Creating the tree trunk surfaces is an exercise in controlled noise. We need to create a map for radius and for theta, which will give us a large cylindrical mesh.

We start with a method for producing fissures. This operates on an initially uniform gray map of probabilities. A randomly selected point is either selected as fissure or not fissure, according to the probability value there. The probabilities for neighbours within a 5x7 window around the pixel are then altered. In either case, neighbours in the horizontal direction become less likely to make the same decision, and vertical neighbours become more likely to make the same decision.

Running this until most pixels have made a decision produces a fairly good map of fissuring, in a restricted range of frequencies. Here is such a map upsampled to 400x400.

To add fissures at other frequencies, we repeat the algorithm on a pyramid of initially gray images, taking a geometrically weighted sum of the results, favoring lower frequencies.


This nearly gives us our final map of radius. All that remains is to flare out the base of the trunk. This is done by adding a random selection of sin curves with amplitude inversely proportional to frequency and height above the ground.

We start with a uniform gradient image for our map of theta. We wish to exaggerate the concavities already present in the radius map, so we add to this the horizontal derivative of the absolute value of the horizontal derivative of the radius map. This function has the following effect on points.


Below is the theta offset map for the above noise.


The actual amount we modify theta by is quite small, as shown by the final theta map, which is nearly a smooth gradient.


These maps determine the geometry of our tree trunk. To this we add a texture map. The texture is generated as geometrically weighted pyramid of noise in 3 channels, which is then used to linearly interpolate between red, brown, and gray. To this small vertical flecks are added to simulate lichen and hanging splinters of bark. The texture map is stretched vertically when it is applied to the tree geometry.


The tree is also given a bump map, which is random noise stretched vertically by a factor on the order of 100. These effects all add to produce the bark image shown below, which is taken from the top left corner of the final image without fog.

REFERENCES:

[Sylvain 2002]
Sylvain Lefebvre and Fabrice Neyret, "Synthesising bark", Eurographics 2002

One of the major technical challenges of the project was to create convincing plants for ground cover in the image. This was accomplished by implementing a parametric 0L-system parser, similar to the  one described by [Prusinkiewicz 03], with the addition of randomized parameters.

The parser takes in a context-free grammar that describes the rough structure of the plant. Each symbol of the grammar is allowed to have one associated floating-point parameter.

The parser starts with an "axiom" (an initial string) explicitly given by the user, and expands this word on every iteration using the productions of the grammar. Each symbol may have several productions, each of which have an associated parameter range. Assuming these ranges are disjoint, the l-system can determine which production to apply to a symbol based on that symbol's current parameter value.

In addition to specifying the new symbols to replace an old symbol, each production must also specify a parameter to go with each new symbol. (If a parameter is not explicitly given, it is assumed to be "0"). The parameter can be described as any linear combination of the old symbol's parameter ("s"), a random U[0,1] parameter ("r"), and a constant.

A sample production would be
A(s) {1,10} -> B(s*2) C(5) D(r) E(s*5.5+r*2-10)

This says that A gets expanded to the string BCDE (with the given parameters) when A's parameter "s" is between 1 and 10 (inclusive).

Once a grammar is defined, the l-system is run through an arbitrary number of iterations to generate a sample plant structure, and is saved to a file. This file is then fed to a plant modeling program, which creates an appropriate PBRT model from it.

In defining the l-system grammar, a few symbols are reserved for explicit commands to the modeler. In particular, six symbols (+,-,&,^,/,\) are reserved to define curvature commands for the plant's stem,  two symbols (#,!) are reserved to specify increments/decrements to the stem width, f(s) is used to draw a stem segment of length s, and l(s) is used to draw a leaf at scale s. Note that these are the only symbols that are read in by the modeler (and they are the only ones needed).

The modeler creates a PBRT file by taking the l-system output, as well as an explicit model for a leaf and a stem segment. The leaf and stem segment models are defined as objects, and then instantiated repeatedly in the full plant model.

The resulting plants can be quite convincing, as seen in the test images. The randomized parameter in particular makes a big difference, since it allows organic-looking growth of the plant without requiring the definition of a massive, complex grammar.

REFERENCES

[Prusinkiewicz 03]
Prusinkiewicz, P., Hanan, J., Hammel, M., Mech, R., "L-systems: from the
Theory to Visual Models of Plants," Course notes from SIGGRAPH 2003.

[Smith 84]
Smith, A.R., "Plants, Fractals, and Formal Languages," ACM SIGGRAPH
Computer Graphics, Proceedings of the 11th annual conference on computer
graphics and interactive techniques, Volume 18 Issue 3, January 1984.

The goal for the fog was to create a variable-density fog region, where the fog was thicker along the ground, as well as clinging to the trees.  In general, this requires the fog density to be a function of the distance to both the ground (a trianglemesh or a heightfield) and the surrounding tree surfaces.

Initially, I planned to implement the shell mapping technique from a paper to be published at Siggraph 2005, by Porumbescu et al: Shell Maps(PDF).   A shell map is an extension of a standard two-dimensional texture, where an offset surface is generated for a triangle mesh object, and the volume enclosed by the original mesh and the offset surface is subdivided into tetrahedra.  Combined with a UV map of the original surface, the tetrahedra can then be used to map a slab of UVW texture space to the space surrounding the object in a consistent piecewise-linear mapping.  For the forest scene, I planned on generating the union mesh of the ground and the tree trunks in the scene, and then expanding the offset surface as much as possible without self-intersection, in order to cover all of space within the shell map and establishing a UVW mapping of all space that related to the scene geometry well.

As in the original paper, I used the offset surface generation code from UNC Chapel Hill [Cohen 96].  The offset surface maintains a 1:1 correspondence between original mesh surface vertices and offset surface vertices, which allows for the generation of triangular prisms from corresponding surface and offset surface triangles.  Since the triangular prisms are not guaranteed to be convex, they cannot directly be used to map between shell space and texture space.  Instead, they must be split into 3 tetrahedra each, while keeping the splits consistent - that is, two matching faces on two prisms must be split along the same line to guarantee a continuous mapping across the face.  Below is the wireframe of the constructed tetrahedral mesh around the Stanford bunny, with a closeup:

However, even though the construction of the shell map is not time-intensive, using it for volume rendering is far more so.  To render a volumetric ray with a single scattering bounce (The singlescattering volume integrator in PBRT does this), we now need to step along the ray in the texture space, where the ray is piecewise linear.  This means that in the world space, we must find all the tetrahedra that are intersected by the ray, transform each intersection segment into UVW space, obtain the transmission and illumination of the volume within each UVW ray segment, and then finally combine all the segment's transmission and luminance values together.  The primary cost here is to find the original ray-tetrahedron intersections.  I used the PBRT built-in octree data structure, adding the ability for the structure to handle ray intersections, to intesect the ray with the shell map tetrahedra.  However, this was still a slow process, with the intersection tests taking several times the time of the actual transmission calculations, and comparable time to the incoming luminance calculations.   Attempts to traverse the shell map data structure itself for tracking the ray ran into difficulties with numerical accuracy when a ray barely entered a tetrahedra (where the entering and exiting times were within the floating-point accuracy of the system and thus not sortable) or when the ray's maxt point corresponded to the inner surface of the shell map and the determination of whether the ray ended inside the tetrahedron was not numerically stable.  The second case was especially common, since the inner surface of the shell map is equivalent to geometry in the scene, and thus commonly represents the endpoint of a ray.  As a result of all of the above, we determined that the shell map would simply be far too slow to work for a more complex scene.  Below is the only succesful image of shell-mapped fog we generated.  The underlying fog is simply an exponential falloff fog, a default PBRT plugin.  It is wrapped around a simple C-shaped triangle mesh:

While this method does clump fog around the tree trunks, the fog does not truly follow the tree surface.  In order to have the fog be a function of true distance to the tree, I decided to cast a ray from P toward the closest tree center, and use the distance to the first intersection as the distance to the tree surface.  Since ray casts are expensive, I only cast the ray if I am within a threshold distance of the tree center - otherwise I simply use the center distance, minus an average tree radius, as the distance to the tree surface.  I also included a transition region in which I linearly blend from the cylindrical approximation to the true raytraced distance function.  This approach works well, since far away from the tree small surface variations have little impact on fog density.  Below is an image generated using ray-cast surface distance fog.  As is clear, the fog now clings along the tree surface well.

Finally, I added height above ground to the density function.  Since the heightfield has significant variation in the scene, I did not use a height approximation far away from the ground.  Instead, a ray is always cast downward from P in order to determine the closest surface below P.  Note that this has an additional benefit in that the ground foliage is also used in the density determination - the fog will appear to cling to the topsides of foliage as well.  Combining all this into a single density function, I obtained the equation:

REFERENCES

[Porumbescu 05]
Serban D. Porumbescu, Brian C. Budge, Louis Feng, Kenneth I. Joy, "Shell Maps,"  To be published in Proceedings of ACM Siggraph 2005.

[Cohen 96]
Cohen, Jonathan, Amitabh Varshney, Dinesh Manocha, Greg Turk, Hans Weber, Pankaj Agarwal, Frederick Brooks, and William Wright.  "Simplification Envelopes," Proceedings of SIGGRAPH 96 (New Orleans, LA, August 4-9, 1996).