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Quadtrees and Mesh Generation Student Lecture in course MATH/CSC 870 Philipp Richter Thursday, April 19th, 2007 Content of Lecture 1) 2) 3) 4) 5) Motivation Introduction to Meshes Introduction to Quadtrees Generating Meshes using Quadtrees Looking further Motivation We will start in the context of VLSI layouts Definition: Very-large-scale integration (VLSI) is the process of creating integrated circuits by combining thousands of transistor-based circuits into a single chip. A VLSI layout of a chip Motivation We will start in the context of VLSI layouts Common Problem in VLSI layout design: Because components are very close together and emit heat, there is the possibility that they influence each other. This can be very hazardous to the function of the chip and can even result in its destruction. Infrared thermal image of a chip Motivation We will start in the context of VLSI layouts There are different ways to solve the problem: Earlier, engineers built a prototype of the chip and then measured the heat emission. Of course, this takes time and money. Today, given modern computing power and knowledge, we can simulate the heat process using finite-element-methods. Motivation We will start in the context of VLSI layouts Finite-element-methods involve subdividing the chip surface in small elements and then assigning a heat emission value to each. It is assumed to be known how neighboring elements influence each other. It is a numerical process that requires the elements to have a triangular shape with angles not below 45° to provide a fast convergence. Introduction to Meshes In our application we seek a triangular mesh. We limit our discussion to the following input: • It is a square with disjoint polygonal components inside (called the domain) • The square has vertices at (0,0), (0,U), (U,0), (U,U), where U is a power of 2 • Coordinates of the components are integers between 0 and U • Orientation of component edges is limited to 0°, 45°, 90° and 135° Introduction to Meshes In our application we seek a triangular mesh. A mesh on some input data Introduction to Meshes In our application we seek a triangular mesh. Now our goal is to compute a triangular mesh of the input with the following properties: • Mesh must be conforming: a triangle must not have vertices of other triangles in the interior of one of its edges • Mesh must respect the input: edges of the components must be contained in the mesh • Mesh triangles must be well-shaped: angles of the triangles must be between 45° and 90° • Mesh must be non-uniform: it must be fine near the edges of the components and coarse far away Introduction to Meshes In our application we seek a triangular mesh. uniform mesh non-uniform mesh Introduction to Meshes In our application we seek a triangular mesh. Why not use a Delaunay Triangulation? It could yield triangles that are not well shaped because in a triangulation we are only allowed to use the input vertices. In a mesh, we can add so called Steiner Points that help us getting a mesh with our desired properties. delaunay triangulation Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Definition: A Quadtree is a tree data structure in which each internal node has up to four children. Every node in the Quadtree corresponds to a square. If a node v has children, then their corresponding squares are the four quadrants of the square of v. Quadtree Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Definition: The leaves of a Quadtree form a Quadtree Subdivision of the square of the root. Definition: The children of a node are labelled NE, NW, SW, and SE to indicate to which quadrant they correspond; NE stands for the north-east quadrant, NW for the north-west quadrant, and so on. Definition: The four vertices at the corners of each square are called corner vertices or corners. Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Definition: Line segments connecting corners are called sides of the square Definition: Line segments contained in the boundary of a square are called edges of the square. Therefore each side contains at least one edge. Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Definition: Quadtree for a set P of points inside a square : : [ x : x' ][ y : y' ] If card(P) <= 1 then the Quadtree consists of a single leaf where the set P and the square are stored. Otherwise, let NE , NW , SW , and SE denote the four x x' y y ' x : y : quadrants of . Let mid and mid , and 2 2 define PNE : p P : px xmid and py ymid , PNW : p P : px xmid and p y ymid , PSW : p P : px xmid and py ymid , PSE : p P : px xmid and p y ymid . Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. The Quadtree now consists of a root node v where the square is stored. Below we shall denote the square stored at v by v . Furthermore, v has four Children: the NE-child is the root of a Quadtree for the set PNE inside the square NE , the NW-child is the root of a Quadtree for the set PNW inside the square NW , the SW-child is the root of a Quadtree for the set PSW inside the square SW , the SE-child is the root of a Quadtree for the set PSE inside the square SE . Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. A recursive algorithm is immediately given by the definition: split the current square into four quadrants, partition the point set accordingly, and recursively construct Quadtrees for each quadrant with its associated point set. http://www.cs.wustl.edu/~suri/cs506/projects/quad.html Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Lemma: The depth of a Quadtree for a set P of points in the plane is at most logs c 32 , where c is the smallest distance between any two points in P and s is the side length of the initial square that contains P. Theorem: A Quadtree of depth d storing a set of n points has Od 1 n nodes and can be constructed in Od 1 n time. Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Algorithm NorthNeighbor(v, T) Input. A node v in a Quadtree T. Output. The deepest node v’ whose depth is at most the depth of v such that v' is a north-neighbor of v n and nil if there is no such node. 1. if v = root(T) then return nil 2. if v = SW-child of parent(v) then return NW-child of parent(v) 3. if v = SE-child of parent(v) then return NE-child of parent(v) 4. NorthNeighbor(parent(v), T) 5. if = nil or is leaf 6. then return 7. else if v = NW-child of parent(v) 8. then return SW-child of 9. else return SE-child of Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. The algorithms WestNeighbor(v, T), SouthNeighbor(v, T) and EastNeighbor(v, T) are similar. Theorem: Let T be a Quadtree of depth d. The neighbour of a given node v in T in a given direction, as defined above, can be found in Od 1 time. Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Algorithm BalanceQuadTree(T) Input. A Quadtree T. Output. A balanced version of T. 1. Insert all the leaves of T into a linear list L. 2. while L is not empty 3. do Remove a leaf from L 4. if has to be split 5. then Make into an internal node with four children, which are leaves that correspond to the four quadrants of . If stores a point, then store the point in the correct new leaf instead. 6. Insert the four new leaves into L. 7. Check if had neighbors that now need to be split and, if so, insert them into L. 8. return T Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Introduction to Quadtrees We will see later that these are at the very heart of our Mesh generation method. Theorem: Let T be a Quadtree with m nodes. Then the balanced version of T has Om nodes and it can be constructed in Od 1 m time. Generating Meshes using Quadtrees Now we can apply what we learned about Quadtrees. Algorithm GenerateMesh(S) Input. A set S of components inside the square [0 : U ] [0 : U ] with the properties stated in the introduction to meshes. Output. A triangular Mesh M that is conforming, respects the input, consists of wellshaped triangles, and is non-uniform. 1. Construct a Quadtree T on the set S inside the square [0 : U ] [0 : U ] with the following stopping criterion: a square is split as long as it is larger than unit size and its closure intersects the boundary of some component. 2. T BalanceQuadTree(T) 3. Construct the doubly-connected edge list for the Quadtree subdivision M corresponding to T. 4. for each face of M 5. do if the interior of is intersected by an edge of a component 6. then Add the intersection (which is a diagonal) as an edge to M. 7. else if has only vertices at its corners. 8. then Add a diagonal of as an edge to M. 9. else Add a Steiner point in the center of , connect it to all vertices on the boundary of , and change M accordingly. 10. return M Generating Meshes using Quadtrees Now we can apply what we learned about Quadtrees. Theorem: Let S be a set of disjoint polygonal components inside the square [0 : U ] [0 : U ] with the properties stated in the beginning of this section. Then there exists a non-uniform triangular mesh for this input that is conforming, respects the input, and has only well-shaped triangles. The number of triangles is O pS logU , where pS is the sum of the perimeters of the components in S, and the mesh can be constructed in O pS log2 U time. Looking further There is more than Mesh generation for Quadtrees Mesh generation is but one application of Quadtrees. They are also used in computer graphics, image analysis, geographic information systems, and many other areas. Typically they are used to answer range queries, but they can be used for other operations as well. Quadtrees have also been applied to hidden surface removal, ray tracing, medial axis transforms, map overlay of raster maps, and nearest neighbour finding. Looking further There is more than Mesh generation for Quadtrees What is a Range Query? Input description: A set S of n points and a query region Q. Problem description: Which points from S lie within Q? Looking further There is more than Mesh generation for Quadtrees There is a distinction between unstructured and structured Meshes. From a mathematical perspective, Meshes are called grids. An unstructured grid is a tessellation of a part of the Euclidean plane or Euclidean space by simple shapes, such as triangles or tetrahedra, in an irregular pattern. A structured grid is a tessellation of the Euclidean plane by congruent quadrilaterals or a space-filling tessellation of rectilinear cuboids. unstructured grid structured grid Looking further There is more than Mesh generation for Quadtrees The generalization of the Quadtree to 3 dimensions is the so called Octree, where each node has 8 children. Minimizing the number of triangles in the mesh under certain conditions is relevant and was subject to recent research. There has been work done on the control of the density of triangles in the mesh. Bibliography - de Berg, M., M. van Kreveld, M. Overmars, and O.Schwarzkopf.Computational Geometry. 2nd ed. Eindhoven (NL), Utrecht (NL): Springer-Verlag, 2000. - "Quadtree." Wikipedia. 17 Apr 2007 <http://en.wikipedia.org/wiki/Quadtree>. - "Very-large-scale integration." Wikipedia. 17 Apr 2007 <http://en.wikipedia.org/wiki/Very-large-scale_integration>. - "Unstructured grid." Wikipedia. 17 Apr 2007 <http://en.wikipedia.org/wiki/Unstructured_grid>. - "Regular grid." Wikipedia. 17 Apr 2007 <http://en.wikipedia.org/wiki/Regular_grid>. Thank you!