Molar Axis Estimation from Computed Tomography Images

Dongxia Zhang, Yangzhou Gan, Zeyang Xia*, Xinwen Zhou, Shoubin Liu, Jing Xiong, and Guanglin Li

Abstract— Estimation of tooth axis is needed for some clinicaldental treatment. Existing methods require to segment the toothvolume from Computed Tomography (CT) images, and thenestimate the axis from the tooth volume. However, they may failduring estimating molar axis due to that the tooth segmentationfrom CT images is challenging and current segmentationmethods may get poor segmentation results especially for thesemolars with angle which will result in the failure of axisestimation. To resolve this problem, this paper proposes a newmethod for molar axis estimation from CT images. The keyinnovation point is that: instead of estimating the 3D axis ofeach molar from the segmented volume, the method estimatesthe 3D axis from two projection images. The method includesthree steps. (1) The 3D images of each molar are projected totwo 2D image planes. (2) The molar contour are segmentedand the contour’s 2D axis are extracted in each 2D projectionimage. Principal Component Analysis (PCA) and a modifiedsymmetry axis detection algorithm are employed to extract the2D axis from the segmented molar contour. (3) A 3D molar axisis obtained by combining the two 2D axes. Experimental resultsverified that the proposed method was effective to estimate theaxis of molar from CT images.

I. INTRODUCTIONIn dental implants, dentists need to know the axes of

neighboring teeth for the determination of allocation anddepth of dental implants [1] [2]. Additionally, in orthodontictreatment planning, tooth axis plays an important referencefactor in planning the position of the brackets [3]. Hence,it is necessary to estimate tooth axis to aid clinical dentalimplants and orthodontic treatment.

In clinics, tooth axis is generally determined manuallyby dentists which is objective and tedious. Only a fewmethods have been developed for the automatic estimation

*Research supported by National Natural Science Foundation of China(No. 51305436), Guangdong Natural Science Funds for DistinguishedYoung Scholar (No. 2015A030306020), Major Project of GuangdongProvince Science and Technology Department (No. 2014B090919002),and Shenzhen High-level Oversea Talent Program (Peacock Plan) (No.KQCX20130628112914284).

D. Zhang is with Shenzhen Institutes of Advanced Technology, ChineseAcademy of Sciences, Shenzhen, China, and also with Harbin Institute ofTechnology Shenzhen Graduate School, Shenzhen, China.

Y. Gan, Z. Xia, and G. Li are with Shenzhen Institutes of AdvancedTechnology, Chinese Academy of Sciences, Shenzhen, China, and also withKey Laboratory of Human-Machine Intelligence-Synergy Systems, Shen-zhen Institutes of Advanced Technology, Chinese Academy of Sciences,Shenzhen, China.

X. Zhou is with the Department of Automation, and Shanghai Key Lab ofNavigation and Location Services, Shanghai Jiao Tong University, Shanghai,China.

S. Liu is with Harbin Institute of Technology Graduate School, Shenzhen,China.

J. Xiong is with Shenzhen Institutes of Advanced Technology, ChineseAcademy of Sciences, Shenzhen, China.

Corresponding author: Zeyang Xia. Email: zy.xia@siat.ac.cn. Phone: +86-755-8639-2218.

of tooth axis from computed tomography (CT) image. Allthese methods [4] [5] first segmented the tooth volume fromthe CT images, and then used principal component analysis(PCA) to estimate tooth axis from the volume.

For the tooth with single-root, it is feasible to estimatethe axes from the corresponding dental pulp [5] as thedental pulp has the similar shape with the tooth and iseasy to be segmented from surrounding hard tissue of tooth.However, for the molars which have multiple-root, the axescannot been extracted from the corresponding dental pulpdue to that molars often have two or three pulp cavities andsome pulp cavities are too small and narrow to be correctlysegmented, and the axis estimation is much dependent onthe tooth segmentation accuracy. Currently, the state of theart tooth segmentation methods [6-8] segment tooth volumefrom CT images mainly using the slice-by-slice segmentationstrategy based on the tooth shape coherence between adjacentslices. They have difficulty in accurately segmenting theseangled molars especially the third molars as the tooth shapecoherence between adjacent slices of the angled molar is low.In addition, the calculated axes by PCA may be inaccurateas the result of PCA is essentially the major axis and maynot be the exact axis of molar.

This study proposes a new method to estimate molar axisfrom CT images. Instead of estimating axis from the 3Dvolume of molar, the method projects the 3D images ofeach molar into two 2D planes, and then segments the molarcontour and extracts the 2D axis from the contour in each2D projection image. Finally, the 3D tooth axis is obtainedfrom the two extracted 2D axes. In the extraction of each2D axis, PCA is first applied to estimate a rough axis, thenthe rough axis is refined applying a modified symmetry axisextraction algorithm.

II. METHODS

The flowchart of molar axis estimation is shown in Fig. 1.The 3D stack of CT images is first projected to two planesto obtain two 2D projection images for each molar. Then,molar contours are segmented from the two 2D projectionimages, and 2D axes are extracted from the segmented molarcontours. Finally, 3D axis of each molar is obtained by com-bining the corresponding two 2D axes. In our methods, thefirst 2D projection image corresponds to the 2D panoramicradiograph including all the teeth which is synthetized byprojecting the CT images along the perpendicular directionof dental arch (in buccal-lingual direction). The second 2Dprojection image is obtained by projecting the CT images’local region which belongs to the molar along the perpendic-ular direction of the 2D axis extracted from 2D panoramic

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Fig. 1. The flowchart of molar axis estimation.

radiograph.

A. Synthetizing 2D Panoramic Radiograph

The 2D panoramic radiograph is synthetized by projectingthe CT images along the perpendicular direction of dentalarch [9-11]. First, the dental arch is determined using thefollowing method: calculating the Maximum Intensity Pro-jection (MIP) of mandible CT images, extracting the largestconnected component of MIP image and using morpholog-ical thinning [12] on the extracted largest connected com-ponent and then, B-spline fitting is applied to the thinningresults to generate dental arch. To obtain the panoramicradiograph, only the voxels within the largest connectedcomponent region of MIP images are projected. The intensityof each pixel in the projection image is calculated usingDiscrete Radon Transform (DRT) method [13].

B. Synthetizing the Second 2D Projection Image

Given the 2D region (the red curve in Fig. 2) and theaxis of molar (the green line in Fig. 2) extracted from the2D panoramic radiograph, the local projection region forthe molar corresponds to the inner region between the twoblue lines which parallel the green axis and represent theleft-most and right-most sides of molar contour, and theprojection direction of the second 2D projection image isperpendicular to the axis. The 2D molar region segmentationand 2D projection axis extraction from the 2D panoramicradiograph is illustrated in the following subsections.

Fig. 2. The determination of the second 2D projectionimage’s projection region and projection direction.

C. Tooth Contour Segmentation from 2D Projection Images

Molar regions are segmented from the 2D panoramicradiograph and local projection images (i.e., the second 2Dprojection image) combining the contour extracting methodproposed in [14] and the region growing algorithm [15].Region growing method where we consider the intensity,the gradient and the total number of the pixels is appliedto the inner region of the extracted contour using [14] tofinally determine the molar contour. The process of extractingcontour using the method in [14] is as following: first, theupper and lower teeth are separated via a smooth curve ob-tained by fitting the gap valleys between the upper and lowerjaw bones. Then molars are isolated from their neighborsusing the same method as upper and lower tooth separation.A radial scan from the molar crown center is performedand the pixels with the maximum value of the probabilitydensity function along the radial line are determined to bethe contour points. The left-most and right-most sides ofcorresponding crown’s contour are used to produce the twofirst points for root contour and root contour is extractedby maximizing the intensity difference between speciallydesigned “inner intensity” and “outer intensity” parameters in[14]. Then, crown contour and its root contour are connectedtogether forming molar’s whole contour where the regiongrowing method which includes some manual interventionto select some seed pixels is applied. The boundary of regiongrowing result is the final segmented molar contour.

D. Calculation of 2D Projection Axis using PCA and Mod-ified Symmetry Axis Extraction Algorithm

To extract 2D axes of segmented molar contours, PCA [16]is first used to roughly estimate the axes and then a modifiedsymmetry axis extraction algorithm is used to refine it.

Just as Fig. 3 (a) shows, the method in [17] extracts humanfacial symmetry axis based on comparing the gray leveldifference of each pair of pixels, i.e., gray level differencehistogram analysis method (GLDH). In [17], the angle be-tween the green line connecting the two pixels in the samecomparing pair and the axis being detected is surely 90◦,i.e., it is a reliable orthogonal axis extraction algorithm.Given the geometric feature of the molar contour on the

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projection images, we modify the orthogonal symmetry axesextraction method in [17] to skew symmetry axes extractionalgorithm, i.e., the angle between the connecting lines of thetwo compared pixels and the axis being detected is not surely90◦ just as Fig. 3 (b) shows and the range of the sharp angleis [60◦, 90◦] determined by us. In Fig. 3 (b), the two pixelsare bisected by the cross point of the blue and the greenlines, where the blue connecting lines are parallel to eachother.

Fig. 3. Modified symmetry axis.

The process of estimating 2D axes is as following: firstly,PCA is used to roughly estimate the molar axis on segmentedmolar region. Then, at the range of [−30◦, 30◦] around thePCA axis, we search the 2D axis using modified symmetryaxis extraction algorithm.

The 2D axes extracted from the 2D panoramic radiographand local projection images are shown in Fig. 4.

Fig. 4. 2D axes extracted from 2D panoramic radiograph andlocal projection images.

E. Synthetizing the 3D Axis from Two 2D Axes by CoordinateTransformation

We establish local coordinate system for each molar. Thez′ axis parallels the z axis of the world coordinate system;y′ is the tangent direction of dental arch; original pointo′ is located at the 3D spatial position of the mass centerof segmented tooth region from 2D panoramic radiograph.Just as Fig. 5 (a) shows, the blue plane and the red planeindividually represent the y′o′z′ plane and the z′o′x′ planeof the local coordinate system. Based on such establishmentof each local coordinate system, the first 2D projection of

each molar is just the projection result on the blue y′o′z′

plane.Just as Fig. 5 (b) shows, the purple plane represents the

second 2D projection plane where the green “axis 1” and“axis 2” locate on. The green “axis 2” (i.e., the 2D projectionaxis on the second projection image) in 3D space is just thereal axis. By homogenous transformation, the real 3D axiscan be transformed from the local coordinate system to theworld coordinate system.

Fig. 5. The 3D axis in local coordinate system.III. EXPERIMENTAL RESULTS AND DISCUSSION

This study was reviewed and approved by Institutional Re-view Board of Shenzhen Institutes of Advanced Technology,Chinese Academy of Sciences. Written informed consents ofthe subjects were obtained.

A. Dataset

Cone beam CT (CBCT) images of ten subjects were usedto validate the proposed method. These tested images wereacquired by a CT scanner (NewTom VG, Italy) with 120kV, 5 mA, a matrix of 624 × 624, a spatial voxel size of0.25 mm, and the time of exposure of 6 s. “Reference axes”manually determined by dentists were regarded as the goldstandard and compared with the results of the algorithm. Theangle and distance between the “reference axes” and axesestimated by the algorithm were evaluated to quantitativelyestimate the accuracy of the method.

B. Molar Axis Estimation Results

The results of molar axis estimated using the proposedmethod are shown in Fig. 6. The results indicate that theestimated axes using the proposed method is similar withthe “reference axes” (the red axes in Fig. 6) in vision. Thequantitative accuracy of axis estimation for the three typesof molars are presented in Table I.

C. Discussion

Compared to existing molar axis estimation methods,the proposed method mainly has two advantages. The firstone refers to the tooth segmentation. Existing molar axisestimation methods [4] [5] need to first segment the volumeof molars from the CT images. The molar segmentation fromCT images is a challenging work due to the similar inten-sity between the tooth and alveolar bone and the complex

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Fig. 6. The 3D molar axes obtained by our method.

topological changes of different root branches. Currently, thestate of the art tooth segmentation methods [6-8] segmenttooth volume from CT images mainly using the slice-by-slicesegmentation strategy based on the tooth shape coherencebetween adjacent slices. To segment these angled molarsespecially the third molars in which tooth shape coherencebetween adjacent slices of the angled tooth is low, thesemethods will get poor results which will result in the failureof molar axis estimation. Instead of estimating the axis fromthe segmented molar volume, this study proposes to generatethe 3D axis of each molar by combining two 2D axesextracted from two 2D projection images. Thus, the proposedmethod only needs to segment tooth contour in two projectedimages for each molar. The projected images generally haveclearer tooth boundary than the original CT images [14], andthe tooth contour is more implementable to be segmented inthe projected images.

TABLE I. THE AVERAGE ANGLE AND DISTANCE DIF-FERENCE

Molars Angle(◦) Distance (mm)

First molars 2.15 ± 0.96 1.08 ± 0.59Second molars 2.31 ± 0.91 1.37 ± 0.67Third molars 2.23 ± 0.87 1.17 ± 0.55

The second advantage of the proposed method is theaxis estimation scheme from the segmented tooth region.Existing methods [4] [5] directly used the PCA to estimatethe tooth axis from the segmented volume. However, the axisestimated by PCA is indeed the major axis which often doesnot correspond to the exact tooth axis. To address this issue,this study proposes to first estimate an axis using PCA, thenrefine the axis applying a modified symmetry axis extractionalgorithm.

In this study, the 3D axis of each molar is obtainedby combining two 2D axes extracted from two projectionimages. It is feasible to project the CT images to more imageplanes and extract the corresponding 2D axes from these2D projection images, and a more accurate 3D axis can beobtained by combining the multiple 2D axes.

IV. CONCLUSIONThis study proposes a new molar axis estimation method

from CT images. Instead of estimating the 3D axis of eachmolar from the segmented 3D molar volume, the method ob-tains the axis from two projection images. It first projects the

3D images of each molar to two 2D image planes, and thensegments the molar contour in each 2D projection image.PCA and a modified symmetry axis extraction algorithm areemployed to extract the 2D axis from the segmented molarcontour. Finally, a 3D tooth axis is obtained from the twoextracted 2D axes. Experimental results validated that theproposed method can estimate satisfying molar axis fromCT images.

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