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Continuous LOD Terrain Meshing Using Adaptive Quadtrees
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Continuous LOD Terrain Meshing Using Adaptive Quadtrees

February 28, 2000 Article Start Previous Page 2 of 4 Next


My meshing algorithm is based on [1], which has also influenced [2] and [3]. There are a few key modifications, but much of the basic approach is the same, and I borrow a lot of the [1] terminology.

There are two parts to meshing. I call the first part Update() and the second part Render(), after [1]. During Update(), we'll decide which vertices to include in the output mesh. Then, during Render() we'll generate a triangle mesh that includes those vertices. I'll start by explaining Update() and Render() for an extremely simple heightfield: a 3x3 grid (Figure 2). To Update() it, we'll look at each of the optional vertices and decide whether to include them in the mesh. Following the terminology of [1], we'll say that if and only if a vertex is "enabled", then we'll use it in the mesh.


Figure 2. A 3x3 heightfield. Dashed lines and vertices are optional in an LOD mesh.

Take as given that the center and corner vertices are enabled. So the task is to calculate the enabled state for each of the four edge vertices, according to some LOD calculation which takes the viewpoint and the vertex attributes into account.

Once we know which vertices are enabled, we can Render() the mesh. It's easy; we just make a triangle fan with the center vertex as the hub, and include each enabled vertex in clockwise order around the outside. See Figure 3 for examples.

Figure 3. Examples of LOD meshes on the 3x3 heightfield.
Disabled vertices in black.

To Update() and Render() an adaptive quadtree heightfield, we extend the above process by starting with that same 3x3 square and recursively subdividing it. By subdividing, we can introduce new vertices, and treat them like we treated the vertices of the original square. In order to prevent cracks, however, we'll have to observe some rules.

First, we can subdivide any combination of the four quadrants. When we subdivide a quadrant, we'll treat the quadrant as a sub-square, and enable its center vertex. For mesh consistency, we will also have to enable the edge vertices of the parent square which are corners of the quadrant (Figure 4). We'll define enabling a square to imply the enabling of its center vertex as well as those corners.

Figure 4. Subdividing the NE quadrant of a square. The gray vertices are already known to be enabled, but the black vertices must be enabled when we subdivide.

Next, notice that an edge vertex in a sub-square is shared with a neighboring sub-square (except at the outside edges of our terrain). So when we enable an edge vertex, we will have to make sure that the neighboring sub-square which shares that vertex is also enabled (Figure 5). Enabling this neighbor square can in turn cause other vertices to be enabled, potentially propagating enabled flags through the quadtree. This propagation is necessary to ensure mesh consistency. See [1] for a good description of these dependency rules.

Figure 5. While updating the NE quadrant, we decide to enable the black vertex. Since that vertex is also shared by the SE quadrant (marked in gray), we must enable that quadrant also. Enabling the SE quadrant will in turn force us to enable the gray vertices.

After we're done with the Update(), we can Render() the quadtree. Rendering is actually pretty simple; the complicated consistency stuff was taken care of in Update(). The basic strategy is to recursively Render() any enabled sub-squares, and then render any parts of the square which weren't covered by enabled sub-squares. (Figure 6) shows an example mesh.

Figure 6. An example mesh. Enabled vertices are marked in black. The gray triangles are drawn by recursive calls to Render() on the associated sub-squares. The white triangle are drawn by the original call to Render().

 Evaluating vertices and squares

 In the above description, I glossed over the part about deciding whether a vertex should be enabled. There are a few different ways to do this. All of them take into account what I'll call the "vertex interpolation error", or vertex error for short. What this is, is the difference in height between the correct location of a vertex, and the height of the edge in the triangle which approximates the vertex when the vertex is disabled (Figure 7). Vertices which have a large error should be enabled in preference to vertices which have a small error. The other key variable that goes into the vertex enable test is the distance of the vertex from the viewpoint. Intuitively, given two vertices with the same error, we should enable the closer one before we enable the more distant one.

Figure 7. Vertex interpolation error. When a vertex is enabled
or disabled, the mesh changes shape. The maximum change
occurs at the enabled vertex's position, shown by the dashed
line. The magnitude of the change is the difference between
the true height of the vertex (black) and the height of the
original edge below the vertex (white). The white point is just
the average of the two gray points.

There are other factors that can be included as well. [1] for instance takes into account the direction from the viewpoint to the vertex. The justification is based on the idea of screen-space geometric error; intuitively the vertex errors are less visible when the view direction is more vertical. [1] goes through the math in detail.

However, I don't think screen-space geometric error is a particularly good metric, for two reasons. One, it ignores texture perspective and depth buffering errors -- even if a vertex does not move in screen space because the motion is directly towards or away from the viewpoint, the vertex's view-space z value does affect perspective-correction as well as depth-buffering. Two, the viewpoint-straight-down case is both an easy case for terrain LOD algorithms, and not a typical case.

In my opinion, there's no point in optimizing for an atypical easy case in an interactive system. The performance of the more typical and difficult case, when the view axis is more horizontal and much more terrain is visible, will determine the minimum system frame-rate and hence the effectiveness of the algorithm.

Instead of screen-space geometric error, I advocate doing a similar test which results in 3D view-space error proportional to view distance. It's really very similar to the screen-space-error test, but without the issues I mention above. It involves only three quantities: an approximation of the viewpoint-vertex distance called the L1-norm, the vertex error, and a detail threshold constant. Here it is:

L1 = max(abs(vertx - viewx), abs(verty - viewy), abs(vertz -  viewz));
enabled = error * Threshold < L1;

You probably recognize the L1-norm, even if you didn't know it had a fancy name. In practice, using the L1-norm instead of the true viewpoint distance will result in slightly more subdivision along the diagonals of the horizontal terrain plane. I've never been able to detect this effect by eye, so I don't worry much about it. [4] and others use view-space-z rather than the L1-norm, which is theoretically even more appropriate than true viewpoint distance. Nevertheless, the L1-norm works like a champ for me, and [3] uses it too.

You can treat the Threshold quantity as an adjust-for-best-results slider, but it does have an intuitive geometric interpretation. Roughly, it means: for a given view distance z, the worst vertex error I'll tolerate is z / Threshold. You could do some view-angle computations and relate Threshold to maximum pixel error, but I've personally never gone past the adjust-for-best-results stage.

So that covers the vertex enabled test. But if you were paying attention earlier, you may also have noticed that I glossed over another point, perhaps more important: during Update(), how do we know whether to subdivide a quadrant or not? The answer is to do what I call a "box test". The box test asks the question: given an axis-aligned 3D box enclosing a portion of terrain (i.e. a quadtree square), and the maximum vertex error contained within that box, and no other information about what's inside the box, is it possible that the vertex enable test would return true? If so, then we should subdivide the box. If not, then there's no reason to subdivide.

The beauty of it is, by doing the box test, we can potentially trim out thousands of vertices from consideration in one fell swoop. It makes Update() completely scalable: its cost is not related to the size of the full dataset, only to the size of the actual data that's included in the current LOD mesh. And as a side benefit, the precomputed vertical box extent can be used during Render() for frustum culling.

The box test is conservative, in that a square's max-error could be for a vertex on the opposite side of the box from the viewpoint, and thus the vertex test itself would/will fail for that actual vertex, whereas the box test might succeed. But once we subdivide, e'll go ahead and do four more, more accurate box tests on the sub-squares, and the penalty for conservatism is fairly small: a few extra vertex and box tests, and a couple extra vertices in the mesh.

Fortunately, given the above simple vertex test, a suitable box test is easy to formulate:

bc[x,y,z] == coordinates of box center
ex[x,y,z] == extent of box from the center (i.e. 1/2 the box dimensions)
L1 = max(abs(bcx - viewx) - exx, abs(bcy - viewy) - exy, abs(bcz - viewz) - exz)
enabled = maxerror * Threshold < L1

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