Geodesics in heat: A new approach to computing distance

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1 Geodesics in heat: A new approach to computing distance based on heat flow Diana Papyan Faculty of Informatics - Technische Universität München Abstract In this report we are going to introduce new method of computing geodesic distance from one point to the specified subset of a given domain, using heat flow. The heat method is simple to implement in comparison with other methods, as it is based on solving standard linear elliptic problems. The other great advantage of this method is that it is possible to apply on any type of domain (point clouds, triangular meshes, polygonal meshes, etc.) since it is using only standard differential operators. At the end of the report we will present a graph which will provide you with information about efficiency of two different methods (Heat method, Fast Marching) of computing geodesic distance. There it is clearly visible that out of those two, heat method is faster then the others and in the same place mean error in most of cases is better then in fast marching. 1 Introduction There exists already several methods to compute geodesics distance, but they are either too slow, or hard to implement or not efficient enough. The heat method can be solution for all of this problems. When applying heat on one point of shape it will spread all over the surface after some time. To describe that heat flow lets take a hot needle and attach it to a point on bunny s surface. The function which will show the amount of heat reaching from source point (hot needle, x) to destination point (y) after time (t) is called heat kernel and is noted as k t,x (y). One of the most famous methods to count geodesic distance using heat kernel is Varadhan s formula 1. Basically the formula uses only pointwise transformation of heat kernel to get distance between any two points. φ(x, y) = lim 4t log k t,x (y) (1) t 0 In this equation (x) represents the source Figure 1: Geodesic distance from heat source to point in Riemannian manifold, (y) - destination the surface point in Riemannian manifold, (t) - the time passed after which we are measuring the amount of heat reaching destination point. The reason why we can t use Varadhan s formula in every case is that it is highly dependent on heat kernel. Any approximation of it will bring to quite big error in distance. You can see that in the Figure 1. Heat method is avoiding this problem by using wider range of inputs. The only requirement for using function as input in heat method is that functions gradient should be parallel to geodesics. Except this, heat method has two major advantages to already existing algorithms. First one is that you can use it on any shape (point cloud, triangular mesh, etc). The other advantage is that it is possible to prefactor parts of solution and use it later, as heat method includes only sparse linear systems. 1

2 Figure 2: Top left is exact reconstruction of heat kernel and under it is counted geodesic distance using that heat kernel. Given an exact reconstruction of the heat kernel (top left) Varadhan s formula can be used to recover geodesic distance (bottom left) but fails in the presence of approximation or numerical error (middle, right), as shown here for a point source in 1D. The robustness of the heat method stems from the fact that it depends only on the direction of the gradient. 2 Existing methods The base approach for solving distance problem is using eikonal equation 2 with boundary condition φ γ = 0 where γ is the subset of the domain. φ(x) = 1 (2) φ is showing the distance from the point where heat was applied to the x point. denotes the gradient of φ function. But this equation is non-linear partially differential hyperbolic equation. Solving it can be really challenging. Two best known methods to solve this equation are Fast Marching and Fast Sweeping. Though Fast Sweeping is easier to implement but it is slower in comparison with Fast Marching. [2] But still both of them are slow and not efficient as they are not reusing any computation, so for each subset of domain you need to compute everything from scratch. Also this method is almost not possible to make parallel as it is using priority queue. So even if you have a big cluster the program still will run on only one processor, which is a huge problem. The heat method is mostly close to Rangarajan and Gurumoorthy method 3, which is using Schrödinger wave equation formalism to solve Eikonal equation. φ = log ψ (3) In equation is stated for Planck s constant. But the problem of this method is that you need to use Fast Fourier Transformation to solve it, which is automatically restricting you to use regular grids only. And also there is no any information on how should be chosen for getting best results. 3 The Heat Method From here on will be stated as negative, semi definite Laplace-Beltrami operator acting on differentiable real valued functions over the Riemannian manifold(m). 2

3 Figure 3: Heat method brief outline: u heat is spreading over domain. u is the gradient of heat function which is point to the the heat source. X is a unit vector field whose direction is opposite to gradients direction. Function φ is final recovered distance. Now lets discuss heat method. Heat Method consists of three main steps which are 1) Solve heat equation u = u 2) Normalize gradient X = u u 3) Solve Poisson equation φ = X φ approximates geodesic distance. It is most precise when time (t) is approaching 0 in heat equation. Also when solving the Poisson equation you re not getting unique answer. All the values should be later shifted so that the smallest number will become 0. So basically in heat method we don t have problem of not having precise value of heat kernel as we did in Varadhans formula, as we only need correct direction of gradient. Also we can see from Eikonal equation that the length of true distance function has always length of 1, so no need to have precise value of gradients magnitude at all. So we need to compute the normalized gradient field and find the closest scalar potential by minimizing M φ X 2. Step descriptions can be seen in Figure Time Discretization For being able to solve first step (heat equation) we need to discretize it in time. For doing that we can use single backward Euler step for some fixed time t. So we need to solve 4 linear equation over the whole domain M. (id t )u t = u 0 (4) Here id is stated for identity operator. Now lets consider elliptic boundary value problem (id t )υ t = 0 on M\γ υ t = 1 on γ υ t is equal to u t up to a multiplicative constant for a point source. υ t also has a close relationship with distance, which is 6 t lim t 0 2 log υ t = φ (6) outside of cut locus. Above mentioned relationship shows that transformation applied to υ t preserves the direction of the gradient, and this means that prerequisites are satisfied for step 2 and 3. (5) 3

4 3.2 Spatial Discretization Heat method can be applied to any type of shape with discrete gradient ( ), divergence ( ) and Laplace operator ( ). Underneath is it shown how it can be done for several types of shapes, but it is not limited to those types, you can apply it to other meshes too. The computational cost of the heat method mostly depends on the intrinsic dimension n of M. The other methods like fast marching requires a grid of the same dimension m as the ambient space. This is a great advantage of heat method when using it in machine learning projects where m can be significantly larger than n Simplicial Meshes Let u R V specify a piecewise linear function on a triangulated surface. Standart discretization of Laplacian at the vertex i is defined by 7 (Lu) i = 1 (cot α i,j + cot β i,j )(u j u i ) (7) 2A i where A i is defined by one third the area of all triangles incident on vertex i, the sum is counted on all neighboring vertices j, and α i,j, β i,j are the angles opposing the corresponding edge. To get feeling of what was said you can see figure It can also be rewritten as matrix operation 8 j L = A 1 L C (8) where A, L C R V V. A is a diagonal matrix containing the vertex areas, L C is named as cotan operator and is the sum part from the equation 7. Heat flow can be computed using equation 9 (A tl C )u = δ γ (9) where δ is a Kronecker delta over γ (Kronecker delta gives the integrated value of a Dirac delta). The gradient in following triangle is expressed as equation 10 u = 1 u i (N e i ) (10) 2A f i where A f is the area of the face, N is its unit normal, e i is the i th edge vector (oriented counterclockwise), and u i is the value of u at the opposing vertex. The divergence associated with i vertex of triangle can be computed with equation X = 1 cot θ 1 (e 1 X j ) + cot θ 2 (e 2 X j ) (11) 2 j where the sum is taken over incident triangles j each with a vector X j. e 1 and e 2 are the two edge vectors of triangle j containing i vertex, and θ 1, θ 2 are the opposing angles. Now lets take b R V, and lets assume that it is a vector of (integrated) divergences of the normalized vector field X. In that case we can compute final distance by solving Poisson equation which is the last step of heat method 12 L C φ = b (12) These solution can be generalized to higher dimensional meshes(tetrahedral meshes) using discrete operators. 4

5 Diana Papyan Figure 4: In the examples it is visible that heat method is working quite good on point clouds. Even if there is some missing points like in left image. And both images don t have any points connecting information Point Clouds Now we need to solve the heat equation for a discrete point sample P Rn of M with no connection information. For a discrete point sample P Rn of M with no connectivity information, there is a recent method for computing Laplacian which is A 1 ν LP C, where Aν is a diagonal matrix of Voronoi areas and LP C is symmetric positive semidefinite. To solve step 2 of heat method (computing vector field) we need to declare new function u : p R which would be height function over approximate tangent planes Tp at each point of point cloud and then count the gradient of a weighted least squares (WLS) approximation of u. Computing tangent planes can be done in different ways, but it is suggested to use moving least squares (MLS) approximation as it is simple to implement. Step 3 of heat method (getting real distance) is solved by positive semi-definite Poisson equation 13. LP C φ = DT Aν X 3.3 (13) Choosing Time Step Accuracy of heat method is highly dependent on time step t which is chosen by person. To go back to 6 equation, we can see that in smooth setting, choosing smaller value of time step would bring to better accuracy in geodesic distance. But the same time lets look into equation 4. The solution for this equation is a function of the combinatorial distance, independent of how we discretize the Laplacian. So if we have a fixed mesh making the t value smaller not always brings to best results. There is an option to over-go that problem by refining the mesh and then decreasing t accordingly. But there is a reason for not doing that, because for getting smoothed approximation of geodesic distance we need to pick large values of t. That is why we will try to find an optimal t which is not too big and not to small. In that case we will get smooth approximation and good accuracy together. Coming up with optimal expression for t is quite difficult because of complexity of analyses involving cut lotus. But practice shows that t = mh2 where h is the mean spacing between adjacent nodes and m is just a positive constant, works quite well in most cases. This estimate is motivated by the fact that h2 is invariant to scaling and refinement. Some experiments were 5

6 made to check which is the smallest possible m which can recover real distance. You can see that in 3.3 and 3.3 figures. Figure 5: The experiment is done with h = 1. The solution approaches the combinatorial distance as t goes to zero So the smallest m which recovers distance with good accuracy is when m = 1. So in all upcoming examples we use 14 expression for time step. t = h 2 (14) This equation is not bringing to good results only if we need to get really smooth approximation of geodesic distance. 3.4 Smoothed Distance Geodesic distance is not smooth at cut locuss points, which are for example points from where there is no unique shortest path to the source. Non-smoothness can bring to numerical difficulty in cases when there will be need to compute derivatives of the distance function, or may simply be undesirable aesthetically. Figure 6: Horizontal axis shows different values of m, vertical axis shows mean error. Best results are reaching when m = 1 There are some other methods to get really smooth distances, which are for example diffusion distance or biharmonic distance, but they are far away from being close to real geodesic distance as indicated by uneven spacing of isolines. You can check 3.4 figures middle part for proof. Figure 7: When the heat is applied from front side of shape it results in non-smooth edges on the opposite side (left). Going to right side those are images with longer t duration. The best smoothed approximations of geodesic distance is reached when t is biggest (right). 6

7 Diana Papyan Figure 8: Top: heat method smoothed geodesic approximation (left) and biharmonic distance (center) both mitigate sharp edges, yet isoline spacing of the biharmonic distance can vary dramatically. Bottom: biharmonic distance tends to exhibit elliptical level lines near the heat source, while heat method smoothed distance maintains isotropic circular profiles as seen in the exact distance (right). But we can reach smooth approximation with heat method too. The trick is to choose big value for t. The results can be seen in 3.4 figure. The computational cost remains the same. As can be seen from figure isolines are evenly spaced for any value of t. That comes from normalization step of heat method (step 2). When using equation t = mh2 meaningful values of m is varying from 1 till 106. When choosing bigger values little visible change can be noticed. 4 Accuracy Tests were done to compare accuracy of fast marching and heat method. In figure 3.4 you can see the visual difference between exact geodesic distance, distance computed with heat method and distance computed by fast marching method. More information about accuracy can 7 Figure 9: Left: exact geodesic distance. Using default parameters, the heat method (middle) and fast marching (right) both produce results of comparable accuracy, here within less than 1% of the exact distance

8 be viewed in 4 figure. References [1] Keenan Crane, Clarisse Weischedel, and Max Wardetzky. Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. [2] Pierre A. Gremaud and Christopher M. Kuster. Computational study of fast methods for the eikonal equation. 8