Triangulations and Applications

Transcription

1 Øyvind Hjelle and Morten Dæhlen Triangulations and Applications Figures algorithms and some equations as an aid to oral exam. November 27, 2007 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

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3 Contents 1 Delaunay Triangulations and Voronoi Diagrams Algorithms for Delaunay Triangulation Data Dependent Triangulations Constrained Delaunay Triangulation Delaunay Refinement Mesh Generation Least Squares Approximation of Scattered Data Approximation over Triangulations of Subsets of Data Existence and Uniqueness Sparsity and Symmetry Penalized Least Squares Smoothing Terms for Penalized Least Squares Approximation over General Triangulations Weighted Least Squares Constrained Least Squares Approximation over Binary Triangulations Numerical Examples for Binary Triangulations References

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5 List of Algorithms 1.1 Circumcircle test LOP, Local Optimization Procedure Simple Delaunay triangulation Radial Sweep recswapdelaunay(edge e i ) LOP for data dependent triangulations Simulated annealing Eliminate u m,with α m < π, from Q ec,l Include e c as an edge in a triangulation recswapdelaunayconstr(edge e i ) Delaunay refinement General multilevel scheme

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7 1 Delaunay Triangulations and Voronoi Diagrams Algorithm 1.1 Circumcircle test 1. if (cos α < 0 and cos β < 0) 2. return FALSE // swap the edge 3. if (cos α > 0 and cos β > 0) 4. return TRUE 5. if (cos αsin β + sin αcos β < 0) 6. return FALSE // swap the edge 7. else 8. return TRUE Algorithm 1.2 LOP, Local Optimization Procedure 1. Make an arbitrary legal triangulation of a point set P. 2. if is locally optimal, stop. 3. Let e i be an interior edge of which is not locally optimal. 4. Swap e i to e i, thus transforming to. 5. Let :=. 6. goto 2.

8 (a) (b) α α Fig Two triangulations of the same point set that satisfy the MaxMin angle criterion. The triangulation in (a) is a Delaunay triangulation. a b a α 1 a α 2 b α 2 b α 1 Fig A neutral case for the MaxMin angle criterion. Only a satisfy the Min- Max criterion. The triangulation b is optimal with respect to the MaxMin angle criterion.

9 p i Fig The Voronoi region of a point p i in the plane. Fig The Voronoi diagram of a set of points in the plane.

10 Fig The straight line dual of the Voronoi diagram which yields a Delaunay triangulation. (a) p 4 (b) p 4 p 3 p 3 p 1 p 1 p 2 p 2 Fig Almost cocircular points p 1, p 2, p 3 and p 4 left, and cocircular points right. In the cocircular case there are two possible solutions when constructing the Delaunay triangulation.

11 Fig A Voronoi diagram and its straight-line dual, the Delaunay triangulation, with some of the circumcircles. p 2 p 1 V ( p 1 ) v V ( p 2 ) V ( p 3 ) C( t ) 1,2, 3 p 4 p 3 Fig The circimcircle C(t 1,2,3) of a triangle in a Delaunay triangulation and the Voronoi regions of p 1, p 2 and p 3. A point p 4 inside C(t 1,2,3) is not in agreement with the definition of a Voronoi diagram.

12 (a) p 4 (b) p 4 b p b + c b+p p 1 p +q c q p 3 p 1 q p c+q c p b 3 p 2 p 2 Fig Illustration for the proof of Lemma??. α α β γ 2α Fig Inscribed angles α on the same arc are equal, and γ < α < β. (a) p 4 (b) V ( p4) e v, p 1 v V ( p ) 2 C( t 1,2,4 ) v V ( p 3 ) p 2 p 3 p 3 Fig Illustration for the proof of Lemma??.

13 Fig Example showing that choosing the diagonal with the minimum length violates the circle criterion. p 4 v v 4 3 β p 3 p 1 v 2 α v 1 p 2 Fig Illustration for circumcircle test based on interior angles. (a) (b) q a a C C δ p p b c b c Fig Illustration for the proof of Theorem??.

14 Fig Voronoi diagram of a point set P on the left, and Voronoi diagram of a subset e P P on the right. r p q s Fig Illustration for the proof of Theorem??.

15 2 Algorithms for Delaunay Triangulation Algorithm 2.1 Simple Delaunay triangulation 1. Compute conv(p). 2. Apply Algorithm?? to the vertices of conv(p) to find an initial triangulation. 3. Apply Algorithm?? to insert points of P, that are interior to conv(p), into. This gives a new triangulation after all points have been inserted. 4. Apply the LOP repeatedly on the edges of until no edge-swap occurs, and thus obtain a triangulation which has all of its edges locally optimal. Algorithm 2.2 Radial Sweep 1. Choose a point p near the centroid of P and connect p by radiating edges to all other points of P, Figure 2.1(a). 2. Sort and order the points {P \p} by orientation and distance from p and connect the ordered sequence by edges as in Figure 2.1(b). The result from this step is a triangulation with a star-shaped domain as seen from p. Triangles may be degenerate since points may have identical orientation relative to p. 3. Form a triangle for each triple of points (p i 1, p i, p i+1) on the boundary of the triangulation. If the edge between p i 1 and p i+1 is outside the existing boundary, include the triangle in the triangulation and update the boundary. Repeat this step until no more triangles can be added. The resulting triangulation has a convex boundary and all points are included in the triangulation, Figure 2.1(c). 4. Apply the LOP repeatedly to the edges until no edge-swap occurs, to obtain a Delaunay triangulation, Figure 2.1(d).

16 10 2 Algorithms for Delaunay Triangulation Algorithm 2.3 recswapdelaunay(edge e i ) 1. if (circumcircletest(e i) == OK) // Algorithm 1.1 in Section?? 2. return 3. swapedge(e i) // the swapped edge e i is incident with p 4. recswapdelaunay(e i,1) // call this procedure recursively 5. recswapdelaunay(e i,2) // call this procedure recursively P 0 0 = P P 1 0 P 1 1 P 2 0 P 2 1 P 2 2 P 2 3 P 3 0 P 3 1 P 3 2 P 3 3 P 3 4 P 3 5 P 3 6 P }{{} }{{} }{{} }{{} }{{} }{{} }{{} 0 0 = (P)

17 (a) (b) p (c) (d) Fig The Radial Sweep algorithm.

18 b α p t b,a,p c t a,b,c e b β C(t a,b,c ) β < α Fig Circle growing from the base line e b to find a point p to form a new triangle with e b. a

19 (a) (b) p (c) (d) p Fig Influence polygons Q p in (a) and (c) when p is inserted interior and exterior to an existing triangulation. Delaunay triangulations after p has been inserted are shown in (b) and (d).

20 (a) (b) p p (c) (d) p p (e) p Fig Swapping procedure when inserting a point p into a Delaunay triangulation. From (b) to the final triangulation in (e), each picture shows the triangulation after one new edge has been swapped.

21 (a) (b) e 1,1 e 1,3 ' e 1 p e 1 p e 1,2 e 3 e 3 e 1,4 e 2 e 2 Fig Starting the recursive swapping procedure. c x e i,3 e i,1 b p ' e i e i,2 C ' e i,4 C a Fig Illustration for Lemma??.

22 (a) (b) C C C C t i p p ' e i e i t i Fig (a): The initial edges are Delaunay. (b): Edges that are swapped to p are Delaunay. 25 O( N 2 ) O( N log N ) O(N ) Fig The curves y = x, y = xlog x and y = x 2 from bottom to top for illustrating the difference between run-time order of O(N), O(N log N) and O(N 2 ).

23 (a) 1 (b) (c) Fig Illustration of a worst case example for incremental Delaunay triangulation algoritms. (a) e u (b) p p R p L e b Fig Merging two Delaunay triangulations in the divide-and-conquer algorithm.

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25 3 Data Dependent Triangulations Algorithm 3.1 LOP for data dependent triangulations 1. Make an arbitrary legal triangulation of a point set P. 2. if is locally optimal, that is, if (??) holds for all interior edges in, stop. 3. Let e i be an interior edge of which is not locally optimal. 4. Swap e i to e i, transforming to. 5. Let :=. 6. goto y x Fig Test function F 1(x, y) = (tanh (9y 9x) + 1) /9.

26 Algorithm 3.2 Simulated annealing 1. do k = 1,..., ntemps 2. t k = r k t 0, 0 < r < 1, e.g., r = do l = 1,..., nlimit 4. while the number of good swaps glimit 5. let be the current triangulation; choose a random edge e i in 6. if e i is swappable 7. let be the result of swapping e i and let d = C p( ) C p( ) be the corresponding change of the global cost function (??). 8. if d < 0, i.e., if the global cost decreases 9. swap e i ( good swap ) 10. else 11. choose a random number θ, 0 θ if θ e d/t k 13. swap e i ( bad swap ) 14. endif 15. endif (a) (b) Fig Delaunay triangulation of grid data and level curves.

27 Fig Triangulation optimized with the LOP algorithm using ABN criterion and l 1 norm. 1 e i 2 e i t k e i 4 e i t l 3 e i Fig Edges that must be checked when considering swapping the edge e i.

28 n (1) n (2) p 1 = ( x1, y1, z1) Q 1 (x,y) θ t 1 t 2 e i Q 2 (x,y) p 2 = ( x2, y2, z2) n = ( n x, n y ) ( 2 x 2, y ) ( 1 x 1, y ) Fig Geometric embedding information used when considering edge-swap of an edge e i in a convex quadrilateral defined by two triangles t 1 and t 2. Q 1(x,y) and Q 2(x,y) are the equations of the planes defined by t 1 and t 2. The dashed lines display the projection of the quadrilateral in the (x, y) -plane. n = (n x, n y) is a unit vector orthogonal to the projection of e i in the (x,y)-plane. Q 1 (x,y) L 1 e i L 2 Q 2 (x,y) y x v (1) γ v (2) Q h Fig The SCO data dependent swapping criterion.

29 (a) (b) e 1 e 2 e 1 ' e 2 (c) ' e 1 ' e 2 Fig A bad swap of e 2 followed by a swap of e 1 which reduces the clobal cost k Fig Probability of making bad swaps in Step 12 of Algorithm 3.2.

30 (a) (b) Fig Triangulation optimized with the simulated annealing algorithm using ABN criterion and l 1 norm. Fig points measured from an interior detail of a car.

31 Fig Delaunay triangulation of the points in Figure 3.10, and level curves.

32 Fig Data dependent triangulation from LOP algorithm, SCO criterion and l 1 norm.

33 Fig Data dependent triangulation from the simulated annealing algorithm, SCO criterion and l 1 norm.

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35 4 Constrained Delaunay Triangulation Algorithm 4.1 Eliminate u m,with α m < π, from Q ec,l 1. while (r 1) 2. Let (u m, w s) be a diagonal in a convex quadrilateral (u m, w s 1, w s, w s+1). 3. Swap (u m, w s) to (w s 1, w s+1) 4. r r 1 Algorithm 4.2 Include e c as an edge in a triangulation 1. while (n >= 1) 2. Find a point u m, 1 m n where α m < π 3. Apply Algorithm 4.1 to u m 4. n n 1 5. Q ec,l (p a, u 1,..., u m 1, u m+1,..., u n, p b ) Algorithm 4.3 recswapdelaunayconstr(edge e i ) 1. if (e i E c) 2. return 3. if (circumcircletest(e i) == true) // Algorithm 1.1 in Section?? 4. return 5. swapedge(e i) // the swapped edge e i is incident with p 6. recswapdelaunay(e i,1) // call this procedure recursively 7. recswapdelaunay(e i,2) // call this procedure recursively

36 (a) (b) (c) (d) Fig (a): A planar straight-line graph G(P, E c). (b): Conventional Delaunay triangulation of the point set P. (c): Constrained Delaunay triangulation of G(P, E c). (d): Illustration of the modified circle criterion for constrained Delaunay triangulation. (a) (b) p 2 e' e θ 1 p 1 θ2 p 3 θ2 e' p 4 Fig The triangulation in (a) with e as an edge is a conventional Delaunay triangulation. The triangulation in (b) is a constrained Delaunay triangulation due to the constrained edge e.

37 (a) p a e c pb e c L Q, (b) p a e c pb e c R Q, Fig (a) The influence region of a constrained edge e c that is inserted into an existing triangulation. (b) the influence polygons Q ec,l and Q ec,r of e c. p' e b t p Y Fig The growing circle reaches a point p first, but p is separated from e b by a constrained edge and cannot form a triangle with e b.

38 e c L Q, (a) e c e c R Q, (b) (c) (d) Fig Retriangulation of the influence region of a constrained edge e c. (a) to (c) show how new triangles are constructed when retriangulating the left influence polygon Q ec,l. The growing circles (dotted) are shown when they have reached a point where a new triangle can be formed with the base line. In (d) the constrained Delaunay triangulation of the whole influence region is shown.

39 (a) e c L Q, u 2 u 4 u α 1 α 6 4 α 5 u 5 u 6 α 1 α 2 α 3 p a u 3 e c pb e c R Q, u 5 (b) α 1 u 1 u 3 u 4 α 3 α 4 α 5 p a u 2 α 2 e c pb (c) u 1 p a α 1 e c pb Fig Illustration for edge insertion and swapping. (b) shows the situation when the point u 1 has been isolated from the influence polygon Q ec,l, and (c) shows the situation when the last edge is swapped and takes on the role as the constrained edge e c.

40 w = u 0 m 1 α m u m w = r u + 1 m+1 e c w 1 w 2... w r Fig Illustration for Lemma??. (a) (b) p e c e c Fig (a): The influence polygon Q p of a point p in a CDT (P, e c) is shown with bold edges. (b): The updated CDT (P p, e c).

41 5 Delaunay Refinement Mesh Generation Algorithm 5.1 Delaunay refinement 1. Make the initial CDT of the PSLG. Remove triangles outside the triangulation domain. 2. while skinny triangles remain // (controls termination) 3. while any segment s is encroached upon 4. SplitSegment(s) 5. Let t be a skinny triangle and v the circumcenter of t. 6. if v encroaches upon any segments s 1, s 2,..., s k // look-ahead 7. for i = 1,..., k 8. SplitSegment(s i) 9. goto else 11. KillTriangle(t) 12. goto 5

42 36 5 Delaunay Refinement Mesh Generation Case 1. (KillTriangle) v is inserted at the circumcenter of a skinny triangle t. The parent node p is chosen to be one of the two endpoints of the shortest edge of t. (Figure 5.9). Case 2. (SplitSegment) v is a node inserted at the midpoint of a segment s that is encroached upon by a node p. Thus p lies inside the diametral circle of s. If more than one node encroaches upon s, assume without loss of generality that p is the closest node to v that encroaches upon s. The shortest edge connected to v, which defines the insertion radius r v, has p as the other endpoint unless p is not yet inserted and thus rejected. This follows from the Delaunay property. Four possible roles of p must be considered under SplitSegment. Case 2a. p is an input node, or p is a node inserted in a segment not incident to s. (Figure 5.10(a)) Case 2b. p is at the circumcenter of a skinny triangle and thus rejected since it encroaches upon s. (Figure 5.10(b), and Step 6 8 of Algorithm 5.1.) Case 2c. p is a node on a segment s incident to s that makes an angle 45 α < 90 with s. (Figure 5.10(c)) Case 2d. As Case 2c with α 45. (Figure 5.10(d))

43 5 Delaunay Refinement Mesh Generation 37 A sample input planar straight line graph (PSLG). Constrained Delaunay triangulation of the PSLG. Encroached segments are bold. One encroached segment has been bisected. A second encroached segment has been bisected. A third encroached subsegment has been bisected. The last encroached subsegment has been bisected and a skinny triangle is found. The skinny triangle s cirumcenter is inserted. Find another skinny triangle. This cirumcenter encroaches upon a segment, and is therefore rejected. Although the vertex was rejected, the segment it encroached upon is still marked for bisection. The encroached segment is split, and the skinny triangle that led to its bisection is eliminated. A circumcenter is successfully inserted, creating another skinny triangle. The triangle s circumcenter is rejected since it encroaches upon a segment. The encroached segment will be bisected. The skinny triangle was not eliminated. Try to insert its circumcenter again. This time, its circumcenter is inserted successfully. Only one skinny triangle is left. The final mesh with no interior angle smaller than arcsin 1 2B Fig The Delaunay refinement algorithm step-by-step with upper bound B = 2 on the circumradius-to-shortest-edge ratio. Illustration and most figure texts from Shewchuk [1].

44 α min r α min r/l l α min Fig Relationship between circumradius-to-shortest-edge ratio r/l and the minimum angle α min of a triangle: r/l = 1/(2 sin α min).

45 (a) (b) t v t (c) (d) v t v Fig (a): A skinny triangle t in a Delaunay triangulation. (b): t s circumcircle. (c): The Voronoi diagram and the circumcenter of t positioned at a Voronoi point. (d): Updated Delaunay triangulation after insertion of a node at t s circumcenter.

46 Fig Spatial graded mesh uppermost, and a uniform mesh below. Illustration from Shewchuk [1].

47 (a) (b) (c) Fig Recursive bisection of a segment that is encroached upon. (a): Two nodes encroach upon the segment initially. (b): After splitting the segment at its midpoint, it is still encroached upon by a node. (c): After the second bisection there is no encroachment. α ' s 1 s 1 ' s 2 q p s 2 Fig Recursive bisection of incident segments that never terminates when α 45.

48 C(t) t p C(s' ) v s' Fig Illustration for Lemma??. s Fig Illustration of local feature size lfs() at some points relative to a planar straight line graph. An arbitrary point in the plane marked with has local feature size equal to the radius of the circle drawn with center at.

49 r v v t l p Fig Case 1 where v is inserted at the center of a skinny triangle t. The parent node p is chosen as one of the endpoints of the shortest edge of t.

50 (a) Input PSLG (b) p p t l r g g r v r p v r v v s s (c) p s (d) p s a l α r v β v a α l β v r v s s (e) p t l rg g r p r v v s Fig Different roles of a parent node p which encroaches upon a segment s. (a): p is on the input PSLG. (b): p is at the circumcenter of a skinny triangle (and rejected). (c) and (d): p is on a segment s incident with s with α 45 and α < 45 respectively. The filled bullet at p indicates the position of p which defines the lower bounds on the insertion radius r v in each case of Lemma??. (e): illustrates the same case as (b), but with p in a position which defines the lower bound on r v.

51 Fig Corner-lopping when angles of the input PSLG are less than 60. Illustration from Shewchuk [2].

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53 6 Least Squares Approximation of Scattered Data S 0 1 ( ) = { f C 0 (Ω) : f ti Π 1, i = 1,..., T }, (6.1) (N 1 (x, y), N 2 (x, y),..., N n (x, y)), N i (v j ) = δ ij, j = 1,...,n, (6.2) N i ( x, y) Ω i v i Fig A basis function N i(x, y) for the function space S 0 1( ) and its (compact) support Ω i. f(x, y) = n = g = n c i N i (x, y), (6.3) i=1 ( g x, g y ). ( g x, g y, 1 ). (6.4)

54 48 6 Least Squares Approximation of Scattered Data n' b c v 2 v 1 g a Fig Triangle patch and normal vector. n = v 1 v 2 = ( η a c a η b c b η c c c, µ a c a µ b c b µ c c c, 2A) (6.5) η a = (y b y c ), η b = (y c y a ), η c = (y a y b ), µ a = (x c x b ), µ b = (x a x c ), µ c = (x b x a ), A = 1 2 ((x b x a )(y c y a ) (y b y a )(x c x a )). (6.6) g = (η a c a + η b c b + η c c c, µ a c a + µ b c b + µ c c c )/2A. (6.7) ( g = 1 n η i c i, 2A i=1 ) n µ i c i. (6.8) i=1

55 6.1 Approximation over Triangulations of Subsets of Data Approximation over Triangulations of Subsets of Data Fig Triangulation generated from a subset of a given data set. V = {v k = (x k, y k )} n k=1. (6.9) f(x k, y k ) z k, k = 1,...,m. m I(c) = (f(x k, y k ) z k ) 2. (6.10) k=1 j=1 k=1 m n I(c) = c j N j (x k, y k ) z k 2 = Bc z 2 2. (6.11) N 1 (x 1, y 1 ) N 2 (x 1, y 1 ) N n (x 1, y 1 ) N 1 (x 2, y 2 ) N 2 (x 2, y 2 ) N n (x 2, y 2 ) B =......, (6.12) N 1 (x m, y m ) N 2 (x m, y m ) N n (x m, y m ) u 2 = ( u u 2 1/2 m).

56 50 6 Least Squares Approximation of Scattered Data I c i = 2 m n c j N j (x k, y k ) z k N i (x k, y k ) = 0, k=1 j=1 i = 1,...,n n m m N i (x k, y k )N j (x k, y k )c j = N i (x k, y k )z k, j=1 k=1 k=1 i = 1,...,n. ( B T B ) c = B T z, (6.13) ( B T B ) m ij = N i (x k, y k )N j (x k, y k ) k=1 ( B T z ) = m N i i (x k, y k )z k. k=1 Fig Least squares approximation to 4500 scattered data points sampled from Franke s test function. The triangulation has approximately 500 nodes. Triangles are Gouraud-shaded. f(x, y) = 3 +(9y 2) 2 4 e (9x 2) (9x+1)2 e 49 9y (9x 7) 2 +(9y 3) 2 2 e 4 1 (9y 7) 2 5 e (9x 4)2. (6.14)

57 6.2 Existence and Uniqueness Existence and Uniqueness B = ( B1 B )..... = N 1 (x n+1, y n+1 ) N 2 (x n+1, y n+1 ) N n (x n+1, y n+1 ). N 1 (x n+2, y n+2 ) N 2 (x n+2, y n+2 ) N n (x n+2, y n+2 ) N 1 (x m, y m ) N 2 (x m, y m ) N n (x m, y m ) (6.15)

58 52 6 Least Squares Approximation of Scattered Data 6.3 Sparsity and Symmetry ( B T B ) m ij = N i (x k, y k )N j (x k, y k ), (6.16) k=1 v j v i Ωi ΩiI Ω j Ω j Fig A possible non-zero off-diagonal element in the system matrix (B T B) ij corresponds to an edge between the vertices v i and v j in the triangulation. It is non-zero if one or more data points fall strictly inside Ω i Ω j. Fig Sparsity pattern of a system matrix B T B for the least squares problem.

59 6.4 Penalized Least Squares Penalized Least Squares Fig Least squares approximation to 4500 scattered data points sampled from Franke s test function. Random noise was added to the z-values of the data points when generating the surface on the right. J (c) = c T Ec, (6.17) m m I(c) = (f(x k, y k ) z k ) 2 + λj (c) = (f(x k, y k ) z k ) 2 + λc T Ec k=1 k=1 = Bc z λct Ec ( B T B + λe ) c = B T z, (6.18) λ d = B T B F / E F. (6.19)

60 54 6 Least Squares Approximation of Scattered Data 6.5 Smoothing Terms for Penalized Least Squares and the thin-plate energy g 2 = [ ( ) 2 g + x [ ( 2 ) 2 ( g 2 ) 2 g x x y ( ) ] 2 g y ( 2 ) 2 ] g y 2 (6.20) (6.21) Membrane energy functional. ( gk g k = x, g ) ( k = 1 n ) n ηi k c i, µ k i c i. y 2A k [ T T ( gk ) 2 ( ) ] 2 J 1 (c) = A k g k 2 gk = A k + x y k=1 k=1 ( T n 2 ( 1 n ) 2 = ηi i) k 4A c + µ k i c i k k=1 i=1 i=1 ( i) T n ( 1 n i) = ηi k 4A c ηj k c n n j + µ k i c µ k j c j k k=1 i=1 j=1 i=1 j=1 T n n ( = η k i ηj k + ) µk i µk j /4Ak c i c j = c T Ec, i=1 i=j k=1 i=1 i=1 where T E ij = (ηi k ηj k + µ k i µ k j)/4a k. k=1 The Umbrella-operator. M k f = 1 n k l Ω k c l c k. M k f = n 1, l = k ρ k l c l, ρ k 1 l = n k, if (v k, v l ) is an edge in l=1 0 otherwise. (6.22)

61 6.5 Smoothing Terms for Penalized Least Squares 55 L L L c k L L c l Fig The umbrella-operator v i Fig Illustration of which triangle vertices that generate non-zero off-diagonal elements in the system matrix together with the vertex v i. The 1 -vertices generate non-zeros with v i by the membrane energy and in the basic least squares problem (matrix B T B); the 1 and 2 -vertices generate non-zeros by the thin-plate energy term; and 1, 2 and 3 -vertices generate non-zeros by the umbrella-operator. J 2 (c) = = = [ n n n ] 2 (M k f) 2 = ρ k i c i k=1 k=1 i=1 k=1 i=1 [ n i] n n ρ k i c ρ k j c j n [ n n j=1 ρ k i ρk j c i c j = i=1 j=1 k=1 i=1 j=1 ] n n E ij c i c j = c T Ec, (6.23) where n E ij = ρ k i ρk j. k=1

62 56 6 Least Squares Approximation of Scattered Data M k = 1 ω kl c l c k, W k l Ω k 1, l = k ρ k ω l = kl W k, if (v k, v l ) is an edge in 0 otherwise. t r t 2 n ek g 2 l t 1 g 1 e k s Fig Stencil for the second order divided difference D 2 kf used to make the discrete thin-plate energy measure. Discrete thin-plate energy functional. g i n ek = ( g i x, g i y ) n e k = g i n ek, i = 1, 2. T k f = g 2 g 1 n ek = n ek ( g2 x g 1 x, g 2 y g 1 y ) n ek = ( g 2 g 1 ) n ek (6.24) g 1 = ( η l c l + η 1 s c s + η 1 t c t, µ l c l + µ 1 s c s + µ 1 t c t) /2Alst, g 2 = ( η r c r + η 2 t c t + η 2 sc s, µ r c r + µ 2 tc t + µ 2 sc s ) /2Arts. (6.25) n ek = (y t y s, x s x t )/L ek, (6.26)

63 6.5 Smoothing Terms for Penalized Least Squares 57 where T k f = β k l = L e k 2A lst i ω(e k ) β k i c i, (6.27) β k r = L e k 2A rts β k s = L e k A tlr 2A lst A rts β k t = L e k A srl 2A lst A rts. E I E I [ n ] 2 J 3 (c) = L ek (T k f) 2 = L ek βi k c i k=1 k=1 k=1 i=1 i=1 E I [ n = L ek βi i] n k c βj k c j j=1 E n n I = L ek βi k βk j c i c j = i=1 j=1 k=1 i=1 j=1 n n E ij c i c j = c T Ec, where E I E ij = L ek βi k βj k. k=1 T z (λ) = 1 m (f λ (x k, y k ) z k ) 2 (6.28) m k=1

64 58 6 Least Squares Approximation of Scattered Data Fig Penalized least squares aproximation of a noisy point set. 6.6 Approximation over General Triangulations Ω l Fig Point set for penalized least squares. The shaded region is a domain of a basis function Ω j that do not cover any data points. x T ( B T B ) x = (Bx) T (Bx) = Bx

65 6.6 Approximation over General Triangulations 59 c T ( B T B + λe ) c = c T ( B T B ) c + λc T Ec = 0, (6.29) Uniqueness with the membrane energy functional. [ T ( gk ) 2 J 1 (c) = A k + x k=1 ( ) ] 2 gk = c T Ec. y c T (B T B)c = (Bc) T (Bc) = Bc 2 2 = 0, (6.30) Uniqueness with the umbrella-operator. J 2 (c) = n (M k f) 2 = k=1 [ n 1 k=1 n k l Ω k c l c k ] 2. c k = 1 c l, n k l Ω k k = 1,...n. (6.31) c k = 1 ω kl c l, W k l Ω k k = 1,...n Uniqueness with the thin-plate energy functional. E I E I J 3 (c) = (T k f) 2 = [( g 2 g 1 ) n ek ] 2 = c T Ec. k=1 k=1

66 60 6 Least Squares Approximation of Scattered Data 6.7 Weighted Least Squares m m I(c) = w k [f(x k, y k ) z k ] 2 = w k k=1 k=1 j=1 2 n c j N j (x k, y k ) z k. I c i = 2 m k=1 w k n c j N j (x k, y k ) z k N i (x k, y k ) = 0, j=1 i = 1,...,n, n m m w k N i (x k, y k )N j (x k, y k )c j = w k N i (x k, y k )z k, i = 1,..., n. j=1 k=1 k=1 ( B T B ) c = B T z, ( B T B ) = m w ij k N i (x k, y k )N j (x k, y k ), i, j = 1,...,n, (6.32a) k=1 ( B T z ) m i = w k N i (x k, y k )z k, i = 1,...,n. (6.32b) k=1

67 6.8 Constrained Least Squares Constrained Least Squares I c i = 2 j=1 k=1 I(c) = k=1 f(x, y) = k=1 Γ = {(x r, y r, z r )} n+γ r=n+1. n+γ f(x, y) = c j N j (x, y). j=1 n c j N j (x, y) + j=1 m n c j N j (x k, y k ) + j=1 m n c j N j (x k, y k ) + j=1 n+γ r=n+1 n+γ r=n+1 n+γ r=n+1 z r N r (x, y). z r N r (x k, y k ) z k. z r N r (x k, y k ) z k n m m N i (x k, y k )N j (x k, y k )c j = N i (x k, y k ) z k where k=1 ( B T B ) c = B T z, n+γ r=n+1 2 N i (x k, y k ) = 0, z r N r (x k, y k ), ( B T B ) = m N ij i (x k, y k )N j (x k, y k ), i, j = 1,...,n, and k=1 ( B T z ) m i = N i (x k, y k ) z k k=1 n+γ r=n+1 z r N r (x k, y k ), i = 1,...,n.

68 62 6 Least Squares Approximation of Scattered Data 6.9 Approximation over Binary Triangulations The general multilevel scheme. Algorithm 6.1 General multilevel scheme 1. for k = 1, 2,3, Find a surface approximation f k. 3. if f k (x i, y i) z i < ɛ, i = 1,..., m, stop. 4. else 5. Refine k locally where ɛ is exceeded to produce k+1, and use f k to make an initial guess for f k+1. Approximation Scheme for Binary Triangulations. 1 v 2,1 2 v 3,1 3 v 5,3 1 v 1,0 Fig Starting with the triangulation 1 and the regular grid Ψ 1 on the left, vertex v 2 3,1 in Ψ 2 is first activated together with its two parents v 1 1,0 and v 1 2,1 in Ψ 1. The resulting triangulation is 2 in the middle. Next, v 3 5,3 in Ψ 3 is activated together with ancestors belonging to both Ψ 1 and Ψ 2 to obtain 3 on the right. Ψ k = { vi,j k } 2 k,2 k. i,j=0,0 c k+1 ij = f k (x i, y j ). S 0 1 ( 1) S 0 1 ( 2) S 0 1 ( h),

69 6.10 Numerical Examples for Binary Triangulations Numerical Examples for Binary Triangulations Approximation of data sampled from Franke s function Approximation of noisy data from Franke s function Approximation of parametrized 3D scattered data. Terrain modeling of mountain area.

70 Fig Approximation to Franke s test function; the resulting binary triangulation imposed on the surface.

71 Fig The same binary triangulation as in Figure 6.14, and input data for numerical examples. The lower left corner corresponds to the nearest corner in Figure Fig Approximation with huge smoothing parameter.

72 Fig Approximation to Franke s function from a data set with noise. Fig Approximation of terrain data and the given hypsographic data.

73 Fig Approximation of terrain data, and triangulation imposed on the surface. Fig Terrain modeling of an area in Jotunheimen, Norway.

74 Fig Approximation of parametrized 3D scattered data.

75 References 1. J. R. Shewchuk. Delaunay Refinement Mesh Generation. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, Available as Technical Report CMU-CS J. R. Shewchuk. Delaunay refinement algorithms for triangular mesh generation. Computational Geometry, 22:21 74, 2002.