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11th Annu. Int.. Con$, 1989, pp. 2017-2018. [15] J. Y. Cheung and S. S. Hull, Jr., “Detection of abnormal electrocardio-

grams using a neural network approach,” in Proc. IEEE Eng. Med. and Biol. Soc. 11th Annu. Int. Con$, 1989, pp. 201.5-2016.

[16] E. Pietka, “Neural nets for ECG classification,” Proc. IEEE Eng. Med and Biol. Soc. 11th Annu. Int. Con$, 1989, pp. 2021-2022.

[17] C. Shim, B. Espinoza-Varas, and J. Y . Cheung, “A PC-based neural network for recognition of different syllables using LPC coefficient difference,” in Proc. IJCNN, vol. 11, June 1990, pp. 185-190.

[18] M. D. Tom, M. F. Tenorio, “A spatio-temporal pattem recognition approach to word recognition,” in Proc. IJCNN, vol. I, June 1990, pp. 351-355.

[19] M. F. Kelly, P. A. Parker, and R. N. Scott, “The application of neural networks to myoelectric signal analysis: a preliminary study,” IEEE Trans. Biomed. Eng., vol. 37, no. 3, pp. 221-230, Mar. 1990.

[20] G. A. Carpenter, S. Grossberg, N. Markuzon, J. H. Reynolds, and D. B. Rosen, “Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps,” IEEE Trans. Neural Networks, vol. 3, no. 5 , pp. 698-713, Sept. 1992.

[21] “MIT/BIH arrhythmia database directory” (CD-ROM), Harvard Univ. and Mass. Inst. of Tech. Div. of Health Sciences and Tech., Cambridge, MA, document BMEC TROlO (revised), July 1992.

[22] T. Y. Lo and P. C. Tang, “A fast bandpass filter for ECG processing,” in Proc. IEEE Eng. Med. Biol. Soc. 4th Annu. Int. Con$, 1982, pp. 2tL21.

[23] N. V. Thakor, J. G. Webster, and W. J. Tompkins, “Estimation of QRS complex power spectra for design of a QRS filter,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 11, pp. 702-706, Nov. 1984.

[24] S. J. Orfanidis, Optimum Signal Processing-An Introduction, 2nd ed. New York: McGraw-Hill, 1988, pp. 2.59-261.

[2S] G. A. Carpenter, S. Grossberg, and D. B. Rosen, “Fuzzy ART: Fast stable learning and categorization of analog pattems by an adaptive resonance system,” Neural Networks, vol. 4, pp. 7.59-771, 1991.

[26] F. M. Ham and S. Han, “Quantitative study of the QRS complex using , fuzzy ARTMAP and the MITBIH arrhythmia database,” in Proc. 1993

INNS Annu. Meet. (World Congress on Neural Network), July 1993, vol. 1, pp. 207-211.

[27] ~, “Cardiac arrhythmia classification using fuzzy ARTMAF’,” in Proc. IEEE Eng. Med. and Biol. Soc. 15th Annu. Int.. Con$, Oct. 1982, pp. 288-289.

TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996

Automated Detection of the Left Ventricular Region in Gated Nuclear Cardiac Imaging

Abd-El-Ouahab Boudraa,* Mohammad Arzi, Jacques Sau, Jacques Champier, Slimane Hadj-Moussa, Jean-Eugbne Besson,

Dominique Sappey-Marinier, Roland Itti, and Jean-Jacques Mallet

Ahstmet- An approach to automated outlining the left ventricular contour and its bounded area in gated isotopic ventriculography is proposed. Its purpose is to determine the ejection fraction (EF), an important parameter for measuring cardiac function. The method uses a modified version of the fuzzy C-means (MFCM) algorithm and a labeling technique. The MFCM algorithm is applied to the end diastolic (ED) Ranre and then the (FCM) is applied to the remaining images in a “box” of interest. The MFCM generates a number of fuzzy clusters. Each cluster is a substructure of the heart (left ventricle, ...). A cluster validity index to estimate the optimum clusters number present in image data point is used. This index takes account of the homogeneity in each cluster and is connected to the geometrical property of data set. The labeling is only performed to achieve the detection process in the ED frame. Since the left ventricle (LV) cluster has the greatest area of the cardiac images sequence in ED phase, a framing operation is performed to obtain, automatically, the “box” enclosing the LV cluster. The EF assessed in 50 patients by the proposed method and a semi-automatic one, routinely used, are presented. A good correlation between the two methods EF values is obtained ( R = 0.93). The LV contour found has been judged very satisfactory by a team of trained clinicians.

I. INTRODUCTION Gated cardiac blood pool imaging is a well-established method for

the assessment of the left ventricular function (LVF) [I]. Many studies have indicated excellent correlations between data derived from gated cardiac imaging and contrast angiography. The quantification of the LVF is of fundamental importance for the diagnosis. It requires the detection of the LV contour in the end diastolic (ED) and end systolic (ES) frames. The aim of the study is to estimate noninvasively the ejection fraction (EF). The parameter is a good indicator of the LVF. The acquisition of the cardiac frames is gated by signal derived ,from the patient’s electrocardiogram. The images are two-dimensional (2-0) projections of the three dimensional count distributions. Our study is limited to the left anterior oblique (LAO) projection, standard view, where the left and right ventricles are well separated.

In the semi-automatic method [2]-[4], the LV boundary is de- lineated manually by the user. This operation results in relatively large inter- and intra-observer variations. A second approach is the automatic one. For example, Duncan ‘ [5] uses directional gradient edge operators and edge point linking scheme. To find complete

Manuscript received November 15, 1993; revised November 3, 1995. Asterisk indicates corresponding author.

*A. E. 0. Boudraa is with the Laboratoire de Biophysique, Facult6 de MCdecine Alexis Carrel, rue Guillaume Paradin 69372 Lyon Cedex 08, France (e-mail: boudra@cimac-res.univ-lyon 1 .fr).

M. Arzi is with Inserm U94, 69500 Bron, France. J. Sau is with Centre de MCcanique, Universit6 Lyon I, 69622 Villeurbanne

Cedex, France. J. Champier, J.-E. Besson, and J. J. Mallet are with the Laboratoire de

Biophysiqne, Facult6 de Mtdecine Alexis Carrel, 69372 Lyon Cedex 08,. France.

S . Hadj-Moussa is with CMC, 25002 Constantine, Alg6rie. D. Sappey-Marinier is with the Laboratoire de Biophysique, Facult6 de

R. Itti is with the Centre de MBdecine NuclBaire, HBpital Neuro-

Publisher Item Identifier S 0018-9294(96)02438-X.

M6decine Lyon-Sud, 69921 Oullins, France.

Cardiologique, 69394 Lyon Cedex 03, France.

0018-9294/96$05.00 0 1996 IEEE

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996 43 1

boundaries, a knowledge-based strategy that incorporates appropriate local and global information in heuristic cost functions is used. Dumay et al. [6] have modified the policy-iteration method [7] to detect the optimal closed path that corresponds with the closed object contour. In a previous paper, we have used the Fourier analysis and the Fuzzy C-Means (FCM) to outline, automatically, the LV region [8]. This method is very dependent on the phase image quality. In patients with heterogeneous ventricular contraction the method fails. In fact, the generated phase images present a ventricular cavity with zones contracting normally and the others are either with late or advance contraction. In this case, it is difficult to separate the ventricular region from the remaining cardiac structures.

In this paper, we propose a new approach to LV detection problem using a modified version of the FCM (MFCM) algorithm and a labeling technique. This approach is fully automated. The method does not make any assumption about the heart shape to guide the detection process. Our aim is to apply the method to a large class of pathological cases corresponding to variable LV shapes (e.g., aneurysm ...). This method avoids the manual outlining of the LV region which is subjective and degrades the reproducibility of the study. The MFCM is used to segment the cardiac image into a given number of clusters (segmentation process) using the grey level as feature. Each cluster is a heart substructure (LV ...). A feature vector is assigned to the cluster in which its fuzzy membership is maximal. Finally, a labeling technique (recognition process) is performed in ED phase to identify which cluster is the LV using as features the spatial coordinates. A new method to estimate the adequate number of clusters is introduced.

11. How MANY CLUSTERS?

In practice, the number of clusters, e, is unknown and in theory it must be 22. The scintigraphic cardiac image is a functional image and not an anatomical one. In the standard view (LAO) only the LV region appears not connected to the surrounding structures. In this kind of images it is impossible to see all anatomical structures such as the great vessels which are superposed. The algorithm needs the adequate number of clusters to work correctly. If c is too 1-e this causes the algorithm to find c fuzzy clusters which not totally correspond to the homogeneous regions of the original frame (splitting process). A similar situation arises when c is less than the optimal value. The algorithm merges the LV cluster and the cluster formed by the right ventricle (RV) and the atria (AT): (RT = RV + AT) into a large cluster formed without boundaries where there are anatomical boundaries (merging process). We cannot choose randomly c without taking account of the information in the image. Furthermore, computer experiments, in nuclear medicine, have indicated that the number of clusters is very image dependent.

There are several relative indexes for comparing clusterings with different number of clusters and deciding which clustering is the “best” 191. But there is no single validity clustering index which provides a satisfactory solution for a variety of cluster structures. Both of the merging and the splitting processus are to be avoided. The new index, proposed for estimating the adequate number of clusters in the image is defined as follows:

B c r l t ( c ) = R ( c ) + QVwt (c )

where Vw,(c) is the ratio of the within-cluster variance to the total, vart and var(li) are the variances of the gray level pixel in the

image and in the lith cluster, respectively. cy is a mixing factor and ll-u, - u,11 is the distance between cluster centroids q and T .

Elcrlt is a combination of two cluster validity indexes Vwt () and R() . Using more than one index may be hepful in determining the optimal number of clusters in the image [lo]. The first term, R() , is connected to the geometrical property of data set. While the second term, V ( ) , takes account of the homogeneity in each cluster. Since the two terms take numerical values lying in two different ranges ( R ( ) >> Vwt()), a mixing factor, a, is introduced to perform a relative normalization between the contribution of each term. Its exact value is of little importance. R ( ) is an increasing function of the number of clusters and its maximum occurs at M 2 clusters ( M x M is the size of the image). In this case each pixel is a cluster centroid. While Vwt() is a decreasing function of the number of clusters and has a maximum at two clusters

Since R(Z) >> Vwt(Z), one may choose Q so that

When the maximum of I , l,,,, is known the mixing factor is

R ( L X ) Vwt(2)

cy = -.

Computer experiments have indicated that the effect of the relative weight, Q, has much more to do with the Scaling of the two terms than their abstract relative importance. Once the scaling issue is settled, the results depend little on the relative importance. The definition of Bcrlt as a weighted sum of a monotonically decreasing function with a monotonically increasing one, guarantees the optimality of c in the range of practical interest.

The decision rule for estimating the true number of clusters is to search for minimum of the Bcr,t(c) curve. A number of clusters, c, is considered as the optimum value, copt, if it verifies the following conditions

where 9 is a threshold value. These conditions allow to identify a “significant” minimum which corresponds to a compact and separate c-partition. 0 < 0 indicates that the point c is located in the decreasing region of Bcrit curve while 0 > 0 signals that the point c is in the increasing region. In other words, the Bcr,t curve exhibits a minimum in point e . An aditional condition, 2), is introduced to determine if the minimum found is “significant” or not. w index is the relative difference between two consecutive numbers of clusters. It signals the presence of a “significant” minimum. The value of w may vary from a kind of images to an other because the contrasts and the intervals of variations are different. The definition of 7 value does not necessarily guarantee finding a global minimum but in general it allows to signal “significant” minimum of practical interest. The value of c is considered “significant” minimum if the value of w is 27.

111. FUZZY CLUSTERING

In hard clustering, each data point is affected to one and only one of the clusters with a degree of membership equal to one. The boundaries between the clusters are well defined. The inadequacy of this model is due to the fact that it does not reflect the description of the real data, where boundaries between clusters might be fuzzy and where a more subtle description of the feature vectors’ affinity to the specific cluster is required [ l l ] . The automatic detection of

432 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996

these boundaries is difficult due to the fuzziness of pixel information. The FCM algorithm is believed to be suitable for overcoming this problem. The FCM algorithm presents the advantage that it has a lesser tendency to get trapped in local minima. It does not make any assumption about the distribution of the patterns in the feature space and the geometry of the clusters.

A. Fuzzy C-Means Algorithm Let RP be the set of p-tuples of reals and X C RP be a subset of n

distinct data points. Every function U: X + [0,1] is said to assign to each z E X its grade of membership in the fuzzy set U. The function U is called a fuzzy subset of X . It is desired to "partition" X by means of fuzzy sets. This is accomplished by defining several fuzzy sets on X such that for each z E X , the sum of the fuzzy memberships of x in the fuzzy subsets is one. We note U& the value of the membership function of the ith fuzzy subset for the kth datum. The FCM algorithm uses iterative optimization to approximate minima of an objective function, J , is defined as follows

X k E arg[max~+&~k)l, Stop}

z k E arg[maxl<z&zk)l if ( 6 5 cl){

if ( c = 3){

goto Step 6)

u Z k +- <&, goto Step 3) .

c +- c + 1, goto Step 2}

else{

Step 6) Compute B,,,, and w ; if (B,,,t > BB,,){

if (w L 17H €1 +- €2, Copt + c - 1, lo + True, goto Step 2.} else{

B L + BcrLt, c + c + 1, goto Step 2)). The proof of the above optimization procedure [Step 1)-Step 5)] is

by Lagrangian multiplier and can be found in Bezdek [12]. It will be omitted here. An other defect measure, A(), may be used [13]. For study, the value of m is set to two [12]. The above MFCM algorithm steps constitnte the Fuzzy,lnst () procedure.

where vi E RP the cluster center of cluster i . The parameter m is the weighting exponent for u ; k and controls the "fuzziness" of the resulting cluster [12]. The larger the value of m the less is the effect of the data points producing uniformly low membership values on the determining the clusters. Consequently, such points tend to be ignored in determining the centers and membership functions. In practice, it is interesting to choose any value of m from the interval { 1.5, 3.0). The optimization of J , produces a fuzzy c partition of the data X = ( 2 1 , . . . , zn}. Local extrema of the objective function are indicative of an "optimal" clustering of the input data. One should obtain high memberships in cluster i for data which are close to the vector U;.

B. Modified Fuzzy C-Means Algorithm While in the FCM algorithm the number of clusters in the data must

be specified prior starting the computing, in the MFCM algorithm the adequate number of clusters is estimated during the clustering process. Consequently, the MFCM like the FCM converges to either a local minimizer or saddle point of the functional J,.

Let lo be a logical value, €1 and €2 two threshold values, c = 3 is the starting number of clusters (see Section V m ) and BrIlt is the value of the cluster validity index corresponding to the adequate number of clusters. €1, is the convergence threshold for the number of clusters c (coarse clustering) and €2 (€2 < €1) is used to refine the fuzzy clustering result for c = Copt (fine clustering) (see Section VIII). For m > 1 Bezdek [12] gave the following necessary conditions (Step I to Step 5) for a minimum of J , Step 1) c + 3, Bc',,t +- 0, lo +- False. Step 2) Choose c fuzzy clusters such that

Step 3) Calculate-the c clusters center

t i o , l l u % k l "

Step 4) Compute new membership values 1

u t k = 2 . - L o cg,, (ddlk 1

Step 5 ) Compute the defect measure A ( k ) = maxl~,<,[lu,k - GL,lcl]

S = m a x ~ ~ k ~ , [ A ( k ) ] if (6 5 €1 and lo = True){

IV. RECOGNITION PROCESS In this high-level processing, image domain knowledge is used

to assign a label to each cluster, thereby generating a semantic description of the image. This a priori knowledge is used to identify which cluster is the LV. To begin the recognition process, the image must be binarized.

A. Thresholding The MFCM affects to each cluster an integer value I ; 1 5 1 5 Copt .

Since the LV and RT have almost the same activity and are spatially very close together, the MFCM assigns them the same value of one. It should be noted that in both LV and RT regions the intensities are the highest in the image. The maximum value of gray levels (Gmax) allows to find the integer 1. The data point having G,,, value belongs to LV region. The LV is always located in the right-hand of the image and shows higher intensity values. Let I m ( M , M ) be the ED image, pp lC( M , M ) the corresponding cartographic image generated by the MFCM algorithm and RL = {LV, RT} the cluster to be isolated.

ThesholdingO I

V ( i , j ) 15 i , j 5 M if ( Im(i , j ) = Gmax){

V ( i , j j 15 ?. j 5 M if ( ~ ( i , j ) # z R L ) else

IC(i,Jj + 1;

ZRL +- IC(i , j j ; Break;}

~ c ( i , j ) +- o

B. Connected Components

The pixels of either LV or RT are closely related. Both LV and RT are connected sets. The connectivity is potentially established in each cluster. This topological attribute is very interesting for extraction of LV cavity. Let x be a bounded subset of an image indexed by (i,~) and P ( Z P , J P ) and Q ( i ~ , j g ) two 1-valued image points. P ( i p , j p ) is associated to the following subsets:

~ ~ ~ ( P ) = ( Q E ~ ; ~ ~ ( P , Q ) = I Z ~ - Z Q ~ + ~ ~ ~ - _ I Q I ~ 1} N2(P) = { Q E r ;&(P ,Q) = max(lip - "1, Ijp - ~ Q I ) 5 1).

The subsets N1(P) and Nz(P) are called the eight-neighborhood and four-neighborhood of P. Two arbitrary points P and Q are eight-neighbors if &(P , Q) 5 1 and four-neighbors if d l (P, Q) 1. 1.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996 433

Fig. 1. Different steps of the proposed method [top left (a)] ED frame (patient 3) in the LAO projection (before filtering). [top middle (b)] Result of the ED frame filtering operation. [top right (c)] Fuzzy clustering result of the filtered frame. [bottom left (d)] Cartographic image thresholding proess. [bottom middle (e)] Labeling of the connected components LV and RT. [bottom right (f)] The fuzzy clustering space of the remaining 15 images to the rectangular window.

C. Labeling After thresholding, IC() is separated into two classes: the class "RL

pixels" and the class "Background pixels" noted S and S" (S' i s the complement of S) respectively. Since LV and RT are connected sets, S may then be represented by two connected components. A labeling technique is used to separate these two connected components. The labeling process consists in assigning a unique label to each connected component in the image. Thus, two one-valued points share the same label if and only if they are in the same connected component. To label S, the cluster RL is associated to dl-connected component (the extension to &-connectivity is obvious). The major steps of the labeling algorithm are:

Step 1)

Step 2)

Step 3)

Step 4)

Labelclust () k + 0; V ( i , j ) 15 i , j 5 M ICk(i , j ) t M * M + 1; If ( IC(i , j ) = 1) 1Ck(i , j ) t ( i - 1) * M + j ; I c + - b + l ; V ( i , j ) 2 5 i , j 5 M ~ ~ " ' ( i , j ) +- min ( l c k ( i , j ) , l c k ( i - 1,j),

If ( ICk(i , j ) = M * M + 1) IC"'(i,j) t ICk(Z,j); V ( i , j ) 2 5 i , j 5 M If (~c" ' ( i , j ) # ~ @ ( i , j ) ) goto Step 2; LLV e 0; V ( i , j ) 2 I: i , j 5 M

If (LLV < ICk+'(i,j)) LLV t Ick+'(i,j);}.

1c"i + l , j ) , I C k ( i , j - 1));

If (ICk+'(i,j) # M * M + 1){

are greater than those of the RT pixels. Since the affected labels are closely related to the connected component position, the resulting LV label, LLv, is greater than the one of the RT. Finally, by simple comparison of the two labels, the LV cavity is totally extracted. Note that the necessary iterations number of labeling ( k ) is found automatically (Step 3) .

V. LV DETECTION METHOD

A. The LV Boundary The MFCM segments the ED frame into copt clusters using as

feature the gray level. The resulting cartographic image IC(M, M ) is binarized and the labeling process performed to achieve the LV outlines detection /*Processing of the ED image: Im(i , j )*/

Fuzzy,l,,t (); /*Fuzzy clustering procedure (c = copt)*/ Thresholdingo; LabelclUst (); Since the LV cluster has the greatest area in the ED phase, LV

contour search procedure is started to delimit the LV boundary [8]. A rectangular window "box" enclosing the LV cluster is obtained. Thus, the clustering of the remaining 15 images is narrowed to this "box". Since the clustering space is reduced, the substructure enclosed within the box is an isolated region (LV) of high intensity against the background of low intensity [8]. Finally, the number of clusters is set to two. In this situation, the labeling process is not necessary.

B. The EF The LV cavity is located in the right part of the image and the RT in the left one. Consequently the position values of the LV pixels

After segmentation and recognition process, the area enclosed by the LV outlines in each frame is computed and the LVF curve

434 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996

TAElLE I

METHOD AND A SEMI-AUTOMATIC METHOD (50 PATENTS) COMPARISON OF EJECTION FRACTIONS ASSESSED BY THE PROPOSED

90

I' Patient EFsa(%) EFfu(%) Patient EFsa(%) EMU(%)

1 50 47 26 55 52 2 57 60 27 48 52 3 32 29 28 55 54 4 55 50 29 49 43 5 41 37 3 0 46 39 6 55 47 31 47 45 7 35 32 32 45 46 8 30 35 33 68 68 9 50 48 34 85 88 10 54 56 35 61 55 11 30 31 36 18 18 12 70 58 37 52 34 1 3 66 67 38 61 66 14 30 39 39 50 46 1 5 34 27 40 43 48 1 6 71 62 41 34 40 17 15 19 42 50 47 18 48 52 4 3 23 30 19 16 21 44 50 34 20 42 47 45 59 53 21 25 25 46 57 50 22 54 48 47 60 60 23 16 18 48 67 70 24 57 61 49 58 58 25 47 38 50 33 40

EFfu and EFsa: The ejection fractions calculated respectively by our methcd and the semi-automatic method.

calculated. The purpose of this curve is to locate the ES frame which will be used to estimate the background noise. The ES frame is located by finding the minimum 'of the LVF curve. To calculate the EF, the largest and the smallest areas enclosed by the LV outlines are computed and the backgrouncl noise substracted. To estimate the background noise, a region of interest (ROI) is constructed displaced to the right of the LV and following the systolic edge. This operation is achieved automatically [8]. The counts in this region are computed and normalized to the LV area [SI.

VI. SEMI-AUTOMATIC EDGE DETECTION METHOD To evaluate the accuracy of the LV EF measurement method, a

comparison with a semi-automatic one, routinely used in clinical laboratories, is performed. To begin, a box (rectangular window) is manually positioned (in our method the box is found automatically) by the user over the LV and its size adjusted. Once the box is positioned and sized, a zero crossing, second derivative edge tracking algorithm is used beginning with the present image until the end of the series. In searching for the eldge of the ventricle, the algorithm starts in the center until a matrix point is reached that satisfies one of two conditions. The first is that a point be a zero point of the two-dimensional second derivativle. The second is that the number of counts at a matrix is less than a threshold value. An edge is detected if a point is a zero crossing or if a point has a count value less than an user selected threshold value. Computer experiments have indicated that the threshold values are about 55% of the maximum count in the box. The edge search is limited to the box.

VII. IGSULTS

The data are acquired in 64 :e 64 matrices (16-bit word mode image), 16 frames per cardiac cycle, and are stored in a nuclear

Clusters number (e)

Fig. 2. Plot of Elcrlt versus number of clusters (Patient 3) .

Fig. 3. substraction.

Automatic placement of the region of interest for background noise

medicine computer system. The obtained images are filtered using a linear filter. This filter has been shown to be more effective in this kind of images [5] , [SI. The algorithm has been run on the LAO view [Fig. l(a)] of from over 50 studies with encouraging results. In higher imaging quality studies (one of which was used as an example throughout the text), the algorithm works consistently well. Fig. 1 shows the different processing steps of the ED image of patient number 2. The linear filtering (size of 3 x 3) has been proved to provide best results [Fig. l(b)]. The clustering [Fig. l(c)] was performed with threshold value e2 = 5 . lop3. A correct clustering of Fig. l(c) data set is obtained With copt equal to eight and the resulting Bc,,t values are plotted in Fig. 2. The relative difference, w, is set to 2.5%. This value has given good fuzzy partitions in a large class of images. We expect that this value may vary from a kind of images to another (e.g., magnetic resonance images, ultrasonic images ...) because the contrasts and the plages of variations are different. This value is application dependent. After thresholding process [Fig. 1 (d)], the obtained connected components (clusters) were labeled [Fig. l(e)]. Fig. l(f) shows the rectangular window, enclosing the LV in the ED, where the remaining 15 images were processed using only the FCM algorithm ( copt = 2). The background correction has been automatically determined. An ROI was drawn at the right side of the LV on the ES frame (Fig. 3) (Patient 10). The average count was 40 cpslpixel. Values of the LV EF obtained by the fuzzy method and the semi-automatic method did not differ significantly from each other (Table I). A good correlahon was found between the two methods: R = 0.93 and y = 0 .87667~ + 4.7265. The LV volume curves of patients 2, 3, and 42 are plotted on the same graph (Fig. 4), the three curves are similar.

'

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 4, APRIL 1996 435

7000 . - Patient 2 (EF=60%)

6000

5000

4000

3000

\ - Patient 3 @F=29%) o ------O- Patient42(EF=47%) /

0 2 4 6 8 1 0 1 2 1 4 16 Frames number

Fig. 4. LVF curves obtained in three patients (2, 3, 42)

We have acquired two sets of 16 images 10 min apart on patient 9. The acquisition times are 10 and 13 min, respectively. This temporal difference is due to radioactivity decreasing. The EF calculated are 50% and 51%. This difference is judged not significant.

VIII. DISCUSSION The main drawback with the FCM algorithm is that the number of

clusters must be fixed before the starting clustering process. Actually, there is no simple way to determine the adequate number of clusters. As another difficulty, the scintigraphic cardiac image is a functional image and not an anatomical one. However, it exhibits at least three clusters (c = 3): the ventricles, the atria, and the background. Finally, we have started the clustering process by setting c to three instead of c = 2 (limit theoretic value).

The number of clusters is image dependent. The optimal number of clusters are, respectively, 10 (w = 7.10%), 8 (w = 8.10%), and 11 (w = 11.50%) for the patients number 2, 3, and 42. In the Fig. 5 we note that the rate is more apparent and the liver is not visible at all. The Bcrlt curve shows a significant minimum at 10 clusters with local minima at 6, 8, and 16 clusters (Fig. 6). The number of clusters of data set and the corresponding values of and w are listed in Table 11. The negative values of w correspond to the decreasing region of the curve (merging process) and the positive values to the increasing one (splitting process). The three local minima values of B,,,t, 2.35%, 1.41%, and 0.82%, are less than 2.5%. A best fuzzy clustering is achieved for w equal to 7.1%. In practice once a value of w greater than 2.5% is obtained we stop the clustering process.

The methodology uses a “coarse-fine” concept. The coarse clus- tering attempts to estimate the number of clusters (cop*) using the threshold value €1. The fine clustering assigns the few data points which remain not well affected to the adequate cluster center using the threshold value €2. Most data points are assigned in the coarse clustering, and give the minimum value of Bcrlt() corresponding to the adequate number of clusters. Computer experiments have indicated that data points reaffected in the fine clustering may lead, in certain cases, to BcrLt(copt) value less than that in the coarse clustering but it remains less than Bc,,t(copt + 1). One may only perform a fine clustering to estimate the value of Copt but this manner of proceeding is computationally very expensive. The threshold values used for the coarse and the fine clustering are €1 = 510-* and €2 = 51OP3, respectively.

The main drawback with the Bcrlt index is computational. It has been noticed that for a very large number of clusters (e.g., c > loo), the computing becomes very important. Fortunately, this is not a serious problem since in practice the feasible number of clusters is much smaller than the number of data points. From our experience

TABLE I1 w VALUES VERSUS NUMBER OF CLUSTERS (PATIENT 2)

C

3

4

5

6

7

8

9

10

11

1 2

Bcrit(c)

87.60

58.51

57.86

42.75

43.77

35.00

35.50

33.69

36.17

37.97

38,30

48,41

50,76

48,6

48,64

49,OO

w

-39.81%

.1.11%

-30.03%

+2.35%

-22.26%

+1.41%

-5.23%

+7.10%

+4.85%

+0,85%

+23,3%

+4,73%

-43,47%

+0,82%

+0.73%

with the scintigraphic images processing, the number of clusters does not exceed 18.

The imaging is performed in the LAO where the two ventricles are we11 separated. The LV appears not connected to the surrounding structures. The proposed method does not use the left lateral (LL) view which is sometimes hepful in detecting LV aneurysm. In the LL projection, where the LV overlaps the RV, the method fails. If the patient is not well positioned, the LV and the RV are not separated. This overlapping may invalide the study.

To find the label of the RL cluster, we searched the value of G,,, in the ED frame. This value corresponds to a “ventricular” data point. To ensure that this pixel lies in the ventricular region, we searched the higher pixel value of each frame of the cardiac cycle, and we took the median value of the obtained data points sequence. The median is a robust parameter and is insensitive to unwanted spike noise values in the monotonically increasing sequence.

The labeling is only performed on the ED frame. Since, the clustering of the remaining fifteen frames is narrowed to the reduced space “box”, the labeling is not necessary. In this case, the number of clusters is set to two (Background, LV). The BCrLt () is only computed in the ED frame to estimate the adequate value Copt . In imposing a fuzzy clustering with two clusters the computation of Bcrlt index is useless. Consequently, the Step 6 is suppressed. We can see that in this case, the MFCM algorithm is nothing but the FCM algorithm. The first image of the series has permitted to search the “box” because in diastole the LV cluster has the greatest area. The clustering of the 15 images was performed in this reduced space. Finally, the size of data to be processed can be largely reduced and computing time saved [8]. One may cluster data points of the remaining images in the whole space ( M x M ) . This implies that the labeling process must be applied 16 times. Consequently the computing time can dramatically

436 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL 43, NO. 4, APRIL 1996

(b) !

Fig. 5. The spleen is well recognized (short arrow).

Study of Patient 2: (a) filtereld ED frame, (b) fuzzy clustering result.

increase: the CPU time is multiplied by a factor of 16. To identify the label assigned by the FCM algorithm to the LV cluster in the “box,” we searched the higher intensity value, G,,,, in this reduced space. Obviously, the value of G,,,,, corresponds to the intensity of a LV region data point. Since the label of each data point is known, after clustering, then the label of the pixel having as intensity G,,, is that of the LV cluster.

Ix. COIVCLUSION

We have described an approach to the automated detection of the LV countours in gated cardiac blood pool imaging. Its purpose

80 -

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2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9

Clusters number (c)

Fig. 6. Plot of BCrlt versus number of clusters (patient 2).

was to determine the EF, a very interesting medical examination. A modified version of the FCM algorithm using a cluster validity index, &it, is proposed. The evaluation of Bcrlt index has permitted to estimate the number of clusters in the ED frame. A labeling is proposed to achieve the detection operation in the ED frame. Finally, a framing operation is performed to obtain, automatically, a rectangular window “box,” enclosing the LV cluster in the ED phase. Since the LV cluster has the greatest area in the ED phase, the clustering of the remaining images is narrowed to this window. Consequently, a significant computation time has been saved. The method does not required any user interaction. One of the criticisms that one can address to this method is that it is limited to the LAO projection. The EF assessed, in 50 patients, by the proposed method correlated well those by a semi-automatic method routinely used in clinical pratice ( R = 0.93). The LV boundary found has been judged very satisfactory by a team of trained clinicians.

ACKNOWLEDGMENT

The authors wish to thank P. L. M. Kerkhof and the anonymous reviewers for their detailed and useful comments which greatly improved the quality of the paper.

REFERENCES

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[2] J. Areeda, E. Garcia, K. Vantrain, D. Brown, A. Wqman, and D. Berman, “A comprehensive method for automatic analysis of restkxercise ventricular function from radionuclide angiocardiography,” in Digital Imaging: Clinical Advances in Nuclear Medicine, P. Esser, Ed. New York Soc. of Nucl. Med., 1982, pp. 241-256.

[3] E. Depuey, J. Almasi, R. Eisner, R. Sonnemaker, and J. Budino, “Ad- vanced nuclear cardiac image processing,” General Electric Company, Medical Systems Division, Miwaukee, WI, internal rep., 1982.

[4] J. H. C. Reiber, “Quantative analysis of left ventricular function from equilibrium gated blood pool sciqtigrams: An overview of computer methods,” Eur. J. Nucl. Med., vol. 10, pp. 97-110, 1985.

[5] J. S. Duncan, “Knowledge directed left ventricular boundary detection in equilibrium radionuclide angiocardiography,” IEEE Trans. Med. h a g . , vol. MI-6, no. 4, pp. 325-336, 1987.

[6] A. C. M. Dumay, M. N. A. J. Claessens, C. Roos, J. J. Gerbrands, and J. H. C. Reiber, “Object delineation in noisy images by a modified policy iteration method,” IEEE Trans. Pattern Anal. Machine Intell., vol. 1, pp. 952-958, 1992.

[7] R. A. Howard, Dynamic Programming and Markov Processes. New York Wiley,, 1970.

[8] A. E. Boudraa, J. J. Mallet, J. E. Besson, S. E. Bouyoucef, and J. Champier, “Left ventricle automated detection method in gated isotopic

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ventriculography using fuzzy clustering,” IEEE Trans. Med. Imag., vol. 12, no. 3, pp. 451465, 1993.

[9] R. C. Dubes, “How many clusters are best?-An experiment,” Pattern Recogn., vol. 20, no. 6, pp. 645-663, 1987.

[lo] J. Mao and A. K. Jain, “Texture classification and segmentation using multiresolution simultaneous autoregressive models,” Pattern Recogn., vol. 25, no. 2, pp. 173-188, 1992.

[I I ] I. Gath and A. B. Geva, “Unsupervised optimal fuzzy clustering,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11. no. 7, pp. 773-781, 1989.

[12] J. C. Bezdek, Pattem Recognition with Fuzzy Objective Function Algo- rithms. New York Plenum, 1981.

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Correction to “Morphological Model of Human Colon Tissue Fluorescence”

George I. Zonios, Robert M. Cothren, Joseph T. Arendt, Jun Wu, Jacques Van Dam, James M. Crawford,

Ramasamy Manoharan,* and Michael S. Feld

Fig. 4 in the above paper’ was printed incorrectly in the February issue of this Transactions. The correct arrangement of the figure is shown at right.

We apologize to the authors and readers for this error.

Manuscript received February 23, 1996. Asterzsk indicates corresponding author.

G. I. Zonios, J. Wu, and M. S. Feld are with the George R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.

*R. Manoharan is with the George R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, Rm. 6-014, Cambridge, MA 02139 USA (e-mail: rmanohar@mit.edu).

R. M. Cothren and J. T. Arendt are with the Department of Biomedical Engineering, The Cleveland Clinic Foundation, Cleveland, OH 44195 USA.

J. Van Dam is with the Gastroenterology Division, Brigham & Women’s Hospital, Boston, MA 021 15 USA.

J. M. Crawford is with the Department of Pathology, Brigham & Women’s Hospital, Boston, MA 021 15 USA.

Publisher Item Identifier S 0018-9294(96)03834-7. ‘G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M.

Crawford, R. Manoharan, and Michael S. Feld, IEEE Trans. Biomed. Eng., vol. 43, no. 2, pp. 113-122, Feb. 1996

I I

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(f) Wavelength (nm)

Fig. 4. Typical intrinsic fluorescence spectre of the various morphological fluorescent structures found in colon tissue, (a) normal lamina propria, (b) adenomatous lamina propria, (c) eosinophil (measured on normal sample), (d) normal submucosa, (e), (f) dysplastic crypt cell. All spectra have been measured on frozen tissue sections, except of the dysplastic crypt cell spectrum in (f) which is measured on fresh tissue obtained using cytology brushing.

0018-9294/96$05.00 0 1996 IEEE