Bản chất của hình ảnh y sinh học (Phần 6)

In order to address thisproblem, smoothed derivatives at each contour point could be estimated byconsidering the cumulative sum of weighted dierences of a certain number ofpairs of point

Analysis of Shap e

Several human organs and biological structures possess readily identiableshapes The shapes of the human heart, brain, kidneys, and several bonesare well known, and, in normal cases, do not deviate much from an \aver-age" shape However, disease processes can aect the structure of organs,and cause deviation from their expected or average shapes Even abnormalentities, such as masses and calcications in the breast, tend to demonstratedierences in shape between benign and malignant conditions For exam-ple, most benign masses in the breast appear as well-circumscribed areas onmammograms, with smooth boundaries that are circular or oval some benignmasses may be macrolobulated On the other hand, malignant masses (can-cerous tumors) are typically ill-dened on mammograms, and possess a rough

or stellate (star-like) shape with strands or spicules appearing to radiate fromShape is a key feature in discriminating between normal and abnormal cellsranges of manifestation, with signicant overlap between their characteris-tics for various categories Furthermore, it should be borne in mind that theimaging geometry, 3D-to-2D projection, and the superimposition of multi-ple objects commonly aect the shapes of objects as perceived on biomedicalimages

We shall study a selection of shape analysis techniques in this chapter Afew applications will be described to demonstrate the usefulness of shapecharacteristics in the analysis of biomedical images

6.1 Representation of Shapes and Contours

The most general form of representation of a contour in discretized space

is in terms of the (x y) coordinates of the digitized points (pixels) alongthe contour A contour withN points could be represented by the series ofcoordinatesfx(n) y(n)g,n= 0 1 2 ::: N;1 Observe that there is no graylevel associated with the pixels along a contour A contour may be depicted

as a binary or bilevel image

6.1.1 Signatures of contours

The dimensionality of representation of a contour may be reduced from two

to one by converting from a coordinate-based representation to distances fromeach contour point to a reference point A convenient reference is the centroid

or center of mass of the contour, whose coordinates are given by

It is obvious that going around a contour more than once generates the samesignature hence, the signature signal is periodic with the period equal toN,the number of pixels on the contour The signature of a contour providesgeneral information on the nature of the contour, such as its smoothness orroughness

Examples: Figures 6.2 (a)and6.3 (a)show the contours of a benign breastThe `*' marks within the contours represent their centroids Figures 6.2 (b)and 6.3 (b) show the signatures of the contours as dened in Equation 6.2

It is evident that the smooth contour of the benign mass possesses a smoothsignature, whereas the spiculated malignant tumor has a rough signature withseveral signicant rapid variations over its period

6.1.2 Chain coding

An ecient representation of a contour may be achieved by specifying the(x y) coordinates of an arbitrary starting point on the contour, the direction oftraversal (clockwise or counter-clockwise), and a code to indicate the manner

of movement to reach the next contour point on a discrete grid A coarserepresentation may be achieved by using only four possible movements: to thepoint at the left of, right of, above, or below the current point, as indicated in

Figure 6.4 (a) A ner representation may be achieved by using eight possiblemovements, including diagonal movements, as indicated in Figure 6.4 (b).The sequence of codes required to traverse through all the points along the

(x, y) _ _

z(0) = x(0) + j y(0) z(N-1) =

x(N-1) + j y(N-1)

z(1) = x(1) + j y(1)

z(2) = x(2) + j y(2)

d(2) d(1) d(0)

d(N-1)

x y

The code is invariant to shift or translation because the starting point

is kept out of the code

To a certain extent, the chain code is invariant to size (scaling) tours of dierent sizes may be generated from the same code by usingdierent sampling grids (step sizes) A contour may also be enlarged by

Con-a fCon-actor ofnby repeating each code element ntimes and maintainingoriginal size by reducing pairs of code elements to single numbers, withapproximation of unequal pairs by their averages reduced to integers

The chain code may be normalized for rotation by taking the rst ference of the code (and adding 4 or 8 to negative dierences, dependingupon the code used)

100 200 300 400 500 600 700 100

500 1000 1500 2000 2500 3000 60

With reference to the 8-symbol code, the rotation of a given contour by

n90oin the counter-clockwise direction may be achieved by adding avalue of 2nto each code element, followed by integer division by 8 Theaddition of an odd number rotates the contour by the correspondingmultiple of 45o however, the rotation of a contour by angles other thanintegral multiples of 90o on a discrete grid is subject to approximation

In the case of the 8-symbol code, the length of a contour is given by thenumber of even codes plusp

2 times the number of odd codes, multiplied

by the grid sampling interval

The chain code may also be used to achieve reduction, check for closure,

Examples: Figure 6.5shows a contour represented using the chain codeswith four and eight symbols The use of a discrete grid with large spacingsleads to the loss of ne detail in the contour However, this feature may beused advantageously to lter out minor irregularities due to noise, artifactsdue to drawing by hand, etc

Consider a functionf(x) Letf0(x) f00(x) and f000(x) represent the rst,second, and third derivatives of f(x) A point of inection of the function or

3 2 3 2 3

3 2 1

2 2 3 2 1 1 1 2 1 0 1 1

3 o

Figure 6.5 (a)curvef(x) is dened as a point wheref00(x) changes its sign Note that thederivation off00(x) requiresf(x) andf0(x) to be continuous and dierentiable

It follows that the following conditions apply at a point of inection:

f00(x) = 0

f0(x)6= 0

f0(x)f00(x) = 0 and

f0(x)f000(x)6= 0: (6.3)Let C = f(x(n) y(n)g n = 0 1 2 ::: N ;1, represent in vector formthe (x y) coordinates of the N points on the given contour The points ofinection on the contour are obtained by solving

6 0 7 4

5

6

6 3 4 4

5 2 2 2 3 1

1 7 o

(b)

FIGURE 6.5

A closed contour represented using the chain code (a) using four directionalcodes as in Figure 6.4 (a), and (b) with eight directional codes as in Figure6.4 (b) The `o' mark represents the starting point of the contour, which istraversed in the clockwise direction to derive the code

x0(n)y000(n);x000(n)y0(n)6= 0 (6.5)wherex0(n) y0(n) x00(n) y00(n) x000(n) andy000(n) are the rst, second, andthird derivatives ofx(n) andy(n), respectively.

Segments of contours of breast masses between successive points of inectiontation because the contours of masses are, in general, not smooth False orirrelevant points of inection could appear on relatively straight parts of acontour whenx00(n) andy00(n) are not far from zero In order to address thisproblem, smoothed derivatives at each contour point could be estimated byconsidering the cumulative sum of weighted dierences of a certain number ofpairs of points on either side of the pointx(n) under consideration as

deriva-the value of mwas varied from 3 to 60 to compute derivatives that resulted

in varying numbers of inection points for a given contour The number ofinection points detected as a function of the number of dierences used wasanalyzed to determine the optimal number of dierences that would providethe most appropriate inection points: the value of m at the rst straightsegment on the function was selected

Examples: Figure 6.6shows the contour of a spiculated malignant tumor.The points of inection detected are marked with `*' The number of inec-tion points detected is plotted in Figure 6.7 as a function of the number ofdierences used (m in Equation 6.6) the horizontal and vertical lines indi-cate the optimal number of dierences used to compute the derivative at eachcontour point and the corresponding number of points of inection that werelocated on the contour

The contour in Figure 6.6 is shown in Figure 6.8, overlaid on the sponding part of the original mammogram Segments of the contours areshown in black or white, indicating if they are concave or convex, respec-tively Figure 6.9 provides a similar illustration for a circumscribed benignmass Analysis of concavity of contours is described inSection 6.4

corre-6.1.4 Polygonal modeling of contours

for segmentation and approximation of curves and shapes by polygons forcomputer recognition of handwritten numerals, cell outlines, and ECG signals.modeling of 2D curves in which the number of segments is to be prespecied

FIGURE 6.6

Contour of a spiculated malignant tumor with the points of inection indicated

by `*' Number of points of inection = 58 See alsoFigure 6.8

0 5 10 15 20 25 30 35 40 0

Number of pairs of differences

FIGURE 6.7

Number of inection points detected as a function of the number of dierencesused to estimate the derivative for the contour inFigure 6.6 The horizontaland vertical lines indicate the optimal number of dierences used to computethe derivative at each contour point and the corresponding number of points

of inection that were located on the contour

FIGURE 6.8

Concave and convex parts of the contour of a spiculated malignant tumor,separated by the points of inection See alsoFigure 6.6 The concave partsare shown in black and the convex parts in white The image size is 770

600 pixels or 37:247:7 mm with a pixel size of 62 m Shape factors

fcc = 0:47, SI = 0:62, cf = 0:94 Reproduced with permission from R.M.Rangayyan, N.R Mudigonda, and J.E.L Desautels, \Boundary modeling andshape analysis methods for classication of mammographic masses", Medicaland Biological Engineering and Computing,38: 487 { 496, 2000 cIFMBE

FIGURE 6.9

Concave and convex parts of the contour of a circumscribed benign mass,separated by the points of inection The concave parts are shown in black andthe convex parts in white The image size is 730630 pixels or 31:536:5mm

with a pixel size of 50 m Shape factors fcc = 0:16, SI = 0:22, cf =

0:30 Reproduced with permission from R.M Rangayyan, N.R Mudigonda,and J.E.L Desautels, \Boundary modeling and shape analysis methods forclassication of mammographic masses", Medical and Biological Engineeringand Computing,38: 487 { 496, 2000 cIFMBE

for initiating the process, in relation to the complexity of the shape This isnot a desirable step when dealing with complex or spiculated shapes of breastpolygon formed by the points of inection detected on the original contour wasused as the initial input to the polygonal modeling procedure This step helps

in automating the polygonalization algorithm: the method does not requireany interaction from the user in terms of the initial number of segments.Given an irregular contourCas specied by the set of its (x y) coordinates,the polygonal modeling algorithm starts by dividing the contour into a set

of piecewise-continuous curved parts by locating the points of inection onthe contour as explained in Section 6.1.3 Each segmented curved part isrepresented by a pair of linear segments based on its arc-to-chord deviation.The procedure is iterated subject to predened boundary conditions so as tominimize the error between the true length of the contour and the cumulativelength computed from the polygonal segments

LetC =fx(n) y(n)g n= 0 1 2 ::: N;1, represent the given contour.Let SCmk SCmk 2 C m = 1 2 ::: M, be M curved parts, each con-taining a set of contour points, at the start of the kth iteration, such that

SC1 k S

SC2 k S

:::S

SCMk  C: The iterative procedure proposed by

1 In each curved part represented by SCmk, the arc-to-chord distance

is computed for all the points, and the point on the curve with themaximum arc-to-chord deviation (dmax) is located

2 If dmax  0:25 mm (5 pixels in the images with a pixel size of 50m

at the point of maximum deviation to approximate the same with apair of linear segments, irrespective of the length of the resulting linearsegments If 0:1mmdmax<0:25mm, the curved part is segmented

at the point of maximum deviation subject to the condition that theresulting linear segments satisfy a minimum-length criterion, which was

4 If the number of polygonal segments following the kth iteration equalsthat of the previous iteration, the algorithm is considered to have con-verged and the polygonalization process is terminated Otherwise, theprocedure (Steps 1 to 3) is repeated until the algorithm converges

The criterion for choosing the threshold for arc-to-chord deviation was based

on the assumption that any segment possessing a smaller deviation is icant in the analysis of contours of breast masses

insignif-Examples: Figure 6.10 (a)shows the points of inection (denoted by `*')and the initial stage of polygonal modeling (straight-line segments) of thecontour of a spiculated malignant tumor (see alsoFigure 6.8) Figure 6.10 (b)shows the nal result of polygonal modeling of the same contour The al-gorithm converged after four iterations, as shown by the convergence plot in

Figure 6.11 The result of the application of the polygonal modeling algorithm

to the contour of a circumscribed benign mass is shown inFigure 6.12

The number of linear segments required for the approximation of a contourincreases with its shape complexity polygons with the number of sides in therange 20;

contours of breast masses and tumors The number of iterations required forthe convergence of the algorithm did not vary much for dierent mass contourshapes, remaining within the range 3;5 This is due to the fact that therelative complexity of the contour to be segmented is taken into considera-tion during the initial preprocessing step of locating the points of inectionhence, the subsequent polygonalization process is robust and computationallyecient The algorithm performed well and delivered satisfactory results onvarious irregular shapes of spiculated cases of benign and malignant masses

6.1.5 Parabolic modeling of contours

masses between successive points of inection as parabolas An inspection

of the segments of the contours illustrated in Figures 6.6 and 6.12 (a) (seealsoFigures 6.8 and6.9)indicates that most of the curved portions betweensuccessive points inection lend themselves well to modeling as parabolas.Some of the segments are relatively straight however, such segments maynot contribute much to the task of discrimination between benign masses andmalignant tumors

Let us consider a segment of a contour represented in the continuous 2D

x(s) y(s)] over the intervalS1 sS2, wheresindicatesdistance along the contour and S1and S2are the end-points of the segment.Let us now consider the approximation of the curve by a parabola Regardless

of the position and orientation of the given curve, let us consider the simplestrepresentation of a parabola asY =AX2in the coordinate space (X Y) TheparameterAcontrols the narrowness of the parabola: the larger the value of

A, the narrower is the parabola Allowing for a rotation of  and a shift of(c d) between the (x y) and (X Y) spaces, we have

x(s) =X(s)cos;Y(s)sin+c

y(s) =X(s)sin+Y(s)cos+d: (6.7)

(a) (b)

FIGURE 6.10

Polygonal modeling of the contour of a spiculated malignant tumor (a) Points

of inection (indicated by `*') and the initial polygonal approximation(straight-line segments) number of sides = 58 (b) Final model after fouriterations number of sides = 146 See also Figure 6.8 Reproduced withpermission from R.M Rangayyan, N.R Mudigonda, and J.E.L Desautels,

\Boundary modeling and shape analysis methods for classication of mographic masses", Medical and Biological Engineering and Computing, 38:

mam-487 { 496, 2000 cIFMBE

0 1 2 3 4 5 6 60

con-\Boundary modeling and shape analysis methods for classication of mographic masses", Medical and Biological Engineering and Computing, 38:

mam-487 { 496, 2000 cIFMBE

(a) (b)

FIGURE 6.12

Polygonal modeling of the contour of a circumscribed benign mass (a) Points

of inection (indicated by `*') and the initial polygonal approximation(straight-line segments) initial number of sides = 14 (b) Final model num-ber of sides = 36, number of iterations = 4 See alsoFigure 6.9 Reproducedwith permission from R.M Rangayyan, N.R Mudigonda, and J.E.L Desau-tels, \Boundary modeling and shape analysis methods for classication ofmammographic masses", Medical and Biological Engineering and Computing,38: 487 { 496, 2000 cIFMBE

We also have the following relationships:

X00(s) = 0 =x00(s)cos+y00(s)sin (6.13)which, upon multiplication with sin, yields

x00(s)sincos+y00(s)sinsin= 0: (6.14)Similarly, we also get

Y00(s) = 2A=;x00(s)sin+y00(s)cos (6.15)which, upon multiplication with cos, yields

2Acos=;x00(s)sincos+y00(s)coscos: (6.16)Combining Equations 6.14 and 6.16 we get

The equations above indicate thaty00(s) andx00(s) are constants with valuesrelated to A and  The values of the two derivatives may be computedfrom the given curve over all available points, and averaged to obtain thecorresponding (constant) values Equations 6.14 and 6.17 may then be solved

simultaneously to obtainandA Thus, the parameter of the parabolic model

is obtained from the given contour segment

Menut et al hypothesized that malignant tumors, due to the presence ofnarrow spicules or microlobulations, would have several parabolic segmentswith large values of A on the other hand, benign masses, due to their char-acteristics of being oval or macrolobulated, would have a small number ofparabolic segments with small values of A The same reasons were also ex-pected to lead to a larger standard deviation ofAfor malignant tumors thanfor benign masses In addition to the parameterA, Menut et al proposed touse the width of the projection of each parabola on to theX axis, with theexpectation that its values would be smaller for malignant tumors than forbenign masses A classication accuracy of 76% was obtained with a set of

54 contours

6.1.6 Thinning and skeletonization

Objects that are linear or oblong, or structures that have branching tomotic) patterns may be eectively characterized by their skeletons Theskeleton of an object or region is obtained by its medial-axis transform or viathe given image needs to be binarized so as to include only the patterns ofinterest Let the set of pixels in the binary pattern be denoted asB, let Cbethe set of contour pixels ofB, and letci be an arbitrary contour point inC.For each pointb inB, a pointci is found such that the distance between thepointbandci, represented asd(b ci), is at its minimum If a second pointck

(anos-is found in C such thatd(b ck) = d(b ci), thenb is a part of the skeleton of

B otherwise,bis not a part of the skeleton

the image has been binarized, with the pixels inside the ROIs being labeled

as 1 and the background pixels as 0 A contour point is dened as any pixelhaving the value 1 and at least one 8-connected neighbor valued 0 Let the8-connected neighboring pixels of the pixelp1 being processed be indexed as

2 4

2 Delete all agged pixels.

3 Do the same as Step 1 above replacing the conditions (c) and (d) with(c')p2 p4 p8= 0

(d')p2 p6 p8= 0

4 Delete all agged pixels

5 Iterate Steps 1;4 until no further pixels are deleted

The algorithm described above has the properties that it does not removeend points, does not break connectivity, and does not cause excessive erosion

Example: Figure 6.13 (a) shows a pattern of blood vessels in a sectionextraction of the tissue for study Figure 6.13 (b) shows the skeleton of theimage in part (a) of the gure It is seen that the skeleton represents thegeneral orientational pattern and overall shape of the blood vessels in theoriginal image However, information regarding the variation in the thicknessstudied the eect of injury and healing on the microvascular structure ofligaments by analyzing the statistics of the volume and directional distribution

of blood vessels as illustrated in Figure 6.13 see Section 8.7.2for details

6.2 Shape Factors

Although contours may be eectively characterized by representations andmodels such as the chain code and the polygonal model described in thepreceding section, it is often desirable to encode the nature or form of acontour using a small number of measures, commonly referred to as shapefactors The nature of the contour to be encapsulated in the measures mayvary from one application to another Regardless of the application, a fewbasic properties are essential for ecient representation, of which the mostimportant are:

invariance to shift in spatial position,

invariance to rotation, and

invariance to scaling (enlargement or reduction)

Invariance to reection may also be desirable in some applications Shape tors that meet the criteria listed above can eectively and eciently representcontours for pattern classication

fac-(a) (b)

FIGURE 6.13

(a) Binarized image of blood vessels in a ligament perfused with black ink age courtesy of R.C Bray and M.R Doschak, University of Calgary (b) Skele-ton of the image in (a) after 15 iterations of the algorithm described in Sec-tion 6.1.6

Im-A basic method that is commonly used to represent shape is to t an ellipse

or a rectangle to the given (closed) contour The ratio of the major axis ofthe ellipse to its minor axis (or, equivalently, the ratio of the larger side

to the smaller side of the bounding rectangle) is known as its eccentricity,and represents its deviation from a circle (for which the ratio will be equal tounity) Such a measure, however, represents only the elongation of the object,and may have, on its own, limited application in practice Several shapefactors of increasing complexity and specicity of application are described inthe following sections

6.2.1 Compactness

Compactness is a simple and popular measure of the eciency of a contour

to contain a given area, and is commonly dened as

Co=P2

whereP andAare the contour perimeter and area enclosed, respectively Thesmaller the area contained by a contour of a given length, the larger will be thevalue of compactness Compactness, as dened in Equation 6.19, has a lowerbound of 4 for a circle (except for the trivial case of zero forP = 0), but

no upper bound It is evident that compactness is invariant to shift, scaling,rotation, and reection of a contour

well as to obtain increasing values with increase in complexity of the shape,

Figure 6.15illustrates a few objects with simple geometric shapes includingscaling and rotation the values of compactness Co and cf for the contours

of the objects are listed in Table 6.1

both denitions of compactness provide the desired invariance to scaling androtation (within the limitations due to the use of a discrete grid)

Figure 6.16illustrates a few objects of varying shape complexity, prepared

cf for the contours of the objects are listed in Table 6.2 It is seen thatcompactness increases with shape roughness and/or complexity

(a) (b) (c) (d) (e)

FIGURE 6.14

Examples of contours with their values of compactnessCoandcf, as dened

in Equations 6.19 and 6.20 (a) Circle (b) Square (c) Rectangle with sidesequal to 1:0 and 0:5 units (d) Rectangle with sides equal to 1:0 and 0:25units (e) Right-angled triangle of height 1:0 and base 0:25 units

FIGURE 6.15

A set of simple geometric shapes, including scaling and rotation, created on

a discrete grid, to test shape factors Reproduced with permission from L.Shen, R.M Rangayyan, and J.E.L Desautels, \Application of shape analysis

to mammographic calcications", IEEE Transactions on Medical Imaging,13(2): 263 { 274, 1994 cIEEE

© 2005 by CRC Press LLC

FIGURE 6.16

A set of objects of varying shape complexity The objects were prepared bycutting construction paper The contours of the objects include imperfectionsand artifacts Reproduced with permission from L Shen, R.M Rangayyan,and J.E.L Desautels, \Application of shape analysis to mammographic cal-cications", IEEE Transactions on Medical Imaging, 13(2): 263 { 274, 1994.c

Statistical moments of PDFs and other data distributions have been utilized

as pattern features in a number of applications the same concepts have been

a 2D continuous imagef(x y), the regular momentsmpqof order (p+q) are