation The result of the L5 lter may be used fornormalization with respect to luminance and contrast

The use of a large sliding window to smooth the ltered images could lead

to the loss of boundaries across regions with dierent texture Hsiao and

Sawchuk
454] applied a modied LLMMSE lter so as to derive Laws' textureenergy measures while preserving the edges of regions, and applied the resultsfor pattern classication.

Example: The results of the application of the operators L5L5, E5E5,and W5W5 to the 128
128 Lenna image in Figure 10.5 (a) are shown in

Figure 7.12 (a){ (c) Also shown in parts (e) { (f) of the gure are the sums

of the absolute values of the ltered images using a 9
9 moving window It isevident that theL5L5 lter results in a measure of local brightness Carefulinspection of the results of theE5E5 andW5W5 lters shows that they havehigh values for dierent regions of the original image possessing dierent types

of texture (edges and waves, respectively) Feature vectors composed of thevalues of various Laws' operators for each pixel may be used for classifyingthe image into texture categories on a pixel-by-pixel basis The results may

be used for texture segmentation and recognition

In an example provided by Laws
452] (see also Pietkainen et al
453]),the texture energy measures have been shown to be useful in the segmen-tation of an image composed of patches with dierent texture Miller andAstley
372, 455] used features of mammograms based upon theR5R5 oper-ator, and obtained an accuracy of 80:3% in the segmentation of the nonfat(glandular) regions in mammograms SeeSection 8.8for a discussion on theapplication of Laws' and other methods of texture analysis for the detection

of breast masses in mammograms

7.5 Fractal Analysis

Fractals are dened in several dierent ways, the most common of which isthat of a pattern composed of repeated occurrences of a basic unit at multiplescales of detail in a certain order of generation this denition includes the no-tion of \self-similarity" or nested recurrence of the same motif at smaller andsmaller scales (see Section 11.9 for a discussion on self-similar, space-llingcurves) The relationship to texture is evident in the property of repeatedoccurrence of a motif Fractal patterns occur abundantly in nature as well

as in biological and physiological systems
456, 457, 458, 459, 460, 461, 462]:the self-replicating patterns of the complex leaf structures of ferns (see Fig-

ure 7.13),the ramications of the bronchial tree in the lung (seeFigure 7.1),

and the branching and spreading (anastomotic) patterns of the arteries in theheart (see Figure 9.20),to name a few Fractals and the notion of chaos arerelated to the area of nonlinear dynamic systems
456, 463], and have foundseveral applications in biomedical signal and image analysis

(a) (d)

FIGURE 7.12

Results of convolution of the Lenna test image of size 128
128 pixels
see

Figure 10.5 (a)]using the following 5
5 Laws' operators: (a)L5L5, (b)E5E5,and (c) W5W5 (d) { (f) were obtained by summing the absolute values ofthe results in (a) { (c), respectively, in a 9
9 moving window, and representthree measures of texture energy The image in (c) was obtained by mappingthe range
;200 200] out of the full range of
;1338 1184] to
0 255]

FIGURE 7.13

The leaf of a fern with a fractal pattern

7.5.1 Fractal dimension

Whereas the self-similar aspect of fractals is apparent in the examples tioned above, it is not so obvious in other patterns such as clouds, coastlines,and mammograms, which are also said to have fractal-like characteristics Insuch cases, the \fractal nature" perceived is more easily related to the notion

men-of complexity in the dimensionality men-of the object, leading to the concept men-of thefractal dimension If one were to use a large ruler to measure the length of acoastline, the minor details present in the border having small-scale variationswould be skipped, and a certain length would be derived If a smaller rulerwere to be used, smaller details would get measured, and the total lengththat is measured would increase (between the same end points as before).This relationship may be expressed as
457]

where l() is the length measured with as the measuring unit (the size ofthe ruler),df is the fractal dimension, andl0 is a constant Fractal patternsexhibit a linear relationship between the log of the measured length and thelog of the measuring unit:

log
l()] = log
l0] + (1;df) log
] (7.33)the slope of this relationship is related to the fractal dimension df of thepattern This method is known as the caliper method to estimate the fractaldimension of a curve It is obvious thatdf = 1 for a straight line

Fractal dimension is a measure that quanties how the given pattern llsspace The fractal dimension of a straight line is unity, that of a circle or a 2Dperfectly planar (sheet-like) object is two, and that of a sphere is three As theirregularity or complexity of a pattern increases, its fractal dimension increases

up to its own Euclidean dimension dE plus one The fractal dimension of ajagged, rugged, convoluted, kinky, or crinkly curve will be greater than unity,and reaches the value of two as its complexity increases The fractal dimension

of a rough 2D surface will be greater than two, and approaches three as thesurface roughness increases In this sense, fractal dimension may be used as

a measure of the roughness of texture in images

Several methods have been proposed to estimate the fractal dimension ofpatterns
464, 465, 466, 467, 464, 468, 469] Among the methods described bySchepers et al
467] for the estimation of the fractal dimension of 1D signals

is that of computing the relative dispersionRD(), dened as the ratio of thestandard deviation to the mean, using varying bin size or number of samples

of the signal For a fractal signal, the expected variation of RD() is

RD() =RD(0) 

0

H ; 1

(7.34)

where 0 is a reference value for the bin size, andH is the Hurst coecientthat is related to the fractal dimension as

(Note: dE = 1 for 1D signals, 2 for 2D images, etc.) The value of H, andhence df, may be estimated by measuring the slope of the straight-line ap-proximation to the relationship between log
RD()] and log()

7.5.2 Fractional Brownian motion model

Fractal signals may be modeled in terms of fractional Brownian motion
466,

467, 470] The expectation of the dierences between the values of such asignal at a position and another at+ ! follow the relationship

E
jf(+ !);f()j]/ j!j H: (7.36)The slope of a plot of the averaged dierence as above versus ! (on a log {log scale) may be used to estimateH and the fractal dimension

Chen et al
470] applied fractal analysis for the enhancement and

classi-cation of ultrasonographic images of the liver Burdett et al
471] derivedthe fractal dimension of 2D ROIs of mammograms with masses by using theexpression in Equation 7.36 Benign masses, due to their smooth and ho-mogeneous texture, were found to have low fractal dimensions of about 2:38,whereas malignant tumors, due to their rough and heterogeneous texture, hadhigher fractal dimensions of about 2:56

The PSD of a fractional Brownian motion signal "(!) is expected to followthe so-called power law as

"(!)/

1

The derivative of a signal generated by a fractional Brownian motion model

is known as a fractional Gaussian noise signal the exponent in the power-lawrelationship for such a signal is changed to (2H;1)

7.5.3 Fractal analysis of texture

Based upon a fractional Brownian motion model, Wu et al
472] dened anaveraged intensity-dierence measureid(k) for various values of the displace-ment or distance parameterkas

The slope of a plot of log
id(k)] versus log
k] was used to estimate H andthe fractal dimension Wu et al applied multiresolution fractal analysis aswell GCM features, Fourier spectral features, gray-level dierence statistics,and Laws' texture energy measures for the classication of ultrasonographicimages of the liver as normal, hepatoma, or cirrhosis classication accuracies

of 88:9%, 83:3%, 80:0%, 74:4%, and 71:1%, respectively, were obtained In

a related study, Lee et al
473] derived features based upon fractal analysisincluding the application of multiresolution wavelet transforms Classica-tion accuracies of 96:7% in distinguishing between normal and abnormal liverimages, and 93:6% in discriminating between cirrhosis and hepatoma wereobtained

Byng et al
448] (see also Peleg et al
464], Yae et al
474], and Caldwell

et al
475]) describe a surface-area measure to represent the complexity oftexture in an image by interpreting the gray level as the height of a function

of space seeFigure 7.14 In a perfectly uniform image of sizeN
N pixels,with each pixel being of size 
 units of area, the surface area would beequal to (N)2 When adjacent pixels are of unequal value, more surface area

of the blocks representing the pixels will be exposed, as shown in Figure 7.14.The total surface area for the image may be calculated as

4
4, etc., and replacing the blocks by a single pixel with the ing average A perfectly uniform image would demonstrate no change in itsarea, and have a fractal dimension of two images with rough texture wouldhave increasing values of the fractal dimension, approaching three Yae et

correspond-al
474] obtained fractal dimension values in the range of
2:23 2:54] with 60mammograms Byng et al
448] demonstrated the usefulness of the fractaldimension as a measure of increased broglandular density in the breast, andrelated it to the risk of development of breast cancer Fractal dimension wasfound to complement histogram skewness (seeSection 7.3)as an indicator ofbreast cancer risk