Bite - In

Dentistry in India has grown by leaps and bounds and our splurge of dental institutions has lead to a tremendous thrust in the R & D departments also. T his is an original fundamental research article presented by Souvik RayChaudhuri, Abir Ranjan Raksit and Suman Chakraborty.

Geometric Modeling of Human Dental Profile using a Bezier Curve Approximation

Introduction

With advent of computer graphics and pertinent simulation techniques, geometric modeling is extensively being used now a days for various medical applications, typically for simulation of various bio-organs. Such applications are of immense use in dental applications as well. In this context, the generation of human dental profile can be of special mention. This is because it is of great importance in systematic processing of sets of artificial teeth, computer-aided prosthetic dentistry, preparation of virtual experimental modules in dental education, and many others. In recent times, there has been a considerable progress in this matter, with involved imaging techniques being used to mimic arrangement of teeth inside human mouth, looking at various angles using axonometric projection. However, such imaging techniques are rather costly, necessitating involved image processing, as well as precision display control.

The aim of the present work is to develop a systematic, yet very simplistic geometric modeling approach for computer generation of centerline of human dental profile, consistent with realistic dental arrangements in the mouth. In order to achieve this goal, we attempt to devise a smooth curve representing the above profile. For this purpose, our target is to generate a spline curve guided by suitable control points, and constrained by the locations of individual teeth positioned on the profile. It is important to mention here that a high order polynomial exactly interpolating the tooth locations is not at all suitable in this context, since it imparts unnecessary oscillations to the profile. On the other hand, an extensively used spline curve in geometric modeling applications (especially in the context of computer graphics) namely, the Bezier curve [1] is ideally suited for this purpose. This is because, with a systematic choice of control points, any kind of regularity or irregularity in the profile can be handled, without sacrificing any smoothness requirements. Such curves have inherent advantages of generating smooth profiles replicating certain key features apparent from the geometric constraints of the curve, without sacrificing the variation-diminishing property [1] as well. Moreover, these curves do not require any explicit slope specification, and they also interpolate the end control points exactly. Additionally, profile-contours conforming to shapes represented by these curves can easily be realized in terms of computer aided manufacturing (CAM) techniques, for accurate, precise and controlled production of bio-dental appliances necessitating trace of the dental profile as a basic input. However, the challenge remains in specification of appropriate control points yielding the requisite nature of the curve. Accordingly in this work, we develop a systematic mathematical algorithm for estimation of suitable control points leading to parametric evaluation of points located on the dental profile itself.

The proposed algorithm is verified from measurements of key dimensions of representative dental profiles as well, in order to adjudge its usefulness. Thus, the algorithm leads to a very convenient, economical and simplistic replication of human dental set. Such an approach is yet to be adapted in a similar context, to the best of the knowledge of the present investigators.

Mathematical Modeling and computer simulation

The first step in mathematical modeling is to develop a systematic algorithm for selection of control points for the Bezier curve. This is of immense importance, since the Bezier curve is smoothly guided by the control points, without exactly interpolating them, except for the end points. As a starting basis for selection of control points, it can be noted that a substantial portion of the dental profile closely resembles to a well-known geometrical form, namely, the parabola (except, may be the end portions, especially at locations in the neighborhood of the so-called 'wisdom-teeth'). We first outline the pertinent geometrical features that correlate the contour of a dental profile with an equivalent parabolic curve in its neighborhood.

The representation of the non-parabolic end portions will be mentioned subsequently. The basic idea is to form a sample space containing numerous possible control points {Pi} as a design basis, such that Pi Î S " i. The parabolic space is designed to be a dominating subspace SP of the whole sample space. We consider a generic parabolic curve of the form x2 = 4ay, Where the vertex is at the origin (located at the junction of the two incisor teeth), and the axis is the y-axis (a line of symmetry that originates from the vertex and goes into the mouth) [2]. Referring to figure (1), where such a profile is sketched, we denote the location of the end of the ith tooth by the point 'i' on the profile (i=1for the first incisor, =2 for the second incisor, =3 for the canine, and so on). Analogously, 'j' represents the location of jth tooth. First, we attempt to obtain a realistic prediction of the parameter 'a' for specification of the parabola. In this context, it can be noted that slope of the parabola at any point (x,y) on it is given by

dy/dx=x/2a, (1)

For practical estimation of the above, a flexible wire is bent after inserting the same into the mouth of the person under investigation, so as to follow the dental profile of the same. On that wire, any three successive teeth (preferably located near the front, for convenience in measurement) are marked as i-1, i and i+1. On withdrawal of the wire from the mouth and development of the profile, typical measurements such as xi and the distance between the teeth i-1 and i+1, symbolized as li-1, i+1, are carefully noted. Using those values, we evaluate

cos?i= (x i+1-x i-1)/l i-1,i+1 (2)

In equation (2), ?i is the scope angle of the straight line segment joining the points i-1 and i+1. By mean value theorem [3], there must be a point located on the curve in between these two, where the tangent is parallel to this chord. We can assume that point as the point 'i' without much of an error (anyway, the exact dental profile is not a parabola, and hence the error involved in the above approximation is therefore, not too significant. The only goal to be achieved here is to get a parabolic curve as an initial bank of control points, which can be used to generate the Bezier representation of the actual dental contour). Thus,

(dy/dx)i ťtanqi (3)

and

a=2(dy/dx)i/xi (4)

In equation (4), xi can measured directly, or indirectly through measurement of li and ai (refer to figure (1) as:

xi=licosai (5)

For averaging over a set of values of 'a' (in order to get an averaged estimate of 'a'), repeated measurements over successive three teeth can be taken. However, this is not a necessity since, only a rough estimation of 'a' is needed for mathematical modeling purpose. Once an estimation for 'a' is obtained, the curve x2= 4ay can be specified. Any point on that curve can now be chosen as a control point out of the set of control points used to obtain a Bezier form of the dental profile. It is interesting to note that actual measurement of points located on the dental profile reveals that except the end locations, a major portion of the profile almost merges with the parabola. This is checked by observing that

xj=?li,j cos?i (6)

(where tan?i=(dy/dx)i ) is almost equal to the directly measured values of xj (without using slope of various points on the parabola), except for the end locations. Therefore, except for those locations, we can start with control points located on the parabolic approximation and utilize the convex-hull property of the Bezier curve [1], which essentially means that the Bezier would always be within the convex hull of the chosen control polygon (i.e., successive straight-line join of control points). The end control points (at locations in vicinity of the positions of the wisdom teeth) are selected as the last points on the tooth-profile. Though these points, in reality, do not lie on the parabola, we get rid of the situation by a special property of the Bezier curve that it exactly interpolates the end control points, and therefore any kind of approximation in the neighborhood is not necessary. For plotting the parametric form of the Bezier curve, a suitable recursion algorithm is used (to be outlined subsequently). Thus, points (x,y) located on the Bezier approximation of the centerline dental profile can be obtained. Getting corresponding values of x and y and taking account the pixel resolution of graphic screen (which is 30pixel/cm for x axis and 28 pixel /cm for y axis for our present study), a scaling transformation is used to multiply the current coordinates to get equivalent values in pixels for turbo C package (version 4) in which the graphic screen inherently assumes a left handed co-ordinate system with origin at the top left corner of the viewport. The locations of points on the center line dental profile for each jaw are obtained by transforming the origin to the pixel location (300,240), and a subsequent plotting using the putpixel function. Regarding the choice of control points, it can be noted that we first draw a Bezier curve using the initial sets of points ascertained earlier. On the same plot exact teeth locations are marked.

Now based on deviations of the generated points on the Bezie curve from actual tooth locations, we use successive small increments or decrements of the control point positions until a nearly converged solution is obtained. At locations where curve becomes highly asymmetric, an additional number of control points is selected locally for exact replication of the profile. Degree of Bezier is ascertained accordingly.

Summary of the overall algorithm

Step 1 : Obtain equation of the approximating parabola, using equations (2) - (4); and hence choose arbitrary sets of control points located on that parabolic curve. Also choose end control points as end locations of the actual tooth profile.

Step 2 : Using the above set of control points, use the following recursion for estimation of points on the Bezier curve as follows:

for i=o,n
for r=1,n
for j=0,n-r
if rŁi, u=b, else u=a
compute
pjr(u)=((b-u)/(b-a)) pjr-1(u)+((u-a)/(b-a)) pj+1r-1(u)
end
end
end

It can be noted here that [a,b] represents the range of the parameter u, which is taken as [0,1], for convenient normalization of the curve. It is also worth mentioning that when r=n, p0n(u)represents a point on the Bezier curve, corresponding to the current value of u.

Step 3 : Check the deviation from points on the above curve to corresponding points located on the actual dental profile. Accordingly, apply slight increment/decrement to the coordinates of control points in the neighborhood. It is important to make sure that the control polygon circumscribes the set of actual points on the dental profile, under any circumstances, so as to make use of convex hull property of the Bezier curve.

In case the control points obtained in step 3 deviate from the previous set of control points (refer to step 1) within a very close tolerance (typically, a relative error of 10 - 4 is acceptable), the Bezier curve drawn represents the centerline dental profile to a reasonably accurate extent, and therefore that is accepted as a converged solution. Otherwise, one has to go back to step 2 and iterate until convergence.

Results and discussion

First, we assess the usefulness of parabolic approximation as a starting design basis for the selection of control points. For that purpose, we calculate the root mean of the square of deviation respective control points positioned on the parabola from the corresponding points located on one symmetric half of the actual profile. For comparison purpose, corresponding x-coordinates are chosen. That deviation, when normalized using the mean of the true x- coordinates, essentially shows that the error is only marginal, except for the end points. We find that the maximum error in the calculation of xi is12.337% (at the end points) which is well within acceptable limits. However, since no parabolic approximation is used for asserting the end controls points, the deviation at other locations is practically imperceptible. The error sample standard deviation is 8.6% and error population standard deviation is 8.3%, which signifies that the error estimation is fairly accurate. For the purpose of brevity, only two case studies are presented in this paper.

Figure 2(a-b) and 3(a-b) represent the Bezier representations derived from the above two case studies, respectively, corresponding to both upper jaw (a) and lower jaw (b). The circled points represent points identified to be located on the actual dental profile. It can be easily seen from the figure that the actual tooth locations are very reliably captured by the equivalent Bezier curve representations. It can be noted that only a little care needs to be devoted in this respect for systematic evaluation of the pertinent control points. While generating the profile, it has been generally observed that control points 12 to 14 in number are altogether sufficient for plotting each symmetric half of the profile.

Conclusions

The algorithm presented above is fairly general, easy to implement and does not necessitate any expensive imaging techniques. A further improvement in this could be achieved by employing a B-Spline approximation for a more precise local control, though it adds to the complexity of the spline generation in some sense. However, for the present purpose, it has been observed from the study that Bezier approximation effectively suffices the purpose of generating the entire dental profile for the purpose of geometric modeling, making sure that the profile passes virtually over all the individual tooth locations of concern in a smooth manner.

References

1. D. F. Rogers and J. A. Adams, Mathematical Elements for Computer Graphics, McGraw-Hill Publishing Co., New York, 1989.

2. S.L.Loney, The Elements of Coordinate Geometry, Macmillan, London, 1962.

3. E. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons Pvt. Ltd., Asia, 1983