Bản chất của hình ảnh y sinh học (Phần 4)

4.4.2 Gray-scale windowing If a given imagefmn has all of its pixel values in a narrow range of graylevels, or if certain details of particular interest within the image occupy anarrow r

Image Enhancement

In spite of the signicant advances made in biomedical imaging techniques overthe past few decades, several practical factors often lead to the acquisition

of images with less than the desired levels of contrast, visibility of detail,

or overall quality In the preceding chapters, we reviewed several practicallimitations, considerations, and trade-os that could lead to poor images.When the nature of the artifact that led to the poor quality of the image

is known, such as noise as explained in Chapter 3, we may design specicmethods to remove or reduce the artifact When the degradation is due to ablur function, deblurring and restoration techniques, described in Chapter 10,

may be applied to reverse the phenomenon In some applications of biomedicalimaging, it becomes possible to include additional steps or modications in theimaging procedure to improve image quality, although at additional radiationdose to the subject in the case of some X-ray imaging procedures, as we shallsee in the sections to follow

In several situations, the understanding of the exact cause of the loss ofquality is limited or nonexistent, and the investigator is forced to attempt toimprove or enhance the quality of the image on hand using several techniquesapplied in an ad hoc manner In some applications, a nonspecic improve-ment in the general appearance of the given image may suce Researchers

in the eld of image processing have developed a large repertoire of image hancement techniques that have been demonstrated to work well under certainconditions with certain types of images Some of the enhancement techniques,indeed, have an underlying philosophy or hypothesis, as we shall see in thefollowing sections however, the practical application of the techniques mayencounter diculties due to a mismatch between the applicable conditions orassumptions and those that relate to the problem on hand

en-A few biomedical imaging situations and applications where enhancement

of the feature of interest would be desirable are:

Microcalcications in mammograms

Lung nodules in chest X-ray images

Vascular structure of the brain

Hair-line fractures in the ribs

respect to their surrounding structures, or low contrast others could be dered not readily visible due to superimposed structures in planar images.Enhancement of the contrast, edges, and general detail visibility in the im-ages, without causing any distortion or artifacts, would be desirable in theapplications mentioned above.

ren-In this chapter, we shall explore a wide range of image enhancement niques that can lead to improved contrast or visibility of certain image fea-tures such as edges or objects of specic characteristics In extending thetechniques to other applications, it should be borne in mind that ad hocprocedures borrowed from other areas may not lead to the best possible oroptimal results Regardless, if the improvement so gained is substantial andconsistent as judged by the users and experts in the domain of application,one may have on hand a practically useful technique (See the July 1972 andMay 1979 issues of the Proceedings of the IEEE for reviews and articles ondigital image processing, including historically signicant images.)

tech-4.1 Digital Subtraction Angiography

In digital subtraction angiography (DSA), an X-ray contrast agent (such as

an iodine compound) is injected so as to increase the density (attenuation ecient) of the blood within a certain organ or system of interest A number

co-of X-ray images are taken as the contrast agent spreads through the arterialnetwork and before the agent is dispersed via circulation throughout the body

An image taken before the injection of the agent is used as the \mask" or erence image, and subtracted from the \live" images obtained with the agent

ref-in the system to obtaref-in enhanced images of the arterial system of ref-interest.Imaging systems that perform contrast-enhanced X-ray imaging (withoutsubtraction) in a motion or cine mode are known as cine-angiography systems.Such systems are useful in studying circulation through the coronary system

to detect sclerosis (narrowing or blockage of arteries due to the deposition ofcholesterol, calcium, and other substances)

Figures 4.1 (a), (b), and (c) show the mask, live, and the result of DSA,respectively, illustrating the arterial structure in the brain of a subject 223,

224, 225] The arteries are barely visible in the live image Figure 4.1 (b)], inspite of the contrast agent Subtraction of the skull and the other parts thathave remained unchanged between the mask and the live images has resulted

in greatly improved visualization of the arteries in the DSA image Figure 4.1(c)] The mathematical procedure involved may be expressed simply as

f = f1 ;
f2 or

f(mn) = f1(mn);
f2(mn) (4.1)wheref1is the live image,f2is the mask image, and
are weighting factors(if required), andf is the result of DSA.

The simple mathematical operation of subtraction (on a pixel-by-pixel sis) has, indeed, a signicant application in medical imaging The technique,however, is sensitive to motion, which causes misalignment of the components

ba-to be subtracted The DSA result in Figure 4.1 (c) demonstrates motionartifacts in the lowest quarter and around the periphery of the image Meth-ods to minimize motion artifact in DSA have been proposed by Meijering et

al 223, 224, 225] Figure 4.1 (d) shows the DSA result after correction ofmotion artifacts Regardless of its simplicity, DSA carries a certain risk ofallergic reaction, infection, and occasionally death, due to the injection of thecontrast agent

4.2 Dual-energy and Energy-subtraction X-ray Imaging

Dierent materials have varying energy-dependent X-ray attenuation cients X-ray measurements or images obtained at multiple energy levels (alsoknown as energy-selective imaging) could be combined to derive informationabout the distribution of specic materials in the object or body imaged.Weighted combinations of multiple-energy images may be obtained to displaysoft-tissue and hard-tissue details separately 5] The disadvantages of dual-energy imaging exist in the need to subject the patient to two or more X-rayexposures (at dierent energy or kV) Furthermore, due to the time lapsebetween the exposures, motion artifacts could arise in the resulting image

coe-In a variation of the dual-energy method, MacMahon 226, 227] describesenergy-subtraction imaging using a dual-plate CR system The Fuji FCR9501ES (Fujilm Medical Systems USA, Stamford, CT) digital chest unit usestwo receptor plates instead of one The plates are separated by a copper lter.The rst plate acquires the full-spectrum X-ray image in the usual manner.The copper lter passes only the high-energy components of the X rays on

to the second plate Because bones and calcium-containing structures wouldhave preferentially absorbed the low-energy components of the X rays, andbecause the high-energy components would have passed through low-densitytissues with little attenuation, the transmitted high-energy components could

be expected to contain more information related to denser tissues than tolighter tissues The two plates capture two dierent views derived from thesame X-ray beam the patient is not subjected to two dierent imaging ex-posures, but only one Weighted subtraction of the two images as in Equa-tion 4.1 provides various results that can demonstrate soft tissues or bonesand calcied tissues in enhanced detail seeFigures 4.2and 4.3

1999 cKluwer Academic Publishers.

Energy-subtraction imaging as above has been found to be useful in tecting fracture of the ribs, in assessing the presence of calcication in lungnodules (which would indicate that they are benign, and hence, need not beexamined further or treated), and in detecting calcied pleural plaques due

de-to prolonged exposure de-to asbesde-tos 226, 227] The bone-detail image in

Fig-ure 4.3 (a) shows, in enhanced detail, a small calcied granuloma near thelower-right corner of the image

FIGURE 4.2

Full-spectrum PA chest image (CR) of a patient See also Figure 4.3 age courtesy of H MacMahon, University of Chicago, Chicago, IL Repro-duced with permission from H MacMahon, \Improvement in detection ofpulmonary nodules: Digital image processing and computer-aided diagnosis",RadioGraphics, 20(4): 1169{1171, 2000 cRSNA

(a) Bone-detail image, and (b) soft-tissue detail image obtained by energy subtraction See alsoFigure 4.2 Images courtesy

of H MacMahon, University of Chicago, Chicago, IL Reproduced with permission from H MacMahon, \Improvement indetection of pulmonary nodules: Digital image processing and computer-aided diagnosis", RadioGraphics, 20(4): 1169{1171,

2000 cRSNA

© 2005 by CRC Press LLC

4.3 Temporal Subtraction

Temporal or time-lapse subtraction of images could be useful in detectingnormal or pathological changes that have occurred over a period of time.MacMahon 226] describes and illustrates the use of temporal subtraction inthe detection of lung nodules that could be dicult to see in planar chest im-ages due to superimposed structures DR and CR imaging facilitate temporalsubtraction

In temporal subtraction, it is desired that normal anatomic structures aresuppressed and pathological changes are enhanced Registration of the images

is crucial in temporal subtraction misregistration could lead to artifacts ilar to those due to motion in DSA Geometric transformation and warpingtechniques are useful in matching landmark features that are not expected tohave changed in the interval between the two imaging sessions 223, 224, 225].Mazur et al 228] describe image correlation and geometric transformationtechniques for the registration of radiographs for temporal subtraction

sim-4.4 Gray-scale Transforms

The gray-level histogram of an image gives a global impression of the presence

of dierent levels of density or intensity in the image over the dynamic rangeavailable (seeSection 2.7for details and illustrations) When the pixels in agiven image do not make full use of the available dynamic range, the histogramwill indicate low levels of occurrences of certain gray-level values or ranges.The given image may also contain large areas representing objects with certainspecic ranges of gray level the histogram will then indicate large populations

of pixels occupying the corresponding gray-level ranges Based upon a study

of the histogram of an image, we could design gray-scale transforms or

look-up tables (LUTs) that alter the overall appearance of the image, and couldimprove the visibility of selected details

4.4.1 Gray-scale thresholding

When the gray levels of the objects of interest in an image are known, orcan be determined from the histogram of the given image, the image may bethresholded to obtain a variety of images that can display selected features ofinterest For example, if it is known that the objects of interest in the imagehave gray-level values greater thanL1, we could create an image for display

255 iff(mn) L1

where f(mn) is the original image g(mn) is the thresholded image to bedisplayed and the display range is 0255] The result is a bilevel or binaryimage Thresholding may be considered to be a form of image enhancement inthe sense that the objects of interest are perceived better in the resulting im-age The same operation may also be considered to be a detection operation seeSection 5.1

If the values less than L1 were to be considered as noise (or features of nointerest), and the gray levels within the objects of interest that are greaterthanL1are of interest in the displayed image, we could also dene the outputimage as

g(mn) = 0 iff(mn)L1

f(mn) iff(mn)L1 : (4.3)The resulting image will display the features of interest including their gray-level variations

Methods for the derivation of optimal thresholds are described inSections

5.4.1, 8.3.2, and 8.7.2

Example: A CT slice image of a patient with neuroblastoma is shown

in Figure 4.4 (a) A binarized version of the image, with thresholding as inEquation 4.2 using L1 = 200 HU, is shown in part (b) of the gure Asexpected, the bony parts of the image appear in the result however, thecalcied parts of the tumor, which also have high density comparable to that

of bone, appear in the result The result of thresholding the image as inEquation 4.3 with L1 = 200 HU is shown in part (c) of the gure Therelative intensities of the hard bone and the calcied parts of the tumor areevident in the result

4.4.2 Gray-scale windowing

If a given imagef(mn) has all of its pixel values in a narrow range of graylevels, or if certain details of particular interest within the image occupy anarrow range of gray levels, it would be desirable to stretch the range ofinterest to the full range of display available In the absence of reason toemploy a nonlinear transformation, a linear transformation as follows could

be used for this purpose:

to the display range of 0 255] see also Figures2.15and 2.16 Image courtesy ofAlberta Children's Hospital, Calgary (b) The image in (a) thresholded at the level

below this threshold appear as black (c) The image in (a) thresholded at the level

of 200HU as in Equation 4.3 Values above 200HU appear at their original level,and values below this threshold appear as black (d) TheHU range of 0 400] hasbeen linearly mapped to the display range of 0 255] as in Equation 4.4 Pixelscorresponding to tissues lighter than water appear as black Pixels greater than

400HU are saturated at the maximum gray level of 255

by simply multiplying the normalized values by 255 Details (pixels) belowthe lower limit f1 will be eliminated (rendered black) and those above theupper limitf2 will be saturated (rendered white) in the resulting image Thedetails within the range f1f2] will be displayed with increased contrast andlatitude, utilizing the full range of display available.

Example: A CT slice image of a patient with neuroblastoma is shown

in Figure 4.4 (a) This image displays the range of ;200400]HU linearlymapped to the display range of 0255] as given by Equation 4.4 The fullrange ofHU values in the image is ;10001042]HU Part (d) of the gureshows another display of the same original data, but with mapping of therange 0400]HU to 0255] as given by Equation 4.4 In this result, pixelscorresponding to tissues lighter than water appear as black pixels greaterthan 400 HU are saturated at the maximum gray level of 255 Gray-levelthresholding and mapping are commonly used for detailed interpretation of

CT images

Example: Figure 4.5 (a) shows a part of the chest X-ray image in

Fig-ure 1.11 (b),downsampled to 512512 pixels The histogram of the image isshown inFigure 4.6 (a) observe the large number of pixels with the gray levelzero Figure 4.6 (b) shows two linear gray-scale transformations (LUTs) thatmap the range 00:6] (dash-dot line) and 0:20:7] (solid line) to the range

01] the results of application of the two LUTs to the image in Figure 4.5 (a)are shown in Figures 4.5 (b) and (c), respectively The image in Figure 4.5(b) shows the details in and around the heart with enhanced visibility how-ever, large portions of the original image have been saturated The image inFigure 4.5 (c) provides an improved visualization of a larger range of tissuesthan the image in (b) regardless, the details with normalized gray levels lessthan 0:2 and greater than 0:7 have been lost

Example: Figure 4.7 (a) shows an image of a myocyte Figure 4.8 (a)

shows the normalized histogram of the image Most of the pixels in the imagehave gray levels within the limited range of 50150] the remainder of theavailable range 0255] is not used eectively

Figure 4.7 (b) shows the image in (a) after the normalized gray-level range

of 0:20:6] was stretched to the full range of 01] by the linear transformation

in Equation 4.4 The details within the myocyte are visible with enhancedclarity in the transformed image The corresponding histogram in Figure 4.8(b) shows that the image now occupies the full range of gray scale available however, several gray levels within the range are unoccupied, as indicated bythe white stripes in the histogram

4.4.3 Gamma correction

Figure 2.6 shows the H-D curves of two devices The slope of the curve isknown as  An imaging system with a large could lead to an image with

(a) (b)

(c)

FIGURE 4.5

(a) Part of a chest X-ray image The histogram of the image is shown in

Figure 4.6 (a) (b) Image in (a) enhanced by linear mapping of the range

00:6] to 01] (c) Image in (a) enhanced by linear mapping of the range

0:20:7] to 01] See Figure 4.6 (b) for plots of the LUTs

input gray level (normalized)

(b)

FIGURE 4.6

(a) Normalized histogram of the chest X-ray image inFigure 4.5 (a) entropy =

7:55bits (b) Linear density-windowing transformations that map the ranges

00:6] to 01] (dash-dot line) and 0:20:7] to 01] (solid line)

(a) (b)

FIGURE 4.7

(a) Image of a myocyte as acquired originally (b) Image in (a) enhanced bylinear mapping of the normalized range 0:20:6] to 01] See Figure 4.8forthe histograms of the images

high contrast however, the image may not utilize the full range of the availablegray scale On the other hand, a system with a small  could result in animage with wide latitude but poor contrast Gamma correction is a nonlineartransformation process by which we may alter the transition from one graylevel to the next, and change the contrast and latitude of gray scale in theimage The transformation may be expressed as 203]

g(mn) = f(mn)]
 (4.5)wheref(mn) is the given image with its gray scale normalized to the range

01], and g(mn) is the transformed image (Note: Lindley 229] provides adierent denition as

Example: Figure 4.9 (a)shows a part of a chest X-ray image Figure 4.10

illustrates three transforms with  = 0:31:0 and 2:0 Parts (b) and (c) ofFigure 4.9 show the results of gamma correction with  = 0:3 and  = 2:0,respectively The two results demonstrate enhanced visibility of details in thedarker and lighter gray-scale regions (with reference to the original image)

input gray level (normalized)

FIGURE 4.10

Gamma-correction transforms with= 0:3 (solid line),= 1:0 (dotted line),and= 2:0 (dash-dot line)

4.5 Histogram Transformation

As we saw inSection 2.7,the histogram of an image may be normalized andinterpreted as a PDF Then, based upon certain principles of informationtheory, we reach the property that maximal information is conveyed whenthe PDF of a process is uniform, that is, the corresponding image has allpossible gray levels with equal probability of occurrence (see Section 2.8)

Based upon this property, the technique of histogram equalization has beenproposed as a method to enhance the appearance of an image 9, 8, 11] Othertechniques have also been proposed to map the histogram of the given imageinto a dierent \desired" type of histogram, with the expectation that thetransformed image so obtained will bear an enhanced appearance Althoughthe methods often do not yield useful results in biomedical applications, andalthough the underlying assumptions may not be applicable in many practicalsituations, histogram-based methods for image enhancement are popular Thefollowing sections provide the details and results of a few such methods

4.5.1 Histogram equalization

Consider an imagef(mn) of sizeM N pixels, with gray levels l= 012::: L;1 Let the histogram of the image be represented byPf(l) as dened inEquation 2.12 Let us normalize the gray levels by dividing by the maximumlevel available or permitted, as r= L;l1, such that 0r1 Letpf(r) bethe normalized histogram or PDF as given by Equation 2.15

If we were to apply a transformation s= T(r) to the random variable r,the PDF of the new variablesis given by 8]

pg(s) =pf(r) dr

ds



r = T ;1 ( s ) (4.7)whereg refers to the resulting imageg(mn) with the normalized gray levels

0s1 Consider the transformation

s=T(r) =

Z r

0 pf(w)dw 0r1: (4.8)This is the cumulative (probability) distribution function of r T(r) has thefollowing important and desired properties:

T(r) is single-valued and monotonically increasing over the interval 0

r1 This is necessary to maintain the black-to-white transition orderbetween the original and processed images

0T(r)1 for 0r1 This is required in order to maintain thesame range of values in the input and output images

pg(s) = pf(r) 1pf(r) r=T;1 ( s )= 1 0s1: (4.9)Thus,T(r) equalizes the histogram of the given image that is, the histogram

or PDF of the resulting image g(mn) is uniform As we saw inSection 2.8,

a uniform PDF has maximal entropy

Discrete version of histogram equalization: For a digital imagef(mn)with a total ofP =MN pixels andL gray levelsrkk = 01:::L;1 0

rk 1, occurringnk times, respectively, the PDF may be approximated bythe histogram

In practical applications, the resulting values in the range 01] have to

be scaled to the display range, such as 0255] Histogram equalization isusually implemented via an LUT that lists the related (skrk) pairs as given

by Equation 4.11 It should be noted that a quantized histogram-equalizingtransformation is likely to contain several segments of many-to-one gray-leveltransformation: this renders the transformation nonunique and irreversible

Example: Figure 4.11 (a)shows a 240288 image of a girl in a snow cave:the high reectivity of the snow has caused the details inside the cave to havepoor visibility Part (b) of the same gure shows the result after histogramequalization the histograms of the original and equalized images are shown

inFigure 4.12 Although the result of equalization shows some of the features

of the girl within the cave better than the original, several details remain darkand unclear

The histogram of the equalized image in Figure 4.12 (b) indicates that,while a large number of gray levels have higher probabilities of occurrencethan their corresponding levels in the original see Figure 4.12 (a)], several graylevels are unoccupied in the enhanced image (observe the white stripes in thehistogram, which indicate zero probability of occurrence of the correspondinggray levels) The equalizing transform (LUT), shown inFigure 4.13,indicatesthat there are several many-to-one gray-level mappings: note the presence ofseveral horizontal segments in the LUT It should also be observed that theoriginal image has a well-spread histogram, with an entropy of 6:93bits due

to the absence of several gray levels in the equalized image, its entropy of

5:8bitsturns out to be lower than that of the original

Figure 4.11 (c) shows the result of linear stretching or windowing of therange 023] in the original image in Figure 4.11 (a) to the full range of 0255].The result shows the details of the girl and the inside of the cave more clearlythan the original or the equalized version however, the high-intensity detailsoutside the cave have been washed out.

Figure 4.11 (d) shows the result of enhancing the original image in ure 4.11 (a) with  = 0:3 Although the details inside the cave are not asclearly seen as in Figure 4.11 (c), the result has maintained the details at allgray levels

Example: Figure 4.14 (a) shows a part of a chest X-ray image part (b)

of the same gure shows the corresponding histogram-equalized image

50 100 150 200 250 0

Input gray level

FIGURE 4.13

Histogram-equalizing transform (LUT) for the image in Figure 4.11 (a) see

Figure 4.12for the histograms of the original and equalized images

though some parts of the image demonstrate improved visibility of features,

it should be observed that the low-density tissues in the lower right-hand tion of the image have been reduced to poor levels of visibility The histogram

por-of the equalized image is shown in Figure 4.15 (a) the equalizing transform

is shown in part (b) of the same gure It is seen that several gray levelsare unoccupied in the equalized image for this reason, the entropy of theenhanced image was reduced to 5:95bits from the value of 7:55bits for theoriginal image

Example: Figure 4.16 (b) shows the histogram-equalized version of themyocyte image in Figure 4.16 (a) The corresponding equalizing transform,shown in Figure 4.17 (b), indicates a sharp transition from the darker graylevels to the brighter gray levels The rapid transition has caused the output

to have high contrast over a small eective dynamic range, and has renderedthe result useless The entropies of the original and enhanced images are

4:96bitsand 4:49bits, respectively

4.5.2 Histogram specication

A major limitation of histogram equalization is that it can provide only oneoutput image, which may not be satisfactory in many cases The user has

(a) (b)

FIGURE 4.14

(a) Part of a chest X-ray image The histogram of the image is shown in

Figure 4.6 (a) (b) Image in (a) enhanced by histogram equalization Thehistogram of the image is shown inFigure 4.15 (a) See Figure 4.15 (b) for aplot of the LUT

no control over the procedure or the result In a related procedure known

as histogram specication, a series of histogram-equalization steps is used toobtain an image with a histogram that is expected to be close to a prespeciedhistogram Then, by specifying several histograms, it is possible to obtain arange of enhanced images, from which one or more may be selected for furtheranalysis or use

Suppose that the desired or specied normalized histogram is pd(t), withthe desired image being represented as d, having the normalized gray levels

t = 012:::L;1 Now, the given imagef with the PDF pf(r) may behistogram-equalized by the transformation

q=T2(t) =Z t

0 pd(w)dw 0t1: (4.13)Observe that, in order to derive a histogram-equalizing transform, we needonly the PDF of the image the image itself is not needed Let us call the(hypothetical) image so obtained ase, having the gray levels q The inverse

Input gray level

(b)

FIGURE 4.15

(a) Normalized histogram of the histogram-equalized chest X-ray image in

Figure 4.14 (b) entropy = 5:95bits (b) The histogram-equalizing mation (LUT) See Figure 4.6 (a) for the histogram of the original image

transfor-(a) (b)

FIGURE 4.16

(a) Image of a myocyte The histogram of the image is shown inFigure 4.8

(a) (b) Image in (a) enhanced by histogram equalization The histogram ofthe image is shown inFigure 4.17 (a) See Figure 4.17 (b) for a plot of theLUT

of the transform above, which we may express as t =T2; 1(q), will map thegray levelsqback to t

Now,pg(s) andpe(q) are both uniform PDFs, and hence are identical tions The desired PDF may, therefore, be obtained by applying the transform

func-T2; 1 tos that is, t =T2; 1(s) It is assumed here thatT2; 1(s) exists, and is

a single-valued (unique) transform Based on the above, the procedure forhistogram specication is as follows:

1 Specify the desired histogram and derive the equivalent PDFpd(t)

2 Derive the histogram-equalizing transformq=T2(t)

3 Derive the histogram-equalizing transform s = T1(r) from the PDF

pf(r) of the given imagef

4 Apply the inverse of the transformT2to the PDF obtained in the ous step and obtaint=T2; 1(s) This step may be directly implemented

Input gray level

(b)

FIGURE 4.17

(a) Normalized histogram of the histogram-equalized myocyte image in

Fig-ure 4.16 (b) (b) The histogram-equalizing transformation (LUT) SeeFigure4.8 (a)for the histogram of the original image

dicult, if not impossible, to derive the inverse transform T2 The ble existence of many-to-one mapping segments in the histogram-equalizingtransformT2, as we saw in the examples inSection 4.5.1,may render inversionimpossible Appropriate specication of the desired PDF could facilitate thedesign of an LUT to approximately representT2; 1 The LUTs corresponding

possi-to T1 and T2; 1 may be combined into one LUT that may be applied to thegiven image f to obtain the desired image din a single step Note that theimage obtained as above may have a histogram that only approximates theone specied

4.5.3 Limitations of global operations

Global operators such as gray-scale and histogram transforms provide simplemechanisms to manipulate the appearance of images Some knowledge aboutthe range of gray levels of the features of interest can assist in the design oflinear or nonlinear LUTs for the enhancement of selected features in a givenimage Although histogram equalization can lead to useful results in somesituations, it is quite common to result in poor images Even if we keep asidethe limitations related to nonunique transforms, a global approach to imageenhancement ignores the nonstationary nature of images, and hence couldlead to poor results The results of histogram equalization of the chest X-ray and myocyte images inFigures 4.14and4.16demonstrate the limitations

of global transforms Given the wide range of details of interest in medicalimages, such as the hard tissues (bone) and soft tissues (lung) in a chest X-ray image, it is desirable to design local and adaptive transforms for eectiveimage enhancement

4.5.4 Local-area histogram equalization

Global histogram equalization tends to result in images where features ing gray levels with low probabilities of occurrence in the original image aremerged upon quantization of the equalizing transform, and hence are lost inthe enhanced image Ketchum 231] attempted to address this problem bysuggesting the application of histogram equalization on a local basis In local-area histogram equalization (LAHE), the histogram of the pixels within a 2Dsliding rectangular window, centered at the current pixel being processed, isequalized, and the resulting transform is applied only to the central pixel theprocess is repeated for every pixel in the image The window provides thelocal context for the pixel being processed The method is computationallyexpensive because a new transform needs to be computed for every pixel.Pizer et al 232], Leszczynski and Shalev 233], and Rehm and Dallas 234]proposed variations of LAHE, and extended the method to the enhancement

hav-of medical images In one hav-of the variations hav-of LAHE, the histogram-equalizing

transforms are computed not for every pixel, but only for a number of lapping rectangular blocks spanning the image The pixels at the center ofeach block are processed using the corresponding transform Pixels that arenot at the centers of the blocks are processed using interpolated versions ofthe transforms corresponding to the four neighboring center pixels The suc-cess of LAHE depends upon the appropriate choice of the size of the slidingwindow in relation to the sizes of the objects present in the image, and of thecorresponding background areas.

nonover-Example: The images in Figures 4.18 (c) and (d) show the results ofapplication of the LAHE method to the image in part (a) of the gure, usingwindows of size 1111 and 101101 pixels, respectively The result ofglobal histogram equalization is shown in part (b) of the gure for comparison.Although the results of LAHE provide improved visualization of some of thedetails within the snow cave, the method has led to gray-level inversion in

a few regions (black patches in white snow areas) this eect is due to thespreading of the gray levels in a small region over the full range of 0255],which is not applicable to all local areas in a given image The overall quality

of the results of LAHE has been downgraded by this eect

4.5.5 Adaptive-neighborhood histogram equalization

A limitation of LAHE lies in the use of rectangular windows: although such

a window provides the local context of the pixel being processed, there is

no apparent justication to the choice of the rectangular shape for the ing window Furthermore, the success of the method depends signicantlyupon proper choice of the size of the window the use of a xed window of aprespecied size over an entire image has no particular reasoning

mov-Paranjape et al 230] proposed an adaptive-neighborhood approach to togram equalization As we saw in Section 3.7.5, the adaptive-neighborhoodimage processing paradigm is based upon the identication of variable-shape,variable-size neighborhoods for each pixel by region growing Because theregion-growing procedure used for adaptive-neighborhood image processingleads to a relatively uniform region, with gray-level variations limited to thatpermitted by the specied threshold, the local histogram of such a region willtend to span a limited range of gray levels Equalizing such a histogram andpermitting the occurrence of the entire range of gray levels in any and everylocal context is inappropriate In order to provide an increased context tohistogram equalization, Paranjape et al included in the local area not onlythe foreground region grown, but also a background composed of a ribbon

his-of pixels molded to the foreground see Figure 3.46 The extent of the localcontext provided depends upon the tolerance specied for region growing, thewidth of the background ribbon of pixels, and the nature of gray-level vari-ability present in the given image The method adapts to local details present

in the given image regions of dierent size and shape are grown for each pixel

(a) (b)

FIGURE 4.18

(a) Image of a girl in a snow cave (240
288 pixels) (b) Result of global histogram

window Results of adaptive-neighborhood histogram equalization with (e) growthtolerance 16 and background width 5 pixels, and (f) growth tolerance 64 and back-ground width 8 pixels Reproduced with permission from R.B Paranjape, W.M.Morrow, and R.M Rangayyan, \Adaptive-neighborhood histogram equalization forimage enhancement",CVGIP: Graphical Models and Image Processing,54(3):259{

267, 1992 cAcademic Press

After obtaining the histogram of the local region, the equalizing transform

is derived, and applied only to the seed pixel from where the process wasstarted The same value is applied to all redundant seed pixels in the region that is, to the pixels that have the same gray-level value as the seed (for whichthe same region would have been grown using a simple tolerance)

In an extension of adaptive-neighborhood histogram equalization to colorimages proposed by Ciuc et al 235], instead of equalizing the local histogram,

an adaptive histogram stretching operation is applied to the local histograms.The enhancement operation is applied only to the intensity of the image undesired changes to the color balance (hue) are prevented by this method

Example: Figure 4.19 shows a simple test image with square objects of ferent gray levels, as well as its enhanced versions using global, local-area, andadaptive-neighborhood histogram equalization The limitations of global his-togram equalization are apparent in the fact that the brighter, inner square onthe right-hand side of the image remains almost invisible The result of LAHEpermits improved visualization of the inner squares however, the artifacts due

dif-to block-wise processing are obvious and disturbing Adaptive-neighborhoodhistogram equalization has provided the best result, with enhanced visibility

of the inner squares and without any artifacts

FIGURE 4.19

(a) A test image and its enhanced versions by: (b) global or full-frame togram equalization, (c) LAHE, and (d) adaptive-neighborhood histogramequalization Image courtesy of R.B Paranjape

his-image in part (a) of the gure The two his-images were obtained using growthtolerance values of 16 and 64, and background width of 5 and 8 pixels Thelarger tolerance and larger background width provide for larger areas of thelocal context to be included in the local histogram The result of global his-togram equalization is shown in part (b) of the gure for comparison Theresults of adaptive-neighborhood histogram equalization provide improved vi-sualization of details and image features both inside and outside the snowcave Furthermore, the result with the larger growth tolerance and back-ground ribbon width is relatively free of the gray-level inversion (black patches

in otherwise white areas) present in the results of LAHE, shown in parts (c)and (d) of the same gure

4.6 Convolution Mask Operators

Filtering images using 33 convolution masks is a popular approach Severalsuch masks have been proposed and are in practical use for image enhance-ment Equation 3.39 demonstrates the use of a simple 33 mask to representthe local mean lter We shall explore a few other 33 convolution masksfor image enhancement in the following sections

4.6.1 Unsharp masking

When an image is blurred by some unknown phenomenon, we could assumethat each pixel in the original image contributes, in an additive manner, acertain fraction of its value to the neighboring pixels Then, each pixel iscomposed of its own true value, plus fractional components of its neighbors.The spreading of the value of a pixel into its neighborhood may be viewed asthe development of a local fog or blurred background

In an established photographic technique known as unsharp masking, thegiven degraded image, in its negative form, is rst blurred, and a positivetransparency is created from the result The original negative and the positiveare held together, and a (positive) print is made of the combination Theprocedure leads to the subtraction of the local blur or fog component, andhence to an improved and sharper image

A popular 33 convolution mask that mimics unsharp masking is given by

2 6 6 4

;

1

8 2 ;

1 8

; 18 ; 18 ; 18

3 7 7 5

g(mn) in the given degraded image, and the dierence between the pixeland the local mean g(mn) The expression is equivalent to the mask inEquation 4.14, with = 1 and the local mean being computed as the average

of the eight neighbors of the pixel being processed Note that because themask possesses symmetry about both thexandy axes, reversal has no eect,and hence is not required, in performing convolution

The relative weighting between the pixel being processed and the localdierence could be modied depending upon the nature of the image and thedesired eect, leading to various values at the central location in the maskgiven in Equation 4.14 Equivalently, dierent values of could be used inEquation 4.15 Because the local dierence in Equation 4.15 is a measure ofthe local gradient, and because gradients are associated with edges, combiningthe given image with its local gradient could be expected to lead to edgeenhancement or high-frequency emphasis

Example: Figure 4.20 (a)shows a test image of a clock part (b) of thesame gure shows the result of unsharp masking using the 33 mask inEquation 4.14 It is evident that the details in the image, such as the numerals,have been sharpened by the operation However, it is also seen that the high-frequency emphasis property of the lter has led to increased noise in theimage

Figures 4.21 (a), 4.22 (a), 4.23 (a),and4.24 (a)show the image of a myocyte,

a part of a chest X-ray image, an MR image of a knee, and the Shapes testimage the results of enhancement obtained by the unsharp masking operatorare shown in parts (b) of the same gures The chest image, in particular, hasbeen enhanced well by the operation: details of the lungs in the dark region inthe lower-right quadrant of the image are seen better in the enhanced imagethan in the original

An important point to observe from the result of enhancement of the Shapestest image is that the unsharp masking lter performs edge enhancement Fur-

in Figure 4.25illustrate the artifact in an enlarged format Although the tifact is not as strongly evident in the other test images, the eect is, indeed,present Radiologists often do not prefer edge enhancement, possibly for thisreason.

ar-Note that the unsharp masking operation could lead to negative pixel values

in the enhanced image the user has to decide how to handle this aspectwhen displaying the result The illustrations in this section were prepared

by linearly mapping selected ranges of the results to the display range of

0255], as stated in the gure captions compression of the larger dynamicrange in the enhanced image to a smaller display range could mute the eect

of enhancement to some extent

4.6.2 Subtracting Laplacian

Under certain conditions, a degraded imageg may be modeled as being theresult of a diusion process that spreads intensity values over space as afunction of time, according to the partial dierential equation 11]

In the initial state att= 0, we haveg(xy0) =f(xy), the original image

At some time instant t = 0, the degraded image g( ) is observed.The degraded image may be expressed in a Taylor series as

2

2 @2g

@t2 ( ) +: (4.18)Ignoring the quadratic and higher-order terms, letting g(xy0) = f(xy),and using the diusion model in Equation 4.16, we get

where fe represents an approximation to f Thus, we have an enhancedimage obtained as a weighted subtraction of the given image and its Laplacian(gradient)

A discrete implementation of the Laplacian is given by the 33 convolution

;50250] out of ;184250].

of the subtracting Laplacian display range ;50200] out of ;130282].

(a) (b)

FIGURE 4.22

(a) Part of a chest X-ray image (b) Result of unsharp masking display range

;30230] out of ;59264] (c) Laplacian (gradient) of the image displayrange ;55] out of ;134156] (d) Result of the subtracting Laplacian display range ;50250] out of ;156328]

(a) (b)

FIGURE 4.23

(a) MR image of a knee (b) Result of unsharp masking display range

;40250] out of ;72353] (c) Laplacian (gradient) of the image displayrange ;5050] out of ;302365] (d) Result of the subtracting Laplacian display range ;50250] out of ;261549]

(a) (b)

FIGURE 4.24

(a) Shapes test image (b) Result of unsharp masking display range

;100250] out of ;130414] See also Figure 4.25 (c) Laplacian ent) of the image display range ;5050] out of ;624532] (d) Result of thesubtracting Laplacian display range ;300300] out of ;532832]

(gradi-(a) (b)

FIGURE 4.25

Enlarged views of a part of (a) the Shapes test image and (b) the result

of unsharp masking see also Figure 4.24 (a) and (b) Observe the enhancement artifact

edge-see also Equation 2.82 and the associated discussion Observe that the netweight of the coecients in the Laplacian mask is zero therefore, the maskperforms a dierentiation operation that will lead to the loss of intensityinformation (that is, the result in an area of any uniform brightness value will

be zero)

Letting the weighting factor = 1 in Equation 4.19, we get the following

33 mask known as the subtracting Laplacian:

2 4

Example: Part (c) of Figure 4.20shows the Laplacian of the test image

in part (a) of the same gure The Laplacian shows large values (positive or