SURFACE RECONSTRUCTION AND SIMPLIFICATION

Transcription

1 NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS DEPARTMENT OF INFORMATICS AND TELECOMMUNICATIOS SECTOR OF THEORETICAL INFORMATICS SURFACE RECONSTRUCTION AND SIMPLIFICATION Alexander Agathos Master Thesis Supervisors: Dr. Theoharis Theoharis Nikos Platis

2 Acknowledgements The witting of a thesis which involve state of the art topics is a very difficult and dangerous process. This kind of thesis is based on various recent publications. This thesis tries to create a path which can be followed by the reader, having a reassurance that he would not be led to dark places or dead ends. The forest of publications is vast and the danger for a researcher to get lost is more than visible. I hope that I have succeeded to create a correct map which can be used by the reader so that he can safely investigate further details. The flawless syntax and use of the Greek Language could not be achieved without the persistence and the correct study of my drafts from my supervisor Nikos Platis. I would like to heartly thank him for the patience he showed so that a correct document from all viewpoints could be created. Also his advices on various technical subjects were correct and led to the improvement of my results. I would like to also thank my parents who put up with me and my ways in this difficult stage of my life. I would also like to thank my second supervisor Dr. Theoharis Theoharis for the moral support he showed me during the last stages of my thesis. Big thabks also to Dr. Panagiotis Stamatopoulos for the additional hard-disk place he supplied me in the University server.

3 CONTENTS Introduction to Surface Reconstruction and Simplification Basic elements of computational and differential geometry Simplices and Simplicial Complexes Voronoi diagram Power diagrams Medial axes Delaunay Triangulation Elements of theoretical Differential Geometry Restricted Voronoi diagrams and Restricted Delaunay Triangulation k-nearest Neighbors(k-NN) of a point Delaunay based Surface Reconstruction Basic Theorems and definitions The crust of a surface Co-Cone based surface reconstruction Parallel implementation of the Co-Cone criterion Extraction of a manifold Extraction of the Power Crust of a surface Alpha Shapes Gabriel Graph Natural Neighbors of a surface Ball Pivot algorithm Surface Reconstruction with Marching Cubes Incremental wrapping of a surface with cubes Adaptive wrapping of a surface with cubes Extraction of a Polygon approximation of the surface Fighting aliasing problems Normal estimation Normal alignment ii

4 4. A sampling methodology Quadrature approximation of a surface Ellipsoid curvature & scaled Ellipsoids Distance measures D, Q Correlation of D & Q Sampling Regions Sampling criteria Local Delaunay triangulation Calculating the Nearest Neighbors of a point Isometric projection of p onto Τ p (M) Local Delaunay Neighbors Triangulation of S Surface Simplification & Hierarchical representation of a surface Philosophies of Surface Simplification Lindstrom & Turk Surface Simplification Garland Surface Simplification References iii

5 Introduction to Surface Reconstruction and Simplification The problem of surface reconstruction is like follows: Let M be the surface that is sampled with a set of points S. We search ways to connect the points of S so that a polygon approximation of M is created. Regarding issues of hardware and software this approximation is preferred to be triangular because it can easily be processed. The sampling of the surface is achieved with a scanner which can acquire the relevant coordinates of the points on the surface it samples. The scanner may also be able to acquire the normal information on the points of the surface it samples. In this thesis we will not assume this information. We will describe in chapters three and four techniques for the calculation of this information. Practically, the way we sample a surface is in accordance with the intuition that on areas where the curvature is high a dense sampling is needed and on areas where the curvature is low a sparse sampling is needed. Theoretically now, from signal processing theory it is known that when we want to sample an analog signal we have always in mind the Nyquist limit that specifies the lower bound on the sampling frequency, i.e. how close should the samples be with each other. On areas with high frequency (high curvature) we should sample with high frequency, meaning that the samples must be close with each other. On areas with low frequency the sampling should be sparse. Therefore the sampling we make on a surface is in accordance with signal theory. In chapters two and four we will describe two sampling theorems which have close relation with the curvature in the area we sample. In general problems occur in the sampling of a surface when it has boundaries. From sampling theory we know that we can reconstruct a signal from its samples if its frequency is bounded. So if we have a sudden break of the signal, such when the surface has boundaries, then we can not reconstruct with a finite sampling the signal. T. Dey [26, 27, 28] propose ways to detect boundaries by examining where the sampling is less than needed. The only restriction is that areas that are not near the boundary should be sampled dense enough, otherwise holes that do not exist might occur. Gopi [51], defines the minimum and maximum distance that the samples should have with each other. The way he defines these measures imposes a non monotonic sampling on the surface because there might be the case that if we increase the sampling instead of improving to actually worsen the reconstructed result. In chapter one we will present basic elements of computational geometry. In chapter two we will present philosophies for Delaunay based surface reconstruction. In chapter three we will study the method of Marching cubes for surface reconstruction. In chapter four we will present a sampling theory which will enable us to reconstruct a surface using a local Delaunay triangulation. 1

6 In order to do surface reconstruction it is required to have a dense sample in order for our results to be correct. Today needs amount to an excess of one million points. When we reconstruct a surface we normally want to process it. For example to see it in real time on various view angles or we may want to send it on line to a terminal (cellular phone). It is necessary to decrease the complexity of the reconstructed surface because the hardware that are going to process these surfaces may not have the power to process such complicated data quickly. In chapter 6 we will present techniques to decrease the complexity of the polygon approximation of a surface. 2

7 Chapter 1 Basic elements of computational and differential geometry In this chapter we will present elements from computational and theoretical differential geometry which will help in the sequel to better understand ideas from surface reconstruction Simplices and Simplicial Complexes Polytope: As a n-dimensional polytope we define the finite intersection of n-dimensionsional half-spaces with a non empty interior. The faces of the polytope are the (n 1)-dimensional polytopes of this polytope. Convex Hull: Let S R n be a set of points. The convex hull is the intersection of all polytopes which contain S. This is the minimum polytope which contain S and it is obviously convex. We will symbolize the convex hull of S as convs. In two dimensions we can imagine the convex hull if we regard the points of S as nails and the convex hull as the polygon which is created if we place an elastic cord around all these nails. An example of a convex hull can be seen in Figure In two dimensions the convex hull is a polygon and in three dimensions a polytope. Figure The convex hull of a set of points on the plane n-simplex: Let S be k+1 points of R n, k n which are linear independent, a k-simplex, σ, is the convex hull of S, convs. λ-face of a simplex: Let S be n+1 points of R n which is a n-simplex σ. Α λ-simplex τ of a set T of λ+1 elements of R n where T S is called a face of σ. This relation is denoted as τ σ. For example, the 3-simplex is a tetrahedron, the 2-simplex is a triangle, the 1-simplex is a line segment and the 0-simplex is a point. 3

8 Simplicial Complex: A simplicial complex K is a collection of simplices such that: 1. If σ Κ and τ σ then τ Κ, so every face of a simplex in K will belong to Κ 2. If σ,τ Κ then τ σ τ,σ or τ σ =, so if σ,τ belong in Κ then they either have no intersection or they share a common face Later we will define the Delaunay triangulation which is a simplicial complex. 1.2 Voronoi Diagram Let S be a set of points in space. The Voronoi diagram of S is a tessellation of the space in cells in which all points are closer to one and only point of S. We denote Vor(S i ) the cell which defines the point S i S and Vor(S) the set of all Voronoi cells. We will name the point which define the cell as its generator. The characteristic of Voronoi cells is that they are convex polytopes. Formally, Vor(S i ) = { x R 3 : S k S i S, d(x, S i ) d(x, S k )}, Vor(S) = t Vor( Si), Relation When the generator of a cell belong to the Convex Hull then the cell is not bounded, so there exist cells of the Voronoi diagram that extent to infinity. In each bounded cell we will call pole the furthest vertex of the polytope from its generator. We will see later that the poles play a significant role in surface reconstruction. An example of a Voronoi cell with its generator and pole can be seen in Figure Si S s Σχήµα The Voronoi cell which is generated by a sample point s S. The Voronoi cell is a convex polytope, its faces are convex polygons. The pole is colored with red color. The Voronoi cells play a significant role in surface reconstruction as we will see later. If S is dense enough then the cells of Vor(S) are long and skinny and they are very important for the creation of the normals on the points of S. The vertices of the Voronoi diagram have a significant property. Each vertex is the center of a maximal sphere, empty in its interior of points of S and on its boundary there are at least four points of S. We call this sphere the Voronoi sphere of the respected Voronoi vertex. Voronoi diagrams are going to be studied in the three dimensional Euclidian space. In the figures we will present two dimensional space. The reader should have always in mind that the properties we describe for three dimensions hold also for two dimensions but with one degree (unit) less. 4

9 When the sample points are in general position then the Voronoi sphere will have exactly four generators on its boundary. The Voronoi sphere of a pole is called a Polar sphere. An example of a Voronoi diagram in two dimensions is shown in the figure below. Figure The Voronoi diagram of seven points on the plane. The Voronoi cells are polygons, each face of a Voronoi cell is a line segment or half-lines and the Voronoi spheres are circles. The center of the circum-circle of three points of S is a Voronoi Vertex. Each edge of the Voronoi diagram is in neighbor relations with exactly three generators. A Voronoi edge or its extension cross the center of the minimum circumsphere of its three neighbor generators, Figure S i S t O S k Figure A Voronoi edge or its extension crosses the center Ο of the minimum circum-sphere of the generators S i, S k, S t. The distance function we used so far (Relation 1.2.1) is the Euclidian. We will define in the next section another useful distance function Power Diagrams A variation to the definition of Voronoi diagrams and very useful for surface reconstruction is the Power Diagrams. The main difference that Power diagrams have with the Voronoi diagrams is that the distance function is biased according to a radius of a sphere with center one of the two points that we take the distance. This additional aspect makes this distance relevant to the point the sphere is centered to. We can see this intuitively by regarding that the point is expanded to a sphere of a given radius with relation to the other point that the distance is examined. By replacing in Relation the distance function with these biased distance functions we construct the Power diagrams. The 3d points of a set S are in general position when there does not exist a line containing more than two sample points, a plane containing more than three points, a sphere containing more than four points of S. In the sequel we will assume that the points of S are in general position. 5

10 A very practical distance function was presented by Amenta [29-31]. Assuming that a point s i is inflated into a sphere of radius ρ i. This distance function is defined as follows: d pow (x, Β si,ρi ) = d 2 (s i, x) ρ i 2, where x is a point in the Euclidian space. We see that d pow is defined by the Euclidian distance subtracting the inflated radius ρ i, so we can observe that d pow is a biased Euclidian distance. This distance function is very practical because it can be used by software packages that construct the Voronoi diagrams if we alter the distance function taking into account the radius ρ i. Power diagrams are the union of convex cells similar to those created in Voronoi diagrams. Also if Bs i,ρ i, Bs j,ρ j are two open spheres of radius ρ i, ρ j with centers the generators s i, s j, which are in neighbor relations, then if Bs i, ρ B i s j, ρ then the j plane which is defined by their common face contains the sector of the intersection of the two balls Bs i,ρ i, Bs j,ρ j. An example of such a tessellation of space can be seen in Figure Medial Axis Figure The Power Diagram of 4 points The Medial axis of a closed surface is defined as the set of all points in space which are equidistant to more than two points of the surface. The definition of the Medial axis for a closed curve on the plane is the same and an example can be seen in Figure d d Figure The red line is the medial axis of the closed curve(black color). 6

11 We can say that the Medial axis gives a skeleton to the surface. The Medial axis provide us a way to approximate the Normals on the Sample Points. This is very important because with this way we can reconstruct the surface without the knowledge of the distribution of the scanned points (uniform or not). If the surface (curve) is smooth then the Medial axis is far away from the surface (curve). In two dimensions if the sampling S of a curve is dense then all the Voronoi vertices of S belong to the Medial axis. In three dimensions this result does not hold. Even if the sampling of a smooth surface is done densely there might exist Voronoi vertices that are very near to the surface, clearly these vertices can not belong to the Medial axis. The poles of the Voronoi diagram are most probable to belong to the Medial axis Delaunay Triangulation The Delaunay triangulation D of a set of points S is a tessellation of the space into triangles with vertices the points of S. As already mentioned the triangle is a 2-simplex. Each triangle s ijk in D has a neighbor point u with which a 3-simplex s ijku can be created, i.e. a tetrahedron. A triangle s ijk belong to the Delaunay triangulation if and only if one of the following hold: 1) s ijk belong to the convex hull of S. 2) s ijk is the common face of two tetrahedral s ijku, s ijkv and u is outside of the circum-sphere of s ijkv. The above rule has a significant practical importance and is used by most of the software packages that calculate the Delaunay triangulation of a point set S. The Delaunay triangulation D of S has a tight dual relation with its Voronoi diagram: 1) The triangles of D have vertices the generators of the Voronoi cells. 2) Each vertex of the Voronoi diagram corresponds to a tetrahedron in D. 3) Each edge of the Voronoi diagram corresponds to a triangle in D. 4) Each face of the Voronoi diagram corresponds to an edge in D. We define as: 3-cell, the Voronoi cell of a point in S 2-cell, the intersection of two 3-cells, i.e. the common face of these two cells 1-cell, the intersection of three 3-cells i.e. the common edge of these three cells 0-cell, the intersection of four 3-cells i.e. the common vertex of these four cells 7

12 Let Τ be a subset of S with T = k + 1, 0 k 3, V T = l κορυφές p T According to what we have said: Vor( p) σ Τ is a k-simplex of D V Τ is a 3-k cell of Vor(S) A definition to check if a triangle S ijk belongs to D can be now expressed: The triangle S ijk will belong to D if and only if each tetrahedron, which has S ijk as a face, has circum-sphere which coincides with the Voronoi sphere of the common vertex of the four cells which have as representatives the vertices of this tetrahedron. From what we have said until now, a triangle D Is either shared by two tetrahedral which are called cojugates. Or is a face one and only tetrahedron. This triangle belongs on the convex hull of S. The above dual relation will help us construct the Voronoi diagram of the point set S having only knowledge of the Delaunay triangulation. We will present now an algorithm to do this. We assume to have a list which will contain the triangles of the triangulation that the algorithm has checked so far and they are faces of tetrahedral whose conjugate tetrahedral have not yet been processed. We will call this list ActiveFaceList and we denote it as AFL. We assume also that for each triangle we have also information about the center of the circum-sphere of the tetrahedral which was checked and has this triangle as a face. This center as we have already said is the common vertex of four cells. Construction of the Voronoi diagram with the aid of the Delaunay triangulation. 1) Find the Delaunay triangulation D of the point set S. 2) For each tetrahedron σ ijkv of D. 1) Calculate the center of the circumscribed sphere, Ο. 2) For each triangle τ of σ ijkv 1) If τ is in AFL 1) The center O which is stored in τ creates a Voronoi edge ΟΟ of the three vertices of τ 2) We erase the triangle from the list 2) Otherwise we store the center Ο concerning this triangle and we insert the triangle into the AFL. 3) The tetrahedral that have remained in AFL belong on the convex hull of S, for every face\ of these tetrahedrals 1) Let Ο be the center which we have stored for this triangle and A the center of the circum-circle of this triangle. If O is outside from the convex hull of S then the direction of the Voronoi edge is opposite of the direction of OA, otherwise it has the same direction. 8

13 If the construction of the Voronoi cells is necessary we can follow a technique described in chapter 2 (paragraph 2.9). What is usually done is to calculate the Delaunay triangulation from the Voronoi diagram. In general the construction of the Voronoi diagram is more difficult than the construction of the Delaunay triangulation. Also the construction of the Delaunay triangulation can be parallelized more easily than the Voronoi diagram. We will describe now the construction of the Delaunay triangulation, mentioning some algorithms and we will last describe a uniform and adaptive parallel philosophy for their construction. In general there exist two philosophies for the construction of the Delaunay triangulation. 1 st Philosophy: Interpolative Construction We first randomize the set of points S. We then proceed with the construction of an initial tetrahedron, large enough so that it contains all points of S. This is a dummy tetrahedron which will be removed after the triangulation is finished. We begin now to insert into the algorithm the points of S one by one. Each time we insert a new point we update the triangulation so that to take account the newly inserted point. This is done by the proper construction of new tetrahedral and the destruction of some of the old. Therefore we can understand the reason why these algorithms are called interpolative. For the sake of completeness we will describe two methods of this philosophy that have prevailed and are very similar with each other. The first is the method of Boissonat [3, 4] and the second is the method of Edelsbrunner and Shah [54]. We will expand more in the methodology of Boissonat since this methodology will help us understand better a technique which will be described in chapter two. Boissonat renew the triangulation after the insertion of a new point by examining the circum-spheres of the tetrahedral which the former contain. These tetrahedral must be destroyed in order to obtain a valid Delaunay triangulation. As we can see in Figure (in two dimensions) a region which is in conflict with the definition of the Delaunay triangulation is created. Figure The shaded region illustrates the area which is conflict with the Delaunay triangulation definition after the insertion of the new point (gray color). Every triangle which is in this area is destroyed. The red lines illustrate the edges of the triangles which are not going to be destroyed. With the purple lines we illustrate the new edges that are going to be created, these edges along with the red ones create the new triangles. 9

14 In three dimensions an area is created consisting from tetrahedral that have to be destroyed. From these tetrahedral we keep only the faces that have a conjugate tetrahedron whose circum-sphere does not contain the newly inserted point. These faces are linked with the newly inserted point creating new tetrahedral. In order for the above process to be handled efficiently, Boissonat defined a data structure which was named Delaunay Tree. The Delaunay tree is a Directed Acyclic Graph (DAG) which keeps the history of the whole process. Specifically the nodes of the Delaunay tree are the already created tetrahedral. Its leaves are the tetrahedral that exist after the insertion of the new point. Each time a new tetrahedron is created it destroys another tetrahedron which existed before the insertion of the new point. This new tetrahedron will be a new leave in the Delaunay tree. Two arcs will end in this leave, one from the tetrahedron that is destroyed and one from the tetrahedron that the Which is conjugate with the tetrahedron which is destroyed. We call Father the tetrahedron which was destroyed to give its place to its child and Step father the tetrahedron which is conjugate to the new tetrahedron. Εdelsbrunner, Shah created another algorithm which has many similarities with Boissonats: From the current triangulation we check the triangles which have the newly inserted point on their outside and link them with the newly inserted point to create tetrahedral. Then it is checked if the newly created tetrahedral fulfill the 2 nd criterion of the Delaunay triangulation, if not then we flip the tetrahedron in a manner seen in the Figure below. Continuing this process it can be proved that we end up into tetrahedron that fulfills the Delaunay property. u u v t z v t z w w Figure Tetrahedral flipping: From left to right we get from two, three tetrahedral (flipping of a triangle into an edge). From right to left we get from three, two tetrahedral (flipping of an edge into a triangle). From left to right we go when s vtz violate the Delaunay property and so from the two tetrahedral s uvtz, s wvtz we go to the three tetrahedral s uwtv, s uwtz, s uwzv. This flipping is called flipping of a triangle into an edge. From right to left we go when the edge s uw violates the Delaunay property and so from the three tetrahedral s uwtv, s uwtz, s uwzv we go to the two tetrahedral s uvtz, s wvtz. This flipping is called flipping of an edge into a triangle. 10

15 2 nd Philosophy: Incremental Process According to this process the tetrahedral of the Delaunay complex are built incrementally one by one without the need to destroy any of the already created tetrahedral. We will use the already mentioned Active Face List [8, 9]. In this list we will store all the faces (triangles) of the tetrahedral that the algorithm has already calculated. This list must be equipped with a fast search function. This AFL is updated when a new triangle, not already included in the list, is inserted. If the inserted triangle is included in the list then we delete this triangle from the list. In order for the above checks to be fast a hash table is usually used. The algorithm that is going to be described in the sequel consists of two basic functions: Create Tetrahedron Function : This function takes as input a face f of the new tetrahedron that is going to be created and will be conjugate with the tetrahedron that it will share this share with. This face looks outside of the already created tetrahedron. We call this space the OuterHalfSpace (OHS(f)) of the face f. We search in the OHS(f) a point which can create with the face f a new tetrahedron that belong in the Delaunay complex. This can be done by taking each point in the OHS(f) and create the circum-sphere of the tetrahedron created by this point and the face f. Let ρ, c be the radius and the center of this sphere. From all these points in OHS(f) we choose the point p to create the new tetrahedron which minimizes the function below: r, c OHS( f ) dd( f, p) =, p P r αλλιώς So this function returns the point described above. It is not hard to see that this function always returns a point if the face f does not belong on the convex hull. If it does then it returns a NULL point. Create new tetrahedron function: This function creates the seed tetrahedron which will be used to generate the rest of the Delaunay complex. We choose a point p approximately in the center of the point space of S. In the sequel we choose the nearest neighbor of p, q, which is surely a Delaunay edge. Then we choose in the area around the edge pq the point r such that the circum-circle of the triangle f pqr has minimum area. Then we call the Create Tetrahedron to get the fourth point and create the seed Tetrahedron. We choose an arbitrary OHS(f pqr ) for this face since no tetrahedron exists to have this triangle as a face. 11

16 To decrease the time complexity of the above functions we split the area in cubes, creating in this way a grid structure which can efficiently decrease the time complexity of these functions (see also Chapter 5). Also we must always make sure that the face looks outside from the tetrahedron that contains this face before inserting it into the AFL. Incremental Delaunay construction algorithm 1. Initially set empty the AFL and initialize a list DT in which we will store the Delaunay Tetrahedral 2. Call the create new tetrahedron function to obtain an initial tetrahedron t 3. DT += t 4. We inform the AFL with all of the faces of t 5. While AFL not empty 1. Extract a face f from the AFL 2. Using the face f using the create new tetrahedron function we create a new tetrahedron t 3. If t NULL (i.e. if the face f does not belong on the convex hull) 1. DT += t 2. For each face f of t different of f we inform the AFL 6. We take as output the list with the Delaunay tetrahedral, DT, and the list of the faces that belong on the convex hull which are stored in the AFL We can now understand the major difference between the interpolative and incremental construction. The incremental construction add constantly new tetrahedral, it does not destroy any, in contrast with the interpolative construction which destroys previously created tetrahedral. Thus concerning issues of parallel implementation of the Delaunay triangulation the incremental philosophy requires less communication between the processors making it more suitable for this purpose. Uniform Parallel Implementation: The constructive algorithm that we have just described can be very easily adjusted to work in parallel [8, 9]: Assuming we have N 3 processors we can separate the Bounding Box of S, BB(S), into a NxNxN grid structure and assign each processor to work on one of the N 3 cubes of this grid structure. Each Processor will have each own AFL and also an ability to execute both of the functions we have described above. Each Processor can create a new Tetrahedron if and only if the face upon which the new tetrahedron is going to be created belongs in the cube that this processor is assigned. Duplicates can not occur if we enforce to keep only the tetrahedral whose left up in front vertex belong in the cube that each processor is assigned to. Adaptive Parallel Implementation: Using octrees, in a way we will discuss in the third chapter, we can construct an adaptive covering of the volume of the BB(S) with N cubes. Assigning these cubes to N Processors we can create the Delaunay triangulation in Parallel. The efficiency of such an algorithm is better compared with the Uniform implementation but the Adaptive implementation is more complicated than the Uniform counterpart. In both cases we assume a Shared Memory parallel system, i.e: All the processors can have access to a main memory where the points are being stored. We can implement both of the above implementations such that there is no communication between the processors apart from the termination period when all of the tetrahedral are collected. 12

17 Terminating what we have to say about the Delaunay complex we are going to give two more definitions: An online Delaunay triangulation algorithm is the algorithm which allows the user to add additional points. The Delaunay triangulation is updated so that it is also the triangulation of these additional points. An offline Delaunay triangulation algorithm is the algorithm that does not allow the user to add additional points to update it. It is not hard to see that the Interpolative philosophy create online algorithms and the Incremental philosophy create offline algorithms Elements of Theoretical Differential Geometry For the sake of completeness we will present basic elements of theoretical differential geometry and topology [55, 56]. Topology: Let X be a space. A sequence of open subsets Γ = {Χ i } i Ι of X is called a topology if and only if: i. t X i k Γ, i k a ii. iii. subsequence of elements of the countable space I. subsequence of elements of the countable space I. Χ i I X i. ik I ik I X ik Γ, i k a Το, X Γ, i.e: Topological Space: A space X equipped with a topology Γ is said to be a topological space. Hausdorff Space: A topological space X is called Hausdorff if for every x 1 x 2 X there exist two open areas U 1, U 2 with U 1 U 2 =. A separation of a topological space is called a pair U, V of open, disjoint subsets of X such as U V = Χ. Connected Topological Space: A topological space is called connected if it has no separation. In the sequel we will always assume that the spaces are topological. Continuous Function: We will call a function f: X Y continuous if for every set Β Υ the set A = f -1 (B) = {x X : f(x) B} is open. Homeomorphism: A continuous function f: X Y is called homoemorphism if it is one-to-one, continuous, onto and its inverse function is also continuous. 13

18 Two spaces X, Y will be called homoemorphic if there exists a homoemorphism f : X Y. We will denote such spaces as X Y and we will say that they have similar behavior. k-manifold: A space X will be called k-manifold if every x X has a neighborhood which is homoemorphic to R k. k-manifold with boundary: A space X will be called k-manifold with boundary if every x X has a neighborhood homoemorphic to R k or H k, where H k is a k-dimensional subspace of R k. A k-manifold is also differential when its homoemorphism on R k is also differential Restricted Voronoi diagrams and Restricted Delaunay Triangulation Let M be a surface and S be its sample points. Let Vor(S), Del(S) be the Voronoi diagram and Delaunay triangulation of S. The restriction of Vor(S) on M is called the bisection of the Voronoi diagram with M, we denote this restriction as Vor M (S) = Vor(S) M. We observe that the Vor M (S) creates on M cells. These cells are the bisection of the cells of Vor(S) with M. We denote these cells as Vor M (p) = Vor(p) M, where p is the generator of the cell. The restriction of the Delaunay triangulation of S on M is the set of the triangles σ pqr of Del(S) where Vor M (p) Vor M (q) Vor M (r). In chapter two in order to reconstruct a surface we search for the restriction of the Delaunay triangulation of its sample points onto this surface. In chapter 5 we will search these triangles on a plane in R 2 homoemorphic to a small area on the surface. In Figure we can see an example of a restricted Voronoi region accompanied with the restricted Delaunay triangles. p u q r t Μ s Σχήµα The restriction of a Voronoi Diagram on M. The triangles belong to the restriction of the Delaunay triangulation on Μ k-nearest Neighbors (k-nn) of a point Let S be a set of points and p a point of the space. We call k-nearest neighbors (k-nn) of a point p the k closest points of S to p. The most popular way to find the k-nn of a point is to use the k-d tree structure [59]. An alternative way is to use the Delaunay triangulation [15, 17]. 14

19 Chapter 2 Delaunay Based Surface Reconstruction The three dimensional Delaunay triangulation of the sample point S gives a total triangulation of the sample point S. This means that it completely fills the space of points with triangles, which we do not wish when we want to reconstruct a surface. Many researchers have invented various heuristic techniques such that to discard the triangles that do not belong on the surface, this is similar to the work of a sculptor when he carves into the marble to create the surface he has in mind. Among all the researchers, Amenta [23, 24] first gave a complete mathematical foundation to support her heuristics. Amenta has proposed three different algorithms for surface reconstruction. The first is the crust algorithm [23, 24] which is an extension to the three dimensions of the algorithm she has proposed in [22]. The second is the cone complement [25] (cocone) algorithm which is far more simple than the crust algorithm. She proposes an idea which is utilized ((in)directly) by various others researchers. Dey [26-28] extends the cocone algorithm to support surfaces which have boundary. The third is the power crust algorithm [29-31]. With this algorithm the surface is no longer a subset of the Delaunay triangulation but a subset of the faces of the power diagram (chapter 1). Apart from the algorithms that Amenta created we will also present the algorithm of Boissonat [35, 36]. Boissonat uses the natural neighbors to determine an approximation of the surface using triangles from the Delaunay complex. The whole construction can find similarities to the idea behind the cocone algorithm. We will also present the algorithm of Adamy [34] which uses a subset of the Delaunay complex, the Gabriel graph, for the reconstruction of the surface. This algorithm also has similarities with the idea behind the cocone algorithm. We will also present the algorithm of Muche [32, 52] which uses alpha shapes in order to reconstruct the surface and we will mention some of the enhancements that Teichmann [33] made to Muche s algorithm which fight some of the sampling discrepancies. We will also present the algorithm of Bernardini [37] which uses a pivot sphere traveling on the sample points creating by this way triangles. 15

20 2.1 Basic Theorems and definitions Amenta first introduced basic theorems and premises that establish a strict mathematical methodology for correct surface reconstruction. Her methodology has become a theoretical base of several researchers. We will describe now several some basic results from her work. We have already defined the medial axis of a closed surface M. Local feature size: We define local feature size of a point x in space, lfs(x), the minimum Euclidian distance of this point from the medial of the closed surface M. We will now prove that the local feature size of a point fulfils the Lipschitz relation. Lemma 2.1.1: For two points p, q on M, lfs(p) lfs(q) d(p,q). Proof: We will first prove that lfs(p) lfs(q) d(p,q). If lfs(q) d(p,q) then we have nothing to prove since lfs(p) 0. If lfs(q) > d(p,q) then the sphere with center q and radius lfs(q) contains p. This sphere also contains the sphere with center p and radius lfs(q) d(p,q), see also Figure p d(p,q) q LFS(q) Figure 2.1.1: The sphere (q, lfs(q)) contains the sphere (p, lfs(q) d(p.q)) Since the sphere (p, lfs(q) d(p,q)) is contained into the sphere (q, lfs(q)) it does not contain any point of the medial axis, thus lfs(p) lfs(q) d(p,q). We can also prove following the above methodology that lfs(q) lfs(p) d(p,q). Assuming a sampling S of the surface M, S is an ε-sampling of M if for every x M there exists p S such that: d(p, x) εlfs(x), ε < 1. The definition of an ε-sampling gives a natural way to do the sampling of a surface depending on its curvature. Specifically if the curvature in a point x of M is high then the medial axis approaches closely the surface in a neighborhood around this point. This means that lfs(x) becomes small, thus the sampling around this point must be done densely. On the contrary if the curvature in a point x of M is low then the medial axis is far away from the surface around this point, this means that lfs(x) is big, thus the sampling can be done sparsely around this area. In the next chapter we will see that Gopi [26-28] also gives a way to do sampling that is closely related to ε-sampling. By d(p,q) we will denote the Eucledian distance between the points p, q. 16

21 On each point of the surface M there exist two maximal spheres, which are called medial spheres. These two spheres are on opposite positions of the surface, i.e. the one is inside and the other outside of the surface, and they are tangent to this point. The characteristic of these spheres is that they do have an empty intersection with the surface M, Figure The radius of one of these spheres is equal to the local feature size of the point that these spheres are tangential to. Μ Figure The medial spheres of a point in a surface Μ. We will follow now the following denotation: If we have a point p of S and a point q of the space then, if there is no confusion, we will denote as y the vector py. Also we will denote as n p the normal vector of the surface on the point p. Lemma 2.1.2: The restriction of the Voronoi cell V p of a point p S on the surface ε M is contained into a sphere of radius lfs( p). 1 ε Proof: Let y V p M. Since y V p, p will be the closest point of y. Since y M and S is an ε-sampling, because of Lemma it will hold that d(p, y) ε lfs(y) ε ε (lfs(p) + d(p,y)) d( p, y) lfs( p). 1 ε Lemma 2.1.3: Let υ be a vertex of the Voronoi cell V p. If d(υ, p) lfs(p) then 1 ε ( υ, n p) 2sin. 2 2ε As we have said in the first chapter, the pole of a Voronoi cell is the vertex of the cell that is furthest from the generator of the cell. Lemma 2.1.4: Let υ be the pole of the Voronoi cell V p, then d(υ,p) lfs(p). Combining the above Lemmas we can say that if υ is the pole of V p then υ approximates in order O(ε) the normal vector of the surface on p. So when the sampling is dense enough then ε << 1, meaning that υ is a good approximation of the normal vector and can be used reliably into applications that require this information. 17

22 2.2. The crust of a surface The algorithm that we are going to describe is the generalization in three dimensions of the algorithm that Amenta [23, 24] proposed for the reconstruction of a curve. The basic steps for the reconstruction of a closed curve from its sample points are: 1. Find the Delaunay triangulation of S and with its help the vertices of Vor(S), P. 2. Find the Delaunay triangulation of S P 3. From the edges of Del(S P) we keep those that their vertices belong to S. The edges that we keep from the above algorithm reconstruct the curve, supposing that a dense sampling has been made. We can regard the vertices of the Voronoi diagram as a filter that selects only the vertices that belong on the curve that we want to reconstruct. This algorithm does not extend easily to three dimensions. The reason is that in two dimension the vertices of the Voronoi diagram of a dense sample set S approximates the medial axes of the curve. In three dimensions this does not hold. There may exist Voronoi vertices close enough to the surface even if the surface is quite smooth, these vertices clearly does not belong on the medial axis. Let υ be the pole of a Voronoi cell V p. We will denote this pole as υ +. We define as the anti-diametrical pole υ - of υ + the Voronoi vertex of V p that is further from the + π generator p and ( υ, υ ), see also Figure υ + p Figure The two anti-diametrical poles of Vor(p) in two dimensions The pole υ + is called positive and its anti-diametrical υ - is called negative. It has been proved by Amenta, Lemma 2.1.3, that the poles of the Voronoi cells berlong on the medial axis of the surface that we want to reconstruct. The pole υ + is not defined when the point p belongs on the convex hull. In that case we regard that the pole υ + lies in infinity and is on the ray that passes through p and has the average direction of those Voronoi edges that stretch into infinity. So we choose as υ + the point far away on this ray. The anti-diametric pole υ - is defined as we have already defined. 18

23 We will describe now the crust algorithm. Crust Algorithm 1. Find the Delaunay triangulation D of S using an online algorithm. Calculate from D the Voronoi vertices. 2. For each point p S 1. Calculate the poles υ +, υ - as we have already described Let the normal vector on p be υ, we denote it as n p 3. Let P be the set of the poles that we have found. We insert the point set P into the online triangulation and by this way we find the Delaunay triangulation of S P. 4. Define a connected list Crust_Triangles, which we initialize as NULL. From the triangles of the Delaunay triangulation we insert into the list Crust_Triangles only the ones that have all of their vertices in S. 5. For each triangle Τ from the Crust_Triangles: 1. We calculate the normal vector n T of the triangle T. 2. For each vertex v i, i = 1 3 of T we calculate the acute angle φ i between n vi and n T. We assume that the indices i have an order which satisfies φ 1 φ 2 φ Assuming an angle θ as a threshold we keep the triangle Τ if φ 1,φ 2 θ and φ 3 2.2θ. Otherwise we delete the triangle from the list. 6. Give as output the Crust_Triangles list. The angle θ which we defined in the algorithm is selected by the user interactively. The triangles that we get from the Crust algorithm usually do not define a manifold. In chapter 2.5 we will describe a way to extract a manifold from the triangle list we obtain from an algorithm. If S consists of n elements we calculate the Delaunay triangulation of 3n points. This complexity will be reduced to n point in the algorithm of the next chapter with a slight increase though of the memory required since we will also store the edges of the Voronoi diagram. We must always have in mind that with P we denote the set of υ - and υ +, apart from those υ + whose generator belong on the convex hull of S. 19

24 2.3 Co-Cone based surface reconstruction As we have already mentioned the positive poles of S define the normal vectors of the surface M on these points. We have also in chapter one that on each Voronoi edge corresponds a triangle from the Delaunay triangulation. In general in Surface reconstruction we search the triangles that belong to the restricted Delaunay triangulation on the surface M. When V p,m V q,m V r,m, where with V p,m we denote the restriction of the Voronoi cell on M. This means that the Voronoi edge which is dual to the triangle σ pqr bisects the surface M. The problem in this kind of thinking is that we do not know the surface M, we only know a sampling of it, so we can not directly test if the Voronoi edge bisect the surface. Amenta [25] defined a way to approximately test if the dual Voronoi edge of a Delaunay triangle truly bisects the surface. Boissonat [35, 36] defined a function which isosurface approximates the surface. By this way he can test if the Voronoi edge bisects this isosurface. Amenta instead of the surface tests the triangle that the Voronoi edge is dual to. As we have already said on each point p of S we can define a normal vector equal to + υ. + υ We define as cone complement area of a point p, the complement of the cone whose major axis has the same direction as υ + π and angle. Denote by Cp this area, the 8 following equation holds: 3 Cp = y R : ( y, υ), Figure π Σχήµα 2.3.1: The cone supplement area of p in two and in three dimensions Cone complement criterion (CCC): The triangle σ prs approximately belong to the restricted Delaunay triangulation of the sample set S on the surface M, if for its dual Voronoi edge e holds: C p e, C r e, C s e. This idea is illustrated in Figure

25 ε p q r Figure The cone complement criterion for the Delaunay triangle σ pqr, the blue lines denote the complement cone on each of the triangle vertices. The line ε is the Voronoi edge which is dual to the triangle σ pqr Co-cone based surface reconstruction algorithm 1. Find the Delaunay triangulation D of S 2. Find the poles of S as we did in paragraph 2.2 and the Voronoi edges of each cell 3. Define a connected list Surface_Triangles which we initilalize as NULL. 4. For each triangle Τ of D: 1. Chech if T fulfils the CCC 1. Surface_Triangles += Τ 5. Give as output the Surface_Triangles. We will describe now a practical way which can assist to check the cone complement criterion efficiently. As we have already mentioned, for a triangle σ pqr the cone complement criterion holds when the three complement cone areas defined on each of the vertices have non empty intersection with the Voronoi edge ε, see also Figure In order to test the criterion we can work as Figure dictates, p + υ Figure The three cases that a line + Perpendicular to υ bisects the Voronoi edge ε The parametric form of the Voronoi edge ε(p 1, P 2 ) takes the form: P = (1-λ)P 1 + λp 2. Setting + P υ P 1 = 0 ( P λ υ PP 1 2) = 0 So we have three cases to test: ε P P 2 p + υ φ (α) 0 λ 1 (β) λ < 0 (γ) λ > 1 P 1 P ε P 2 + P1υ λ = + PP υ 1 2 p + υ φ P 1 ε P P 2 21

26 (α) 0 λ 1. In this case the criterion holds. π (β) λ < 0. In this case the criterion holds only if φ = ( P, P1) 8 π (γ) λ > 1. In this case the criterion holds only if φ = ( P, P2) 8 We will now describe various Lemmas and Theorems which can help to extract useful conclusions. Lemma 2.3.1: Let y be a point of the Voronoi cell V p such as y δlfs(p), then 1 ε 1 ε y n p sin + sin. δ (1 ε) (1 ε ) Lemma 2.3.2: Let V p be a Voronoi cell of Vor(S) and V p,μ its restriction on M. For a π point y V p,m the acute angle y n ε, ε 2 p < 0.4. Proof: Since y V p,m belong to V p the closest point of y on M is p. Since y M and S is ε-sampling, py ε lfs(y). Also because of the Lipschitz property of the local lfs( p) ε feature size, lfs(y) lfs(p) + py lfs ( y), thus, py lfs( p). Let 1 ε 1 ε y Vp, M be another (different from y) point of the cell. For this point it will also hold ε 2ε that py lfs( p), thus yy lfs( p). Using Lemma we prove what 1 ε 1 ε we want. Theorem 2.3.1: The triangles of the Delaunay triangulation that belong to the restricted Delaunay triangulation on the surface belong on the complex of triangles we acquire as output from the Co-Cone algorithm if ε 0.4, θ π/8. Proof: Let ε be the Voronoi edge which is dual to a triangle that belongs to the restriction of the Delaunay triangulation on M. Let y be the intersection of M with the Voronoi edge ε (or its extension). Let υ be the positive pole of V p, the surface on p, and α the acute angle υ, see Figure n p n p the normal of np υ α y ε p Figure

27 π It suffices to prove that α + Ο(ε). From Lemma we know that 8 π 1 ε y n p ε. From Lemma we know that υn p 2sin. 2 1 ε So we know now that the output of the Co-Cone algorithm is a superset of the restricted Delaunay triangulation on M. These triangles should be further filtered so as to acquire the restricted Delaunay triangulation. Detecting boundaries Until now we have only described ways to reconstruct only closed surfaces. We have not yet described how to deal with a surface with boundaries. The area of the surface near the boundaries is sharp, this means that we need infinite sampling which we do not have. Dey [26-28] proposed an automated method for the detection of areas which are not sufficiently sampled and thus are possible candidates of areas near a boundary. This method has a constrain, the areas that are not near boundaries must be sampled sufficiently so that no falsely boundaries are detected. Let p S and C p be the cone complement area of the Voronoi cell V p. The set Ν p = { q S : C p V q } is called the set of the cone complement neighbors of p. As we have already mentioned, if S is an ε-sampling then the Voronoi cells are long and skinny. Dey found heuristic parameters to check if the cells are long and skinny. These parameters are the radius and height of the cell. We define as radius of the cell V p the quantity r p = max{ y : y C p }. We can practically find the radius using the methodology described in Figure υ p Convex Hull We define as height of the cell V p the quantity h p = υ p Convex Hull Let ρ, α be two constants, the point p S will be called flat if the following two criterions hold: 1. Ratio Criterion: r p ρh p. 2. Normal Vector Criterion: q where p N q, (,u) poles of V p, V q. υ α, where υ, u are the Typical values for the above constants are ρ = 1.3ε & α = 0.14 rad. Among the above two criterions the most important is the first one because it tests if the cell is long and skinny. The second criterion enforces the smoothness of the surface in the area we reconstruct. 23