Economy

The dynamic noise implementation presented here requires the following measures over static noise:

• The table holding permutation values must be independent of that holding gradient lookup values.
• The dot products evaluated at grid points must be scaled.
• Time changes iteratively, not as a function parameter.
• Gradient values must be updated whenever a time step occurs.

The true cost of these new requirements is small and is further diminished due to constraints and provisions already inherent in target systems. For example, a hardware implementation of the static noise function is well-suited to pipelining; therefore, it may be advantageous to place the permutation table in a processing stage independent of the table holding gradient vectors, in order to load-balance stages in the pipeline. Notice also that the gradient direction values require fewer bits per entry than do the permutation values, so that a separate gradient direction table does not require much additional memory.

As an approximation, we can think of the dynamic noise function as being half the computational complexity of a comparable improved noise function. A static improved noise function on a regular grid in N-space is usually computed as an interpolation of two noise functions in (N-1)-space (except with grid-point evaluations using vectors in N-space). By reducing the dimensional requirement by one dimension, less than half of the computations are required for a dynamic noise evaluation.

It is expected that dynamic noise benefits from GPU vector hardware operations, because such operations could be used to perform the grid-point dot products and scaling, and that matrix operations might be used to perform groups of vector operations for even better savings.

Conclusion

This algorithm is quite simple and useful, and can produce convincing results for most noisy effects, although the particular implementation given here is likely not suitable for very highend visuals due to coarse timestepping and efficient integer math. (For a critique of Perlin noise in general, see [7].) The believed enhancements that dynamic noise provides are reduced computational complexity by over half, better localized noise density conservation (so there is more of a sense of flow), and no question of slice orientation or ordered holes appearing from use of a higher dimension. This method also provides smoothly variable flow velocity when rotating a slice through a higher dimension. While the need for independence between the gradient table and the permutation table is required in this dynamic noise implementation (unlike the reusable table of static noise), a pipelined hardware implementation begs this independence anyway.

A set of dynamic noise C++ classes (1D, 2D, and 3D) can be found in source code below, along with a separate Windows demonstration binary. The executable displays layered dynamic noise in various places around a scene. These files are provided for your convenience. There is no guarantee about any behavior on any system - use and modification is allowed within your own liability. noise.zip

Thanks and happy coding!

References

[1] Perlin, K. An image synthesizer. 1985.
http://portal.acm.org/citation.cfm?id=325247

[2] Perlin, K. Making noise. 2000.
http://www.noisemachine.com/talk1/index.html

[3] Perlin, K. Improving noise. 2002.
http://mrl.nyu.edu/~perlin/paper445.pdf

[4] Perlin, K. A sheet of simplex noise. 2006.
http://mrl.nyu.edu/~perlin/homepage2006/simplex_noise/index.html

[5] Gustavson, S. Simplex noise demystified. 2005.
http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf

[6] Olano, M. Modified noise for evaluation on graphics hardware. 2005.
http://www.cs.umbc.edu/~olano/papers/mNoise.pdf

[7] Cook, RL, et al. Wavelet noise. 2005.
http://portal.acm.org/citation.cfm?id=1073204.1073264