Topology Changes for Surface Meshes in Active Contours

Transcription

1 Topology Changes for Surface Meshes in Active Contours Jochen Abhau University of Innsbruck, Department of Computer Science FSP Conference Vorau, November 2006

2 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

3 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

4 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

5 3D Image Segmentation In medical imaging, 3D ultrasound voxel images are taken. One wants to find a segmenting contour, which visualises an object inside the image. The topology of the object is not known a priori.

6 The Balloon Evolution Start: user places small triangulated sphere inside the object he wants to segment. sphere explicitly represented, evolved over time move points outwards along normal direction gradient-based segmentation criterion

7 Need of Topology Changes Problem: During the evolution process, the contour can self-intersect, and topology changes have to be triggered.

8 (Partial) Solution for Topology Changes Algorithm: (1) Introduce a marker array for voxels already being inside the mesh. detection of self-intersections is fast and easy. (2) Open the mesh where self-intersections were detected, by deleting some vertices, edges and faces. (3) Close the mesh: This is the difficult part of the topological adaptation. In 3D, no robust direct algorithm exists.

9 Cyst mesh, stop with marker array

10 (Partial) Solution for Topology Changes Algorithm: (1) Introduce a marker array for voxels already being inside the mesh. detection of self-intersections is fast and easy. (2) Open the mesh where self-intersections were detected, by deleting some vertices, edges and faces. (3) Close the mesh: This is the difficult part of the topological adaptation. In 3D, no robust direct algorithm exists.

11 Cyst mesh, open near regions

12 (Partial) Solution for Topology Changes Algorithm: (1) Introduce a marker array for voxels already being inside the mesh. detection of self-intersections is fast and easy. (2) Open the mesh where self-intersections were detected, by deleting some vertices, edges and faces. (3) Close the mesh: This is the difficult part of the topological adaptation. In 3D, no robust direct algorithm exists.

13 Cyst mesh, some iterations after closure

14 (Partial) Solution for Topology Changes Algorithm: (1) Introduce a marker array for voxels already being inside the mesh. detection of self-intersections is fast and easy. (2) Open the mesh where self-intersections were detected, by deleting some vertices, edges and faces. (3) Close the mesh: This is the difficult part of the topological adaptation. In 3D, no robust direct algorithm exists.

15 Comparison with Previous Work Lachaud, Montanvert (1999): Heuristics, no guarantee for correctness Bischoff, Kobbelt in 2D (2004): Vertices restricted to lie on edges of 2D grid. A., Scherzer (to appear 2007) : Reconstruction step after evolution.

16 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

17 Motivation Given: A mesh with several opened regions (yellow), where self-intersections occured Goal: Connect them for a topology change

18 Motivation Just in mind: Forget about the rest of the mesh and Connect the open regions to caps

19 Motivation Just in mind: Forget about the rest of the mesh and Connect the open regions to caps

20 Motivation Goal now: Add triangles to the pieces, such that the arising complex is homeomorphic to S 2 and no intersections occur. Condition splits into a combinatorial part and a geometric part.

21 Motivation Goal now: Add triangles to the pieces, such that the arising complex is homeomorphic to S 2 and no intersections occur. Condition splits into a combinatorial part and a geometric part.

22 Motivation Result: the opened parts are connected now topologically correct, without self-intersections

23 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

24 The Situation We are given an opened mesh with boundary points P 1,..., P λ1, P λ1 +1,... P λ1 +λ 2,... P λ1 +λ 2 +1,..., P P k i=1 λ i determining the k polygons along which was cut. The vector λ = (λ 1,..., λ k ) partitions d = k λ i, λ 1... λ k 3. i=1 R 3

25 Notations Let for i = 1,..., k and λ a partition of d µ 0 = 0, µ i = i λ l l=1 Λ i = {µ i 1 + 1,..., µ i } = i-th cut area ( ) µi µ ω i = i 1 µ 1 S(Λ µ i µ i µ i i ) S d ω = ω 1... ω k

26 cut areas Λ 1 = {1, 2, 3, 4, 5}, Λ 2 = {6, 7, 8, 9}, yellow

27 Notations Let for i = 1,..., k and λ a partition of d µ 0 = 0, µ i = i λ l l=1 Λ i = {µ i 1 + 1,..., µ i } = i-th cut area ( ) µi µ ω i = i 1 µ 1 S(Λ µ i µ i µ i i ) S d ω = ω 1... ω k

28 ω 1 maps as on the left, ω 2 as on the right hand side

29 Notations Let for i = 1,..., k and λ a partition of d µ 0 = 0, µ i = i λ l l=1 Λ i = {µ i 1 + 1,..., µ i } = i-th cut area ( ) µi µ ω i = i 1 µ 1 S(Λ µ i µ i µ i i ) S d ω = ω 1... ω k

30 Mappings Describe Spheres Consider mappings M λ = {π : {1,..., d} {1,..., d} π(λ i ) A mapping π M λ generates a complex S(π) out of P 1,..., P d in three steps: (1) Choose points Q 1,..., Q k distinct from P 1,..., P d arbitrarily and triangulate the caps. l i Λ l for all i}

31 Q 1 and Q 2 are chosen, caps arise

32 Mappings Describe Spheres Consider mappings M λ = {π : {1,..., d} {1,..., d} π(λ i ) A mapping π M λ generates a complex S(π) out of P 1,..., P d in three steps: (1) Choose points Q 1,..., Q k distinct from P 1,..., P d arbitrarily and triangulate the caps. l i Λ l for all i} (2) Insert the faces [P i, P ω(i), P π(i) ] to connect the cut areas.

33 face [P 5, P ω(5)=1, P π(5) ] was inserted, here π(5) = 6

34 Mappings Describe Spheres Consider mappings M λ = {π : {1,..., d} {1,..., d} π(λ i ) A mapping π M λ generates a complex S(π) out of P 1,..., P d in three steps: (1) Choose points Q 1,..., Q k distinct from P 1,..., P d arbitrarily and triangulate the caps. l i Λ l for all i} (2) Insert the faces [P i, P ω(i), P π(i) ] to connect the cut areas. (3) Insert another 2k 4 faces

35 and this is the possible end

36 Mappings Describe Spheres Consider mappings M λ = {π : {1,..., d} {1,..., d} π(λ i ) A mapping π M λ generates a complex S(π) out of P 1,..., P d in three steps: (1) Choose points Q 1,..., Q k distinct from P 1,..., P d arbitrarily and triangulate the caps. l i Λ l for all i} (2) Insert the faces [P i, P ω(i), P π(i) ] to connect the cut areas. (3) Insert another 2k 4 faces

37 Questions Two questions arise: How can one find mappings which give rise to a sphere, i.e. S(π) = S 2 What is the structure of these good mappings?

38 Computation of good mappings A mapping π M λ is a good mapping, if S(π) is a homological 2-sphere each edge is shared by two faces Note that the homology criterion can be checked by number theory software (GAP, Pari) is useful for theoretical considerations about the structure of all good mappings only depends on λ and not on P 1,..., P d is not computed during the balloon evolution can be used as formula generator

39 Computation of good mappings, Examples For λ 1 = λ 2 = λ 3 = 3, ( ) π = is a good mapping.

40 Structure of good mappings A good mapping, generates a family of further good mappings by group actions. The groups of rotations or exchanges of cut areas act by conjugation, reflections acts by reversing the orientation.

41 the numbering of the cut-areas is reversed

42 Fast Triangle-Triangle Intersection Test Combinatorics of the mesh is controlled, but the inserted triangles can intersect. use fast triangle-triangle intersection test of Möller: compute plane equation for one triangle check if points of other triangle lie on same side by signed distances do the same vice versa compute intersection line and project onte largest axis compute the intervals for both triangles

43 Fast Triangle-Triangle Intersection Test Combinatorics of the mesh is controlled, but the inserted triangles can intersect. use fast triangle-triangle intersection test of Möller: compute plane equation for one triangle check if points of other triangle lie on same side by signed distances do the same vice versa compute intersection line and project onte largest axis compute the intervals for both triangles

44 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

45 Algorithm for Topology Changes Algorithm for topology changes in the mesh: (1) Determine π s such that S(π) S 2 by formula, subdivision or data base. (2) Operate by reflections on the mappings, to obtain the desired orientation. (3) Operate by exchanges and rotations and obtain (all) possible topology changes. (4) Take a π for the topology change such that no self-intersections in the mesh are produced and some distance criterion is minimized.

46 Algorithm for Topology Changes Algorithm for topology changes in the mesh: (1) Determine π s such that S(π) S 2 by formula, subdivision or data base. (2) Operate by reflections on the mappings, to obtain the desired orientation. (3) Operate by exchanges and rotations and obtain (all) possible topology changes. (4) Take a π for the topology change such that no self-intersections in the mesh are produced and some distance criterion is minimized.

47 Algorithm for Topology Changes Algorithm for topology changes in the mesh: (1) Determine π s such that S(π) S 2 by formula, subdivision or data base. (2) Operate by reflections on the mappings, to obtain the desired orientation. (3) Operate by exchanges and rotations and obtain (all) possible topology changes. (4) Take a π for the topology change such that no self-intersections in the mesh are produced and some distance criterion is minimized.

48 Algorithm for Topology Changes Algorithm for topology changes in the mesh: (1) Determine π s such that S(π) S 2 by formula, subdivision or data base. (2) Operate by reflections on the mappings, to obtain the desired orientation. (3) Operate by exchanges and rotations and obtain (all) possible topology changes. (4) Take a π for the topology change such that no self-intersections in the mesh are produced and some distance criterion is minimized.

49 Algorithm for Topology Changes Algorithm for topology changes in the mesh: (1) Determine π s such that S(π) S 2 by formula, subdivision or data base. (2) Operate by reflections on the mappings, to obtain the desired orientation. (3) Operate by exchanges and rotations and obtain (all) possible topology changes. (4) Take a π for the topology change such that no self-intersections in the mesh are produced and some distance criterion is minimized.

50 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

51 Results not completely implemented but seems to be stable and fast, espescially for complicated topology changes

52 Torus Example Stopped before self-intersection

53 Torus Example Open the close parts

54 Torus Example λ 1 = 6, λ 2 = 4, combine them

55 Torus Example End of the balloon evolution

56 Cyst Example End of the balloon evolution

57 Outline 1 Introduction 2 Topological Adaptation Algorithm Overview of the Idea The Computational Model Algorithm for Topology Changes 3 Results and Outlook Results Outlook

58 Current Work, Outlook There are several open problems: How many generators need to be checked to perform a correct topology change? Can we restrict the test to permutations, where possible? How can a self-intersection and edge length check be implemented already during the generation of the combination mappings? Can we compute a formula for all generators and all λ s?

59 Main References S. Bischoff and L. Kobbelt. Snakes with topology control. The Visual Computer. Vol 20, pp , J. O. Lachaud and A. Montanvert. Deformable Meshes with Automated Topology Changes for Coarse-to-fine 3D Surface Extraction. lachaud/articles/1999- mia/revuemia.html T. Möller. A Fast Triangle-Triangle Intersection Test. Journal of Graphics Tools, 2/2, pp 25-30, 1997.

60 The End Thank you for your attention.