Implementation of Integral based Digital Curvature Estimators in DGtal

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1 Implementation of Integral based Digital Curvature Estimators in DGtal David Coeurjolly 1, Jacques-Olivier Lacaud 2, Jérémy Levallois 1,2 1 Université de Lyon, CNRS INSA-Lyon, LIRIS, UMR5205, F-69621, France. 2 Université de Savoie, CNRS LAMA, UMR5127, F-73776, France. david.coeurjolly@liris.cnrs.fr, jacques-olivier.lacaud@univ-savoie.fr, jeremy.levallois@liris.cnrs.fr Abstract In many geometry processing applications, differential geometric quantities estimation suc as curvature or normal vector field is an essential step.in[1],weavedefined curvature estimators on digital sape boundaries based on Integral Invariants. In tis paper, we focus on implementation details of tese estimators. Keywords: Digital geometry, curvature estimation, integral invariants 1 Introduction In many sape processing applications, te estimation of differential quantities on te sape boundary suc as normal vectors, curvatures or principal directions are crucial. Wen evaluating a differential estimator on discrete or digital data, because of digitization approximation, we need a way to matematically link te estimated quantity to te expect Euclidean one. In Digital Geometry, we usually consider multigrid convergence principles: wen te sape is digitized on a grid wit resolution tending to zero, te estimated quantity sould converge to te expected one [2]. In [1], we ave proposed digital versions of integral invariant estimators for wic convergence results can be obtained wen te kernel size tends to zero. In tis paper, we first remind te general principle of [1]. Ten we focus on implementation details of tese estimators in DGtal library, and finally we sow some results. 1.1 Principle In geometry processing, interesting matematical tools ave been developed to design differential estimators on smoot surfaces based on integral invariants. Te principle is simple: we move a convolution kernel along te sape surface and we compute integrals on te intersection between te sape and te convolution kernel, as follow in dimension 3: V r (x) def = B r(x) χ(p)dp (1) were B r (x) is te Euclidean ball of radius r, centeredatx and χ(p) te caracteristic function of X. In dimension 2, we simply denote A r (x) suc quantity (represented in orange color on Fig. 1). Tis work as been mainly funded by DigitalSnow ANR-11-BS researc grants 27

2 Br(x) X Br(x) x x x x x π X (ˆx) ˆx X X X Figure 1: Integral invariant computation in dimension 2 (left) and 3(middle), and notations in dimension 2 (rigt) Autors of [4][5] ave demonstrated tat some integral quantities provide interesting curvature information wen te kernel size tends to zero. Indeed, tanks to Taylor expansion at x of te surface X approximated by a parametric function y = f(x) in 2d and z = f(x, y) in 3d and wit a fixed radius r, we obtain convergent local curvature estimators κ r (X, x) and H r (X, x) of quantities κ(x, x) and H(X, x) respectively: κ r (X, x) def = 3π 2r 3A r(x) r 3, Hr (X, x) def = 8 3r 4V r(x) πr 4 (2) κ r (X, x) =κ(x, x)+o(r), Hr (X, x) =H(X, x)+o(r) (3) were κ r (X, x) is te 2d curvature of X at x and H r (X, x) is te 3d mean curvature of X at x. In [1], we were interested in studying te beavior of integral invariants in digital geometry using a multigrid framework wit digitization on grids wit grid step tending to zero. We define a digital version of Eq. (2) and (3) estimators. 0 <<r, ˆκ r (Z, x, ) def = 3π 2r 3 Area(B r/ ( 1 x) Z, ) r 3. (4) were ˆκ r is an integral digital curvature estimator of a digital sape Z Z 2 at point x R 2 and step. B r/ ( 1 x) Z, ) means te intersection between Z and a Ball B of radius r digitized by centered in x. In te same way, we ave in 3d: 0 <<r, Ĥ r (Z,x,) def = 8 3r 4 Vol(B r/ ( 1 x) Z,) πr 4. (5) were Ĥr is an integral digital mean curvature estimator of a digital sape Z Z 3 at point x R 3 and step. We ave demonstrated tat Eq. (4) and (5) are multigrid convergent. Wit ypotesis about te sape geometry and te radius of te convolution kernel, we guarantee a teoretical convergence speed in O( 1 3 ), confirmed wit experimental results. We ave also discussed about an integral digital Gaussian curvature and principal curvature directions estimators obtained by digital approximation of te covariance matrices on Eq. (1) integral. Convergence results rely on te fact tat digital moments converge in te same manner as volumes [3]. Note tat as oter convolution estimators, results of our algoritm depend on te size of te convolution kernel. 2 Implementation in DGtal We ave implemented integral invariant curvature estimators in DGtal library. DGtal is an opensource C++ library tat provides geometry structures, algoritms and tools for digital data (ttp://libdgtal.org). Project DGtal gaters in a unified setting many data structures and algoritms of digital geometry and related fields (digital topology, image processing, discrete geometry, aritmetic). For example, DGtal provides us parametric or implicit multigrid sape construction in dimension 2 and 3, as well as object loaders. Furtermore, we benefit from available implementations of existing 2d curvature estimators. It allows us to make comparisons easy. 28

3 In our framework, objects are embedded into a Kalimsky space topological model. Tis topological structure gives us cells from 0 to n dimension (in 3d, cells are called pointels, linels, pixels and voxels for respectively cells of dimension 0, 1, 2 and 3). We call surfels cells of n 1 dimension and spel cells of n dimension. Kalimsky cells are defined into a Kalimsky Domain (cubical grid). For more information about te topology in DGtal, please refer to documentation (ttp://libdgtal.org/doc/stable/packagetopology.tml) 2.1 User interface Te user part is rater simple by using only IntegralInvariantMeanCurvatureEstimator or IntegralInvariantGaussianCurvatureEstimator. Tese classes are parametrized by a Kalimsky space of a digital sape and a functor wo return a value for a given Kalimsky cell. Indeed, te χ(p) caracteristic function is defined as a functor (ere wit values in {0, 1} for eac spel of te Kalimsky space). At initialization (init()), we specify te current grid step, and an Euclidean radius r e in order to construct te convolution kernel. As described in Sect. 3, tis parameter r e determines te level of feature detected by te estimator. For te evaluation (eval()), te user as two possibilities: evaluate te curvature at a specific surfel of te digitized sape surface, or at a range of surfels. Tis coice can be important because optimizations are available wit te second option (see in Sect. 2.2). For te first one, te estimator return a curvature value at te given surfel. If you coose te second possibility, te estimator will try to optimize computations by using previous results tanks to displacement kernel masks. But it requires to set a range of 0-adjacent surfels. If te range of surfels does not follow tat rule, no optimization is performed. 2.2 Implementation details Te following paragrap gives details on IntegralInvariantMeanCurvatureEstimator and IntegralInvariantGaussianCurvatureEstimator classes. To sum up, we need a convolution kernel wic will be integrated along a digitized sape boundary. At initialization, we first create a convolution kernel (Ball2D in 2d, Ball3D in 3d) of Euclidean radius r e and digitized at te grid step. DigitalSurfaceConvolver is in carge of centering te convolution kernel on te surfel on te digitized sape border. Its objective is to compute B r/ ( 1 x) Z and B r/( 1 x) Z from Eq. (4) and (5). At evaluation, it will place te convolution kernel in order to lie is center wit te cell to estimate. After, it will iterate on all spels of te kernel, and compute te integral: V (x) = spel B r/ ( 1 x) Z f(x)g(s x) (6) were x is te current spel, f is te sape functor (set by user), g is te kernel functor (return 1 on all spels from te kernel). IntegralInvariantMeanCurvatureEstimator uses tis result to get Area(B r/ ( 1 x) Z, ) in 2d and Vol(B r/ ( 1 x) Z,) in 3d. Wit Eq. (4) and (5), it finally returns curvature quantities. Using tis approac, we obtain a computation cost in O((r/) d )per surface element. For IntegralInvariantGaussianCurvatureEstimator, te initialization is te same but at evaluation DigitalSurfaceConvolver computes a covariance matrix of B r/ ( 1 x) Z. Eigenvectors and eigenvalues analysis of tis covariance matrix elps us to extract principal curvatures information. We can easily obtain mean or Gaussian curvature from tem. Wen we move te convolution kernel to 0-adjacent cells, we see tat only few cells need to be updated from te previous convolution. As seen in te top left illustration of Fig. 2, only te green and te red part are interesting, te grey part is te same from te previous result. Tis effect can produce a lot of computations, so we only need to remove te green part, and add te red one (by additivity of te convolution). 29

4 r/ ( 1 x + δ) Z, ) = Area(B r/ ( 1 x) Z, ) Area(B 1 Area(B1r/ ( x) Z, ) 1 + Area(B2r/ ( x + δ) Z, ). were δ is a translation vector in 0-adjacency, B1r/ is te mask to remove, and B2r/ te mask to add (bot depending to δ). We pre-compute at initialization of estimators a set a displacement masks from te full convolution kernel (see Fig. 2 at rigt for an example in 2d). At evaluation, we use te full kernel for te first computation. If te next surfel is adjacent to te previous one, we use a pair of masks (depending to te sift) to remove/add interested quantities to te last. In te worst case, all surfels set by user to te estimator are not adjacent and it will compute te curvature always wit te full convolution kernel, wic means no optimizations. Te computation cost per surface element can be reduced from O((r/)d ) (size of te complete kernel) to O((r/)d 1 ), and te extra-cost of pre-computing displacement masks is meaningless: for example, wit a full 2d kernel support size of 91893, te optimization built 8 supplementary masks of size 450. Figure 2: Illustration of te optimization wit partial masks of a 2d ball for a given. On te top left: Illustration of a displacement of a full kernel from x to x + δ, on te bottom left: full 2d kernel support, on te rigt: 0-adjacent 2d partial displacement masks of full kernel support. 3 Results For all results presented in tis section, yellow color is te igest curvature value, blue is te lowest curvature value, and red means in-between max and min values. In Fig. 3, we can see in order mean and Gaussian curvature mapped in a blobby cube1. We can also extract principal curvature information. As mentioned before, te user needs to coose a convolution kernel size re, wic will influence te level of details captured (see Fig. 3) and computation time. Euclidean radius re of a convolution kernel determines te level of details of estimated curvature. It is an intuitive beavior, because a smaller convolution kernel will be more influenced by small features, but will ignore large features and dependent to te noise level. A larger convolution kernel will result te opposite effect. 1 Implicit surface is 81x4 + 81y z 4 45x2 45y 2 45z 2 6 = 0. 30

5 Figure 3: Illustration of curvature estimation.from left to rigt: mean curvature and Gaussian curvature (back color means zero curvature surfels), first and second principal curvature directions mapped on a blobby cube on a blobby cube, 2d curvature mapped on aflower2d, 3dGaussian curvature on Al wit a convolution kernel size of 14 and wit a convolution kernel size of 7. References [1] D. Coeurjolly, J.-O. Lacaud, and J. Levallois. Integral based curvature estimators in digital geometry. In Discrete Geometry for Computer Imagery, [2] D. Coeurjolly, J.-O. Lacaud, and T. Roussillon. Digital Geometry Algoritms, Teoretical Foundations and Applications of Computational Imaging, volume 2 of LNCVB, capter Multigrid convergence of discrete geometric estimators, pages Springer, [3] R. Klette and J. Žunić. Multigrid convergence of calculated features in image analysis. Journal of Matematical Imaging and Vision, 13: , [4] H Pottmann, J Wallner, Q Huang, and Y Yang. Integral invariants for robust geometry processing. Computer Aided Geometric Design, 26(1):37 60, [5] H Pottmann, J Wallner, Y Yang, Y Lai, and S Hu. Principal curvatures from te integral invariant viewpoint. Computer Aided Geometric Design, 24(8-9): ,

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