Tree Bits

In an earlier post I mentioned that I was using a Space Colonization algorithm to grow trees. This was more about producing an internal representation of the tree, for instance a list of line segments describing the tree trunk and branches. Still somehow the tree needs to materialize into voxels. In this post I will describe how I have done it so far.

If you remember another post, at some point I had decided to go into a full isosurface representation of the world. This means that for any point of the 3D space a density value is defined. If the value is positive, the point is inside a solid. If it is negative, it is outside. And if it is zero, it means this point is in the surface and should be visualized.

The question is, how to create a density function that will result in a tree?

You will find two different entities in your typical tree: the leaves and the branches --which include the trunk as some sort of over-sized branch. It made sense that the tree density function was the sum of two different density functions, one for the leaves and one for the branches.

For the trunk and branches, the colonization algorithm was producing a series of connected segments. This was just a collection of start and end points. First I needed to compute the girth of each branch.


Long time ago Leonardo DaVinci noticed a rule about tree branches. It seems that when a branch splits into two or more branches, the cross-section areas of the new branches add up to the area of the original branch.

At the right you can see a doodle I took from his handbook that illustrates this point.

This is also called the pipe model. It kind of makes sense when you think most trees just need to move their juices up and down.


With this in mind, it is possible to back-track the hierarchy of connected segments starting from the thinnest and adding up the areas down to the trunk. A the end of this process each segment will have its starting and end diameters defined.

That was the easy part, but this is still far from a real density function.

If you remember yet another post, I was computing all the density functions using OpenCL kernels. Each instance of the kernel would take a point in 3D space and compute the density function.

I wrote a new kernel for a density function based on a list of segments. It is fairly straightforward to compute the value, as the following image shows:

Here the large dot is the point in 3D space we need to compute. The line coming out of this point is the shortest distance from the point to the segment. If you subtract the radius of the cone at that position from the length of this line you will end up with the distance from the point to the surface of the cone. This distance is the value of the density function for the field.

The tricky part is making this fast. In theory each point should be tested against each segment. For a high density grids and large number of segments the naive approach is just too slow.

The first obvious optimization is to avoid testing all segments. If only segments that are reasonably close are  tested it would make a big difference. So I stored  the segments in an octree and tested only those segments that were in the neighbor cells to the point.

To speed up access to the octree in OpenCL, I made sure I was moving the octree data into local memory before going over the list of segments. You can see this trick in the ATI Stream OpenCL examples, specifically the one about solving the N-body problem. This case is similar to the N-body, only that some bodies are points are some other are segments.

Now, this produced segments that were far too smooth. For pipes or columns they would be great. Tree trunks and branches are not that regular. I realized I needed to add some noise to the segment surfaces.

This was a rather small change. Instead of using the regular segment radius (noted as "r" in the image before), we can affect this radius by a 3D noise. For this to work, you need to pick the right coordinates to evaluate the noise function. A good approach is to use the point where the shortest line intersects the smooth cone volume, that is, the same point we were using  before to compute the density field. Using these coordinates we evaluate the noise function. This returns a value we can use to offset the radius of the cone and produce a new distance to the volume.

In the following image you can see the effect this produces. By playing with the noise type and properties you can achieve different effects. Instead of a noise function you could also use cellular fields like the ones described by Worley.

This works for branches, roots and trunks. What about the other half of the problem, the leaves?

I chose to address this in a peculiar way. Maybe my eyesight is not alright, but when looked from a distance, a forest looks more like a field of clouds than a massive collection of individual leaves. You can see the leaves as some sort of noise in the profile of the crowns. One of my goals is to have individual objects that are too distant still blend properly in the background, instead of just disappearing from the view.

Now this works only if the trees are healthy and full of leaves. It is a limitation of my approach. I will see how far I can go with this before considering an alternative. It becomes very apparent when you are really close to the leaves that they are just a really noisy volume.

I have an idea to address this, which is to render additional billboards on the client based on the crown polygons. This is similar to what most games do when rendering grass. I would render tree leaves instead.

With that out of the way, we can start thinking about the tree crown as a cloud. Its definition is quite simple, just a list of ellipsoids. When a branch ends, the colonization algorithm inserts a point in the cloud. The diameter of the ellipsoid at that point is a function of the size of the tree and how distant this point is from the main branches.

If enough ellipsoids are created they eventually blend into some sort of cloud. The density function of this cloud is computed in a kernel very similar to the one used for the segments.

Adding a lot of noise helps making the crown more interesting. This noise needs large low frequency amplitudes but also a lot of high frequency detail too. The low frequencies break the appearance of the base ellipsoids while the high frequency resemble the look of the individual leaves.

The following image shows the results:

I know there is a lot of room for improvement. I still plan to revisit the generation of trees. I want more realistic trunks and roots, and also create new types of trees.

As I said before, individual leaves will be rendered by instancing on the client. This should improve a lot the realism of this method. What you have seen here is rather the content generation side of it. I also plan to revisit the global illumination solution to add light scattering in the tree crowns.

Until then, I hope this will inspire you to make some voxel trees of your own.