fractal modeling and segmentation for the

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 6, DECEMBER 1997 785

Fractal Modeling and Segmentation for theEnhancement of Microcalcifications

in Digital MammogramsHuai Li, Member, IEEE, K. J. Ray Liu,* Senior Member, IEEE, and Shih-Chung B. Lo, Member, IEEE

AbstractThe objective of this research is to model the mam-mographic parenchymal, ductal patterns and enhance the mi-crocalcifications using deterministic fractal approach. Accordingto the theory of deterministic fractal geometry, images can bemodeled by deterministic fractal objects which are attractors ofsets of two-dimensional (2-D) affine transformations. The iteratedfunctions systems and the collage theorem are the mathemat-ical foundations of fractal image modeling. In this paper, amethodology based on fractal image modeling is developed toanalyze and model breast background structures. We show thatgeneral mammographic parenchymal and ductal patterns can bewell modeled by a set of parameters of affine transformations.Therefore, microcalcifications can be enhanced by taking thedifference between the original image and the modeled image.Our results are compared with those of the partial waveletreconstruction and morphological operation approaches. Theresults demonstrate that the fractal modeling method is aneffective way to enhance microcalcifications. It may also be ableto improve the detection and classification of microcalcificationsin a computer-aided diagnosis system.

Index TermsEnhancement, fractal modeling, mammograms,microcalcifications.

I. INTRODUCTION

BREAST cancer is the most frequently occurring cancerand one of the leading causes of death among women [1],[2]. However, there is clear evidence that early diagnosis andsubsequent treatment can significantly improve the chance ofsurvival for patients with breast cancer [1][4]. Mammographyis the most effective method for the detection of early breastcancer [2], [4]. But studies have shown that radiologists donot detect all breast cancers that are retrospectively detectedon the mammograms [5], [6]. Because of the subtle andcomplex nature of the radiographic findings associated withbreast cancer, errors in radiological diagnosis can be attributedto human factors such as varying decision criteria, distractionby other image features, and simple oversight. Studies suggest

Manuscript received April 15, 1996; revised March 25, 1997. This work wassupported in part by the National Science Foundation under NYI Award MIP-9457397 and the U. S. Army under Grant DAMD17-93-J-3007. The AssociateEditor responsible for coordinating the review of this paper and recommendingits publication was L. P. Clarke. Asterisk indicates corresponding author.H. Li is with the Odyssey Technologies Inc., Jessup, MD 20794 USA.*K. J. R. Liu is with the Department of Electrical Engineering, University

of Maryland at College Park, College Park, MD 20742 USA (e-mail: [email protected]).S.-C. B. Lo is with the Department of Radiology, Georgetown University

Medical Center, Washington, DC 20007 USA.Publisher Item Identifier S 0278-0062(97)08961-1.

that these errors may occur even with experienced radiologists[5], [6]. In order to increase diagnostic efficiency, computer-assisted schemes based on advanced image processing andpattern recognition techniques can be used to locate andclassify possible lesions, thereby alerting the radiologist toexamine these areas with particular attention. Moreover, thesecomputer-assisted schemes can improve the performance ofthe automatic computer-aided diagnosis systems, which canserve as a prereader to the radiologist and give the radiolo-gist the second opinion in the diagnosis.Microcalcifications are considered to be important signs of

breast cancer. It has been reported that 3050% of breast can-cers detected radiographically demonstrate microcalcificationson mammograms [7], and 6080% of breast carcinomas re-veal microcalcifications upon histologic examinations [8]. Thehigh correlation between the presence of microcalcificationsand the presence of breast cancers indicates that accuratedetection of microcalcifications will improve the efficacy ofmammography as a diagnostic procedure. The task of detectionof microcalcifications for the diagnosis of breast cancer isa difficult one. Dense breasts, improper technical factors, orsimple oversight by radiologists may contribute to the failureof detecting microcalcifications.Given a mammogram, there are three major problems in

analyzing and detecting microcalcifications.1) Microcalcifications are very small. On mammograms,

they appear as tiny objects which can be described asgranular, linear, or irregular. According to the literature,the sizes of microcalcifications are from 0.11.0 mm,and the average diameter is about 0.3 mm [7]. Smallones (ranging 0.10.2 mm) can hardly be seen on themammogram due to their superimposition on the breastparenchymal textures and noise.

2) Microcalcifications often appear in an inhomogeneousbackground describing the structure of the breast tissue.Some parts of the background, such as dense tissue, maybe brighter than the microcalcifications in the fatty partof the breast.

3) Some microcalcifications have low contrast to the back-ground. In other words, the intensity and size of themicrocalcifications can be very close to noise or theinhomogeneous background.

The above reasons make microcalcifications relatively dif-ficult to detect. Especially, some subtle case, such as faint

02780062/97$10.00 1997 IEEE

786 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 6, DECEMBER 1997

microcalcifications which have small sizes and are superim-posed on dense breast regions, are very difficult to detect, evenfor experienced radiologists. Consequently, computer-assisteddetection of microcalcifications has aroused a great deal ofinterest. Different approaches have been proposed for en-hancing and segmenting microcalcifications, including variousfiltering and local thresholding methods [9][12], mathemat-ical morphology [13], [14], neural networks [15][17], thestochastic models [18], [19], the stochastic fractal model [20],pyramidal multiresolution image representation [21][23], andthe contour-based approach [24]. We noted that most of theenhancement techniques used in the past research works notonly enhanced microcalcifications, but also enhanced back-ground structure and noise. Our basic idea is that if wecan tell the different properties of disease patterns (such asmicrocalcifications) and background patterns in both spatialand frequency domains, then we can separate the wholeimage into different layers using different models accordingto the difference in patterns. One layer only contains dis-ease pattern information. The other layer contains nondiseaserelated background information. Hence, the disease patternwill be enhanced by taking the background layer from theoriginal image. In our previous study, we employed partialwavelet reconstruction and morphological operation to removethe background information, thereby enhancing microcalci-fications. The results were used to test the computer-aideddiagnosis system (CADx), and improved the performance ofCADx [15].Recently, both stochastic and deterministic fractal-based

techniques have been applied in many areas of digital imageprocessing, such as image segmentation, image analysis [20],[30][33], image synthesis, computer graphics [34], [35], [38],and texture coding [36], [37]. Based on the deterministicfractal theory, images can be modeled by deterministic fractalobjects which are attractors of sets of two-dimensional (2-D)affine transformations [38], [40], [41]. In other words, imagecontext can be constructed by a set of model parameters whichrequire fewer bits to describe than the original image. Themathematical theory of iterated function systems (IFS), alongwith the collage theorem, constitutes the broad foundationsof fractal modeling and coding. In this work, we proposethe deterministic fractal model to model the mammographicbackground and to enhance microcalcifications. We observedthat microcalcifications are visible as small objects whichappear to be added to the mammographic background. Someof them are bright, some are faint. Microcalcifications can becharacterized as different shapes. But compared with breastbackground tissue, they have less structure. On the otherhand, the mammographic parenchymal and ductal patterns inmammograms possess structures with high local self-similaritywhich is the basic property of fractal objects. These tissuepatterns can be constructed by fractal models, and be takenout from the original image, as such the microcalcificationinformation will be enhanced. The results are very encour-aging compared with those of partial wavelet reconstructionand morphological operation methods. We anticipate that theproposed fractal approach is very helpful for radiologists todetect the microcalcifications, and also facilitates the evalua-

tion procedures in a mammographic computer-aided diagnosissystem.The remainder of this paper is organized as follows. In

Section II, the general enhancement techniques and the ideabehind our enhancement scheme are described. The theoryand algorithm of fractal modeling are presented in Section III.Also, in this section, the enhancement of microcalcificationsbased on fractal modeling approach is formulated. Evaluationresults are given and discussed in Section IV. Finally, thispaper is summarized in Section V.

II. ENHANCEMENT TECHNIQUESImage enhancement refers to attenuation, or sharpening,

of image features such as edges, boundaries, or contrast tomake the processed image more useful for analysis. Imageenhancement includes gray-level and contrast manipulation,noise reduction, background removal, edge crisping and sharp-ening, filtering, interpolation and magnification, pseudocolor-ing, and so on. For a specific application, the enhancementtechnique used may be different. The greatest difficulty inimage enhancement is quantifying the evaluation criteria forenhancement. Image enhancement techniques can be improvedif the enhancement criteria can be stated precisely. Oftensuch criteria are application dependent. In the following, wesummarize general enhancement techniques used in mammo-graphic images, and in Section III we describe our proposedfractal approach to the enhancement of microcalcifications,which are the important disease pattern on mammograms. Thedefinitions of the criteria used in our study and more detailexplanation of using these criteria are given in Section IV.

A. Conventional Enhancement TechniquesUnsharp masking [26], spatial filtering [27], region-based

contrast enhancement [29], and multiscale analysis [23] arethe most useful techniques to enhance mammographic features.But, most of the enhancement techniques used in past researchenhanced not only microcalcifications, but also backgroundstructure and noise. Therefore, these kinds of enhancementwere not microcalcification-oriented.1) Enhancement by Contrast Stretching: The simplest meth-

od of increasing the contrast in a mammogram is to adjust themammogram histogram so that there is a greater separationbetween foreground and background gray-level distributions.Denoting the input image gray level by , and the output gray-scale values by , the rescaling transformation is ,where can be any designing function. Equation (1) showsa typical contrast stretching transformation of the gray-leveldistribution in the mammogram [42]

(1)

where the slope and are chosen greater than unity inthe region of stretch, the parameters and can be obtainedby examining the histogram of the original mammogram, andis the maximum gray level of the original mammogram.

LI et al.: ENHANCEMENT OF MICROCALCIFICATIONS IN DIGITAL MAMMOGRAMS 787

2) Enhancement by Histogram Modeling: Histogram mod-eling techniques modify an image so that its histogram has adesired shape. This is useful in stretching the low contrastlevels of mammograms with narrow histograms. A typicaltechnique in histogram modeling is histogram equalization.Let us consider the mammogram histogram as a probabilitydistribution. Based on the information theory, the uniformdistribution achieves the maximum entropy which contains themost information. Therefore, if we redistribute the gray levelsto obtain a histogram as uniform as possible, the mammograminformation should be maximized [25], [42].3) Convolution Mask Enhancement: Convolutional mask-

ing is one of the most commonly used methods formammogram enhancement. Unsharp masking and Sobel-gradient operations are two examples. The processed imageis sharper because low-frequency information in the mammo-gram is reduced in intensity while high-frequency details areamplified [26], [27].4) Fixed-Neighborhood Statistical Enhancement: The en-

hancement techniques we stated above are global-basedapproaches. For some mammograms which contain inhomo-geneous background, local-based enhancement techniques canhave better performance. Local enhancement techniques usestatistical properties in the neighborhood of a pixel to estimatethe background, suppress it, and increase local contrast [28].5) Region-Based Enhancement: The above techniques can

all be classified as either fixed-neighborhood or global tech-niques. They may adapt to local features within a neighbor-hood, but do not adapt the size of the neighborhood to localproperties. Many medical images, including mammograms,possess clinically defined image features within a region ofinterest. These features can vary widely in size and shape, andoften cannot be enhanced by fixed-neighborhood or globaltechniques. Thus, there is a need for adaptive-neighborhoodtechniques which adaptively change the size of regions in agiven image and enhance the regions with respect to theirlocal background [29].

B. Enhancement by Background RemovalIn this paper, our goal is to enhance the visibility and

detectability of microcalcifications. Background removal isconsidered a necessary procedure. Background removal is adirect method of reducing the slowly varying portions of animage, which in turn allows increased gray-level variation inimage details. It is usually performed by subtracting a low-pass filtered version of the image from itself. Morphologicalprocessing [13] and partial wavelet reconstruction [15], [22]are two methods of estimating the image background thathave been used successfully for this purpose. We will sum-marize these two methods in Sections II-B. In Section III,we propose a novel enhancement technique by backgroundremoval method, which is based on modeling the backgroundstructure using the fractal model, and subtracting this modeledimage from the original image. The performance of the fractalapproach will be compared with those of the morphologicaland wavelet methods.1) Morphological Operations: Morphological operations

can be employed for many image processing purposes,

including edge detection, segmentation, and enhancementof images [13], [14]. The beauty and the simplicity of themathematical morphology approach come from the fact that alarge class of filters can be represented as the combination oftwo simple operations on images: the erosion and dilation.Let denote the set of integers and denote adiscrete image signal, where the domain set is given by

and the range set byA structuring element is a subset

in with a simple geometrical shape and size. Denoteas the symmetric set of and

as the translation of by where Theerosion and dilation can be expressed as [43]

(2)(3)

Opening and closing are defined as [43](4)(5)

A gray-value image can be viewed as a 2-D surface in athree-dimensional (3-D) space. Given an image, the openingoperation removes the objects, which have size smaller thanthe structuring element, with positive intensity. With an ap-propriate structuring element (it is usually considered to bethe maximal size of microcalcifications), the spots includingmicrocalcifications can be recovered by taking the residualimage of the opening

(6)It is appropriate to ignore the negative values on the residualimage because negative value has nothing to do withthe objects of interest, so we take

(7)This approach belongs to the class of image feature enhance-ment by background removal.2) Partial Wavelet Reconstruction: It has been demon-

strated [44], [45] that if the filters and whichare associated with certain mother wavelet and motherscaling function , are given, one can decompose the digitalsignal at scale via the following recursive equations:

(8)

As a contrary process, can be reconstructed by

(9)

It is convenient to view the decomposition as passing asignal through a pair of filters and with the impulse


responses and and downsampling the filtered signalsby two, where and are defined as

(10)The pair of filters and correspond to the halfbandlowpass and highpass filters. The reconstruction procedure isimplemented by upsampling the subsignals and andfiltering with and , respectively, and adding thesetwo filtered signals together. Usually, the signal decompositionscheme is performed recursively to the output of the lowpassfilter It leads to the pyramid wavelet decomposition. Thus,the wavelet transform provides a multiresolution filter-bankdecomposition of a signal with a set of orthonormal bases.The 2-D wavelet transform can be formed by the tensor

product of two one-dimensional (1-D) wavelet transformsalong the horizontal and vertical directions [45] if the 2-Dwavelet filters are separable. The corresponding 2-D filtersequences can be written as

(11)where the first and second subscripts denote the lowpass orhighpass filtering in the and directions, respectively.With the 2-D wavelet filters, the image can be decomposed

into specific subimages which contain information in differentfrequency regions. Therefore, one can reconstruct the specificinformation by partially selecting specific subimages. Forexample, in order to enhance microcalcifications in a high-frequency region, one can reconstruct a filtered version of themammogram by ignoring the subimages which represent thelow-frequency background [15], [22].

III. FRACTAL MODELING ENHANCEMENT

A. Theoretical BackgroundLet us first define an affine transformation in a mathematical

fashion [38].Definition 1: An affine transformation can be

written as , where is an matrixand is an offset vector. Such a transformation will becontractive exactly when its linear part is contractive, and thisdepends on the metric used to measure distances. If we selecta norm in then is contractive when

(12)

Let denote a metric space of digital images, whereis a set, is a given metric. Given a

complete metric space we can define the metric spacewhere is the space of compact subsets of

and the distance between twosets and is the Hausdorff distance, which is characterizedin terms of the metric Under these conditions, it can beshown that the metric space is complete according to theHausdorf metric [38], [39]. Now let be an originalimage to be modeled. The task is to construct a contractiveimage affine transformation defined from to itself,

for which is an approximated fixed point which is called anattractor. In other words, we wish to findsatisfying the requirement

(13)

such that

(14)

where is a tolerance which can be set to different valuesaccording to different applications. The scalar is called thecontractivity of It is shown in Theorem 1 that can be aset of contractive mappings i.e., According tothe deterministic fractal theory, a set of contractive mappings

is the main part of an iterated function system (IFS). Thedefinition of IFS and Theorem 1 are given as follows [38],[39].Definition 2: An iterated function system (IFS) consists of

a complete metric space with a finite set of contractionmappings , with respective contractivity factors, for , and its contractivity factor is

Theorem 1: Let be an IFS withcontractivity factor Then the transformation

defined by for all , is acontraction mapping on the complete metric spacewith contractivity factor Its unique fixed point, or attractor,

, obeys and is givenby for any denotesiterations of the mapWith the definition of IFS and Theorem 1, one can state the

important property of IFS in the following theorem.Theorem 2 (The Collage Theorem): Let be a com-

plete metric space. Let be given, and letbe given. Choose an IFS with

contractivity factor , so that

(15)

Then , for all , where is theattractor of the IFS.The proof of the Collage Theorem can be found in [38].

The Collage Theorem shows that, once an IFS is found, i.e.,is known such that is satisfied, then from anygiven image and any positive integer one can get

(16)

Since we see that after a number of iterations, theconstructed image will be close visually to theoriginal imageThe key point of fractal modeling is to explore the self-

similarity property of images. Real-world images are seldomself-similar, so it is impossible to find a transformation foran entire image. But almost all real images have a local self-similarity. We can divide the image into small blocks, and


for each block find a corresponding So finally, we candefineNow we introduce a mathematical representation for

digital gray-level images. Letrespectively, then for any

digital gray-level image we haveLet and be subsets of

such that andWe call the range squares, and the domain squares. Wedefine a set of mixing functionssuch that using an affine mapping. So, canbe defined as

(17)

where is a scaling factor and is an offset factor; they areblockwise constants on each Also, let us denote asthe restriction of the function to the set The goal is: foreach a and aresought such that

(18)

is minimized. In practice, we use as the mean squareroot metric. Let be two digital images, then themean square root metric is given by

(19)

B. Algorithm Implementation1) Fractal Modeling: Given an pixels gray-

levels digital image, let be the collection of subsets offrom which the are chosen, and let be the

collection of subsets of from which the are chosen.The set is chosen to consist of 8 8, 16 16, 32 32 pixelsof nonoverlapping subsquares of The set consistsof 16 16, 32 32, 64 64 pixels of overlapping subsquaresof That is, only domains with a block side twicethat of the ranges are allowed, resulting in contraction inthe - plane. Therefore, each range pixel in correspondsto a 2 2 pixel area in the corresponding domain Theaverage of the four domain pixel intensities is mapped to thearea of the range pixel when computing Now, for each

, search for all of to find a which minimizes (18),that is, find the part of the image that looks most similar to thatof Note that each can be rotated to four orientationsand flipped and rotated into four other orientations.

Minimization of (18) can be divided into two steps. First,it is necessary to find the optimal and for Foreach we compute the optimal and using theleast squares estimation method. From (18), we construct anunconstrained optimization problem as follows:

(20)where

(21)for allThus, (20) can be rewritten as

(22)

Through solving and we get theoptimal values of and as shown in (23) at the bottomof the page

(24)

where is the total number of pixels in We put into(22), and obtain the minimum error Then, we set a uniformtolerance , and select the best , such thatSuppose there is a cluster of microcalcifications or some

single isolated ones on the image block above , our intentionis to find an area on which the image has a similarstructure as on , but does not have similar microcalcificationpatterns. Then when a difference between the original imageand modeled image is taken, the microcalcifications will beenhanced. This means that when searching for , the suitable

should not cover the region of In our algorithm,for each given , we constrain the search way of by

The modeling process is summarized in the followingalgorithm.Step 1) Initially, the range squares are chosen to be

nonoverlapping subsquares of size 32 32. Asearch is then performed for the domain squareswhich best minimized (22) and satisfied theconstraint of by

Step 2) If the value of (22) is less than a predeterminedtolerance, then the corresponding and arestored and the process is repeated for the nextrange square. If not, the range square is subdivided

(23)


(a) (b) (c)

(d) (e) (f)Fig. 1. The modeling and enhancement results of the simulated texture image and one real mammogram using the fractal modeling approach. (a) Originalimage, (b) modeled image, (c) enhanced image, (d) original mammogram, (e) modeled mammogram, and (f) enhanced mammogram.

(a) (b)Fig. 2. The effects on the modeled image with different tolerances and block sizes. (a) The plot of MSE between the original and modeled mammogramwith different block size Ri

; = 10.0. (b) The plot of MSE between the original and modeled mammogram with different tolerance ; Ri

= 8.

into four equal squares. This quadtreeing process isrepeated until the tolerance condition is satisfied, ora range square of minimum size (here we set 8 8pixels) is reached.

Step 3) The process is continued until the whole image ismodeled. A choice of along with a correspond-

ing and determines the on Once allare found, we can define such that

where and is the blocknumber of

Step 4) Finally, based on the Collage Theorem, the mod-eled image can be easily obtained by performing


(a) (b) (c) (d)Fig. 3. The enhancement results of the simulated spots on normal breasttissue background: (a) original ROI, (b) enhancement by the fractal approach,(c) enhancement by the wavelet approach, and (d) enhancement by themorphological approach.

TABLE IENHANCEMENT EVALUATION OF PHANTOM IMAGE

TABLE IIBACKGROUND NOISE EVALUATION

the iteration for any starting image of the same sizeaccording to and The iteration stops whilethe predetermined tolerance between the originalimage and modeled image is achieved.

2) Enhancement of Microcalcifications: Based on theabove algorithm development, we can enhance microcalcifi-

TABLE IIICONTRAST EVALUATION

cations by using the fractal modeling approach. Let bethe original image, and be the modeled image afteriterations. The procedure is summarized as follows.Step 1) First, we take the difference operation between

and

(25)where is the residue image.

Step 2) It is appropriate to ignore the negative value ofthe difference image because negative partof does not contain any information aboutspots (including microcalcifications) brighter thanthe background, so we take

(26)where is the enhanced image from whichbackground structures are removed.

Step 3) Image contains useful signals and noises.Below a certain threshold any signal is consid-ered unreliable. The threshold is estimated fromthe image itself as times the global standard de-viation of the noise in an image Thus, thevalue of is the same for all images, but dependson each individual image. can be determined bya two-step estimation process. First the standarddeviation of the whole image is taken, andthe initial threshold is chosen to be about 2.5


Fig. 4. The 1-D profiles of original spots embedded on normal breast tissue background and enhanced results by the fractal, wavelet, and morphologicalapproaches. The corresponding images is shown in Fig. 3.

times this global standard deviation. Second, onlythose pixels in which the gray values are below theinitial threshold are used to recalculate the standarddeviation of the noise. This is a simplified versionof a robust estimation of the standard deviation ofnoise [46]. The final threshold is determined byadjusting the value of so that no subtle cases aremissed using human judgement. In our study, wefound empirically that is a suitable choice.The final enhanced image is

(27)

IV. EVALUATION RESULTS AND DISCUSSIONThirty real mammograms with clustered and single mi-

crocalcifications were chosen as testing images. The areasof suspicious microcalcifications were identified by a highlyexperienced radiologist. The selected mammograms were dig-itized with an image resolution of 100 m 100 m perpixel by the laser film digitizer (Model: Lumiscan 150). Theimage sizes are 1792 2560. Each image is 12 b/pixel. Inthis study, we selected 512 512 regions of interest (ROIs)which contain microcalcifications. In addition, we generatedone simple image based on jigsaw function using computer.The simulated image has a simple periodical texture patternand has a cluster of spots and a single spot embedded in thesimulated background structure. This is a suitable example totest the fractal approach.

(a) (b) (c) (d)Fig. 5. The enhancement results of film defects on selected ROIs onmammograms: (a) original ROI, (b) enhancement by the fractal approach,(c) enhancement by the wavelet approach, and (d) enhancement by themorphological approach.

A. Evaluation of EnhancementIn order to evaluate the enhancement results of different ap-

proaches, we computed the contrast, the contrast improvementindex (CII), the background noise level, the peak signal-to-noise ratio (PSNR), and the average signal to noise ratio(ASNR). The definitions of these indexes are given in thefollowing. All computations were based on the selected local


(a) (b) (c) (d)Fig. 6. The enhancement results of clustered microcalcifications on selectedROIs on mammograms: (a) original ROI, (b) enhancement by the fractalapproach (c) enhancement by the wavelet approach, and (d) enhancement bythe morphological approach.

ROIs, which contain microcalcifications as well as film arti-facts, in the original images, fractal enhanced images, waveletenhanced images, and morphological enhanced images.The contrast of an object is defined by [29]

(28)

where is the mean gray-level value of a particular objectin the image, called the foreground, and is the mean gray-level value of a surrounding region, called background. Thisdefinition of contrast has been used to evaluate the enhance-ment in many papers [23], [29]. We computed the contrastof specific ROIs by manually selecting the foreground whichcontains microcalcifications and background with the help of

TABLE IVCII EVALUATION

the radiologist. In order to perform this task, we wrote aprogram which can let the radiologist trim the local ROIsand choose the foreground with adaptive window size. Forthe regions which contained clustered microcalcifications, allmicrocalcifications were selected with different window sizes(3 3 7 7 pixels), depending on the different size ofmicrocalcifications, and the rest of area was considered as thebackground. For the regions which contained single microcal-cification, the microcalcification was selected as above, and thesurrounding region, which had the size three times larger thanthe size of the foreground, was considered as the background.The positions and sizes of all foreground and background aresame for the three different approaches.A quantitative measure of contrast improvement can be

defined by a CII [23]

CII (29)

where and are the contrasts for the ROIsin the processed and original images, respectively.The background noise level can be measured by the standard

derivation in the background region which is defined as

(30)


Fig. 7. The 2-D surfaces of original and enhanced clustered microcalcifications on one selected ROI of mammograms correspond to the first row of Fig. 6.

where is the gray-level value of a surrounding backgroundregion, and is the total number of pixels in the surroundingbackground region.The evaluation of using the contrast is not sufficient in

our study. The definition of the contrast does not include thebackground noise information. Suppose we decrease the graylevels of all pixels by adjusting the window level linearly whenwe display digital mammograms (radiologists always do thisway when they look at digital mammograms), will increasebecause remains same but decreases. In thissituation, the noise level of background does not change. If thebackground is quite smooth, we can still claim that this simplesubstration operation is a special enhancement. Because, eventhough remains same, the object is more noticeableat low gray-level background according to the property ofhuman visual system [47]. But if the background has largevariety (i.e., the noise level is high), the evaluation of usingthe contrast is not suitable. Since our work focused onspecific microcalcification enhancement and the more inter-esting work for radiologists is to enhance microcalcificationsembedded in inhomogeneous and variable background, wedefined two new evaluation indexes, the PSNR and the ASNR.These definitions were based on the general medical physicsmeasurement and accepted by radiologists for the detection ofmicrocalcifications [48], [49].The PSNR in our work is defined as

PSNR (31)

where is the maximum gray-level value of a foreground.The ASNR in our work is defined as

ASNR (32)

B. Results and DiscussionWe have applied the fractal modeling approach to all real

mammograms and the simulated images. Fig. 1 shows themodeled and enhanced results of the simulated image andone of the real mammograms. As we can see in Fig. 1(b)and (e), the background structure in the simulated image andthe general mammographic parenchymal and ductal patternsin mammograms were well modeled. In Fig. 1(c) and (f), wecan see that all small less-structured objects, which includeclusters of microcalcifications, single microcalcifications, andfilm defects (such as artifacts caused by scratches on the screenor film emulsion), were clearly enhanced. One issue we shouldmention is that the fractal modeling approach needs enormouscomputations according to the algorithm implementation. Forthe 512 512 images we used, the computation time is about2 min using Dec Alpha Workstation.In our study, we found that the block size of and

predetermined tolerance are two very important parameterswhich can affect the modeling process. We have tried different

and based on all tested images. Fig. 2 shows the curvesof the mean square error (MSE) between the original andmodeled mammogram with different and As we cansee in Fig. 2(a), with fixed too large block size would resultin visible artificial edge effects on the modeled image, whichwould increase background noises in the residue image. On theother hand, an of too small size would have less-structuredinformation, therefore, making it difficult to search for thecorrect A similar situation occurred when we chose InFig. 2(b), we can see that with fixed too large wouldintroduce more noise and wrong structures on the modeledimage. But, too small would result in no solution of thesearch process. In our experiment, we found empirically that


TABLE VPSNR EVALUATION

the suitable block size of is from 32 32 to 8 8, and therange of is from 1.010.0.In our experiments, we found that the sharp edge informa-

tion was also enhanced [it is clearly seen on the simulatedtexture image in Fig. 1(c)], as well as the bright spots on theimage. It implies that this algorithm can also be used for edgedetection. On the other hand, it also implies that the IFS fractalcompression scheme cannot effectively encode less-structuredobjects such as microcalcifications, film artifacts, and shapeedges. But this method can effectively encode image patternswhich have high local self-similarity such as mammographicbreast tissue background. The reasons that the IFS cannoteffectively model the less-structured objects are as follows:(1) The searching criterion for mapping the domain region

to range region is to minimize the least square errorto the certain tolerance. For the which contains the less-structured objects, such as sharp edges, the contribution ofthese less-structured objects to the least square error is small.Therefore, the which contains similar structure as the

, but without the less-structured objects, can be found tosatisfy the searching criterion. (2) In our proposed modelingalgorithm, the searching constraint of by wasadded in the encoding procedure. The purpose of the constraintis to find an area on which the image has a similar structureas on , but does not have similar less-structured patterns. Onthe other hand, it also explains why the less-structured objectscannot be modeled. In our study, we used this property and

separated microcalcifications as compression errors. For thepurpose of compression, there are a lot of research [50], [51]on combining fractal method with other compression schemes,this is not an issue in this paper.For the purpose of evaluating the performance of our

proposed fractal enhancement method, we chose for com-parison two similar enhancement techniques of backgroundremoval: the morphological and partial wavelet reconstructionmethods which were described in Section II. In the mor-phological approach, a disk with a diameter of 11 pixelswas chosen as the structuring element This is consid-ered to be the maximal size of microcalcifications (1 mm)on our testing mammograms. In the partial wavelet recon-struction method, we investigated the wavelet decomposi-tion of mammograms which contained microcalcifications atdifferent levels by using Daubechies eight-tap orthonormalwavelet filters [44]. We found that all high frequency noisewhich included film defects was mainly located at the firstlevel of decomposed subimages. On the other hand, all mi-crocalcification information was shown in the second andthird levels of decomposed subimages. The low-frequencybackground structure of the mammogram was concentratedlylocated in the fourth and higher levels. A similar reportwas also in [22]. In our study, we decomposed mammo-grams into four levels, and partially selected subimages inthe second and third levels to reconstruct a filtered versionof the image. After reconstruction, the microcalcifications


TABLE VIASNR EVALUATION

were enhanced and low-frequency background structure wasremoved.A thresholding algorithm, which was described in

Section III, was applied to reduce unreliable noise (thevery low contract noise related to the film granularity or dueto the subtraction operations) in the fractal, morphological andwavelet approaches. Since some subtle microcalcificationsare embedded in very inhomogeneous background, thesemicrocalcifications may be missed after thresholding. So, weused local thresholding based on local gray-level statistics(mean and standard derivation) of image pixels within aspecified window to improve the enhancement results inthe ROIs.Except for the real mammograms and the simulated image,

we created a phantom mammogram by adding three smallspots embedded on the normal breast tissue background.As shown in Fig. 3(a) and its 1-D profile [Fig. 4(a)], theintensities of spots are almost comparable to the intensity ofthe background, the intensity of the left spot is even lowerthan its surrounding background. This is a typical subtle casewhich is easy to be missed by radiologists. After processingall images by the three background removal methods, we cutthe local small blocks, which contained microcalcifications andfilm defects, as the ROIs, from the corresponding original andprocessed images. The image patches used for one comparisoncase are in the same location for all original and processedimages. The sizes of ROIs are 64 64 pixels. Fig. 3 shows

the enhancement results of ROIs in the phantom image. Fig. 4shows the corresponding 1-D profiles of Fig. 3. Fig. 5 showsthe enhancement results of file defects in the mammograms.Fig. 6 shows the enhancement results of clustered and singlemicrocalcifications in the mammograms. The first, second,third, and fourth columns in Figs. 3, 5, and 6 correspondto original ROIs, fractal enhancement, wavelet enhancement,and morphological enhancement, respectively. The resultsindicated that all three approaches removed the background,and in turn enhanced less-structured spots, including micro-calcifications and film defects. We noted that even for thespots embedded in the bright background (such as densetissues), the enhancement results were still very promising.Furthermore, we observed that the fractal and morphologicalapproaches can remove more background structures than thewavelet approach does, especially for those ROIs with verylow contrast compared with the surrounding background (forexample, see in the last row of Fig. 6). But the waveletapproach can preserve the overall shape of spots better thanthe other two approaches. This phenomenon is also clearlyobserved in Figs. 4 and 7. For example, the normalizedintensity of the left spot in Fig. 4 increased more by usingthe fractal and morphological approaches than those of usingthe wavelet approach. But, compared with the profile of theoriginal case, the wavelet method reserved the shape of theprofile better than those of the other two approaches. It isprobably because the wavelet transform has the good time-


frequency localization property, so the wavelet method cankeep the dim margins of microcalcifications better than thoseof the other two approaches.In order to quantitatively measure the enhancement perfor-

mance with different approaches, we computed the contrast,the CII, the noise level, the PSNR, and the ASNR. Tables IVIshowed the evaluation results. The values listed in each row inthese tables were computed based on the image patches whichhave the same location for all original and processed images.As we can see from Tables IIVI, different regions have sig-nificant differences in evaluation results. These depend on thecontrast of microcalcifications to the background, the density,and variety of the background in the original image. In Tables Iand II, it is shown that the noise levels of all enhancementROIs by these three approaches were much lower than theoriginal ROIs. It is reasonable, because background structureswere removed. Among these three approaches the noise levelof the fractal approach was the lowest. From the other tables,we can see that the averaged results of the contrast, the CII, thePSNR, and the ASNR of the fractal approach were better thanthose of the wavelet and morphological approaches. All resultsobtained in this study are very encouraging, and indicatethat the fractal modeling and segmentation method is aneffective technique to enhance microcalcifications embeddedin inhomogeneous breast tissues.

V. CONCLUSIONIn this paper, we proposed a pattern-dependent enhance-

ment algorithm based on the fractal modeling scheme. Theproposed approach was applied to enhance microcalcificationsin mammograms. We compared the enhancement results withthose based on morphological operations and partial waveletreconstruction methods. Our study showed that in terms ofcontrast, CII, PSNR, and ASNR, the fractal approach wasthe best, compared to the other methods. The noise levelin the fractal approach was also lower than the other twomethods. These results demonstrated that the fractal modelingmethod is an effective way to extract mammographic patternsand to enhance microcalcifications. Therefore, the proposedmethod may facilitate the radiologists diagnosis of breastcancer. We expect that the proposed fractal method can alsobe used for improving the detection and classification ofmicrocalcifications in a computer system.

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