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 the scene by means of a tree data structure known as a BSP tree.
Originally, this approach was proposed in 3D computer graphics to increase the rendering efficiency by precomputing the BSP tree prior to low-level rendering operations. Some other applications include performing geometrical operations with shapes (constructive solid geometry) in CAD, collision detection in robotics and 3D video games, and other computer applications that involve handling of complex spatial scenes.
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In computer graphics it is desirable that the drawing of a scene be done both correctly and quickly. A simple way to draw a scene is the painter's algorithm: draw it from back to front painting over the background with each closer object. However, that approach is quite limited, since time is wasted drawing objects that will be overdrawn later, and not all objects will be drawn correctly.
Z-buffering can ensure that scenes are drawn correctly and eliminate the ordering step of the painter's algorithm, but it is expensive in terms of memory use. BSP trees will split up objects so that the painter's algorithm will draw them correctly without need of a Z-buffer and eliminate the need to sort the objects; as a simple tree traversal will yield them in the correct order. It also serves as a basis for other algorithms, such as visibility lists, which attempt to reduce overdraw.
The downside is the requirement for a time consuming pre-processing of the scene, which makes it difficult and inefficient to directly implement moving objects into a BSP tree. This is often overcome by using the BSP tree together with a Z-buffer, and using the Z-buffer to correctly merge movable objects such as doors and characters onto the background scene.
BSP trees are often used by 3D video games, particularly first-person shooters and those with indoor environments. Probably the earliest game to use a BSP data structure was Doom (see Doom engine for an in-depth look at Doom's BSP implementation). Other uses include ray tracing and collision detection.
Binary space partitioning is a generic process of recursively dividing a scene into two until the partitioning satisfies one or more requirements. The specific method of division varies depending on its final purpose. For instance, in a BSP tree used for collision detection, the original object would be partitioned until each part becomes simple enough to be individually tested, and in rendering it is desirable that each part be convex so that the painter's algorithm can be used.
The final number of objects will inevitably increase since lines or faces that cross the partitioning plane must be split into two, and it is also desirable that the final tree remains reasonably balanced. Therefore the algorithm for correctly and efficiently creating a good BSP tree is the most difficult part of an implementation. In 3D space, planes are used to partition and split an object's faces; in 2D space lines split an object's segments.
The following picture illustrates the process of partitioning an irregular polygon into a series of convex ones. Notice how each step produces polygons with fewer segments until arriving at G and F, which are convex and require no further partitioning. In this particular case, the partitioning line was picked between existing vertices of the polygon and intersected none of its segments. If the partitioning line intersects a segment, or face in a 3D model, the offending segment(s) or face(s) have to be split into two at the line/plane because each resulting partition must be a full, independent object.
Since the usefulness of a BSP tree depends upon how well it was generated, a good algorithm is essential. Most algorithms will test many possibilities for each partition until they find a good compromise. They might also keep backtracking information in memory, so that if a branch of the tree is found to be unsatisfactory, other alternative partitions may be tried. Thus producing a tree usually requires long computations.
BSP trees are also used to represent natural images. Construction methods for BSP trees representing images were first introduced as efficient representations in which only a few hundred nodes can represent an image that normally requires hundreds of thousands of pixels. Fast algorithms have also been developed to construct BSP trees of images using computer vision and signal processing algorithms. These algorithms, in conjunction with advanced entropy coding and signal approximation approaches, were used to develop image compression methods.
BSP trees are used to improve rendering performance in calculating visible triangles for the painter's algorithm for instance. The tree can be traversed in linear time from an arbitrary viewpoint.
Since a painter's algorithm works by drawing polygons farthest from the eye first, the following code recurses to the bottom of the tree and draws the polygons. As the recursion unwinds, polygons closer to the eye are drawn over far polygons. Because the BSP tree already splits polygons into trivial pieces, the hardest part of the painter's algorithm is already solved - code for back to front tree traversal.[1]
traverse_tree(bsp_tree* tree, point eye) { location = tree->find_location(eye); if(tree->empty()) return; if(location > 0) // if eye in front of location { traverse_tree(tree->back, eye); display(tree->polygon_list); traverse_tree(tree->front, eye); } else if(location < 0) // eye behind location { traverse_tree(tree->front, eye); display(tree->polygon_list); traverse_tree(tree->back, eye); } else // eye coincidental with partition hyperplane { traverse_tree(tree->front, eye); traverse_tree(tree->back, eye); } }
BSP trees divide a region of space into two subregions at each node. They are related to quadtrees and octrees, which divide each region into four or eight subregions, respectively.
| Name | p | s |
|---|---|---|
| Binary Space Partition | 1 | 2 |
| Quadtree | 2 | 4 |
| Octree | 3 | 8 |
where p is the number of dividing planes used, and s is the number of subregions formed.
BSP trees can be used in spaces with any number of dimensions. Quadtrees and octrees are useful for subdividing 2- and 3-dimensional spaces, respectively. Another kind of tree that behaves somewhat like a quadtree or octree, but is useful in any number of dimensions, is the kd-tree.
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