Binary space partitioning

In computer science, binary space partitioning (BSP) is a method for recursively subdividing a space into convex sets by hyperplanes. This subdivision gives rise to a representation of objects within the space by means of a tree data structure known as a BSP tree.

Binary space partitioning was developed in the context of 3D computer graphics,[1][2] where the structure of a BSP tree allows spatial information about the objects in a scene that is useful in rendering, such as their ordering from front-to-back with respect to a viewer at a given location, to be accessed rapidly. Other applications include performing geometrical operations with shapes (constructive solid geometry) in CAD,[3] collision detection in robotics and 3-D video games, ray tracing and other computer applications that involve handling of complex spatial scenes.

Overview

Binary space partitioning is a generic process of recursively dividing a scene into two until the partitioning satisfies one or more requirements. It can be seen as a generalisation of other spatial tree structures such as k-d trees and quadtrees, one where hyperplanes that partition the space may have any orientation, rather than being aligned with the coordinate axes as they are in k-d trees or quadtrees. When used in computer graphics to render scenes composed of planar polygons, the partitioning planes are frequently (but not always) chosen to coincide with the planes defined by polygons in the scene.

The specific choice of partitioning plane and criterion for terminating the partitioning process varies depending on the purpose of the BSP tree. For example, in computer graphics rendering, the scene is divided until each node of the BSP tree contains only polygons that can render in arbitrary order. When back-face culling is used, each node therefore contains a convex set of polygons, whereas when rendering double-sided polygons, each node of the BSP tree contains only polygons in a single plane. In collision detection or ray tracing, a scene may be divided up into primitives on which collision or ray intersection tests are straightforward.

Binary space partitioning arose from the computer graphics need to rapidly draw three-dimensional scenes composed of polygons. A simple way to draw such scenes is the painter's algorithm, which produces polygons in order of distance from the viewer, back to front, painting over the background and previous polygons with each closer object. This approach has two disadvantages: time required to sort polygons in back to front order, and the possibility of errors in overlapping polygons. Fuchs and co-authors[2] showed that constructing a BSP tree solved both of these problems by providing a rapid method of sorting polygons with respect to a given viewpoint (linear in the number of polygons in the scene) and by subdividing overlapping polygons to avoid errors that can occur with the painter's algorithm. A disadvantage of binary space partitioning is that generating a BSP tree can be time-consuming. Typically, it is therefore performed once on static geometry, as a pre-calculation step, prior to rendering or other realtime operations on a scene. The expense of constructing a BSP tree makes it difficult and inefficient to directly implement moving objects into a tree.

BSP trees are often used by 3D video games, particularly first-person shooters and those with indoor environments. Game engines utilising BSP trees include the Doom engine (probably the earliest game to use a BSP data structure was Doom), the Quake engine and its descendants. In video games, BSP trees containing the static geometry of a scene are often used together with a Z-buffer, to correctly merge movable objects such as doors and characters onto the background scene. While binary space partitioning provides a convenient way to store and retrieve spatial information about polygons in a scene, it does not solve the problem of visible surface determination.

Generation

The canonical use of a BSP tree is for rendering polygons (that are double-sided, that is, without back-face culling) with the painter's algorithm. Each polygon is designated with a front side and a back side which could be chosen arbitrarily and only affects the structure of the tree but not the required result.[2] Such a tree is constructed from an unsorted list of all the polygons in a scene. The recursive algorithm for construction of a BSP tree from that list of polygons is:[2]

  1. Choose a polygon P from the list.
  2. Make a node N in the BSP tree, and add P to the list of polygons at that node.
  3. For each other polygon in the list:
    1. If that polygon is wholly in front of the plane containing P, move that polygon to the list of nodes in front of P.
    2. If that polygon is wholly behind the plane containing P, move that polygon to the list of nodes behind P.
    3. If that polygon is intersected by the plane containing P, split it into two polygons and move them to the respective lists of polygons behind and in front of P.
    4. If that polygon lies in the plane containing P, add it to the list of polygons at node N.
  4. Apply this algorithm to the list of polygons in front of P.
  5. Apply this algorithm to the list of polygons behind P.

The following diagram illustrates the use of this algorithm in converting a list of lines or polygons into a BSP tree. At each of the eight steps (i.-viii.), the algorithm above is applied to a list of lines, and one new node is added to the tree.

Start with a list of lines, (or in 3-D, polygons) making up the scene. In the tree diagrams, lists are denoted by rounded rectangles and nodes in the BSP tree by circles. In the spatial diagram of the lines, the direction chosen to be the 'front' of a line is denoted by an arrow.
i. Following the steps of the algorithm above,
  1. We choose a line, A, from the list and,...
  2. ...add it to a node.
  3. We split the remaining lines in the list into those in front of A (i.e. B2, C2, D2), and those behind (B1, C1, D1).
  4. We first process the lines in front of A (in steps ii–v),...
  5. ...followed by those behind (in steps vi–vii).
ii. We now apply the algorithm to the list of lines in front of A (containing B2, C2, D2). We choose a line, B2, add it to a node and split the rest of the list into those lines that are in front of B2 (D2), and those that are behind it (C2, D3).
iii. Choose a line, D2, from the list of lines in front of B2. It is the only line in the list, so after adding it to a node, nothing further needs to be done.
iv. We are done with the lines in front of B2, so consider the lines behind B2 (C2 and D3). Choose one of these (C2), add it to a node, and put the other line in the list (D3) into the list of lines in front of C2.
v. Now look at the list of lines in front of C2. There is only one line (D3), so add this to a node and continue.
vi. We have now added all of the lines in front of A to the BSP tree, so we now start on the list of lines behind A. Choosing a line (B1) from this list, we add B1 to a node and split the remainder of the list into lines in front of B1 (i.e. D1), and lines behind B1 (i.e. C1).
vii. Processing first the list of lines in front of B1, D1 is the only line in this list, so add this to a node and continue.
viii. Looking next at the list of lines behind B1, the only line in this list is C1, so add this to a node, and the BSP tree is complete.

The final number of polygons or lines in a tree is often larger (sometimes much larger[2]) than the original list, since lines or polygons that cross the partitioning plane must be split into two. It is desirable to minimize this increase, but also to maintain reasonable balance in the final tree. The choice of which polygon or line is used as a partitioning plane (in step 1 of the algorithm) is therefore important in creating an efficient BSP tree.

Traversal

A BSP tree is traversed in a linear time, in an order determined by the particular function of the tree. Again using the example of rendering double-sided polygons using the painter's algorithm, to draw a polygon P correctly requires that all polygons behind the plane P lies in must be drawn first, then polygon P, then finally the polygons in front of P. If this drawing order is satisfied for all polygons in a scene, then the entire scene renders in the correct order. This procedure can be implemented by recursively traversing a BSP tree using the following algorithm.[2] From a given viewing location V, to render a BSP tree,

  1. If the current node is a leaf node, render the polygons at the current node.
  2. Otherwise, if the viewing location V is in front of the current node:
    1. Render the child BSP tree containing polygons behind the current node
    2. Render the polygons at the current node
    3. Render the child BSP tree containing polygons in front of the current node
  3. Otherwise, if the viewing location V is behind the current node:
    1. Render the child BSP tree containing polygons in front of the current node
    2. Render the polygons at the current node
    3. Render the child BSP tree containing polygons behind the current node
  4. Otherwise, the viewing location V must be exactly on the plane associated with the current node. Then:
    1. Render the child BSP tree containing polygons in front of the current node
    2. Render the child BSP tree containing polygons behind the current node

Applying this algorithm recursively to the BSP tree generated above results in the following steps:

The tree is traversed in linear time and renders the polygons in a far-to-near ordering (D1, B1, C1, A, D2, B2, C2, D3) suitable for the painter's algorithm.

Timeline

See also

References

  1. 1 2 Schumacker, Robert A.; Brand, Brigitta; Gilliland, Maurice G.; Sharp, Werner H (1969). Study for Applying Computer-Generated Images to Visual Simulation (Report). U.S. Air Force Human Resources Laboratory. p. 142. AFHRL-TR-69-14.
  2. 1 2 3 4 5 6 7 Fuchs, Henry; Kedem, Zvi. M; Naylor, Bruce F. (1980). "On Visible Surface Generation by A Priori Tree Structures". SIGGRAPH '80 Proceedings of the 7th annual conference on Computer graphics and interactive techniques. ACM, New York. pp. 124–133. doi:10.1145/965105.807481.
  3. 1 2 Thibault, William C.; Naylor, Bruce F. (1987). "Set operations on polyhedra using binary space partitioning trees". SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques. ACM, New York. pp. 153–162. doi:10.1145/37402.37421.

Additional references

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