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Signal Processing 87 (2007) 2407–2416

www.elsevier.com/locate/sigpro

HDF: Heat diffusion fields for polyp detection inCT colonography

E. Konukoglu1, B. Acar�

Electrical and Electronics Engineering Department, Bogazic- i University, 34342 Bebek, Istanbul, Turkey

Received 17 January 2006; received in revised form 6 March 2007; accepted 28 March 2007

Available online 19 April 2007

Abstract

Computer aided detection (CAD) in computed tomography colonography (CTC) aims at detecting colonic polyps that

are the precursors of cancer. We propose to use heat diffusion patterns for polyp detection and identification in CTC. The

underlying idea of the proposed method is to utilize the heat diffusion process, which is closely related to the diffusion

filtering, to generate a vector field (HDF) that is correlated with the shape of the colon wall. The HDF and a local voxel

distribution parameter were used for detection and identification. We evaluated the method on real patient CTC data

acquired from eight volunteers. 100% sensitivity is achieved at 0.53 false positives per patient for polyps X8mm and 78%

sensitivity is achieved at 0.53 false positives per patient when all polyps X5mm are considered.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Computer aided detection; Computer aided diagnosis; CT colonography; Polyp detection; Diffusion filtering; Diffusion

pattern; Polyp enhancement

1. Introduction

Computed tomography colonography (CTC) is aminimally invasive technique that employs X-rayCT imaging of the abdomen and pelvis followingcleansing and air insufflation of the colon. Origin-ally proposed in the early 1980s [1], it becamepractical in the early 1990s following the introduc-tion of helical CT and advances in computergraphics [2]. Currently available multislice helicalX-ray CT scanners are capable of producing

e front matter r 2007 Elsevier B.V. All rights reserved

gpro.2007.03.021

ing author. Tel.: +90212 3596465;

72465.

sses: ender.konukoglu@gmail.com (E. Konukoglu),

du.tr, burak.acar@ieee.org (B. Acar).

/www.vavlab.ee.boun.edu.tr (B. Acar).

ress: INRIA, Sophia-Antipolis, France.

hundreds of high resolution (o1mm cubic voxel)images in a single breath hold. Conventionalexamination of these source images is rather time-consuming and the detection accuracy is unavoid-ably limited by human factors such as attentionspan and eye fatigue. Several visualization andnavigation techniques have already been proposedto help the radiologists [3–11]. However, computeraided detection (CAD) tools are envisioned toimprove the efficiency and the accuracy beyondwhat can be achieved by visualization techniquesalone [12–21].

Several studies have investigated CAD for CTC.Vining et al. used abnormal colon wall thickness todetect colonic polyps [12]. Summers et al. used themean, Gaussian and principal curvatures of thecolon surface and showed good preliminary results

.

ARTICLE IN PRESSE. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–24162408

for phantom and patient data [13,14]. Kiss et al.used surface normals along with sphere fitting [16],while Yoshida et al. used both the pre-segmentedsurface differential characteristics captured by ashape index, and the gradient vector field of the CTdata [17,18]. Paik et al. proposed the surface normaloverlap algorithm based on the observation that forlocally spherical and hemispherical structures, largenumbers of surface normals intersect near thecenters of these structures [19]. To improve speci-ficity, Gokturk et al. used triples of randomlyoriented orthogonal cross-sectional images of pre-detected suspicious structures which are thenclassified by support vector machines [20], whileAcar et al. modeled the way radiologists utilize 3Dinformation as they are examining a stack of 2Dimages [21].

We propose the heat diffusion field (HDF)method as a way to detect and characterize aprotruding structure represented as a surface in 3D.The HDF method is a 3D detection and character-ization method built on the well-known diffusionfiltering framework. The diffusion process has (i) aregularizing effect due to the inherent smoothing ofdiffusion; (ii) an enhancement effect since the HDFutilizes the structure of the collision of propagatingheat fronts that carry structural information fromthe whole polyp surface, in other words, thesingularities of HDFs are summaries of colonsurface characteristics in a local neighborhood.The enhancement is due to the sinks generated bypropagating iso-temperature fronts. The sphericalsymmetry of the diffusion pattern, represented bythe HDF, is used for identification. The idea waspreviously proposed in [22]. The HDF method isbased on detecting and characterizing those sinks, inconjunction with local voxel distribution aroundthem. The only parameter of the HDF method is theconstant diffusion coefficient. We have evaluatedthe method for different diffusion coefficients onreal patient data set acquired from eight patients,using the FROC analysis.

2. Method

The primary observation is that if the colonlumen (the inner volume of the colon wall) isthought to be initially heated to a constant level,then the heat diffusion process would generate localheat diffusion pattern singularities (sinks) near thecenters of protruding structures. The diffusionpattern can be described by a vector field VðrÞ; r ¼

½x y z�T that is generated by tracking the propaga-tion of the iso-temperature surfaces along theirnormal directions. V would have the aforemen-tioned singularities near the centers of protrudingstructures. It is thus used to detect suspiciouslocations. Both the vector field geometry and localvoxel distribution around such detected points areused to identify the polyps.

2.1. Segmentation

The colon lumen in each subvolume is segmentedby simple thresholding followed by morphologicalfiltering as follows: the CT data is converted to abinary volume (SðrÞ) by thresholding at 350HU(CTðrÞ4350HU) SðrÞ ¼ 1 o:w: SðrÞ ¼ 0). SðrÞ ¼

0 are the air voxels. Next, the isolated 1’s in SðrÞ areremoved and 26-neighborhood morphological clos-ing followed by a thinning operation is performedon the binary volume S to filter out segmentationnoise using MatlabTM ’s built-in functions. ThusS ¼ 1 represents the tissue while its complementrepresents the colon lumen.

2.2. The HDF computation

The algorithm starts with the segmented colonlumen, frjSðrÞ ¼ 0g, set to a constant temperature,T0 ¼ 1. It is then let cool down. The diffusionprocess and the computation of the HDF vectorfield are governed by the following equations:

Heat diffusion eq.:qTðr; tÞ

qt¼ r �DrTðr; tÞ, ð1Þ

Motion eq.:qTðr; tÞ

qt¼ �rTðr; tÞ � vðr; tÞ, ð2Þ

where Tðr; tÞ stands for the temperature at position r

at time t. Eq. (1) relates the rate of change in Tðr; tÞthroughout the domain, to the divergence ofthe gradient of Tðr; tÞ. D is the diffusion coefficientand governs the rate of diffusion. Eq. (2), onthe other hand, relates this change in temperatureto an instantaneous vector field vðr; tÞ. vðr; tÞrepresents the motion of the iso-temperaturesurface along its normal direction at the givenposition and time.

Equating Eqs. (1) and (2), we get,

� r �DrTðr; tÞ ¼ rTðr; tÞ � vðr; tÞ

Tðr 2 fSðrÞ ¼ 0g; t ¼ 0Þ ¼ T0. ð3Þ

ARTICLE IN PRESSE. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–2416 2409

We can define the vector field VðrÞ, the HDF, as

VðrÞ ¼

Z t

0

jrTðr; tÞjvðr; tÞdt ¼

Z t

0

rTðr; tÞ

jrTðr; tÞj

� ��ð�r �DrTðr; tÞÞdt; Tðr; tÞoT ð4Þ

It is the integral of instantaneous field vðr; tÞweighted by the temperature gradient over time(Eq. (4)). The inequality makes sure that we createthe vector field by following the motion of theleading heat front, which gives the diffusion pattern.As a result only the leading heat front willcontribute to VðrÞ. Since the T value will be low inthe heat front we set T ¼ 0:1 as a relatively lownumber.

The geometrical characteristics of V will be usedduring the detection and the identification of colonicpolyps together with a parameter of voxel distribu-tion, as will be explained in Sections 2.3 and 2.4.

The rate of linear isotropic diffusion is deter-mined by the D parameter. Since the HDF is avector field constructed by time integral of themotion vectors of iso-temperature surface (heatfronts), D has a direct effect on the computed HDF.A too small D would result in incomplete HDF,whereas a too large D would cause over-smoothing,thus the HDF would not be descriptive. Theoptimal D depends on the polyp size range ofinterest. As explained in Section 3.3, we haveexperimented with several D values to show itseffect empirically.

Numerical computation of VðrÞ is done bydiscretizing Eq. (4), where the continuous integra-tion is replaced by discrete summation over a finitenumber of time intervals of length Dt (which istaken as Dt ¼ 0:1 considering the trade-off betweenthe accuracy of the time discretization and thecomputational load). The diffusion term in Eq. (4) issolved numerically using the alternating-direction

implicit (ADI) method [23]. ADI is a variation of theCrank–Nicholson scheme and uses time division forefficient and unconditionally stable solutions. Thediscrete isotropic diffusion equation for one-third ofDt is

Tnþ1=3 � Tn ¼D

d2

Dt

3ðd2xTnþ1=3 þ d2yTn þ d2zTnÞ, (5)

d2xTn ¼ Tniþ1;j;k � 2Tn

i;j;k þ Tni�1;j;k

and likewise for d2yTn; d2zTn,

where the x dimension is implicit and others areexplicit. d is the spatial resolution (0.6mm for our

data set). Eq. (5) is repeated three times to computeT Dt seconds later, keeping one dimension implicitat a time. Note that the time step is divided by 3 inEq. (5). The numerical differentiation was doneusing a Gaussian derivative kernel with s ¼ 0:6mmand �2s kernel support.

Fig. 1 shows the heat distribution at differentstages of the diffusion in the vicinity of a 5.5mmpolyp. The heat diffusion pattern singularity aroundthe polyp center as time proceeds is apparent.

2.3. Polyp detection

A perfectly spherical polyp would yield aspherically symmetric singularity in the center ofthe polyp center. Due to shape variations of colonicpolyps (deviation from the sphere) singularitiesformed in vicinities of polyp centers may not beperfectly spherically symmetric.

The differential characteristics of V are summar-ized by its Jacobian matrix J. For VðrÞ (a 3Dvector field), J and its characteristic equation aredefined as

J ¼

qV x

qx

qV x

qy

qVx

qz

qV y

qx

qVy

qy

qVy

qz

qV z

qx

qV z

qy

qVz

qz

2666666664

3777777775, ð6Þ

jJ� lIj ¼ l3 þ Pl2 þQlþ R ¼ 0, ð7Þ

P ¼ �traceðJÞ; Q ¼ 12ðP2 � traceðJ2ÞÞ; R ¼ �jJj,

D ¼ R2 þ ð4P3 � 18PQÞRþ 4Q3 � ðPQÞ2,

where D is the discriminant of the third ordercharacteristic polynomial given above and l’s arethe eigenvalues.

The sinks sought correspond to the vector fieldpoints where the eigenvalues of J have all negativereal parts close to each other. Their closeness (i.e.the spherical symmetry of VðrÞ) can be measuredusing the fractional anisotropy (FA) parameterwhich is defined as

FA ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða1 � aÞ2 þ ða2 � aÞ2 þ ða3 � aÞ2

ða21 þ a22 þ a23Þ

s,

a ¼a1 þ a2 þ a3

3, ð8Þ

ARTICLE IN PRESS

Fig. 1. (Color online) The contours of moving heat fronts clearly mark a singularity created at the center of a 5.5mm polyp on a fold. (a)

Surface rendering of the colon wall, (b) the initial temperature field (t ¼ 0), (c,d) the temperature fields as diffusion process proceeds. The

contours marked are T ¼ ½0:01�; 0:05�; 0:1�� heat fronts.

E. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–24162410

where ai ¼ RealðliÞ; i ¼ f1; 2; 3g. FA ¼ 0 forperfectly spherically symmetric singularities. ThusFA represents the geometrical information em-bedded in the HDF: smaller the FA is, moreprobable the structure is a polyp, provided thataio0; i ¼ 1; 2; 3, i.e. the condition for the singular-ity to be a sink.

The 3D watershed transform (WT) is usedto segment the 3D buckets (watershed basins in3D) in the 3D FA map generated as explainedabove. The minimum FA point in each bucket ismarked as a hit. This stage is summarized inAlgorithm 1.

Algorithm 1. The HDF geometry based detectionalgorithm

1. Compute ai ¼ RealðliÞ; i ¼ f1; 2; 3g for the whole

volume.

2. Mark all points with aio0; i ¼ f1; 2; 3g as Valid

Points.

3. Compute the FA values for all Valid Points and

assign FA ¼ �1 for all Non-Valid Points meaning

not to be segmented as watershed basins

4. Segment the FA volume created in Step 3 using the

watershed transform (WT) [24]. WT labels the FAbuckets in 3D.

ARTICLE IN PRESSE. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–2416 2411

5. Exclude the FA buckets that touch the volume

boundaries.

6. Mark the minimum FA points in each bucket as a

HDF Hit as long as it is in the tissue.

2.4. Polyp identification

The FA parameter (spherical symmetry) isespecially useful in discriminating folds from polypswhich is the bottleneck for improved CAD perfor-mance. The elongated structure of the folds resultsin a VðrÞ that is 2D symmetric only on the planeperpendicular to the fold’s main axis. This meansminifjaijg ffi 0 which results in high FA values.However, FA alone is not sufficient to eliminate allnon-polyp hits. There are hits with low FA valuesjust because there are nearby air voxels distributedaround somewhat uniformly, like at the junction offolds. To overcome this problem, we perform a localanalysis of the colon wall around the hitsand compute the Triangulation Uniformity (TU)parameter. TU is used together with FA foridentification.

The idea behind TU is that there should be airvoxels around a polyp that are direction-wiseuniformly distributed over a spherical surface patch(polyp surface). A complete spherical surface is notconsidered as the polyps are attached to the colonwall. We mark the azimuth and the elevation anglesof the closest air voxels within a certain distance(15mm is used as an arbitrary choice2) in M

directions (M ¼ 100 as an arbitrary choice3) on aunit sphere. We perform a Delaunay Triangulationof the surface defined by these points and use thenumber of triangles and the trimmed variance of theareas of these triangles to measure this as

Ai; i ¼ 1; . . . ;K : triangle areas

~A ¼ fsmaller 90% of A0isg,

s2A ¼ VARð ~AÞ,

TU ¼K

1þ s2A. ð9Þ

The largest 10% of Ai’s is excluded from thevariance calculation as these large triangles wouldcorrespond to the attachment of the polyps to thecolon wall. TU values for polyps would be larger

2This limit is related to the radius of the maximum polyp size

we are interested in.3Higher values of M would give more accurate description of

the colon surface, however, increase the computational complexity.

than those for non-polyp structures because s2Awould be lower in the latter case.

3. Experiments

3.1. Evaluation methodology

We used free response receiver operator char-acteristic (FROC) curves to evaluate the perfor-mance of our algorithm [25]. FROC curves show thetrade-off between sensitivity (detection rate of truepositives, TPs) and the detection rate of falsepositives (FPs). They are especially suitable for theperformance evaluation of detection algorithms asopposed to pure classification algorithms as the setof negatives is not well-defined. In other words, allpoints in the 3D data except the inner regions ofpolyps are potential negatives. Since this corre-sponds to using a large number in the denominatorfor the conventional definition of specificity, itwould be misleading.

The gold standard was generated by a radiologistwith 8 years of experience in CTC, who marked thecenters of fiber optic colonoscopy (FOC) confirmedvisible polyps in CTC data and measured theirdiameters using a custom built computer program.Those center points and the diameter measure-ments specify spherical regions. All hits in such asphere are considered as TPs associated with the samepolyp.

We sorted all of the detected points starting fromthe most spherically symmetric one (lowest FA) inorder. We then went through this list starting fromthe top, keeping the top most one and eliminatingall of the other points down the list, closer to thathit more than the radius of the smallest polyp ofinterest (x), which is set by the user. The underlyingidea is that the centers of two polyps of interest(assumed to be a hemispherical structure in general)cannot be closer to each other more than x. We alsolimited x to be larger than or equal to 1.5mm. Thisprocess also eliminates multiple hits which mayoccur due to using overlapping subvolumes. Thisgrouping strategy is only based on the a prioripreference that the user (the radiologist) would havemade regarding the size of polyps he/she isinterested in, simulating a clinical application.

The Mahalanobis distance on the 2D ðFA;TUÞspace is used for FROC analysis. We defined adecision function C as

CðrÞ ¼ dðr; q0Þ � dðr; q1Þ, (10)

ARTICLE IN PRESSE. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–24162412

where r is a 2D feature vector of ðFA;TUÞ pairs, q0is the mean feature vector of negatives (FP hits ofthe detector), q1 is the mean feature vector ofpositives (TP hits of the detector) and the Mahala-nobis distance d is defined as

dðr; qÞ ¼ ðr� qÞTC�1ðr� qÞ, (11)

where C is the auto-covariance matrix of the set ofpoints of which q is the mean.

The FROC analysis was carried out by varying athreshold � over the range of C and counting thepolyps and non-polyps among the hits with C4�(i.e. among the hits labeled as positive by thedecision function). C ¼ � defines a hyperplaneseparating the two groups in R2. For each �, thenumber of non-polyps gives the FP (FP rate) andthe number of polyps divided by the total number ofpolyps gives the sensitivity (TP rate). Multipledetections within a single polyp are considered asa single TP.

3.2. Data

We used CTC data acquired from eight patients.The patients were scanned with single- or multi-detector helical CT (GE HiSpeed/CTi or Light-Speed, General Electric Medical Systems,Milwaukee, WI) with effective section width of2.5–3.75mm and 50% overlapping reconstruction.All data were interpolated to an isotropic 3D grid of0:6� 0:6� 0:6mm3. Immediately following CTscanning, each patient also underwent opticalcolonoscopy (OC). These results were correlated tomanual detection results on the CT images done bya radiologist blinded to the CAD results and with 10years of experience in CTC. The radiologist markedthe centers of OC confirmed polyps and measuredtheir diameters using a custom built computerprogram. There were 7 large polyps (X10mm)and 11 small polyps (X5 and o10mm) seen andmarked manually in four of eight patients. The dataset is the same as the one used in [19].

3.3. Results

We have conducted experiments considering threesets of polyps: the polyps with a diameter X5mm,X8mm and X10mm, each set being a subset of thepreceding one. For each set, we used 10 different D

values. The FROC computation was done asexplained in Section 3.1. Inspection of these FROCcurves showed a clear trend in the overall performance

as a function of D. These performance trends did notshow a significant dependence on the polyp size, yetthe performance itself decreased when we consideredthe smaller polyps together with the bigger ones.

Fig. 2 shows these FROC curves for all D valuesand all polyp sets. All plots indicate that theoptimum value for D is in the range of 2pDp5.Too low (D ¼ 1) and too high (D45) D valuescause significant decrease in performance, esp. inmaximum sensitivity. Based on a closer inspectionof the maximum sensitivity and the correspondingFP rates for D ¼ 2; 3; 4; 5, as listed in Table 1, wechose D ¼ 2 as the optimum diffusion coefficient forall polyp sizes considered.

Fig. 3 shows the FROC curves for D ¼ 2 for allpolyp sets. The associated scatter graph in the 2Dfeature of TU vs FA shows the distribution ofpolyps and non-polyps among all the detectedpoints. The curves depict the iso-C lines thatare used for polyp identification and FROCcomputation. As expected, the non-polyps havehigher FA and lower TU values compared to thepolyps.

The algorithm is implemented using MatlabTM7:0and run on an IntelTM PentiumTM4 based PC with2.4GHz clock speed and 1GB RAM. Due tocomputational limits the algorithm was run onsmall overlapping cubes of 30mm� 30mm�30mm, covering the colon surface. The overlappingis 15mm on each side of the subvolume so that allpolyps smaller than 15mm in diameter would becontained in at least on subvolume. It tookapproximately 25 sec to process one subvolume.

4. Discussion

The HDF method introduces the concept ofdiffusion filtering for the CAD applications in CTC.It aims at exploiting the spherical geometry ofcolonic polyps using a regularizing process such asheat diffusion. The principal idea is to generate aflow that would result in a vector field withspherically symmetric sinks (singularities) corre-sponding to polyps, where the feature that measuresthe spherical symmetry is the FA. Considering thestructure of the HDF’s singularities is an efficientway of representing the geometry of the colon wallin the local neighborhood. The inherent regulariza-tion of diffusion, on the other hand, providesimmunity against surface irregularities that may bedue to segmentation noise. The distribution ofvoxels around the detected points is considered via

ARTICLE IN PRESS

Table 1

The table lists ‘‘maximum sensitivity : the corresponding FP rate per patient’’ for different polyp size groups and three different D values, the

best choices according to the FROC curves in Fig. 1

Polyp size : # of polyps D ¼ 2 D ¼ 3 D ¼ 4 D ¼ 5

X10mm : 7 100% : 0.53 100% : 0.50 100% : 1.08 100% : 1.08

X8mm : 10 100% : 0.53 100% : 0.75 100% : 1.08 100% : 1.08

X5mm : 18 78% : 0.53 72% : 0.75 83% : 1.08 83% : 1.08

D ¼ 2 has the best performance when all polyp groups are considered.

0 0.5 1 1.5 2 2.5 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FROC Curves for 18 Polyps with Diameter >= 5mm

False Positive per Patient

Sensitiv

ity

D=1

D=2

D=3

D=4

D=5

D=6

D=7

D=8 D=9D=10

0 0.5 1 1.5 2 2.5 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FROC Curves for 10 Polyps with Diameter >= 8mm

False Positive per PatientS

ensitiv

ity

D=1

D=2D=3

D=4

D=5

D=6

D=7

D=8 D=9

D=10

0 0.5 1 1.5 2 2.5 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FROC Curves for 7 Polyps with Diameter >= 10mm

False Positive per Patient

Sensitiv

ity

D=1

D=2D=3

D=4

D=5

D=6

D=7

D=8 D=9

D=10

a b

c

Fig. 2. FROC curves are given for three sets of polyps (X5mm,X8mm,X10mm) and 10 different values of the diffusion coefficient D in

Eq. (1). Except for the first case where all polyps with a diameter X5mm are considered, the HDF method achieved 100% sensitivity for

some D values. Considering the maximum sensitivity and the corresponding FP rates for all cases, we can identify D ¼ 2, 3 and 4 as the

best choices for all polyp sizes.

E. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–2416 2413

ARTICLE IN PRESS

Polyps >=5mm

Polyps >=8mm

Polyps >=10mm

0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

False Positive per Patient

Se

nsitiv

ity

FROC Curves D=2.0

0 20 40 60 80 100 120 140 160

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FA

TU

FA vs TU for the Detected Points (D=2)

PolypsNon-polyps

Fig. 3. The FROC curves for three different polyp groups are given for D ¼ 2 with the corresponding 2D scatter graph in the feature

space, TU vs FA. The contours on the scatter graph mark the iso-C curves (Eq. (3.1)), as explained in Section 3.1. The solid line is the

C ¼ 0 curve.

4Gaussian function is known to be the kernel of linear isotropic

diffusion, with its variance related to the diffusion coefficient and

the diffusion time.

E. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–24162414

a second feature called the TU. The distribution ofthe detected points in the 2D feature space showsthe complementary nature of FA and TU para-meters and the clear distinction between TPs andFPs. Neither FA nor TU alone would be sufficientto separate polyps from non-polyps as effective.

The experiments conducted with polyps ofdifferent sizes and different diffusion coefficients(D values) suggests D ¼ 2 as the optimum choice.With this choice of D, the HDF’s performance ispractically independent of polyp size for all polypswith a diameter X8mm, whereas the maximumsensitivity decreases to 78% when we also consid-ered the polyps with a diameter X5 and o8mm.Too low and too high D values result in significantdecrease in maximum sensitivity. For the formercase, the low D values causes insufficient diffusionand the HDF is not constructed properly. In otherwords, the propagating heat fronts cannot span theinner volume of polyps, unable to generate thevector field singularities that HDF method relies on.In the latter case, too fast diffusion causes fastpropagation of the heat fronts, thus avoiding aproper construction of HDF by Eq. (2).

The data set we have used is the same as the oneused by Paik et al. in [19], which makes it possible tomake a direct comparison. The 0:53 FP/patient at100% sensitivity achieved by the HDF method is asignificant improvement over the performance ofthe SNO technique reported in [19], which was 7

FP/patient at 100% sensitivity. This corresponds toa 92% improvement in specificity without any lossof sensitivity. Yoshida et al., on the other hand, hadreported 2 FP/patient at 100% sensitivity on adifferent data set in [17]. Although it is not possibleto make a direct comparison, the HDF’s perfor-mance suggests an improvement over these resultsas well.

The drawback of the current implementation ofHDF is its computational cost. Yet, since themethod uses linear diffusion only, a faster imple-mentation of the method based on simple Gaussiansmoothing is possible.4 This would increase thecomputation speed greatly, eliminating the currentdrawback. The computation time can also befurther decreased by using optimized software.

5. Conclusion

We introduced the idea of heat diffusion for CADin CTC and used the well-known diffusion filteringframework for this. The HDFs are computed suchthat they have symmetric sinks (vector fieldsingularities) at polyp centers. Using a regularizingprocess, i.e. diffusion, we achieved the mapping ofthe colon-wall geometry in a local neighborhood to

ARTICLE IN PRESSE. Konukoglu, B. Acar / Signal Processing 87 (2007) 2407–2416 2415

the geometry of a vector field at a singularity (thesink), which is one of the properties of HDF thatsets the difference from other techniques such ascolon wall curvature maps. The propagating heatfronts, originating from the surface of polyp, collidearound its center providing an enhancement effect.

We have also combined the results of thismapping with the TU feature to obtain a 2D featurespace where the separation between false and truepositives is improved. The TU parameter measuresthe spatial distribution of air voxels around asuspicious point.

The feature space (FA-TU space) plots demon-strate a clear separation of the polyps and the non-polyps as well as the complementary nature of thetwo features used. The identification performancecan also be improved significantly with a well-trained classifier on a larger data set. Using otherPDE based image processing techniques for polypenhancement is also a promising approach and it isleft for future research.

Acknowledgments

This research was in part supported by TUBI-TAK KARIYER-DRESS (104E035) project grantand by NIH R01-CA72023 grant via StanfordUniversity, Department of Radiology, CA, USA.We would like to express our sincere gratitude toProf. Christopher F. Beaulieu and Prof. SandyNapel from Stanford University, Department ofRadiology, 3D Laboratory (http://3dradiology.stanford.edu/), CA, for their contributions espe-cially in data collection and clinical evaluationphases. We would also like to acknowledge Prof.David S. Paik from Stanford University for hisinvaluable comments.

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