BSP
From DmWiki
A Binary Space Partitioning Tree (or BSP Tree) is a data structure that is used to organize objects within a space. Within the field of computer graphics, it has applications in hidden surface removal and ray tracing. A BSP tree is a recursive sub-division of space that treats each line segment (or polygon, in 3D) as a cutting plane which is used to categorize all remaining objects in the space as either being in "front" or in "back" of that plane. In other words, when a partition is inserted into the tree, it is first categorized with respect to the root node, and then recursively with respect to each appropriate child.
BSP
Overview A Binary Space Partitioning (BSP) tree represents a
recursive, hierarchical partitioning, or subdivision, of
n-dimensional space into convex subspaces. BSP tree construction
is a process which takes a subspace and partitions it by any
hyperplane that intersects the interior of that subspace. The
result is two new subspaces that can be further partitioned by
recursive application of the method.
A "hyperplane" in n-dimensional space is an n-1 dimensional object
which can be used to divide the space into two half-spaces. For
example, in three dimensional space, the "hyperplane" is a plane.
In two dimensional space, a line is used.
BSP trees are extremely versatile, because they are powerful
sorting and classification structures. They have uses ranging from
hidden surface removal and ray tracing hierarchies to solid
modeling and robot motion planning.
Example
An easy way to think about BSP trees is to limit the discussion to
two dimensions. To simplify the situation, let's say that we will
use only lines parallel to the X or Y axis, and that we will
divide the space equally at each node. For example, given a square
somewhere in the XY plane, we select the first split, and thus the
root of the BSP Tree, to cut the square in half in the X
direction. At each slice, we will choose a line of the opposite
orientation from the last one, so the second slice will divide
each of the new pieces in the Y direction. This process will
continue recursively until we reach a stopping point, and looks
like this
+-----------+ +-----+-----+ +-----+-----+
| | | | | | | |
| | | | | | d | |
| | | | | | | |
| a | -> | b X c | -> +--Y--+ f | -> ...
| | | | | | | |
| | | | | | e | |
| | | | | | | |
+-----------+ +-----+-----+ +-----+-----+
The resulting BSP tree looks like this at each step
a X X ...
-/ \+ -/ \+
/ \ / \
b c Y f
-/ \+
/ \
e d
Other space partitioning structures
BSP trees are closely related to Quadtrees and Octrees. Quadtrees
and Octrees are space partitioning trees which recursively divide
subspaces into four and eight new subspaces, respectively. A BSP
Tree can be used to simulate both of these structures.
How do you build a BSP Tree?
Overview
Given a set of polygons in three dimensional space, we want to
build a BSP tree which contains all of the polygons. For now, we
will ignore the question of how the resulting tree is going to be
used.
The algorithm to build a BSP tree is very simple:
1. Select a partition plane.
2. Partition the set of polygons with the plane.
3. Recurse with each of the two new sets.
Choosing the partition plane
The choice of partition plane depends on how the tree will be
used, and what sort of efficiency criteria you have for the
construction. For some purposes, it is appropriate to choose the
partition plane from the input set of polygons. Other applications
may benefit more from axis aligned orthogonal partitions.
In any case, you want to evaluate how your choice will affect the
results. It is desirable to have a balanced tree, where each leaf
contains roughly the same number of polygons. However, there is
some cost in achieving this. If a polygon happens to span the
partition plane, it will be split into two or more pieces. A poor
choice of the partition plane can result in many such splits, and
a marked increase in the number of polygons. Usually there will be
some trade off between a well balanced tree and a large number of
splits.
Partitioning polygons
Partitioning a set of polygons with a plane is done by classifying
each member of the set with respect to the plane. If a polygon
lies entirely to one side or the other of the plane, then it is
not modified, and is added to the partition set for the side that
it is on. If a polygon spans the plane, it is split into two or
more pieces and the resulting parts are added to the sets
associated with either side as appropriate.
When to stop
The decision to terminate tree construction is, again, a matter of
the specific application. Some methods terminate when the number
of polygons in a leaf node is below a maximum value. Other methods
continue until every polygon is placed in an internal node.
Another criteria is a maximum tree depth.
Pseudo C++ code example
Here is an example of how you might code a BSP tree:
struct BSP_tree {
plane partition;
list polygons;
BSP_tree *front,
*back;
};
This structure definition will be used for all subsequent example
code. It stores pointers to its children, the partitioning plane
for the node, and a list of polygons coincident with the partition
plane. For this example, there will always be at least one polygon
in the coincident list: the polygon used to determine the
partition plane. A constructor method for this structure should
initialize the child pointers to NULL.
void Build_BSP_Tree (BSP_tree *tree, list polygons) {
polygon *root = polygons.Get_From_List ();
tree->partition = root->Get_Plane ();
tree->polygons.Add_To_List (root);
list front_list,
back_list;
polygon *poly;
while ((poly = polygons.Get_From_List ()) != 0)
{
int result = tree->partition.Classify_Polygon (poly);
switch (result)
{
case COINCIDENT:
tree->polygons.Add_To_List (poly);
break;
case IN_BACK_OF:
backlist.Add_To_List (poly);
break;
case IN_FRONT_OF:
frontlist.Add_To_List (poly);
break;
case SPANNING:
polygon *front_piece, *back_piece;
Split_Polygon (poly, tree->partition, front_piece, back_piece);
backlist.Add_To_List (back_piece);
frontlist.Add_To_List (front_piece);
break;
}
}
if ( ! front_list.Is_Empty_List ())
{
tree->front = new BSP_tree;
Build_BSP_Tree (tree->front, front_list);
}
if ( ! back_list.Is_Empty_List ())
{
tree->back = new BSP_tree;
Build_BSP_Tree (tree->back, back_list);
}
}
This routine recursively constructs a BSP tree using the above
definition. It takes the first polygon from the input list and
uses it to partition the remainder of the set. The routine then
calls itself recursively with each of the two partitions. This
implementation assumes that all of the input polygons are convex.
One obvious improvement to this example is to choose the
partitioning plane more intelligently. This issue is addressed
separately in the section, "How can you make a BSP Tree more
efficient?".
How do you partition a polygon with a plane?
Overview
Partitioning a polygon with a plane is a matter of determining
which side of the plane the polygon is on. This is referred to as
a front/back test, and is performed by testing each point in the
polygon against the plane. If all of the points lie to one side of
the plane, then the entire polygon is on that side and does not
need to be split. If some points lie on both sides of the plane,
then the polygon is split into two or more pieces.
The basic algorithm is to loop across all the edges of the polygon
and find those for which one vertex is on each side of the
partition plane. The intersection points of these edges and the
plane are computed, and those points are used as new vertices for
the resulting pieces.
Implementation notes
Classifying a point with respect to a plane is done by passing the
(x, y, z) values of the point into the plane equation, Ax + By +
Cz + D = 0. The result of this operation is the distance from the
plane to the point along the plane's normal vector. It will be
positive if the point is on the side of the plane pointed to by
the normal vector, negative otherwise. If the result is 0, the
point is on the plane.
For those not familiar with the plane equation, The values A, B,
and C are the coordinate values of the normal vector. D can be
calculated by substituting a point known to be on the plane for x,
y, and z.
Convex polygons are generally easier to deal with in BSP tree
construction than concave ones, because splitting them with a
plane always results in exactly two convex pieces. Furthermore,
the algorithm for splitting convex polygons is straightforward and
robust. Splitting of concave polygons, especially self
intersecting ones, is a significant problem in its own right.
Pseudo C++ code example
Here is a very basic function to split a convex polygon with a plane
void Split_Polygon (polygon *poly, plane *part, polygon *&front, polygon *&back ) {
int count = poly->NumVertices (),
out_c = 0, in_c = 0;
point ptA, ptB,
outpts[MAXPTS],
inpts[MAXPTS];
real sideA, sideB;
ptA = poly->Vertex (count - 1);
sideA = part->Classify_Point (ptA);
for (short i = -1; ++i < count;)
{
ptB = poly->Vertex (i);
sideB = part->Classify_Point (ptB);
if (sideB > 0)
{
if (sideA < 0)
{
// compute the intersection point of the line
// from point A to point B with the partition
// plane. This is a simple ray-plane intersection.
vector v = ptB - ptA;
real sect = - part->Classify_Point (ptA) / (part->Normal () | v);
outpts[out_c++] = inpts[in_c++] = ptA + (v * sect);
}
outpts[out_c++] = ptB;
}
else if (sideB < 0)
{
if (sideA > 0)
{
// compute the intersection point of the line
// from point A to point B with the partition
// plane. This is a simple ray-plane intersection.
vector v = ptB - ptA;
real sect = - part->Classify_Point (ptA) / (part->Normal () | v);
outpts[out_c++] = inpts[in_c++] = ptA + (v * sect);
}
inpts[in_c++] = ptB;
}
else
outpts[out_c++] = inpts[in_c++] = ptB;
ptA = ptB;
sideA = sideB;
}
front = new polygon (outpts, out_c);
back = new polygon (inpts, in_c);
}
A simple extension to this code that is good for BSP trees is to
combine its functionality with the routine to classify a polygon
with respect to a plane.
Note that this code is not robust, since numerical stability may
cause errors in the classification of a point. The standard
solution is to make the plane "thick" by use of an epsilon value.
How do you remove hidden surfaces with a BSP Tree?
Overview
Probably the most common application of BSP trees is hidden
surface removal in three dimensions. BSP trees provide an elegant,
efficient method for sorting polygons via a depth first tree walk.
This fact can be exploited in a back to front "painter's
algorithm" approach to the visible surface problem, or a front to
back scanline approach.
BSP trees are well suited to interactive display of static (not
moving) geometry because the tree can be constructed as a
preprocess. Then the display from any arbitrary viewpoint can be
done in linear time. Adding dynamic (moving) objects to the scene
is discussed in another section of this document.
Painter's algorithm
The idea behind the painter's algorithm is to draw polygons far
away from the eye first, followed by drawing those that are close
to the eye. Hidden surfaces will be written over in the image as
the surfaces that obscure them are drawn. One condition for a
successful painter's algorithm is that there be a single plane
which separates any two objects. This means that it might be
necessary to split polygons in certain configurations. For
example, this case can not be drawn correctly with a painter's
algorithm
+------+
| |
+---------------| |--+
| | | |
| | | |
| | | |
| +--------| |--+
| | | |
+--| |--------+ |
| | | |
| | | |
| | | |
+--| |---------------+
| |
+------+
One reason that BSP trees are so elegant for the painter's algorithm
is that the splitting of difficult polygons is an automatic part
of tree construction. Note that only one of these two polygons
needs to be split in order to resolve the problem.
To draw the contents of the tree, perform a back to front tree
traversal. Begin at the root node and classify the eye point with
respect to its partition plane. Draw the subtree at the far child
from the eye, then draw the polygons in this node, then draw the
near subtree. Repeat this procedure recursively for each subtree.
Scanline hidden surface removal
It is just as easy to traverse the BSP tree in front to back order
as it is for back to front. We can use this to our advantage in a
scanline method method by using a write mask which will prevent
pixels from being written more than once. This will represent
significant speedups if a complex lighting model is evaluated for
each pixel, because the painter's algorithm will blindly evaluate
the same pixel many times.
The trick to making a scanline approach successful is to have an
efficient method for masking pixels. One way to do this is to
maintain a list of pixel spans which have not yet been written to
for each scan line. For each polygon scan converted, only pixels
in the available spans are written, and the spans are updated
accordingly.
The scan line spans can be represented as binary trees, which are
just one dimensional BSP trees. This technique can be expanded to
a two dimensional screen coverage algorithm using a two
dimensional BSP tree to represent the masked regions. Any convex
partitioning scheme, such as a quadtree, can be used with similar
effect.
Implementation notes
When building a BSP tree specifically for hidden surface removal,
the partition planes are usually chosen from the input polygon
set. However, any arbitrary plane can be used if there are no
intersecting or concave polygons, as in the example above.
Pseudo C++ code example
Using the BSP_tree structure defined in the section, "How do you
build a BSP Tree?", here is a simple example of a back to front
tree traversal:
void Draw_BSP_Tree (BSP_tree *tree, point eye) {
real result = tree->partition.Classify_Point (eye);
if (result > 0)
{
Draw_BSP_Tree (tree->back, eye);
tree->polygons.Draw_Polygon_List ();
Draw_BSP_Tree (tree->front, eye);
}
else if (result < 0)
{
Draw_BSP_Tree (tree->front, eye);
tree->polygons.Draw_Polygon_List ();
Draw_BSP_Tree (tree->back, eye);
}
else // result is 0
{
// the eye point is on the partition plane...
Draw_BSP_Tree (tree->front, eye);
Draw_BSP_Tree (tree->back, eye);
}
}
If the eye point is classified as being on the partition plane, the
drawing order is unclear. This is not a problem if the
Draw_Polygon_List routine is smart enough to not draw polygons
that are not within the viewing frustum. The coincident polygon
list does not need to be drawn in this case, because those
polygons will not be visible to the user.
It is possible to substantially improve the quality of this
example by including the viewing direction vector in the
computation. You can determine that entire subtrees are behind the
viewer by comparing the view vector to the partition plane normal
vector. This test can also make a better decision about tree
drawing when the eye point lies on the partition plane. It is
worth noting that this improvement resembles the method for
tracing a ray through a BSP tree, which is discussed in another
section of this document.
Front to back tree traversal is accomplished in exactly the same
manner, except that the recursive calls to Draw_BSP_Tree occur in
reverse order.
How do you accelerate ray tracing with a BSP Tree?
Overview
Ray tracing a BSP tree is very similar to hidden surface removal
with a BSP tree. The algorithm is a simple forward tree walk, with
a few additions that apply to ray casting.
How do you perform boolean operations on polytopes with a BSP Tree?
Overview
There are two major classes of solid modeling methods with BSP
trees. For both methods, it is useful to introduce the notion of
an in/out test.
An in/out test is a different way of talking about the front/back
test we have been using to classify points with respect to planes.
The necessity for this shift in thought is evident when
considering polytopes instead of just polygons. A point can not be
merely in front or back of a polytope, but inside or outside.
Somewhat formally, a point is inside of a polytope if it is inside
of, or in back of, each hyperplane which composes the polytope,
otherwise it is outside.
Incremental construction
Incremental construction of a BSP Tree is the process of inserting
convex polytopes into the tree one by one. Each polytope has to be
processed according to the operation desired.
It is useful to examine the construction process in two
dimensions. Consider the following figure
A B
+-------------+ | | | | | E | F | +-----+-------+ | | | | | | | | | | | | +-------+-----+ |
D | C |
| |
| |
+-------------+
H G
Two polygons, ABCD, and EFGH, are to be inserted into the tree. We
wish to find the union of these two polygons. Start by inserting
polygon ABCD into the tree, choosing the splitting hyperplanes to
be coincident with the edges. The tree looks like this after
insertion of ABCD
AB
-/ \+
/ \
/ *
BC
-/ \+
/ \
/ *
CD
-/ \+
/ \
/ *
DA
-/ \+
/ \
* *
Now, polygon EFGH is inserted into the tree, one polygon at a time.
The result looks like this
A B
+-------------+ | | | | | E |J F | +-----+-------+ | | | | | | | | | | | | +-------+-----+ |
D |L |C |
| | |
| | |
+-----+-------+
H K G
AB
-/ \+
/ \
/ *
BC
-/ \+
/ \
/ \
CD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
/ * \ \
EJ KH \
-/ \+ -/ \+ \
/ \ / \ \
/ * / * \
LE HL JF
-/ \+ -/ \+ -/ \+
/ \ / \ / \
* * * * FG *
-/ \+
/ \
/ *
GK
-/ \+
/ \
* *
Notice that when we insert EFGH, we split edges EF and HE along the
edges of ABCD. this has the effect of dividing these segments into
pieces which are inside ABCD, and outside ABCD. Segments EJ and LE
will not be part of the boundary of the union. We could have saved
our selves some work by not inserting them into the tree at all.
For a union operation, you can always throw away segments that
land in inside nodes. You must be careful about this though. What
I mean is that any segments which land in inside nodes of side the
pre-existing tree, not the tree as it is being constructed. EJ and
LE landed in an inside node of the tree for polygon ABCD, and so
can be discarded.
Our tree now looks like this
A B
+-------------+ | | | | | |J F | +-------+ | | | | | | | | | +-------+-----+ |
D |L |C |
| | |
| | |
+-----+-------+
H K G
AB
-/ \+
/ \
/ *
BC
-/ \+
/ \
/ \
CD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
* * \ \
KH \
-/ \+ \
/ \ \
/ * \
HL JF
-/ \+ -/ \+
/ \ / \
* * FG *
-/ \+
/ \
/ *
GK
-/ \+
/ \
* *
Now, we would like some way to eliminate the segments JC and CL, so
that we will be left with the boundary segments of the union.
Examine the segment BC in the tree. What we would like to do is
split BC with the hyperplane JF. Conveniently, we can do this by
pushing the BC segment through the node for JF. The resulting
segments can be classified with the rest of the JF subtree. Notice
that the segment BJ lands in an out node, and that JC lands in an
in node. Remembering that we can discard interior nodes, we can
eliminate JC. The segment BJ replaces BC in the original tree.
This process is repeated for segment CD, yielding the segments CL
and LD. CL is discarded as landing in an interior node, and LD
replaces CD in the original tree. The result looks like this
A B
+-------------+ | | | | | |J F | +-------+ | | | | | L | +-------+ |
D | |
| |
| |
+-----+-------+
H K G
AB
-/ \+
/ \
/ *
BJ
-/ \+
/ \
/ \
LD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
* * \ \
KH \
-/ \+ \
/ \ \
/ * \
HL JF
-/ \+ -/ \+
/ \ / \
* * FG *
-/ \+
/ \
/ *
GK
-/ \+
/ \
* *
As you can see, the result is the union of the polygons ABCD and EFGH.
To perform other boolean operations, the process is similar. For
intersection, you discard segments which land in exterior nodes
instead of internal ones. The difference operation is special. It
requires that you invert the polytope before insertion. For simple
objects, this can be achieved by scaling with a factor of -1. The
insertion process is then cinducted as an intersection operation,
where segments landing in external nodes are discarded.
How do you perform collision detection with a BSP Tree?
Overview
Detecting whether or not a point moving along a line intersects
some object in space is essentially a ray tracing problem.
Detecting whether or not two complex objects intersect is
something of a tree merging problem.
Typically, motion is computed in a series of Euler steps. This
just means that the motion is computed at discrete time intervals
using some description of the speed of motion. For any given point
P moving from point A with a velocity V, it's location can be
computed at time T as P = A + (T * V).
Consider the case where T = 1, and we are computing the motion in
one second steps. To find out if the point P has collided with any
part of the scene, we will first compute the endpoints of the
motion for this time step. P1 = A + V, and P2 = A + (2 * V). These
two endpoints will be classified with respect to the BSP tree. If
P1 is outside of all objects, and P2 is inside some object, then
an intersection has clearly occurred. However, if P2 is also
outside, we still have to check for a collision in between.
Two approaches are possible. The first is commonly used in
applications like games, where speed is critical, and accuracy is
not. This approach is to recursively divide the motion segment in
half, and check the midpoint for containment by some object.
Typically, it is good enough to say that an intersection occurred,
and not be very accurate about where it occurred.
The second approach, which is more accurate, but also more time
consuming, is to treat the motion segment as a ray, and intersect
the ray with the BSP Tree. This also has the advantage that the
motion resulting from the impact can be computed more accurately.
How do you handle dynamic scenes with a BSP Tree?
Overview
So far the discussion of BSP tree structures has been limited to
handling objects that don't move. However, because the hidden
surface removal algorithm is so simple and efficient, it would be
nice if it could be used with dynamic scenes too. Faster animation
is the goal for many applications, most especially games.
The BSP tree hidden surface removal algorithm can easily be
extended to allow for dynamic objects. For each frame, start with
a BSP tree containing all the static objects in the scene, and
reinsert the dynamic objects. While this is straightforward to
implement, it can involve substantial computation.
If a dynamic object is separated from each static object by a
plane, the dynamic object can be represented as a single point
regardless of its complexity. This can dramatically reduce the
computation per frame because only one node per dynamic object is
inserted into the BSP tree. Compare that to one node for every
polygon in the object, and the reason for the savings is obvious.
During tree traversal, each point is expanded into the original
object.
Implementation notes
Inserting a point into the BSP tree is very cheap, because there
is only one front/back test at each node. Points are never split,
which explains the requirement of separation by a plane. The
dynamic object will always be drawn completely in front of the
static objects behind it.
A dynamic object inserted into the tree as a point can become a
child of either a static or dynamic node. If the parent is a
static node, perform a front/back test and insert the new node
appropriately. If it is a dynamic node, a different front/back
test is necessary, because a point doesn't partition three
dimesnional space. The correct front/back test is to simply
compare distances to the eye. Once computed, this distance can be
cached at the node until the frame is drawn.
An alternative when inserting a dynamic node is to construct a
plane whose normal is the vector from the point to the eye. This
plane is used in front/back tests just like the partition plane in
a static node. The plane should be computed lazily and it is not
necessary to normalize the vector.
Cleanup at the end of each frame is easy. A static node can never
be a child of a dynamic node, since all dynamic nodes are inserted
after the static tree is completed. This implies that all subtrees
of dynamic nodes can be removed at the same time as the dynamic
parent node.
(note) I could'nt format this properly within the wiki and shall try to fix it at some other point in time.
