THE GEOMETRIC HEAT EQUATION AND SURFACE FAIRING

Transcription

1 THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING ANREW WILLIS BROWN UNIVERSITY, IVISION OF ENGINEERING, PROVIENCE, RI 02912, USA 1. INTROUCTION This paper concentrates on analysis and discussion of the heat equation as it pertains to smoothing of geometric shapes and its relationship to the problem of surface fairing. The geometric heat equation distorts a given shape in order to obtain scale-space representation of a shape. This scale-space provides a complete description of the original shape in terms of small to large scale structures. These structures may then be interrogated by recognition algorithms in an attempt to classify the shape. In computer science, researchers have been working on methods for eliminating noise from mesh data obtained via 3 measurements. In this case one wishes to eliminate the noise present in the surface measurements in order to obtain an improved approximation of the true surface. This process is called surface fairing. These two seemingly different problems actually have many common goals. This report will focus on a generic implementation of the geometric heat equation and a particular implementation of surface fairing suggested by Gabriel Taubin [8]. Each method will be explained and their inter-relationships will be made clear. These relationships will be supported via experimental results obtained from 3 measurements on a set of human faces. The input data consisted of 3 mesh data sets obtained via a 3 laser range scanner, specifically the ShapeGrabber product of Vitana Corporation [1]. 2. THE GEOMETRIC HEAT EQUATION In 1986, Mokhtarian et. al. described a way to modify the standard heat equation for heat conductivity as a method of smoothing shapes in order to extract a multi-scale shape representation of a given object [6]. In doing so, one modifies the original boundary of the shape,, by moving the contour in the direction of the normal by some positive factor. multiplied by the signed curvature of the contour. (2.1) Here controls the rate of the geometric diffusion, or equivalently, curvature flow. Both of these phrases describe in different words or terminologies what is happening to the boundary. The phrase geometric diffusion is derived from the terminology of heat conduction and indicates that the curvature at each point along the boundary is being dissipated among it s neighbors in the same sense that heat dissipates through a conductive substrate by exchanging energy among its neighbors. The phrase curvature flow is derived from the terminology of differential geometry. In this case, the curvature function is defining the extent of motion or flow which the contour experiences in the direction of the normal vector field. In practice, the contour is represented approximately by a set of linear segments specified by ordered points. This provides a polygonal approximation of the actual 2 shape. From this we compute and via standard numerical methods as applied to equations provided in textbooks on Classical ifferential Geometry such as Struik s [7]. Of note here is the fact that we are using methods of differential geometry to represent the boundary. This alternative representation of the shape makes computation somewhat more difficult. Since each value is differential with respect to the boundary moving frame, they depend upon local regions of the selected parametrization of the curve. This makes the numerical solutions of the PE more difficult to compute since it implies that the grid is non-uniform for each iteration and also may have varying domains for any two sequential timesteps. Hence, constant care must be taken to preserve the spatial structure of the curve. As time passes, the geometric heat equation causes the curvature signal of the boundary, parametrized according to arc length,, to converge to a constant. This implies that the boundary will converge to a circle, and for large enough, this circle eventually reduces to a point 1. In the same way, the standard 1-dimensional PE for heat conduction (2.2) decreases the curvature at each point of the signal with increasing time. (2.2) "!#! $ % " & Hence we observe the same phenomenon with as observed with. However, since the actual values of the heat signal correspond to the curvature signature of the boundary, the net effect of this heat& conduction results in modification of the shape boundary. Hence, by using the heat equation on a differential 1 representation of shape, we have combined classical physics and differential geometry to generate a method for evolving shape contours. 1 In 3 this phenomenon is referred to as shrinkage and is considered to be an undesirable effect. 1

2 !! % $ THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 2 I = 0 I = I = I = I = I = FIGURE 2.1. The Geometric Heat Equation : parabolic curves are superimposed on evolved surfaces for dataset Agnes.mat. Here denotes the number of integration steps necessary to compute the solution. This method also extends to 3 surfaces in Monge form, i.e. of the form %. Since this is a function graph, we may apply the standard methods of elementary differential geometry to obtain representations for curvature. For the face data, we had the benefit of a uniform spatial grid in, which allowed for direct computation of the corresponding form of the geometric heat equation: % Here, the height function at a point % %! % %! % %! % % is modified proportional to the mean curvature. Since the RHS of the equation above differs from the classical definition of mean curvature by only % % [2]. This has the implication that the above equation is equivalent to the exact geometric heat equation for a Monge patch with a non-constant dissipation term! "#. Consequently, application of this PE will smooth the surface according to the value of the mean curvature using a non-uniform timestep scheme. This PE was implemented and executed with the initial condition % being several examples of face surfaces. The results of these experiments over time is provided in Fig. (2.1). As demonstrated in [5], visualization of surface evolution can be enhanced by tracking the evolution of specific lines of curvature on the face. In particular, parabolic lines of maximum curvature and minimum curvature on the face surface are indicated on each of the plots. These correspond to surface regions where one of the two principal directions %'&! % %!#! " %

3 THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 3 I = 6000 I = I = I = I = I = FIGURE 2.2. The Geometric Heat Equation : parabolic curves are superimposed on evolved surfaces for dataset Agnes.mat. Here denotes the number of integration steps necessary to compute the solution. or % is close to zero. Initially, we attempted to compute parabolic lines with respect to the function graph % This may be accomplished by looking for points on the surface which satisfy the relation!#!!. However, in many instances the face was not oriented such that the parabolic lines would lie along areas of importance. Hence, the only meaningful parabolic curves could be extracted via coordinate-free methods using %'&! or %. One can see that due to the necessarily small timestep required by using the forward Euler integration method [4], the geometric heat equation evolves the surface very slowly. For this reason, additional results are provided in Fig. (2.2) showing the extension of the surface evolution up to integration iterations. One could attempt to align the face such that the face is more orthogonal to the plane which should improve the results obtained via the function graph method of computing parabolic lines. Additionally, there are two possible ways of enforcing boundary conditions. The experiments below show results obtained by enforcing vonneumann style boundary conditions, i.e.! for all points on the boundary of the domain. It may be interesting to experiment with irichlet style boundary conditions, i.e. % at all boundary points for some constant. 3. SURFACE FAIRING The problem of surface fairing is somewhat different than that discussed in 2. Here, the object is to eliminate unwanted noise or geometry from a surface which may be a-priori known or unknown. In the case of a known surface, one may wish to amplify specific aspects of the surface while attenuating others for the various purposes. In general, surface fairing is used as a pre-processing step prior to computation or extraction of higher order shape statistics. In these cases, the fairing step allows for more accurate, and consequently more stable, numerical computations based on the surface mesh. Examples of such computations are extraction of surface normals, principal curvatures, and geodesic curves along the surface.

4 ', % 2 = % THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 4 The critical difference between the concepts outlined in 2 and those of surface fairing is that we are attempting to extract a more accurate estimate of the true underlying geometry given a surface mesh. The surface mesh is an approximate description of the true surface via a connected set of 3 points,. Each 3 point or vertex of the mesh corresponds to an actual measurement of the true 3 surface and the connections between the 3 points form planar surface patch approximations of the true surface between vertices. These points may be distributed in a non-uniform manner about the surface and the only requirement on the graph is that it be fully connected (i..e each point must have at least 1 connecting edge to another neighboring point ). The problem of surface fairing has a compelling relationship with the geometric heat equation from 2. For surface fairing methods, the de-facto standard method is called Laplacian smoothing. This method is discussed in Taubin s paper [9] and is considered insufficient due to the shrinkage effect. In order to understand the term Laplacian smoothing we must first define the Laplacian operator and understand some of it s properties and specifically how this operator is related to the solution of the heat equation The Laplacian Operator and the Heat Equation in 1. We start by examining the standard heat equation as discussed in 2. The RHS of this equation may be re-written in terms of the Laplacian operator: "!#!. The corresponding finite difference expression for the Laplacian operator applied to the function at point, time is given by!!!!. This can be computed simply by defining the discrete Laplacian kernel filter,! " $#, and performing convolution: By observing that the filter! "!!! "& is clearly a high-pass filter, it is initially difficult to understand why the Laplacian operator would be associated with any smoothing effect. Yet, by examining the numerical solution of the heat equation we are finally able to establish a connection between the Laplacian operator and smoothing. In order to solve the 1 heat equation, we may integrate the PE using a forward Euler finite difference equation, where all known terms are collected to the RHS of the equation (here ' denotes the timestep). (3.1) )( '* Using the identity matrix and the! " operators, we may re-express (3.1) as (3.2) below. )( (3.2) '+! " & Finally we see that the solution to our partial differential equation now appears numerically the same as filtering. Further, one may simply add the two filters '+! " 10 #-, ' /.. This kernel corresponds to a scaled version of the well known Gaussian kernel 0 for some values of. This is expected since the Gaussian kernel is a Green s function of the heat equation. As ' the diffusion process is accelerated, or equivalently, solutions are computed with larger time steps. Consequently, the solution to this system is obtained by smoothing the signal via convolution with some specific realization of a scaled Gaussian filter. For this reason, the Laplacian operator, which is essentially a second derivative operator has the unlikely and rather inappropriate association with the smoothing effect of Gaussian filtering. This is commonly referred to as Laplacian smoothing, or equivalently, Gaussian smoothing The Shrinkage Effect. The shrinkage effect is analogous to the phenomenon observed in the geometric heat equation where the boundary eventually converges to the shape s center of mass. In the case of explicit integration, at each iteration we applying an averaging filter. which causes each of the grid points move in the direction of their statistical average. It is interesting to note that without external forces applied (i.e. boundary conditions are not forced) 687:9<; this system will eventually converge to the average of all the points as time goes to is given by (3.3). infinity, i.e. for 32, the solution on the domain 4 5 (3.3) 5 For the curvature signature of the boundary, this causes the shape to converge approximately to a circle which then shrinks in size until it finally converges to to the shape s center of mass. This analogy continues to hold true in 3 and the resulting effect of monotonically decreasing perimeter (2) or area (3) is referred to as the shrinkage effect The Laplacian Operator and the Geometric Heat Equation in 2. For the 2 heat equation, this method may be trivially extended via the kernel (3.4) which is often used in image processing applications [10]. (3.4)! " BAC However, when we attempt to apply the Laplacian operator in the case of the 2 geometric heat equation, we encounter a problem which entirely relates to the form of the data. As mentioned in 2 we simply wish to apply the 1 heat equation to the curvature signature of the 6 => &? %

5 0 THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 5 boundary,. This incurrs several numerical nuisances for computation which add to the complexity of computing a stable solution quickly. Specifically, at every time step one must re-compute the normal field and curvature for every point on the boundary. However, if we apply an approximate geometric Gaussian filter to the spatial location of the points, we may compute an approximate solution without the need of additional estimates of curvature and arc length. This is a huge computational advantage for both speed and stability of the computed solution while sacrificing accuracy. Taubin s method of surface fairing suggests one possibility for this filter :. '... The surface may then be smoothed by applying this filter to the boundary (2) or surface (3). For the sake of simplicity, the smoothing algorithm given applies to the smoothing of 2 contours. This algorithm is then modified trivially for direct use on 3 surfaces. # enote the vertices of the boundary curve,, as the ordered set of 2 points. For any given point, the neighborhood of the point is defined as all points which share a common edge with. In this case, since the points are ordered, the neighborhood of ". In 2, the fairing algorithm works as follows: (1) For each point compute the midpoint of the segment connecting the neighborhood as shown in (3.5). (3.5) " (2) Compute the weighting function for the point. This is an arbitrary function which defines the magnitude of the curvature flow at the point. However, the weighting function should have unit gain, i.e., and operate solely upon those points within the neighborhood of. In his paper, Taubin suggests using of. (3) Move in the direction of by a factor of '. (4) Move 0 backwards along the same trajectory by a factor of., where is the number of points in the neighborhood Steps 1-3 above describe the process of standard Laplacian smoothing or, given an appropriate weighting function, the geometric heat equation from 2. Specifically, we have equivalence when the weighting function from Step 2 is an estimate of the curvature such as, where ( ( is the change in the exterior angle of the tangent and is the change in arc length. Step 4 is an attempt to reduce shrinkage. In the problem of surface fairing, one wishes to preserve the volume of the object. Hence, an additional step is necessary after applying the approximate geometric heat equation. This step essentially expands the smoothed surface in the direction of the original surface The 3 Extension : Laplacian Smoothing. The extension of the above algorithm to 3 meshes is trivial. In the case of a surface mesh, instead of computing the midpoint of Step 1, one computes the centroid of all the points in the neighborhood. Step 2 is identical, save for the fact that the neighborhood may now consist of many 3 points, all of which share a common edge with the point of interest. Steps 3 and 4 also generalize directly, where the vectors, and are now 3-dimensional vectors. Since the emergence of this method in 1995, there have been many researchers which have suggested a wide range of possible weighting functions for step 2 of the algorithm. Many of these functions attempt to approximate surface curvature in some way using various combinations of features obtained in the neighborhood of the vertex such as edge lengths, triangle areas or interior / exterior angles of triangles. A list of several of the most important functions is provided in [9]. Unfortunately, this method as defined in Taubin s paper will always either converge to a point, the centroid of the data, or diverge to infinite size depending upon the relative values of ' and. This effect is entirely analogous to the effects of the heat equation summarized by equation (3.3). Later work by esbrun et. al. at Caltech [3] demonstrated an implicit integration method (backward Euler), for solving (3.1) which is unconditionally stable. However, if one wishes to maintain a shape of constant volume the method becomes non-linear and leads to some considerable computational complexity. However, the fact that Taubin s algorithm may be computed quickly and easily for general 3 surfaces of arbitrary topology makes his method extremely powerful. Fig. (3.1) shows results obtained by applying Taubin s surface fairing method to a 3 face dataset. The six figures show how the surface evolves for the weighting function mentioned in Step 2. It is clear that the approximate smoothing evolves the surface in a distinctly different manner. However, the smoothing operation appears to proceed at a much faster rate than observed with the geometric heat equation. It is interesting to note that this averaging process allows for more uniform dissipation due to the definition of the weighting function. For example, in Figs (2.1) and (2.2) the eyes of the surface model are surrounded by very well defined parabolic curves which are almost circular in shape. These shapes remain almost unchanged for the entire duration of the surface evolution. This may be explained by the fact that by definition, the mean curvature at a parabolic point is equal to one of the principal curvatures alone. If this curvature is small, the surface evolution in these regions will proceed extremely slowly. Contrarywise, the surface fairing algorithm does not respect the mean curvature in computing the magnitude of the normal flow. Here, the magnitude of the curvature flow is given roughly by average exterior angle of the connected edges to the given vertex. Hence, one can see that the corresponding parabolic curves quickly evolve as the eyes are smoothed. The lines of curvature seem to combine and split as the surface evolves in both cases destroying some lines of curvature and creating new ones in other instances as saddle points are formed between regions which were previously well separated by non-parabolic points. One example of this can be seen in Fig (3.2) for the surface evolutions and. Fig. (3.2) shows the continuation of Taubin s

6 THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 6 I = I = I = I = I = I = FIGURE 3.1. Smoothing parameters are ' Surface Fairing : parabolic curves are superimposed on the evolved surfaces for the dataset Agnes.mat., denotes the number of integration steps. and surface method after many iterations. For these results, stability was maintained via a surface area normalization step which followed the standard algorithm. 4. CONCLUSIONS Although the objectives of the geometric heat equation and the surface fairing algorithms are somewhat different, it is clear that the underlying theory is very much the same. In essence, the two methods described here are exactly the same given a specific conditions on the surface fairing algorithm of 3. Additionally, results were provided which demonstrated the differences between the two methods. Here, we demonstrated that the approximate method of surface fairing does not as accurately compute curvature flows over the sample mesh. This results in a different surface evolution which can be clearly tracked by the observation of parabolic ridges on the surface. REFERENCES [1] Shapegrabber inc. [2] Wolfram research web site, 2. [3] M. esbrun, M. Meyer, P. Schroder, and A. Barr. Implicit fairing of irregular meshes using diffusion and curvature flow. In SIGGRAPH, [4] B. Gustafsson, H. O. Kriess, and J. Oliger. Time ependent Problems and ifference Methods. John Wiley and Sons Inc., Canada, [5] Peter Hallinan, Gaile Gordon, A.L. Yuille, P. Giblin, and. Mumford. Two-and-Three imensional Patterns of the Face. A.K. Peters Ltd., Natick, MA., USA, [6] F. Mokhtarian and A. K. Mackworth. Scale-based description and recognition of planar curves and two-dimensional shapes. IEEE Trans. on Pattern Anal. Machine Intell., 8(1):34 43, [7] irk J. Struik. Lectures on Classical ifferential Geometry, chapter Elementary Theory of Surfaces. over, NY, 19. [8] Gabriel Taubin. A signal processing approach to fair surface design. In SIGGRAPH, [9] Gabriel Taubin. Geometric signal processing on polygonal meshes. In EUROGRAPHICS, 0. [10] E. Trucco and A. Verri. Introductory Techniques for 3- Computer Vision, chapter Appendix A.2. Prentice Hall, New Jersey, USA, 1998.

7 THE GEOMETRIC HEAT EQUATION AN SURFACE FAIRING 7 I = 400 I = I = I = I = I = FIGURE 3.2. Surface Fairing : parabolic curves are superimposed on the evolved surfaces for the dataset Agnes.mat., denotes the number of integration steps. Smoothing parameters are ' and