Priority Program 1253

Transcription

1 Deutsche Forschungsgemeinschaft Priority Program 1253 Optimization with Partial Differential Equations N. D. Botkin, K.-H. Hoffmann, N. Mayer, L. Kovacs, M. Eder, S. Raith and A. Volf Application Oriented Methods for Computing Shortest Paths and Principal Curvatures on Triangle Surfaces April 2014 Preprint-Number SPP

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3 Application oriented methods for computing shortest paths and principal curvatures on triangle surfaces N. D. Botkin 1, K.-H. Hoffmann 1, N. Mayer 2, L. Kovacs 3,4, M. Eder 3,4, S. Raith 3, and A. Volf 3 1 TUM*, Mathematics M6, Garching, Germany 2 TUM, Mechanical engineering LCC, Garching, Germany 3 TUM, Computer Aided Plastic Surgery (CAPS), Garching, Germany 4 TUM, Klinikum rechts der Isar, Klinik und Poliklinik für Plastische Chirurgie und Handchirurgie, München, Germany * TUM = Technische Universität München Abstract In recent years, imaging techniques and three-dimensional (3-D) surface scans in plastic and aesthetic surgery become more and more popular. The data generated by these methods can be used for planning of operations and clinical evaluation of the surgical success. Conventional aesthetic assessments are purely empirical and based on the subjective evaluation through the surgeon or the patient itself, which may be very individual. Commonly accepted objective and quantifiable evaluation approaches do not exist. Preliminary objective 3-D evaluation methods are based on the simple rightto-left side comparison of distances, volumes, or surface areas. It is desirable to develop an objective evaluation index that takes into account a set of relevant measurements and collect them with weights in the so-called Symmetry Index. This would allow to use the objective analysis of surgical results, which would make the doctor-patient communication more constructive. It should be noticed that the Symmetry Index should contain the intrinsic characteristics of 3-D surfaces coming from the topology and differential geometry such as geodesic distances and curvatures. Another important problem is setting characteristic anatomical points (called landmarks) on 3-D face and body scans to estimate the surgical results. The problem here botkin@ma.tum.de hoffmann@ma.tum.de mayer@lcc.mw.tum.de l.kovacs@lrz.tum.de m.eder@lrz.tum.de stefan.raith@caps.me.tum.de alexander.volf@caps.me.tum.de

4 is that the process of landmark positioning is manual, which is very cumbersome and time-consuming, depends on the particular evaluator, and therefore is only partially reproducible. Thus, an objective, standardized, and reproducible method for the fully automatic anthropometric analysis of three-dimensional data sets is required. There is the believe that a stable algorithm for the computation of curvature tensors can solve this challenging problem. Finally, the problem of sparse representation of aesthetic regions by semantic surface models containing additional meta-information about the anatomical peculiarities of these regions is important and challenging. The development of such semantic surface models requires a semi-analytical description of triangle surface regions. Here, local maps and parametric shape functions can be used. Fitting these objects to the specific regions is then carried out by the variation of the corresponding parameter. The decomposition of the surface into the aesthetic regions and construction of the shape functions should be based on the analysis of curvature. Roughly speaking, the areas with the main curvature signatures (+ +) - convex, (- -) - concave, and (- +) - saddle-type, have to be distinguished. The calculation of semantic basis functions can be done using the kriging method that analytically approximates large data sets with low numerical effort. This paper considers two problems: computation of shortest paths on triangle surfaces and calculation of principal curvatures at vertices of triangle surfaces. Existing methods compute shortest paths along edges so that the result may be completely unacceptable. We propose a technique that allows us to compute smoother and shorter paths using intermediate admissible directions and linear interpolation of potential functions. Conventional methods that compute the curvature in nodes of triangle surfaces are based on processing only adjacent vertices. They are non stable because triangle meshes contain frequently oscillating errors in positions of the vertices. We propose a simple practical method that accounts for larger neighborhoods of vertices. The kriging method aimed to the sparse representation of specific surface parts (aesthetic regions) is sketched. Preliminary numerical results are presented. 1 Introduction In the last decade, discrete differential geometry became very popular because of its application to visualization of technical and medical objects. The basic aim of the discrete differential geometry is to extend geometrical characteristics inherent to smooth manifolds to the case of polyhedron surfaces. We consider two problems: computation of shortest paths on triangle surfaces and calculation of principal curvatures at vertices of triangle surfaces. There are many algorithms for computing shortest paths between two points on graphs (see e.g. a very exhaustive survey [1]). Nevertheless, these methods are not perfectly applicable to practical problems because shortest paths along edges may be very jagged. Moreover, such paths may lie far away from geodesic lines impressed by the observer. We propose a technique that allows us to compute smoother and shorter paths using intermediate admissible directions and linear interpolation of the potential function. This approach is related to the numerical treatment of time independent Hamilton-Jcobi equations arising from optimal 2

5 control theory. As for computing the curvature in nodes of a triangle surface, there are several fundamental principles of extension of the notion of curvature to discrete objects. For example, the Gaussian total curvature at a vertex is given by K = 2π j α j, where α j are the angles between the edges adjacent to the vertex. The mean curvature vector at a vertex p i is given by the cotan formula H i = 1/2 j (cot α ij + cot α ji )(p i p j ) (Pinkall and Polthier, see e.g. [4]), where α ij and α ji are angles opposite to the edge p i p j. There are works that propose formulaes for computing the principal curvatures at a vertex using the edges adjacent to this vertex (see e.g. [2, 3]). Nevertheless, these results are not perfectly applicable to practical problems where triangle surfaces arise from scans, say, of human faces. Such scans yield clouds of points that are processed to obtain triangle surfaces representing human faces. It is clear that such triangle meshes contain frequently oscillating errors in positions of the vertices, which restricts application of local methods that use only adjacent vertices. We propose a simple practical method that accounts for larger neighborhoods of vertices. 2 Computing a shortest path between two points 2.1 Shortest path along edges In this subsection, basic ideas of computing shortest paths are outlined. Consider a triangle surface and denote the set of all vertices (resp. all points) by S (resp. by Σ). The majority of existing algorithms are based on the direct or indirect construction of the potential function d(s), s S, that express the distance between any vertex s and a fixed vertex s f along edges (see Fig. 1). Thus, d(s) = dist(s, s f ) along edges. It is easily to prove that the potential function satisfies the following optimality condition: d(s f ) = 0, d(s) = min l L(s) {d(s + l) + l }, s s f. (1) Here, L(s) denotes the set of all edges adjacent to the vertex s, i.e. the set of all admissible directions (see Fig. 1), l denotes the length of the edge l. Such a nonlinear equation can be solved using the following iterative relaxsation procedure: d 0 (s) = 0 d n (s f ) = 0 d n+1 (s) = min l L(s) {dn (s + l) + l }, s s f. (2) s, n, 3

6 L(s) = {l 1, l 2,..., l 5 } Figure 1: A triangle surface with a selected vertex s f. The set L(s) consists of edges adjacent to s. It easily to prove the following properties of the sequence d n (s): i) d n (s) d(s) s, ii) d n+1 (s) d n (s) s. These properties follow from the monotonicity of the operator q Π[q] defined as Π[q](s) := min l L(s) {q(s + l) + l }, i.e. if q 1 (s) q 2 (s) for all s, then Π[q 1 ](s) Π[q 2 ](s) for all s. Thus, conditions i) and ii) together with relations (1) and (2) imply that d n (s) monotonically converges to d(s) for all s. Construction of a shortest path between s and s f is shown in Fig. 2. Assume that the path is already constructed to the vertex s i. To obtain the next vertex, chose an optimal direction l min as a minimizer in min {d(s i +l)+ l } and set s i+1 = s i +l min. If the minimizer l L(s i ) is not unique, choose anyone. Remark 1 Note that the potential function d( ) is used only in a small neighborhood of a shortest path. Therefore, the construction of the potential function can be restricted to a subregion containing all vertices of a shortest path, which essentially speedups the calculation. Moreover, the computations at different vertices in (2) are completely independent, and therefore very efficient parallelization is possible. 4

7 Figure 2: Construction of a shortest path along edges. 2.2 Shortest path in the case of large sets of admissible directions Now, large sets of admissible directions will be used to compute a modified potential function at the vertices. Fig. 3 shows different ways of choosing admissible directions. On the picture to the left, the ends of admissible directions lie on the edges of the cluster about s, whereas, on the picture to the right, the ends of admissible directions lie inside the cluster. In both cases, if the end point of an admissible direction does not meet a vertex, the linear interpolation is used to compute a value in such a point. Figure 3: Choice of admissible directions. The ends of admissible directions may lie on the edges of the cluster about s (see the picture to the left), or they may lie inside the cluster (see the picture to the right). 5

8 Consider the following iterative procedure: d 0 (s) = 0 d n (s f ) = 0 d n+1 (s) = min l L(s) { d n (s + l) + l }, s s f, (3) where L(s), s S, are large sets of admissible directions, and denotes the operation of linear interpolation of a grid function. It is easily to prove that the sequence d n (s) satisfies properties i) and ii), and therefore d n (s) monotonically converges to some ˆd(s) for all s. Construction of a shortest path between s and s f is explained in Fig. 4. Assume that the path is already constructed to a point s i (not necessary a vertex). To obtain the next point, chose an optimal direction l min as a minimizer in min l L(s i ) { ˆd(si + l) + l } and set s i+1 = s i + l min. Here, L(s), s Σ, are large sets of admissible directions when constructing the path. Remember that Σ is the set of all points of the triangle surface. s, n, Figure 4: Construction of a shortest path between s and s f using intermediate directions and linear interpolation of the potential function ˆd. 3 Computing principal curvatures Assume that the principal curvatures are to be computed at a vertex s labeled by a large green circle in Fig. 5. Consider the first order cluster consisting of triangles adjacent to s (the bounding vertices of this cluster are shown in red). The second order cluster consists 6

9 of all triangles adjacent to the first order cluster (the bounding vertices are shown in blue). Moreover, it is easily to define the third order cluster, and so on. Compute a normal vector n s at s as a weighted sum of the normal vectors to the triangles adjacent to s and draw a cutting plane passing through n s. It is clear that such a plane is uniquely defined by the rotation angle about n s. Let a value of the rotation angle be fixed. To compute the curvature corresponding to this angle, find all intersection points of the cutting plane with the edges of the first, second, and third order clusters (such points are shown in yellow in Fig. 5). Consider the cutting plane and the intersection points on it (see Fig. 6), chose a local reference system in the cutting plane, and construct a polynomial (reasonable degrees are 2,3,4, and 5) that provides the best approximation of the intersection points in the sense of weighted minimum squares. That is, P (x) = a 0 + a 1 x + a 2 x a m x m, and the coefficients are obtained from the condition min a 0,a 1,...a m N ) 2, w j (P (x j ) y j j=1 where w j are some weights, and (x j, y j ) are the coordinates of the intersection points in the local reference system chosen on the cutting plane. The curvature of the polynomial graph is then given by k = 2a 2 /(1 + a 2 1). This value approximates the curvature of the triangle surface at s for a fixed rotation angle of the cutting plane. Rotating the cutting plane about the normal n s, we compute the principal (maximal and minimal) curvatures. 7

10 n s s s Figure 5: The first and second order clusters (red and blue vertices, respectively) about the vertex s, a normal vector n s at s, and the intersection points (yellow) of the cutting plane and the edges of the clusters. Figure 6: Two cutting planes, the corresponding intersection points, and the graphs of approximating polynomials are shown. The picture to the left (resp. to the right) corresponds to the maximal (resp. minimal) value of the curvature. 4 Preliminary numerical results Figure 7 shows results of computing shortest paths. The red line represents a shortest path along edges. The picture at the upper right explains the appearance of jags in the path. It is seen that there are not good admissible direction in the shown area. The green line is constructed using a global refinement of the triangle grid. In this case, the computation time essentially increases. The blue line is constructed using larger sets of admissible directions. 8

11 The picture at the down right shows that the path can cross edges in this case. Note that this path is 15% shorter than the path along edges and really looks like the geodesic line. The method works also well for relatively rough triangle meshes (see Fig. 8). Figure 7: Computation of shortest paths on a triangle surface: along edges (the red line), along edges but using a global refinement of the triangle mesh (the green line), and using larger sets of admissible directions (the blue line). The numbers of vertices and triangles of the surface are about and , respectively. 9

12 Figure 8: Computation of shortest paths on a relatively rough triangle surface. A shortest path along edges is given by the red line, and the green line shows a shortest path computed using larger sets of admissible directions. The number of vertices of the surface is about Figure 9 presents numerical experiments with rough, inhomogeneous, and very distorted triangle meshes. The picture to the left shows two shortest paths along edges. The corresponding paths computed using larger sets of admissible directions are shown in the picture in the middle. Note that the red line is essentially better in the case of larger sets of admissible directions, whereas the green line avoids to cross large distorted triangles. This effect is caused by the interpolation error that is large in the case of big distorted triangles. The way to improve that is related to a modification of the optimality principal when the constructed path meets a big distorted triangle. In the case of relatively homogenous meshes, the construction of shortest paths using larger sets of admissible directions gives good results even in the case of very distorted triangles (see the picture to the right). 10

13 Figure 9: Numerical experiments with rough, inhomogeneous, and very distorted triangle meshes. Shortest paths along edges (to the left) and the corresponding paths computed using larger sets of admissible directions (in the middle) are presented. The computation of a shortest path for a relatively homogeneous but very distorted triangle mesh using larger sets of admissible directions is shown on the picture to the right. Figures 10, and 11 show results of computing the curvature. Figure 10: Computing the maximal (to the left) and mean (to the right) curvatures in the case of an asymmetric face. 11

14 saddle type concavity convexity Figure 11: Computing the Gaussian curvature (to the left) and the classification of vertices according to the signs of the principal curvatures (to the right). Figure 12: Landmarks (see [5]) describing indicative points of a face. 12

15 Figure 13: Distance measurement between landmarks as described by Farkas [5] with the new shortest path method. Figure 14: Localization of the left mouth landmark with the minimum of the maximal curvature. 13

16 Figure 15: Localization of the right mouth landmark with the minimum of the maximal curvature. Figure 16: The result of the localization of the mouth landmarks. 14

17 Surface representation using kriging models Reconstructing free-form surfaces from 3-D measurements is a central problem in many applications. A vary appropriate and commonly used tool here are kriging models. It should be noticed that these models have been developed as estimators of randomly disturbed data. Nevertheless, they work well with deterministic computer models (see e.g. [6]). Kriging models show very good flexibility to approximate many different and complex response functions. They are also a good choice for analytic representation of grid surfaces using few parameters. The mathematical form of a kriging model with, e.g. scalar response, see (4), has two parts. The first part is a linear regression of the data with k regressors modeling the trend over the domain. Most engineering applications use a constant trend model and rely upon the second part of the model to pull the response surface through the observed data by quantifying the correlation of nearby points. ŷ( X) = k β i f i ( X) + Z( X). (4) i=1 Here X is a p-dimensional input vector, e.g. a set of parameters of a computer PDE-based model, f i known functions (polynomials), β i regression coefficients, and Z( X) a model of a Gaussian and stationary random process with zero mean and covariance: cov(x 1, x 2 ) = σ 2 R(x 1, x 2 ). (5) The process variance, σ 2, scales the spatial correlation function R(x 1, x 2 ) that controls the smoothness of the resulting kriging model, the influence of other nearby points, and the differentiability of the response surface by quantifying the correlation between the two observations x 1 and x 2. There are many potential functions that can be used to quantify the correlation between the observations. For example, R(x 1, x 2, θ) = exp ( θ (x 1 x 2 ) 2), θ > 0. (6) The range parameter, 1/ θ, indicates the distance at which the influence equals e 1 = or approximately 37%. To construct a multivariate correlation function, a univariate correlation function is used for each of the p input dimensions, and a product correlation rule is used: p R( X 1, X 2, θ) = R(x 1,i, x 2,i, θ i ). (7) Therefore, a kriging model contains the unknown parameters i=1 γ = { β, σ, θ} to be determined from the n-vector, Y, of observations. The most popular method of estimating kriging model parameters is Maximum Likelihood Estimation (see [6]). The logarithm of 15

18 the multivariate Gaussian likelihood function is given by: l( γ Y ) = n 2 ln[2πσ2 ] 1 2 ln[ R ] 1 2σ 2 ( Y F β) T R 1 ( Y F β), (8) where R is a matrix quantifying the correlation of all of the observations, R ij = R( X i, X j, θ), i, j = 1,...n, and F a matrix of f( X) evaluated at each observation, F ij = f j (X i ), i = 1...n, j = 1...k. Minimization of this function yields relations β = (F T R 1 F ) 1 F T R 1 Y, σ 2 = 1 n ( Y F β) T ( Y F β). (9) Notice that the matrix R in (9) depends on the parameter set γ. To obtain a closed solution, the relations (9) can be substituted into (8), and the last can be minimized over θ. This procedure is being iteratively repeated (see [6]). There is a wide list of software for constructing kriging models (see e.g. [7] for the toolbox DACE). This toolbox provides regression models with polynomials of orders 0, 1, and 2 and several correlation models: Gaussian, see (6), exponential, spherical, cubic, etc. The next figure shows the usage of a kriging estimator for the prediction of outputs of a physical model. The toolbox DACE (see [7]) was utilized for the construction of a kriging estimator. Figure 17: Three surfaces representing outputs of a physical model. Each surface is computed using a kriging estimator whose regression part utilizes polynomials of the first order. The kriging estimator itself is constructed on the basis of 20 observations, simulations of the physical model with different parameter sets. The black dots represent the values obtained from the model simulations but not involved into the construction of the kriging estimator. One can see a very good prediction. 16

19 Conclusion The results presented in this paper are targeted for medical applications. First, the computation of smooth shortest paths on triangle surfaces is supposed to be used for the development of a Symmetry Index providing the objective analysis of surgical results. The measurement of the geodesic distances between characteristic anatomical points, landmarks, on 3-D face and body scans provides intrinsic characteristics of aesthetic regions. Second, the computation of curvatures for sampled directions at each node allows us to develop an objective, standardized, and reproducible method for fully automatic positioning of land marks. And finally, the problem of sparse representation of aesthetic regions by semantic surface models can be solved using the decomposition of the surface into curvature specific regions (convex, concave, and saddle-type) and applying kriging models to represent them analytically. References [1] Giorgio Gallo. Shortest Path Algorithms. Annals of Operations Research 13(1988)3-79 [2] Gabriel Taubin. Estimating the tensor of curvature of a surface from a polyhedral approximation. In Proceedings of Fifth International Conference on Computer Vision (ICCV 95), [3] Meyer, M., Desbrun, M., Schröder, P. and Barr A.H. (2003), Discrete differentialgeometry operators for triangulated 2-manifolds. In H.-C. Hege and K. Polthier, editors, Visualization and Mathematics III, pp , Springer-Verlag, Heidelberg. [4] Max Wardetzky. Convergence of the Cotangent Formula: An Overview (p ). In. Discrete Differential Geometry (ebook). Alexander I. Bobenko, Peter Schröder, John M. Sullivan, Günter M. Ziegler (Eds.) Birkhäuser [5] L. Kovacs, A. Zimmermann, G. Brockmann, H. Baurecht, K. Schwenzer-Zimmerer, N. A. Papadopulos, M. A. Papadopoulos, R. Sader, E. Biemer, and H. F. Zeilhofer, Accuracy and Precision of the Three-Dimensional Assessment of the Facial Surface Using a 3-D Laser Scanner, IEEE Transactions on medical imaging, 25 (2006) No. 6. [6] J. D. Martin and T. W. Simpson. Use of Kriging Models to Approximate Deterministic Computer Models. AIAA Journal, 43(4): , [7] Søren N Lophaven, Hans Bruun Nielsen, Jacob Søndergaard. DACE a Matlab Kriging Toolbox. Technical report, Technical University of Denmark, Informatics and Mathematical Modelling,