Building Fractal Planets
[From the Fall 1994 George Washington U. EECS Newsletter]
Some example planets.
What does it take to build a world? This is the central question
of my research. My overarching goal is the creation from first algorithmic
principles of an entire planet, well-defined everywhere and at all scales,
with visual complexity, appearance, and beauty similar to Earth, and to
bring that model to real-time performance. Needless to say, this undertaking
subsumes a large number of interesting and challenging elements. These
include developing our capabilities in visual realism, models of natural
phenomena, computational efficiency in such models, and algorithmic art.
I am confident that there are enough challenges involved to keep me busy
for the rest of my days.
A planet, at the scales of ordinary human experience, is defined by
its landscapes. Landscapes are in turn defined by the form of the land,
the lighting, the current state of the atmosphere, and by the life forms
found within it. My research encompasses the first three, terrain, lighting,
and atmospherics; peculiarities of taste and predilection lead me to eschew
modeling life forms, leaving them to others to perfect. There is no accounting
for taste, and "I love landscapes!"
A
landscape set on the face of the above planet -- no kidding!
(See the
mpeg movie below.)
A One-Paragraph Autobiography
A truly fortuitous convergence of circumstances has lead me to this undertaking.
First, there was my personal affinity for landscapes: a beautiful landscape
evokes a deep spiritual passion in me; it always has. Then came my brief,
abortive education in the visual arts (with the guidance of my loving father,
I quickly realized this was not a promising way to make a living!) Next
came my extensive self-education in the natural sciences -- I have always
loved science, and keep abreast of it as an avocational interest. Then,
in my late twenties, came the necessity of Getting Serious About My Career,
and my choice of computer science as a field where I could work with scientists
without having to choose picayune specialization in a particular field.
I have always sought "the big picture" and been prone to regarding details
as tedious and obfuscatory. It seems that advanced courses in the sciences
quickly devolve into boring details; hence I was loathe to commit to any
given scientific field (my first love, astrophysics, being even more future-free
than art). Despite this negative attitude towards details, I have a reductionist
mind. Computer science appeals enormously because we can understand the
system under study from first principles. Having chosen computer science
for practical job-market reasons, I soon found (to my considerable surprise)
that I love it! Next came my romance with computer graphics which, equally
unexpectedly, enthralled me enough that I became a very narrow specialist
within the field of computer science. Finally, in the luckiest stroke of
all, I was hired by Mandelbrot to work with him at Yale as a computer graphics
programmer. Within weeks my predilection for landscape renderings showed
itself, Benoit gave me free reign to do as I saw fit and, as they say,
"the rest is history."
A
classic image from what Mandelbrot calls the "Romantic era" of fractal
landscapes.
Fractal Terrain Models
All successful synthetic terrain models for computer graphics are fractal:
That is, they feature complexity resulting from the repetition of form
over a variety of scales. The complexity resulting from this repetition
of form over many scales leads to the odd idea of fractal dimension: a
spatial dimension which is intermediate between the familiar integer-valued
(i.e., 1, 2, and 3) dimensions we're used to dealing with. Most fractal
terrains are based on a fractal function called fractional Brownian motion
or fBm for short. FBm is simply a sum of randomly phase-shifted sine waves,
the amplitude of which varies with frequency as 1/fß for
1<=ß<=3. FBm has a jagged trace which resembles the skyline
of a mountain range. Mandelbrot observed this and reasoned that extending
the function to two (-point-something, if you insist) dimensions would
result in a surface resembling mountainous terrain. He did so and presto!
fractal mountains were born.
A
mountain made from a variety of fBm; fractal dimension approximately 2.2.
The clouds, water, and snowline are also modeled with fBm.
Note that there is no scientific basis, i.e., linkage to first physical
principles, in this choice of terrain models. It is an example of what
I call an ontogenetic model: a model based on visible morphological
character. Mandelbrot and I take a lot of flak for this seemingly ad-hoc
approach to modeling Nature. Benoit maintains that visual reasoning has
its place in math and science; I take the more easily-defensible position
that computer graphics is engineering, not science, and that therefore
the end justifies the means. As natural philosophers--scientists looking
for pattern in Nature wherever we can find it--we both vigorously maintain
that the elegance and descriptive power of such a simple model of potentially-unlimited
complexity recommends it strongly. It certainly withstands a shave with
Occam's Razor, the principle that the simpler model is the preferred model,
better than any known or conceivable "physical" model.
Multifractals
FBm is, by design, statistically homogeneous and isotropic. That is, while
it is not exactly the same in any two places, it has the same "feel" everywhere.
Real terrains are more complex than that. In computer graphics we use a
variety of bastardized schemes to approximate fBm; when I started working
with him in 1987 Mandelbrot was interested in ameliorating some of the
documented artifacts inherent in efficient fBm-approximation algorithms.
[3] Following his lead I developed some conjectures of my own concerning
the morphology of terrain which lead to models which incorporate heterogeneity,
e.g., valleys which are smoother than peaks. [4] I had not yet heard of
the term at the time, but these models were multifractals: heterogeneous
fractals the heterogeneity of which is the same over a variety of scales.
Multifractal models I'm currently developing preserve the elegance of fBm
to a large degree (only one more parameter is added) while extending fBm's
expressive power: A multifractal terrain model can, for instance, readily
provide plains, rolling foothills, and alpine mountains all in one surface
patch. Turbulence has long been known to be composed of a hierarchy of
eddies; hence it is fractal. Furthermore, turbulence is multifractal [1]
; it is one of my current research goals to use my multifractals to construct
improved ontogenetic models of turbulence for computer graphics.
A
multifractal terrain patch. Note the heterogeneity of the terrain.
Erosion
Some of the most salient features in landscapes are due to fluvial erosion,
or running water. These features are formed by drainage of potential from
an area, and as such are context-sensitive. Computer scientists know that
context-sensitive algorithms are expensive to compute, hence river networks
are difficult to include in practical terrain modeling schemes. One of
my contributions has been the introduction of physical models of erosion
[4] to create such terrain features. Mandelbrot always spurned these dynamic
simulations--"Not enough bang for the buck!" he say --but they are one
of the more promising lines of my research. It turns out that having water
flow downhill and sit still in lakes and ponds requires numerical
schemes to solve some rather nasty partial differential equations; for
expertise in this I defer to my colleague Ed Bolton from the Yale Department
of Geophysics. Professor Bolton and I spent the month of August working
on this at NRL; we are now ready to move on to full-scale simulations of
the erosion over geologic time. Expected results of this line of research
include animations of orogenesis (mountain-building) and the formation
of badlands-style canyons, as well as experimental validation of published
erosion laws from the literature of fluvial geomorphology. We also hope
to use static erosion gradient fields along with multifractal terrain models
to inform improved stochastic interpolation to add detail to measured terrain
data sets.
A
synthetic fractal terrain patch before erosion simulation.
The
same terrain patch after erosion simulation.
Atmospherics
Landscape painters have known for hundreds of years that the primary visual
cue indicating really large scale is what they call aerial perspective:
the bluing and loss of contrast of objects viewed at a distance through
the atmosphere. (For an example of early use of aerial perspective see
the landscape appearing behind the Mona Lisa.) Aerial perspective is the
result of two processes: Rayleigh and Mie scattering. Rayleigh scattering
is what makes the sky blue, while Mie scattering makes the sky whitish
on a hazy day. Both are scientifically described by mathematics too complex
to use for production image synthesis; part of my research is in developing
efficient ontogenetic approximations of them. Efficiency of scattering
is in turn affected by the local density of the atmosphere. Another part
of my research involves constructing ontogenetic geometric atmospheric
density distribution (GADD) models. These models need to be continuous,
realistic (e.g., curving around the surface of a planet), and efficiently
integrable. Both scattering and GADD models are critical to obtaining the
aerial perspective necessary for realistic landscape renderings.
Aerial
perspective provides a sense of scale in synthetic landscape imagery.
Proceduralism
A distinguishing aspect of my methodology in image synthesis is
proceduralism.
I define proceduralism as the practice of distilling complex behaviors
into terse algorithmic descriptions, which are accessed via lazy evaluation
(that's a standard term in computer science, not my own pejorative!) This
year I co-authored the first book on the topic. [2] The way I use it, proceduralism
amounts to an artistic process which is ideally suited to the computer:
It abstracts copious detail into a compact set of instructions for its
reproduction. The process of reproduction is both simple and tedious--therefore
it's ideal work for a computer. Pithiness in description (as recommended
by Occam's razor) is traded for exhaustive calculation, as only abstractions
are stored and all specifics must be constructed from them.
I'm
currently teaching a course on procedural methods for computer graphics.
The students have made lots of neat images...
As a proceduralist, I generally constrain my images to be the exact
output of a computer program, that program being as terse as possible.
From the standpoint of formal logic, this implies that the images are theorems
which have been derived in a formal system. Formal systems are perforce
deterministic. Yet as an artist I am an
expressionist, and I claim
that my images represent a sort of spiritual self-expression. This juxtaposition
of determinism, which if universal precludes free will, and self-expression,
which is among the highest manifestations of free will, marks the peculiar
artistic process I practice. One of my theses is supporting the claim that
this artistic process is more different than other creative processes for
the visual arts, than any other new process or medium ever to come along.
My thesis on this topic is illuminated in the essay Formal
Logic & Self-Expression.
An image inspired by recurring end-of-the-world dreams I had as a child.
I
claim that this image, among others, represents a sort of spiritual self-expression
on my part, as an artist. This is, of course, about as scientifically weak
a claim as one can make, as no one can verify it but the claimant.
The Future of This Work
So far, all of my finished images have taken on the order of minutes or
hours to compute on a fast workstation. This work will not come to full
fruition until it is interactive, until it has been realized with real-time
performance. Getting from hours per frame to the tens of frames per second
required for real-time requires speeding things up by several orders of
magnitude. This represents a formidable engineering challenge for the years
to come. It will require mapping certain algorithms into hardware, the
maximization of simplifying assumptions, and as many advances in algorithmic
efficiency as can be found. The result will be real time VR (virtual reality)
scenarios of unprecedented realism and beauty. You may, for example, be
able to pilot your own starship through a synthetic universe, exploring
synthetic planets that you find along your way.
In
low orbit above the planet Gaea, approaching for adventures unknown...
The
Gaea Zoom -- my first animation of a virtual world.
It is my professional opinion that we have barely begun to scratch the
surface of the problem of reproducing in synthetic imagery the visual complexity
that confronts us all day, every day: You'd be hard put to find a scene
as visually simple as the best, most realistic computer graphic representation
of the same scene. It is bringing this visual complexity to synthetic imagery
that drives my research, and imbuing it with as much beauty as I can engineer
that drives my artwork. I look forward to a long and fruitful career in
pursuing these goals.
References
[1] Constantin, P., I. Procacchia, and K.R. Sreenivasan, Isoscalar Contours
in Turbulence, Phy. Rev. Lett., 1991, 67, pp. 1739.
[2] Ebert, D.S.,
ed., Textures and Modeling: A Procedural Approach, ed., 1994, Academic
Press, Cambridge, MA.
[3] Mandelbrot, B.B., Fractal landscapes without creases and with rivers,
in The Science of Fractal Images, H.O. Peitgen and D. Saupe, Editor, 1988,
Springer-Verlag, New York, pp. 243-260.
[4] Musgrave, F.K., C.E. Kolb, and R.S. Mace, The Synthesis and Rendering
of Eroded Fractal Terrains, Computer Graphics, July, 1989, 23:3, pp. 41-50.
A
landscape set on the face of the above planet -- no kidding!
(See the
mpeg movie below.)
A
classic image from what Mandelbrot calls the "Romantic era" of fractal
landscapes.
A
mountain made from a variety of fBm; fractal dimension approximately 2.2.
The clouds, water, and snowline are also modeled with fBm.
A
multifractal terrain patch. Note the heterogeneity of the terrain.
A
synthetic fractal terrain patch before erosion simulation.
The
same terrain patch after erosion simulation.
Aerial
perspective provides a sense of scale in synthetic landscape imagery.
An image inspired by recurring end-of-the-world dreams I had as a child.
I
claim that this image, among others, represents a sort of spiritual self-expression
on my part, as an artist. This is, of course, about as scientifically weak
a claim as one can make, as no one can verify it but the claimant.
In
low orbit above the planet Gaea, approaching for adventures unknown...