ŠVOČ 2015 Bratislava. 3D point cloud surface reconstruction by using level set methods Rekonštrukcia plôch z mračien bodov pomocou level-set metód

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1 ŠVOČ 015 Bratislava 3D oint cloud surface reconstruction by using level set metods Rekonštrukcia lôc z mračien bodov omocou level-set metód Meno a riezvisko študenta: Škola: Fakulta: Ročník, Program/Odbor štúdia: Vedúci ráce: Katedra: Bc. Balázs Kósa Slovenská tecnická univerzita Stavebná fakulta. ročník,. stueň MPM rof. RNDr. Karol Mikula, DrSc. KMDG máj 015

2 Abstrakt V rámci ráce sme vytvorili matematický model a numerickú metódu na rekonštrukciu lôc z 3D mračien bodov omocou tzv. level-set metódy. Prezentovaná metóda rieši rekonštrukciu lôc výočtom distančnej funkcie k útvaru, ktorý je rerezentovaný mračnom bodov, s oužitím tzv. Fast Sweeing Metódy a riešenia advečnej rovnice s krivostnou časťou, ktorá vytvorí evolúciu očiatočnej odmienky do finálneo stavu. Pre numerickú diskretizáciu sme navrli novú bezodmienečne stabilnú metódu, ktorá využíva semi-imlicitnú co-volume scému re krivostnú časť a imlicitný uwind re advektívnu časť modelu. Metóda bola narogramovaná v jazyku C, a testovaná na rerezentatívnyc ríkladoc a komlexnýc reálnyc dátac. Predkladaná ráca ŠVOČ tvorí odstatnú časť dilomovej ráce autora. Abstract In tis work we created a matematical model and numerical metod for surface reconstruction from 3D oint cloud data, using te level-set metod. Te resented metod solves surface reconstruction by te calculation of te distance function to te sae, reresented by te oint cloud, using te so called Fast Sweeing Metod, and te solution of advection equation wit curvature term, wic creates te evolution of an initial condition to te final state. For te numerical discretization of te model we suggested a novel unconditionally stable metod, in wic te semi-imlicit co-volume sceme is used in curvature art and imlicit uwind sceme in advective art. Te metod was imlemented in te rogramming language C and tested on reresentative examles as well as comlex real data. Te resented work is an essential art of te final tesis of te autor.

3 Contents 1 Introduction 4 Matematical formulation 4 3 Numerical solution Calculation of te distance function Numerical sceme for advection equation wit curvature term Time discretization Satial discretization Calculating te coefficients of te linear system Calculation of te initial condition Numerical results 18 5 Conclusions 1

4 1 Introduction Te aim of our work is to create a reliable numerical metod wic can easily create comuterized 3D models from oint cloud data tat resembles te original object as muc as ossible. Tese tye of data can be obtained by 3D scanning or by otogrammetric metods. Paers as [1, ] ave sown us tat for tese tye of tasks te level-set metod is suitable. We follow basic ideas from tese aers, but we take a different aroac in te solution of te artial differential equation resented ere. In te next section we will fully resent our metod and its numerical discretization and solution as well as results wic we acieved so far. Matematical formulation Te level set metod, wic we are using is based on te solution of te advection equation wit te curvature term u t d u δ u (x, t) Ω [0, T ] ( ) u = 0 u (.1) were v = d is te advective velocity defined by te gradient of te distance function d, te arameter δ [0, 1] before te curvature art determines its influence to te result and Ω is te comutational domain. Tis equation is couled wit omogeneous Neumann boundary conditions and an initial condition wic we will discuss later in tis aer. 3 Numerical solution For te numerical solution of te model created from oint cloud data, denoted by Ω 0 Ω and determined by equation (.1) te following stes ave to be executed. First we ave to calculate te distance function to te oint cloud, ten we ave to find a surface containing Ω 0 wic will be te initial condition for te generation of te final solution of te equation. Te final model will be reresented by an isosurface of te calculated function u (x) wit value

5 3.1 Calculation of te distance function For te calculation of distance function we use te Fast sweeing metod, as introduced in [3], wic solves te Eikonal equation wit boundary conditions wic in our case as te following form d (x) = f (x) x Ω d (x) = 0 x Ω 0 Ω (3.1) were f (x) = 1. For introducing te metod we will use te following notation. x i,j,k will be used for te grid oint of Ω, is te size of te edges of a grid cell and d i,j,k denotes te numerical solution at x i,j,k. Te discretization of (3.1) at interior grid oints is done according to te Godunov uwind difference sceme: [ (di,j,k d x min ) ] [ (di,j,k d y min ) ] [ (di,j,k d z min ) ] = i = 1,..., I 1, j = 1,..., J 1, k = 1,..., K 1 d x min = min (d i,j 1,k, d i,j1,k ) (3.) d y min = min (d i 1,j,k, d i1,j,k ) d z min = min (d i,j,k 1, d i,j,k1 ) (x) x, x > 0 = 0, x 0 At te boundary of Ω we use one sided difference. Te initialization of d(x) is done te following way. For every grid cell wic contains a oint from te oint cloud we calculate te exact distance between te vertexes of te element and te oint and set te values of d i,j,k to te calculated distance. For te values to be correct we ave to ceck if tere is a smaller distance for te vertexes in te neigboring grid cells. Te obtained values will be fixed in te main rocess of te algoritm. Tis way we enforce te boundary condition d (x) = 0 x Ω 0 Ω. At all te oter grid oints a large ositive number is assigned to d i,j,k. After te initialization is done te algoritm continues wit Gauss-Seidel iterations wit alternating sweeing orderings. For eac non fixed grid oint x i,j,k we comute te solution of (3.) from te neigboring values d i,j 1,k, d i,j1,k, d i 1,j,k, d i1,j,k, d i,j,k 1, d i,j,k1 and ten we udate d i,j,k if te solution is smaller tan te current value. For tree dimension we swee te comutational domain wit eigt alternating orderings: 5

6 1. i = 1 : I, j = 1 : J, k = 1 : K. i = I : 1, j = 1 : J, k = 1 : K 3. i = I : 1, j = J : 1, k = 1 : K 4. i = I : 1, j = J : 1, k = K : 1 5. i = I : 1, j = 1 : J, k = K : 1 6. i = 1 : I, j = 1 : J, k = K : 1 (3.3) 7. i = 1 : I, j = J : 1, k = K : 1 8. i = 1 : I, j = J : 1, k = 1 : K Te unique solution, denoted by x, to te equation [ (x a1 ) ] [ (x a ) ] [ (x a3 ) ] = (3.4) were a 1 = d x min, a = d y min, a 3 = d z min can be found as follows. We order a 1, a, a 3 in increasing order. For generality we assume a 1 a a 3. Tere is an integer, 1 3, suc tat x is te unique solution tat satisfies (x a 1 ) (x a ) (x a 3 ) = and a < x < a 1 (3.5) To find x we start wit = 1. If x = a 1 a ten x = x. Oterwise we ave to find te solution of te quadratic equation (x a 1 ) (x a ) = tat satisfies x > a. We always take te maximum of te two solutions as our x. If x a 3 ten x = x. If we still doesn t ave a x wic satisfies all te conditions as te tird ste we comute te solution of te quadratic equation (x a 1 ) (x a ) (x a 3 ) = wic will satisfy (3.5). 3. Numerical sceme for advection equation wit curvature term Now tat te distance function is calculated we roceed wit te discretization of te equation (.1). We will do tis analogically to te discretization used in [4] Time discretization For te time discretization we ave to coose uniform discrete time ste, denoted by τ. We can relace te time derivative in (.1) wit a backward difference. Ten we can formulate our semi-imlicit time discretization te following way: 6

7 Let τ be a fixed number and u 0 te initial surface of our model. Ten at every discrete time t n = nτ, n = 1,..., N we searc for te function u n as te solution of te equation u n u n 1 d u n δ ( ) u n 1 u n = 0 (3.6) τ u n Satial discretization Our model consists of a 3D grid, wic is built of voxels wit cubic sae and an edge size. We will interret te satial discretization of te level set function u as te numerical values u i,j,k at te voxel centers. In order to easily calculate te gradient of te level-set equation u n 1 in every time ste of (3.6) we induct a 3D tetraedral grid into te voxel structure and take a iecewise linear aroximation of u (x) on suc a grid. Tis way we obtain a constant value of te gradient for eac tetraedron, by wic we can construct a simle and clear fully discrete system of equations. Figure 1: Our initial voxel grid cell wit a tetraedral grid cell Te 3D tetraedral finite element grid is created wit te following aroac. Every voxel is divided into six yramid saed element wit base surface given by te voxels walls and vertex by te voxel center. Eac one of tese yramids is joined wit te neigboring yramids wit wom tey ave a mutual base surface. Tese newly formed octaedrons are ten slit into four tetraedrons as seen in Figure 1. In our new grid T te level-set function will be udated only at te centers of te voxels, tey will reresent so called degree of freedom (DF) nodes. For te tetraedral grid we construct a co-volume mes, wic will consist of cells associated only wit DF nodes of T. We denote all neigboring DF nodes q of wit C. 7

8 Te DF nodes q are all connected to te DF node by a mutual edge of four tetraedrons, wic is denoted by σ q wit te lengt q. Eac co-volume is bounded by a lane for every q C wic is erendicular to σ q and is denoted by e q. Te set of tetraedrons wic ave σ q are denoted by ε q. For every T ε q c T q is te area of te intersection of e q and T. N will be te set of tetraedrons tat ave DF node as a vertex. On tis grid u will be a iecewise linear function. Ten we can use te notation u = u (x ), were x denotes te coordinates of DF node. Now tat we ave all te notations wic are needed we can begin te derivation of te satial discretization of (3.6). We will do tis by using te following modified form of te equation: were v = d. u n u n 1 v u n = δ ( ) u n 1 u n τ u n 1 (3.7) AS te first ste we will integrate (3.7) over every co-volume. u n u n 1 dx v u n dx = δ ( ) u n 1 u n dx (3.8) τ u n 1 For te first art of left and side of (3.8) we get te aroximation in te form u n u n 1 dx = m () un u n 1 τ τ (3.9) were m () is a measure in R d of te co-volume. To derive te second art of te left and side we use te following aroac. Tis art can be written in an equivalent form by v u = (uv) u v v u n dx = (u n v) dx u n v dx Using te divergence teorem we get (u n v) dx u n v dx = q C u n q e q v n dσ q C u n e q v n dσ u n v n dσ u n v n dσ = (3.10) were u n q is te value of te level-set function on te surface e q. We substitute e q v n dσ in (3.10) wit v q wic by solving te integral will ave te value v q = qv n. By tis we finally get v u n dx = q C v q ( u n q u n 8 ) (3.11)

9 In te uwind aroac te set C can be divided into C = C in C out, were C in = {q C, v q < 0} wic consists of te inflow boundaries and C out = {q C, v q > 0} consisting of te outflow boundaries. By dividing C we can set te values u n q to u n q q C in side of (3.11) to and to u n if q C out. After tese modification we can rewrite te sum rigt and q C v q ( u n q u n ) = q C in ( ) v q u n q u n q C out ( ) v q u n u n As we see te outflow art will be zero and after we rewrite te inflow art for simler imlementation we get te final form of te second art of te left and side of te equation (3.8) v u n dx = q C min (v q, 0) ( u n q u n ) if (3.1) Now wat remains is te discretization of te rigt and side of (3.8). Again we use te divergence teorem to get δ ( ) u n 1 u n dx = δ u n 1 1 u n u n 1 dσ (3.13) e q C q u n 1 n Te integral art 1 u n dσ and e q u n 1 n u n 1 from (3.13) will be aroximated numerically using iecewise linear reconstruction of u n 1 on te tetraedral grid T, tus we get δ u n 1 q C T ε q c T q 1 u n 1 T M = u n 1 = m (T ) m () T N and te final form of te equation (3.7) will be un q u n q u n 1 T m () un u n 1 min (v q, 0) ( u n q u τ ) n = q C δ M 1 un q u n u n 1 T q q C T ε q c T q (3.14) From tis form we are able to derive te system of linear equations wic we will solve at every time ste. For te linear equations we will define te regularized gradients by u T ε = ε u T (3.15) 9

10 After we arrange all te arts of te equation (3.14) we get te following coefficients a n q = min (v q, 0) δ M q u n 1 (3.16) T T ε q c T q tus we can formulate our semi-imlicit co-volume sceme: Let u 0, = 1,..., M be given discrete initial values of te level-set function. Ten, for n = 1,..., N we look for u n, = 1,..., M, satisfying u n τ m () aq n 1 q N ε ( u n q u n ) = u n 1 (3.17) Wit addition of te omogeneous Neumann boundary conditions to our fully discrete sceme we obtain a system of linear equations for wic we can declare te following statement. Teorem. Tere exists unique solution (u n 1,..., u n M ) of (3.17) for any τ > 0, ε > 0, and for every n = 1,..., N. Te system matrix is a strictly diagonally dominant M-matrix. For any τ > 0, ε > 0, te following L stability olds: min u 0 min u n max u n max u 0, 1 n N. (3.18) Proof. First we will rove te inequality for te maximum. Let u n max be te maximum of te time ste n acieved in te DF node. If we aoint tis value to te equation (3.17) we get: u n max τ m () aq n 1 q N ( u n q u n max) = u n 1. Since te coefficient a n 1 q 0 and u n q u n max, tus ( u n q u n max) is eiter zero or negative, te second art of te left and side is non-negative. Terefore u n max u n 1, so we can write max u n max u n 1 (3.19) To rove te inequality for te minimum, we can aly te same argumentation. If we aoint value u n min = min (u n 1,..., u n M ) to (3.17) te second art will be non-ositive and u n 1 u n min, tus we can write tat min u n 1 min u n (3.0) If we aly (3.19) and (3.0) to every time ste 1 n N we get min u 0 min u n max u n max u 0 10

11 by wic we roved our teorem. Te number of time stes N is determined by te difference of te solution in te current and te revious time ste in discrete L norm. Te comutation is stoed if tis difference is less tan te rescribed tolerance, wic we usually set to Ten te stoing time T = Nτ. 3.3 Calculating te coefficients of te linear system Now tat we formulated our sceme for te easy relicability and simlicity of imlementation we will write te co-volume sceme in a "finite-difference notation". We will associate our 3D co-volume wit te index trilet (i, j, k), were i reresents te y axis, j te x axis and k te z axis. Analogically te values u n will be associated wit u n i,j,k. In eac co-volume, te set N consist of 4 tetraedrons on wic we will comute absolute value of gradient denoted by G l i,j,k, l = 1,..., 4. Furtermore to kee te formulas as comreensible as ossible we will introduce a notation seen in Figure. P 6 P67 P 7 P56 T0 P 5 P78 P58 P 8 P37 P15 E0 S0 P48 P 3 P 1 P34 P14 P 4 P 6 P67 P 7 P56 P 5 W0 P6 N0 P37 P15 P P3 P1 B0 P 3 P 1 P34 P14 P 4 Figure : Notation for te additional oints of a grid cell used for te easier formulation of te coefficient comutation 11

12 Now we will exlain wat tese new symbols mean and ow teir values are comuted. For every vertex of a square cubic element we use P 1,..., P 8 to denote te average values of u n at tese oints, wic are calculated te following way: P 1 = 1 8 (u i,j,k u i,j 1,k u i 1,j,k u i 1,j 1,k u i,j,k 1 u i,j 1,k 1 u i 1,j,k 1 u i 1,j 1,k 1 ) P = 1 8 (u i,j,k u i,j 1,k u i1,j,k u i1,j 1,k u i,j,k 1 u i,j 1,k 1 u i1,j,k 1 u i1,j 1,k 1 ) P 3 = 1 8 (u i,j,k u i,j1,k u i1,j,k u i1,j1,k u i,j,k 1 u i,j1,k 1 u i1,j,k 1 u i1,j1,k 1 ) P 4 = 1 8 (u i,j,k u i,j1,k u i 1,j,k u i 1,j1,k u i,j,k 1 u i,j1,k 1 u i 1,j,k 1 u i 1,j1,k 1 ) P 5 = 1 8 (u i,j,k u i,j 1,k u i 1,j,k u i 1,j 1,k u i,j,k1 u i,j 1,k1 u i 1,j,k1 u i 1,j 1,k1 ) P 6 = 1 8 (u i,j,k u i,j 1,k u i1,j,k u i1,j 1,k u i,j,k1 u i,j 1,k1 u i1,j,k1 u i1,j 1,k1 ) P 7 = 1 8 (u i,j,k u i,j1,k u i1,j,k u i1,j1,k u i,j,k1 u i,j1,k1 u i1,j,k1 u i1,j1,k1 ) P 8 = 1 8 (u i,j,k u i,j1,k u i 1,j,k u i 1,j1,k u i,j,k1 u i,j1,k1 u i 1,j,k1 u i 1,j1,k1 ) We will also need te average values between eac of tese oints for wic we use tese notations: P 1 = 1 (P 1 P ) P 14 = 1 (P 1 P 4) P 15 = 1 (P 1 P 5) P 3 = 1 (P P 3) P 6 = 1 (P P 6) P 34 = 1 (P 3 P 4) P 37 = 1 (P 3 P 7) P 48 = 1 (P 4 P 8) P 56 = 1 (P 5 P 6) P 58 = 1 (P 5 P 8) P 67 = 1 (P 6 P 7) P 78 = 1 (P 7 P 8) Te values at te oints were te edges σ of te tetraedral elements intersects te lains e q, for every q C, are marked as N0, S0, E0, W 0 for te corresonding cardinal directions and B0, T 0 for bottom and to. Tese values are calculated as te average between two neigboring co-volumes. N0 = 1 (u i,j,k u i1,j,k ) S0 = 1 (u i,j,k u i 1,j,k ) E0 = 1 (u i,j,k u i,j1,k ) W 0 = 1 (u i,j,k u i,j 1,k ) T 0 = 1 (u i,j,k u i,j,k1 ) B0 = 1 (u i,j,k u i,j,k 1 ) 1

13 Wit tese new oints we are ready to derive te values G l i,j,k, l = 1,..., 4 wit some simle equations. Generally te G l i,j,k can be calculated te following way G l ) ( ui,j,k x ( ui,j,k y ) ( ui,j,k z ) (3.1) were te derivatives are calculated on te l t tetraedron of te co-volume. According to tis formula and Figure for te tetraedrons intersected by te bottom surface of a voxel grid cell we get: Analogically for te to surface db = u i,j,k u i,j,k 1 (P ) ( ) P 1 B0 P 1 G 1 db 0.5 (B0 ) ( ) P 14 P 4 P 1 G db 0.5 (P ) ( ) 3 P 4 P 34 B0 G 3 db 0.5 (P ) ( ) 3 B0 P 3 P G 4 db 0.5 dt = u i,j,k1 u i,j,k (P ) ( ) 6 P 5 T 0 P 56 G 5 dt 0.5 (T ) ( ) 0 P 58 P 8 P 5 G 6 dt 0.5 (P ) ( ) 7 P 8 P 78 T 0 G 7 dt 0.5 (P ) ( ) 67 T 0 P 7 P 6 G 8 dt 0.5, 13

14 for te nort wall te sout wall te east wall dn = u i1,j,k u i,j,k ( ) ( ) P 7 P 6 P 67 N0 G 9 dn 0.5 ( ) ( ) P 37 N0 P 7 P 3 G 10 dn 0.5 ( ) ( ) P 3 P N0 P 3 G 11 dn 0.5 ( ) ( ) N0 P 6 P 6 P G 1 dn, 0.5 G 13 G 14 G 15 G 16 G 17 G 18 G 19 G 0 ds = u i,j,k u i 1,j,k ( ) ( ) P 8 P 5 P 58 S0 ds 0.5 ( ) ( ) P 48 S0 P 8 S4 ds 0.5 ( ) ( ) P 4 P 1 S0 P 14 ds 0.5 ( ) ( ) S0 P 15 P 5 P 1 ds, 0.5 de = u i,j1,k u i,j,k (P ) ( ) 3 P 4 E0 P 34 de 0.5 (P ) ( ) 37 E0 P 7 P 3 de 0.5 (P ) ( ) 8 P 7 P 78 E0 de 0.5 (E0 ) ( ) P 48 P 8 P 4 de 0.5, 14

15 te west wall dw = u i,j,k u i,j 1,k (P ) ( P 1 W 0 P 1 G 1 dw 0.5 ) (P ) ( ) 6 W 0 P 6 P G dw 0.5 (P ) ( ) 6 P 5 P 56 W 0 G 3 dw 0.5 (W ) ( ) 0 P 15 P 5 P 1 G 4 dw 0.5 Now tat we can calculate u n 1 ε from (3.16) we will determine te values v q. As T mentioned earlier v q = qv n and wit v = d by evaluating for te six directions, we get vt (d i,j,k1 d i,j,k ), vb (d i,j,k d i,j,k 1 ) vn (d i1,j,k d i,j,k ), vs (d i,j,k d i 1,j,k ) ve (d i,j1,k d i,j,k ), vw (d i,j,k d i,j 1,k ) Ten we can construct te coefficients b τ 3 t τ 3 n τ 3 s τ 3 e τ 3 w τ 3 min (vb i,j,k, 0) δ M i,j,k 4 min (vt i,j,k, 0) δ M i,j,k 4 min (vn i,j,k, 0) δ M i,j,k 4 min (vs i,j,k, 0) δ M i,j,k 4 min (ve i,j,k, 0) δ M i,j,k 4 min (vw i,j,k, 0) δ M i,j,k ε ( G l i,j,k l=1 8 1 ε ( G l i,j,k l=5 1 l=9 16 l=13 0 l=17 4 l=1 ) ) 1 ε ( G l i,j,k 1 ε ( G l i,j,k 1 ε ( G l i,j,k ) ) ) 1 ε ( G l i,j,k )

16 were M i,j,k is ( M 1 4 ε 4 and we define te diagonal coefficients as l=1 G l i,j,k ) c 1 b i,j,k t i,j,k n i,j,k w i,j,k s i,j,k e i,j,k so we can define for DF node corresonding to (i, j, k) te equation c i,j,k u n i,j,k b i,j,k u n i,j,k 1 t i,j,k u n i,j,k1 n i,j,k u n i1,j,k s i,j,k u n i 1,j,k e i,j,k u n i,j1,k w i,j,k u n i,j 1,k = u n 1 i,j1,k (3.) Wen we collect te equations for all DF nodes and take into account Neumann boundary conditions we get te linear system wic we ave to solve. For te solution of tis system we coose te SOR (Successive Over Relaxation) iterative metod. We start te iterations by setting u n un 1 i,j,k, ten in every iteration l = 1,... we use te following two ste rocedure: Y = (u n(0) i,j,k b i,j,ku n(l) i,j,k 1 t i,j,ku n(l 1) i,j,k1 n i,j,ku n(l 1) i1,j,k s i,j,k u n(l) i 1,j,k e i,j,ku n(l 1) i,j1,k w i,j,ku n(l) u n(l) un(l 1) i,j,k ω ( Y u n(l 1) i,j,k i,j 1,k )/c i,j,k We define squared L norm of residuum at current iteration by R l = i,j,k (c i,j,k u n(l) i,j,k b i,j,ku n(l) i,j,k 1 t i,j,ku n(l) i,j,k1 n i,j,ku n(l) i1,j,k s i,j,k u n(l) i 1,j,k e i,j,ku n(l) i,j1,k w i,j,ku n(l) i,j 1,k un(0) i,j,k ) Te iterative rocess is stoed if R l < T OL. ) 3.4 Calculation of te initial condition As mentioned before, tis metod needs an initial condition u 0 (x), wic will be deformed to get te solution, tat is te final form of te model. Teoretically any initial surface tat contains te oint cloud data set could be used, but an otimal initial guess is crucial for te efficiency of te metod. We can find tis otimal surface by identifying all te oints for wic te value of te distance function is greater or equal to a arameter β. For simlicity let us call tese oints, exterior oints. To find all tese oints we will use te following algoritm: 16

17 Mark all oints on te borders of te grid as exterior and add tem to te set E. For every oint in te set E ceck all neigboring oints in te grid. If te neigboring oint isn t an exterior oint and is distance from te oint cloud is greater or equal to β add it to end of set E and mark as exterior. Continue until you get to te last oint of E. Wen we found all te exterior oint we set u 0 (x) to be equal 0 at all exterior oint and 1 at all te oter oints. Wit tis aroac and te rigt arameter β we can find an initial surface close to te final sae as seen on te Figure 3. Figure 3: Examles for te initial condtition used in our metod 17

18 4 Numerical results After we imlemented te metod in te rogramming language C we tested it on reresentative examles as well as real data. In tis section we resent some of our results. Tese examles are a good dislay of te quality of our metod. Figure 4 and 5 illustrate test examles. Tese were used for te verification of te correct beavior of our metod during te imlementation ase. Te oint cloud data was generated wit te corresonding arametric equations of te objects. Tese reresentative examles were created on a grid containing cells. We can see tat for tese tests wit suc a sarse grid we already got good results. Figure 4: First test object. On te left we see te oint cloud data, in te middle te oint cloud wit te final model and on te rigt te final model only. Figure 5: Second test object. On te left we see te oint cloud data, in te middle te oint cloud wit te final model and on te rigt te final model only. On Figure 6 and 7 we can see real life data. Tese items were arcaeological finds and te oint cloud scans were rovided by Jana Haličková from te Monuments Board 18

19 of te Slovak reublic to wic we exress our great tanks. On Figure 6 we can see a bracelet. Te model was calculated on a grid wit cells. On Figure 7 we can see a sealer. Te model was calculated on a grid wit cells. Tis model as a very interesting surface structure, wic confirms te accuracy of te metod. Figure 9 sows an angel statue wit te numerical results calculated on a grid. For te visualisation of all results we used te software Paraview. Figure 6: Arcaeological finds: bracelet. On te to we see te oint cloud data, in te middle our final result and on te bottom te final result wit triangulated surface. 19

20 Figure 7: Arcaeological finds: sealer. On te left we see te oint cloud data, in te middle and te rigt te final result from different viewoints. Figure 8: Details of te sealer wit triangulated surface. 0

21 Figure 9: Angel statue. On te left we see te original object, on te rigt te result of reconstruction by our metod. 5 Conclusions In tis work we resented our aroac of surface reconstruction from oint cloud data. We formulated te matematical model, derived te time and satial discretization and rovided te reader wit an exact descrition of te numerical solution. We also resented some of our numerical results as an examle. Our results sow tat for smooter object a sarce grid already sows good result, but for a model wit more detail we need more grid oints. 1

22 References [1] J. Haličková, K. Mikula, Level Set Metod for Surface Reconstruction (LSMSR) and its Alication in Survazing. Journal of Surveying Engineering, submitted 013. [] H. K. Zao, S. Oser, B. Merriman, M. KANG, Imlicit and Non-arametric Sae Reconstruction from Unorganized Points UsingVariational Level Set Metod. Comuter Vision and Image Understanding Vol. 80 (000) [3] H. K. Zao, A fast sweeing metod for Eikonal Equations. Matematics of Comutation, 004. [4] S. Corsaro, K. Mikula, A. Sarti, F. Sgallari, Semi-Imlicit Covolume Metod In 3D Image Segmentation. SIAM Journal on Scientific Comuting, 006.