A Comparative Analysis of Depth Computation of Leukaemia Images using a Refined Bit Plane and Uncertainty Based Clustering Techniques

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1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 15, No 1 Sofa 2015 Prnt ISSN: ; Onlne ISSN: DOI: /cat A Comparatve Analyss of Depth Computaton of Leukaema Images usng a Refned Bt Plane and Uncertanty Based Clusterng Technques Swarnalatha Purushotham, B. K. Trpathy School of Computng Scence and Engneerng, VIT Unversty, Vellore , Inda Emals: pswarnalatha@vt.ac.n trpathybk@vt.ac.n Abstract: Several mage segmentaton technques have been developed over the years to analyze the characterstcs of mages. Among these, the uncertanty based approaches and ther hybrds have been found to be more effcent than the conventonal and ndvdual ones. Very recently, a hybrd clusterng algorthm, called Rough Intutonstc Fuzzy C-Means (RIFCM) was proposed by the authors and proved to be more effcent than the conventonal and other algorthms appled n ths drecton, usng varous datasets. Besdes, n order to remove nose from the mages, a Refned Bt Plane (RBP) algorthm was ntroduced by us. In ths paper we use a combnaton of the RBP and RIFCM to propose an approach and apply t to leukema mages. The am of the paper s twofold. Frst, t establshes the superorty of our approach n medcal dagnoss n comparson to most of the conventonal, as well as uncertanty based approaches. The other objectve s to provde a computer-aded dagnoss system that wll assst the doctors n evaluatng medcal mages n general, and also n easy and better assessment of the dsease n leukaema patents. We have appled several measures lke DB-ndex, D-ndex, RMSE, PSNR, tme estmaton n depth computaton and hstogram analyss to support our conclusons. Keywords: RBP, Otsu thresholdng, hstogram analyss, RMSE, PSNR, clusterng, conventonal methods, C-Means, RIFCM, depth computaton, leukaema mages. 126

2 1. Introducton Leukaema s a cancerous condton characterzed by a large quantty of abnormal whte blood cells n the body. It affects blood formng cells n the body. Leukaema ntates n the bone marrow and spreads to other parts of the patent s body. There are two types of leukema, namely acute and chronc. In chronc leukaema, the leukaema cells multply slowly comng from mature, abnormal cells. Acute leukaema develops from early cells, called blasts, whch are young cells that get dvded frequently. The causes of leukema can arse abruptly or slowly and the symptoms are fever, weakness, nfectons, etc. Doctors can dagnose leukaema from the results of several tests rangng from blood tests to spnal taps. A sample of blood s examned to check for changes n the chromosomes of the lymphocytes. Several tests are conducted concernng the functonng of dfferent organs, n order to confrm whether a person s affected by leukaema or not. Treatment for leukaema vares greatly dependng on ts type and the stage of the dsease. Many tmes leukaema s treated by one or more types of treatments. These are chemotherapy, radaton therapy, bologc therapy, surgery and hematopoetc cells or bone marrow transplant, based on the serousness of the dsease. Except hematopoetc transplant whch s a procedure to restore the normal marrow producton that has been destroyed by treatment wth hgh doses of antcancer drugs or radaton, others do not need any method of replacement. Hence, classfcaton plays a major role n the dagnoss of the dsease and the treatment s based on ts degree of development. Medcal magng s becomng wdely prevalent for dagnoss and analyss of a vared number of dseases. Image processng technques are now beng extended for early and effcent detecton of blood cancer or leukaema. As above noted, leukaema affects the blood formng cells and t results n the producton of abnormal leucocytes or Whte Blood Cells (WBC) n a large amount. Snce the number of these cells ncreases, they hnder the producton of the Red Blood Cells (RBC) and prolferate to the bone marrow [1]. When the number of abnormal blood cells ncreases n the bone marrow, they overflow and enter the blood vessels, leadng to the transfer of these cells to other organs and causng mult-organ falure. The presence of a large number of whte blood cells ndcates the presence of leukaema. Conventonal methods use manual countng of the blood cells from the blood smear of the patent. Ths method s very slow. Moreover, the techncans need to start all over agan f they are nterrupted nbetween. To overcome ths problem, we need a feasble soluton whch s not only faster than the tradtonal methods, but t s also accurate and cost-effcent. In ths paper we shall use an algorthm called a RBP flter algorthm for the removal of dstortons from leukaema mages and enhancng them. Ths algorthm dvdes the mage n a set of 8 bt planes, each one correspondng to a bt poston rangng from 0 to 7. Ths slced mage s used for enhancement. 127

3 The edge detecton methods can be used for segmentng leukaema mages nto edges. There are four dfferent conventonal methods Canny (C), Sobel (S), Zero- Cross (Z) and Robert-Cross (R). False edge detecton generates thn or thck lnes ndcatng the presence of nose. The Refned Bt Plane (RBP) algorthm flters out useless nformaton, by preservng the sgnfcant structural propertes n a leukaema mage. These enhanced leukaema mages are helpful for applcatons n depth computaton [2], mage reconstructon [3], etc. Depth computaton s carred out by fndng the thrd dmenson of a clustered Leukaema mage usng conventonal methods. The clustered Leukaema mage s necessary for further processng durng mage reconstructon. The damaged portons can be dentfed by fndng the ntensty values n the regon of extracton and comparng them wth the threshold value. If the ntensty value s less than the threshold value then t s consdered damaged [4, 5]. Snce the conventonal methods are not sutable to handle data havng nherent uncertanty, FCM, IFCM, RCM and ther hybrd means (Rough Fuzzy C-Means) RFCM and RIFCM clusterng technques are used n ths paper. To be more precse, n ths paper we use nne clusterng technques (both conventonal and uncertanty based) along wth the RBP algorthm for better segmentaton of leukaema mages. We also carry out the segmentaton process wthout RBP. Compared to all the other eght methods, the Rough Intutonstc Fuzzy C-Means wth a Refned Bt Plane (RIFCMRBP) yelds better segmented mages. Depth computaton s essental to determne the damaged portons of a leukaema mage. The orgnal mage on depth computaton takes more tme than the mages enhanced by usng RBP. Thus, the tme complexty for dagnosng leukaema mages usng RIFCMRBP method s less. The skeleton of the paper s as follows. References survey of the problem under study s carred out n Secton 2. Ths dscusson also covers bref ntroductons to RBP method, the conventonal and uncertanty based clusterng technques. Secton 3 deals wth defntons of the concepts used n the paper. In Secton 4 we descrbe our proposed methodology, conventonal technques, a refned bt plane flter and the varous C-Means technques, ncludng RIFCM. We also ntroduce the DB and Dunn accuracy measures. In Secton 5 we present a comparatve analyss of all the nne clusterng technques based on the expermental results, takng standard nput mages. In Secton 6 we deal wth depth computaton whch leads to the measure of tme complexty. In Secton 7 we provde concludng observatons and also present some possble future enhancements. We comple the detals of the source materals consulted durng the preparaton of ths work as references. 2. References survey In human blood, the leukocytes consst of lymphocytes, monocytes, eosnophls, basophles and neutrophls. These are dfferentated by ther cytoplasmc granules, stanng propertes of the granules, sze of the cell, the proporton of the nuclear to the cytoplasmc materal and the type of nucleolar lobes. There may be an ncrease 128

4 or decrease n the number of leukocytes due to nfecton or acute stress. In [6] an automatc recognton system of leukocytes whch wll extract the leukocyte regon from a blood smear mage whch reduces the generaton of leukocytes, has been developed. A cluster s a collecton of data elements that are smlar to each other but dssmlar to elements n other clusters. A wde number of clusterng algorthms have been proposed to sut the requrements n each applcaton feld. Cluster analyss s a popular data mnng method beng appled to felds lke web mnng [7], bology ([8, 9]), mage processng [10] and market segmentaton [11]. Clusterng algorthms help to dscover the ntrnsc structural complexty of a dataset. Image segmentaton s carred out by usng clusterng technques. In mage segmentaton, the basc features whch can be segmented are mage lumnance ampltude for a monochrome mage, color components for a color mage and edges and texture of the mage. Medcal mages are used for study of the anatomcal structure, dagnoss, treatment plannng or locatng tumors, etc. In [12] a comparatve study s made for acute leukaema mage segmentaton usng HIS and RGB colour space. In [13] a novel Shadowed C-Means (SCM) clusterng approach s appled on segmentng leukaema cells n blood mcroscopc mages and the proposed algorthm has been establshed to be fast and robust n segmentng staned blood mcroscopc mages n the presence of outlers. To mprove the accuracy of segmentaton, an Interest-based Orderng Scheme (IOS) for fuzzy morphology on Whte-Blood-Cells (WBC) has been proposed n [14]. In [15], usng a refned bt plane method, a lossless hybrd algorthm based on a smple choosy scan order s presented for medcal mages, leukaema mages, etc. Hard C-means algorthm [16], whch s the frst among the data clusterng algorthms, assgns each object to one cluster. It exhbts the advantage of fast convergence rate, but stll has some dffcultes n dealng wth overlappng or skewed data dstrbutons problems [17]. However, the uncertanty nature of realworld datasets has necesstated the development of algorthms generatng overlappng clusters. Perhaps the earlest algorthm havng ths characterstc s the fuzzy C-means algorthm developed by B e z d e k n 1981 [18]. In ths case an object can be assgned nto many clusters wth dfferent membershp values. Rough set theory provdes another approach to deal wth uncertanty and the frst algorthm usng rough sets for data clusterng, called rough C-means was developed by L n g r a s and W e s t [19]. Based on the rough set theory [20], a cluster s strctly defned by means of two approxmatons, called lower approxmaton and upper approxmaton of a cluster. Rough set theory, whch was supposed to be a rval for fuzzy set theory at the begnnng, was shown to be actually a complement to t by Dubos and Prade [21]. In fact, they developed the hybrd models of rough fuzzy sets and fuzzy rough sets. Later on, an ntegrated technology called Rough Fuzzy C-Means (RFCM) algorthm was proposed n [22]. An applcaton of ths method was developed to analyze MRI medcal mages [23]. Although rough set-based clusterng algorthms 129

5 have been consdered effcent soft computng methods, a prevalent problem that appears s the parameter selecton. Snce no pror knowledge s gven, an approprate parameter may not be easly chosen. No matter what the actual dstrbuton of each cluster s, a constant weghted parameter s always set manually. The noton of ntutonstc fuzzy sets, ntroduced by A t a n a s s o v [24] generalzes the noton of fuzzy sets. Usng t, the IFCM algorthm was ntroduced by C h a r a and C h e n [25]. Recently, T r p a t h y et al. [26] have ntroduced the RIFCM algorthm and studed ther propertes through some expermental analyss. It s establshed, usng some accuracy measurng ndces, lke the Dunn ndex and DB ndex, that RIFCM outperforms all the exstng algorthms n ths aspect. In ths paper we use conventonal methods, as well as mprecson based models lke FCM, IFCM, RCM and ther hybrd models lke RFCM and RIFCM to the orgnal leukaema mages, as well as ther fltered versons obtaned by usng the algorthm called the RBP algorthm. It s establshed that the performance of RIFCMRBP s better than other technques usng several technques ncludng effcency measurng ndces lke DB-ndex and D-ndex. Besdes, t has the least tme complexty among all the algorthms consdered. It may be noted that another applcaton of RIFCMBRP for analyss of satellte mages has been dscussed n [27]. 3. Defntons of concepts In ths secton we ntroduce some defntons to be used throughout the paper. In ths paper we shall use the models of a Fuzzy Set (FS), an Intutonstc Fuzzy Set (IFS), a Rough Set (RS), a Rough Fuzzy Set (RFS) and a Rough Intutonstc Fuzzy Set (RIFS). We ntroduce these concepts below. Frstly, we defne the noton of a fuzzy set, ntroduced by Z a d e h [28]. Defnton 3.1. A fuzzy set A defned over a unversal set U s charactersed by ts membershp functon μa : U [0,1], such that each x U s assocated wth ts membershp value μ A() x, whch les wthn the unt nterval [0, 1]. The noton of a fuzzy set was extended to defne the concept of an ntutonstc fuzzy set by A t a n a s s o v [24], as follows. Defnton 3.2. An ntutonstc fuzzy set A defned over a unversal set U s charactersed by two functons, called membershp and non-membershp functons, gven by μa, ν A: U [0, 1], such that each x U s assocated wth two real numbers μ () and ν ( x) lyng n the unt nterval [0, 1] such that 130 A x 0 < μ ( x) + ν ( x) 1. A A A It s worth notng that every ntutonstc fuzzy set A s assocated wth a functon π defned through membershp and non-membershp functons, such that A x U, π () x = 1 μ () x ν () x. A A A

6 The noton of a rough set was ntroduced by P a w l a k [20] as another model to handle uncertanty n data. It s defned as follows. Let U be a unverse and P be a famly of equvalence relatons defned over U. The tuple (U, P) s called a knowledge base. Each R P denotes the set of equvalence classes generated by R by U/R. For any K P, IND(K) denotes the ntersecton of all the equvalence relatons n K and t s tself an equvalence relaton over U. We defne IND( P ) = {IND( K ): K P}. Defnton 3.3. For any X U and R IND( P ) we defne two sets RX = { x U [ x] X} and RX = { x X [ x] X φ} R R I, called lower approxmaton of X and upper approxmaton of X wth respect to R. If R X = RX, we say that X s R-defnable, otherwse X s sad to be rough wth respect to R. Combnng the notons of a fuzzy set and rough sets together, Dubos and P r a d e [21] have defned the notons of a rough fuzzy set and fuzzy rough set. We present below the noton of a rough fuzzy set. Defnton 3.4. Let (U, R) be an approxmaton space and U/ R= { X1, X2,..., X n }. Then for any X F(U), RX and RX,the lower and upper approxmatons of X wth respect to R are fuzzy sets n U/R. That s, RX, RX : U / R [0,1] such that for all j = 1, 2,..., n; RX( X ) = nf μ ( y) and RX ( X ) = sup μ ( y). j y X X j j y X j X If R X RX, then the par ( RX, RX) s called a rough fuzzy set assocated wth X wth respect to R Conventonal edge detecton technques The problem of false edge detecton generates thn or thck lnes, flters out useless nformaton, by preservng the sgnfcant structural propertes n a leukaema mage. We can have four conventonal edge detecton technques Sobel, Canny, Robert Cross and Zero-Cross [29], whch are appled for medcal mages Otsu thresholdng, statstcal methods and hstogram analyss Thresholdng s computatonally nexpensve and fast. It s one of the oldest segmentaton methods and s stll wdely used n smple applcatons. Usng range values or threshold values, the pxels are classfed and clustered usng ether of the thresholdng technques, lke global and local thresholdng Statstcal methods Dfferent mathematcal methods can be appled on the bt-plane to get the enhanced leukaema mage [30, 31]. In the paper, the root mean square error and peak sgnal nose rato have to be computed for better nterpretaton of a leukaema mage. 131

7 RMSE and PSNR values The Root Mean Square Error (RMSE) between an nput mage and an output mage s a measure of the level of nformaton loss. If the sze of the nput mage (x, y) and the output mage o(x, y) s M x N, then the RMSE can be computed by usng the formula M 1N RMSE = ( xy (, ) oxy (, )). MN x= 0 y= 0 On the other hand, the Peak Sgnal-to-Nose Rato (PSNR) s closely related to RMSE and can be computed from t. But the lower value of RMSE and hgher value of PSNR ndcate better results Hstogram analyss A hstogram s a graphcal representaton of the dstrbuton of data. In an mage processng context, the hstogram of an mage normally refers to a hstogram of the pxel ntensty values. Ths hstogram s a graph, showng the number of pxels n an mage for each dfferent ntensty value found n the mage. An mage hstogram s a type of a hstogram that acts as a graphcal representaton of the tonal dstrbuton n a dgtal mage. It plots the number of pxels for each tonal value. By lookng at the hstogram for a specfc mage, a vewer wll be able to judge the entre tonal dstrbuton at a glance. A hstogram can be appled to revew graphcally the mages resultng n terms of allocaton and devaton by representng the gray levels of the mage to be enhanced DB ndex and D ndex The Davs-Bouldn (DB) and Dunn (D) ndces are two of the basc performance analyss ndces. They help n evaluatng the effcency of clusterng. The results are dependent on the number of clusters one requres Davs-Bouldn (DB) ndex The DB ndex s defned as the rato of the sum of the wthn-cluster dstance to the between-cluster dstance. It s formulated as gven n [22] as follows: c 1 sv ( ) + sv ( ) k (1) B = max k for 1 <, k< c. c = 1 d( v, vk) The wthn-cluster dstances, s( v ) clusterng approach. Heren, dv (, v) 1/ 2 have been formulated ndependent of any k s the dstance between the centres v and vk of the -th and the k-th clusters. The am of ths ndex s to mnmze the wthn cluster dstance and maxmze the between cluster separaton. Therefore, a good clusterng procedure must gve the value of DB ndex as low as possble [26], respectvely. 132

8 Dunn (D) ndex Smlarly to the DB ndex, the D ndex s used for dentfcaton of clusters that are compact and separated. It s computed by usng dv (, v) k (2) D = mn mn k for 1 < k,, l < c. max sv ( ) It ams at maxmzng the between-cluster dstance and mnmzng the wthncluster dstance. Hence, a greater value for D ndex proves to be more effcent. 4. Methodology In ths secton we descrbe the methodology used n ths paper. Frst we ntroduce the refned bt plane algorthm n the next subsecton Refned bt plane algorthm START Step 1. Input mage: Read the Leukaema mage n a jpeg format of sze ( ) Step 2. Convert the RGB mage to a gray scale mage Step 3. Apply contrast enhancement usng the hstogram equalzaton for better qualty of the mage Step 3.1. Perform contrast enhancement by checkng the valdty of an array Step 3.2. Hstogram normalzaton usng hnorm = hnorm*(varablel(z)/sum(hnorm)); Step 4. Bt planes extracton on hstogram equalzaton Step 4.1. Use a loop to terate 1:8 planes Step Assgn btplanes of the mage to x{} Step 5. If ( bt plane reconstructon after extracton of the 8 bt planes) Step 5.1. Assgn the best bt plane based on probablty as B=zeros(sze(m))... Output Image: Step 5.2. Dsplay the refned best bt plane (enhanced mage) Step 6. Output mage: A refned bt plane leukaema mage STOP The bt plane algorthm s used to partton the mage nto 0-7 slces. For example, the slced value n terms of bts s equvalent to 155 n decmal. Pre-processng of leukaema mages s an mportant step n the process of detecton and extracton of the sgnfcant features n them [30, 31]. If a bt on the n-th bt plane on an m-bt dataset s set to 1, t contrbutes a value of 2 (m n), otherwse t contrbutes nothng. In ths algorthm contrast enhancement s appled usng hstogram equalzaton to mprove the qualty of an mage before slcng t nto 0-7 planes. 133

9 4.2. Exstng algorthms In ths secton we present the fve uncertanty based clusterng technques to be used n ths paper Fuzzy C-Means FCM s an algorthm proposed by B e z d e k [18]. In fuzzy clusterng (also referred to as soft clusterng), the data elements can belong to more than one cluster, a set of membershp levels s assocated wth each element. They ndcate the strength of the assocaton between that data element and a partcular cluster. Fuzzy clusterng s a process of assgnng these membershp levels, and then usng them to assgn data elements to one or more clusters. Step 1. Assgn ntal cluster centers or means for c clusters. Step 2. Calculate the dstance between the data objects and centrods usng the Eucldean dstance formula (3) d( x, y) = ( x y ) + ( x y ) ( x y ). (4) (5) Step 3. Generate the fuzzy partton matrx or membershp matrx U: 2/( m 1) 1 c d k, f d j > 0; μk = j = 1 d jk 1, otherwse. Step 4. The cluster centrods v are calculated usng the formula v = N j= 1 N ( μ ) j= 1 j m ( μ ) Step 5. Calculate the new partton matrx by usng Steps 2 and 3. Step 6. If then stop, else repeat from Step Rough C-Means RCM algorthm was ntroduced by L n g r a s and W e s t [19] to descrbe a cluster by ts centrod and ts lower and upper approxmaton. The elements n the lower approxmaton are certan elements and are assgned a hgher weght low than the elements n the upper approxmatons, whch s denoted by up. An element can belong to two upper approxmatons. The algorthm s n fve steps. Step 1. Assgn ntal means for c clusters. Step 2. Compute usng ( ) Step 3. Let and be the maxmum and next to the maxmum membershp values of the object to cluster centrods and. If then j x m j n n

10 and, and cannot be a member of any lower approxmaton. Else. Step 4. Calculate the new cluster means by usng xk BU x k xk BU BU x k wlow + wup f BU φ and ( BU BU ) φ; BU BU BU v = xk BU BU k BU BU xk BU BU xk x f BU = φ and ( BU BU ) φ; otherwse. Step 5. Repeat from Step 2 untl a termnaton condton s met or untl there are no more assgnments of objects Intutonstc fuzzy C-Means The IFCM proposed by C h a r a and S n e h [25] brngs nto account the hestaton functon that helps n ncreasng the accuracy of clusterng. Step 1. Assgn the ntal cluster centers or means for c clusters. Step 2. Calculate the dstance between the data objects and centrods, usng formulae (3) and (5) respectvely. Step 3. Generate the fuzzy partton matrx or membershp matrx U: Compute μk usng the formula (4) Step 4. Compute the hestaton matrx usng (6) 1. Step 5. Compute the modfed membershp matrx U usng Step 6. The cluster centrods are calculated usng formula (4). Step 7. Calculate a new partton matrx by usng Steps 2 up to 5. Step 8. If then stop, otherwse repeat from Step Rough fuzzy C-Means RFCM s an algorthm proposed by M t r a [32] and M a j et al. [22]; t combnes the concepts of the rough set theory and fuzzy set theory. The concepts of lower and upper approxmatons n a rough set deals wth uncertanty, vagueness and ncompleteness, whereas the concept of a membershp functon n a fuzzy set helps n enhancng and evaluatng of overlappng clusters. Algorthm s n fve steps. Step 1. Assgn the ntal means for c clusters. Step 2. Compute usng (4). 135

11 Step 3. Let and be the maxmum and next to maxmum membershp values of the object to cluster centrods and. If then and, and cannot be a member of any lower approxmaton. Else Step 4. Calculate the new cluster means by usng m xk BU x k x BU BU μ x k k k wlow + wup f BU φ and ( BU BU ) φ; BU m xk BU BU μ k v = m xk BU BU μ k x k m f BU = φ and ( BU BU ) φ; xk BU BU μ k xk BU xk BU otherwse. Step 5. Repeat from Step 2 untl a termnaton condton s met or untl there are no more assgnments of objects Rough ntutonstc fuzzy C-Means The RIFCM, T r p a t h y et al. [26] uses the concept of rough sets and ntutonstc fuzzy sets. The steps that are to be followed n ths algorthm are eght. Step 1. Assgn the ntal means for c clusters by choosng any random c objects as a cluster. Step 2. Calculate usng the Eucldean dstance formula (3). Step 3. Compute matrx by fndng the values of μk usng (4). Step 4. Compute usng (6). Step 5. Compute usng and normalze. Step 6. Let and be the maxmum and next to maxmum membershp values of the object to the cluster centrods and : If then and and cannot be a member of any lower approxmaton; else. Step 7. Calculate the new cluster means by usng the formula below: 136

12 m xk BU x k xk BU BU μ k x k w low + wup m f BU φ and BU BU φ ; BU x (7) k BU BU μ k v = m xk BU BU μ x k k m f BU φ and BU BU φ; xk BU BU μ k x k BU x k else. BU Step 8. Repeat from Step 2 untl a termnaton condton s met or untl there are no more assgnments of objects. Fg. 1. Block dagram explanng the dfferent steps of the computatonal procedure Fg. 1 has four prmary steps; the nput of leukaema mages, ether usng the RBP to flter out the noses n them or passng the orgnal mage as t s, applyng dfferent clusterng technques (both conventonal and C-Means), computng dfferent measures of effcency and the computaton of depth of the clustered mages, whle measurng the tme taken for ths process. The purpose of usng RBP (ntroduced by us n an earler paper) on leukaema mages s to reaffrm the necessty of ths flterng technque n gettng better mages whle processng the mages further. The use of nne clusterng technques s to show the effcency of C- means clusterng technques over the conventonal technques used for ths purpose on one hand, and on the other hand to confrm the superorty of the RIFCM (developed by us n a paper publshed n 2013) over the other C-Means algorthms n the process, for medcal mages lke those of leukaema. Fnally, the depth computaton provdes better vsualty to assess the qualty of the two dmensonal mages. Whle generatng the depths for the clustered mages, we also keep a track 137

13 on the tme taken. Ths s another measure to establsh the superorty of the RIFCM algorthm, whch we observe to be provdng the best mages and also takes mnmum tme for the generaton of the depth map. In Table 1 we present the epslon values when the four conventonal clusterng technques are used for segmentng the leukaema mages and the correspondng dagram representaton usng a bar dagram provded n Fg. 2 to make the comparson easer. The abbrevatons WORBP and WRBP are used to represent Wthout a Refned Bt Plane and Wth a Refned Bt Plane respectvely. By WORBP we mean that the orgnal mage s used wthout flterng. By WRBP we mean that the mage s fltered by usng the RBP algorthm before gvng t as an nput. The epslon, whch forms the upper bound of the relatve error, s extremely useful n determnng the convergence tolerances of many teratve numercal algorthms. Image thresholdng performs a central role n mage segmentaton. In Otsu Threshold/Epslon, we exhaustvely search for the epslon defned as a weghted sum of varances of the two classes: ths mnmzes the ntra-class varance (the varance wthn the class). Otsu shows that mnmzng the ntra-class varance s the same as maxmzng the nter-class varance. Otsu thresholdng/epslon s used to make each cluster as tght as possble. Thus t mnmzes ther overlap. Below we compute the epslon values for dfferent clusterng methods. The lower the epslon value, the better the clusterng method s. Table 1. Epslon values for an orgnal leukaema mage and the refned one after applyng conventonal clusterng technques Conventonal Clusterng methods WORBP WRBP Input mage Canny Robert Sobel Zero-cross Epslon Values WORBP WRBP Input Image Canny Robert Sobel Zero cross Conventonal Clusterng Methods 138 Fg. 2. Dagram representaton of the results n Table 2 usng a bar dagram The results of a smlar knd of analyss for the fve dfferent C-Means clusterng algorthms are presented n Table 2 and the correspondng bar dagram s presented n Fg. 3. It can be observed that for all clusterng algorthms the epslon values are lower when RBP s used than when RBP s not used. Ths s true even n the case of an orgnal mage.

14 Table 2. Epslon values for the mages usng dfferent C-Means clusterng methods C-Means Algorthms WORBP WRBP Input Image FCM RCM RFCM IFCM RIFCM Epslon Values Input Image 102 FCM RCM RFCM IFCM RIFCM WORBP WRBP C Means Algorthms Fg. 3. Dagram representaton of the epslon values obtaned n Table 2 RBP Image (a) Canny (b) Conventonal methods Sobel (c) Zero-Based (d) Robert-Based (e) Fg. 4. Orgnal leukaema mage and mages after applyng conventonal methods wthout RBP Orgnal Image (a) Canny (b) Conventonal methods Sobel (c ) Zero-Based (d) Robert-Based (e) Fg. 5. Leukaema mage after usng RBP and the mages after applyng conventonal methods to t 139

15 Orgnal mage (a) C-Means methods wthout flterng FCM (b) RCM (c ) RFCM (d) IFCM (e) RIFCM (f) Fg. 6. Orgnal leukaema mage and mages after applyng C-Means methods RBP Image (a) C-Means Methods usng RBP for flterng FCM (b) RCM (c ) RFCM (d) IFCM (e) RIFCM (f) Fg. 7. Leukaema mage after usng RBP and the mages after applyng C-Means methods to t In the followng representaton n the form of Table 3, we present the leukaema mages for all the nne methods (four conventonal methods and fve C-Means methods). In the frst column the mages were not fltered, whle n the second column RBP flterng was appled before clusterng. Table 3 also presents the hstogram assocated wth each of the clustered mage. Analyss of these hstograms provdes the tonal dstrbuton n a leukaema mage. As the hstogram plots the number of pxels for each tonal value, by lookng at t a doctor can judge the entre tonal dstrbuton at a glance, whch determnes whether a patent s sufferng from cancer or not. It s observed that n ths case the RIFCM algorthm also provdes a better hstogram n the case when an mage, fltered by usng RBP, s gven as an nput. RMSE/PSNR Values Orgnal Image RMSE Values Orgnal Image PSNR Values RBP Image RMSE Values Input Image and Conventonal Technques RBP Image PSNR Values Fg. 8. Graphcal representaton of the values obtaned n Table 4 usng a bar dagram 140

16 Table 3. Hstogram analyss of a leukema mage and mages obtaned after applyng the conventonal and the C-Means Clusterng methods: wthout applyng RBP, (a); after applyng RBP, (b) Orgnal Image and mages Orgnal Image applyng RBP after applyng dfferent Hstogram and mages after applyng Hstogram clusterng technques Analyss dfferent clusterng technques Analyss to t Orgnal Image RBP Image Canny Canny Sobel Sobel Zero- Cross Zero-Cross Robert- Cross Robert- Cross FCM FCM RCM RCM RFCM RFCM IFCM IFCM RIFCM RIFCM 141

17 Table 4. RMSE and PSNR values of a leukaema mage when conventonal clusterng technques are used: ) the orgnal mage ) the mage after applyng RBP flterng Conventonal methods Orgnal mage RBP mage RMSE values PSNR values RMSE values PSNR values Input mage Canny Robert Sobel Zero-Cross Table 5. RMSE and PSNR values of ) orgnal leukaema mages ) the mage after applyng RBP obtaned after segmentaton usng dfferent C-Means algorthms C-Means & the proposed method WORBP WRBP DB-Index Dunn Index DB-Index Dunn Index FCM RCM RFCM IFCM RIFCM RMSE / PSNR Values FCM RCM RFCM IFCM RIFCM Clusterng Technques Leukema mage WORBP RMSE Values Leukema mage WORBP PSNR Values Leukema mage WRBP RMSE Values Leukema mage WRBP PSNR Values Fg. 9. Graphcal representaton of the values obtaned n Table 5 usng a bar dagram It could be observed from Table 5 and Fg. 9 that there s no unformty n segmentng the orgnal leukaema mage whereas for the fltered mage, usng RBP, RIFCM provdes the best results n the form of lower values for both the RMSE and PSNR. 142

18 Table 6. DB and Dunn Index values for the leukaema mage when clustered usng the fve C-Means algorthms: ) the orgnal mage ) the mage fltered usng RBP Clusterng technques used Leukaema mage WORBP Leukaema mage WRBP RMSE -Values PSNR-Values RMSE-Values PSNR-Values FCM RCM RFCM IFCM RIFCM DB and Dunn Index Values FCM RCM RFCM IFCM RIFCM WORBP DB Index WORBP Dunn Index WRBP DB Index WRBP Dunn Index C Means Method Fg. 10. Graphcal representaton of the values obtaned n Table 6 usng a bar dagram It s clear from Table 6 and Fg. 10 that for RIFCM method both the DB value and D value are the best (mnmum for the DB case and maxmum for the D case). Ths observaton s true for both cases;.e., when the orgnal mage s taken wthout applyng any flterng technque and when the mages are frst fltered by usng the RBP technque. Above we have analyzed the effcences of the conventonal clusterng technques and the C-Means algorthms separately when appled to leukaema mages. Besdes, we have gven the orgnal mage, as well as the mage fltered by usng RBP n order to establsh that the fltered mages gve better results n all the cases taken separately. The measures were the RMSE and PSNR values and the DB and D ndces. However, when the two Tables 4 and 5 are combned, the results confrm the superorty of WRBPRIFCM for clusterng of the mages wth respect to all other methods consdered n ths paper, whch s vsually supported by combnng Fgs 8 and

19 5. Depth computaton In ths secton, we deal wth another measure to show the superorty of the RIFCM method when appled to fltered leukaema mages. Ths s the depth computaton of the mages, whch provdes the tme requred to reconstruct the mages. The mages obtaned after the depth map generaton provde a better vew and hence asssts the doctors n the dagnoss of the dsease n a better way. That s why, n ths secton we deal wth ths aspect for all the nne clusterng algorthms dealt so far. The depth computaton of varous clusterng technques takng the orgnal mage as an nput and the mages obtaned after flterng by usng the RBP technque s made. The results are shown n Table 7 and Fg. 11. Here also the output of the RIFCM method yelds mproved clusterng of a leukaema mage and the tme of computaton tme n seconds for ths method s the lowest one. Table 7. Tme complexty of Depth Computaton of the leukaema mage after applyng the nne methods (usng RBP or not usng RBP) Clusterng approaches used Orgnal mage RBP fltered mage (tme n seconds) (tme n seconds) Input mage Canny Robert Sobel Zero-cross FCM RCM RFCM IFCM RIFCM In Fg. 11 the frst group of bars corresponds to the case when the orgnal mages and the second group correspond to the case when the fltered mages usng the RBP technque are gven as an nput. Tme n Seconds Wthout RBP ( tme n seconds) Wth RBP (tme n seconds) Clusterng Approaches Used Input Image Canny Robert Sobel Zero cross FCM RCm RFCM IFCM RIFCM Fg. 11. A graph representng the tme complexty n seconds of Depth Computaton for the leukaema mage after applyng the nne methods (usng RBP or not usng RBP) 144

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