Meeting Ken Perlin
Noise Generation
Perlin Noise
Simplex Noise
Abstract
Can you make the same random noise again?
We will understand the basics of procedural noise generation: generating random noise-like numbers that allow us to model and create very realistic-looking textures, such as wood, terrain, mountains, and clouds.
Perlin Noise based Textures
Ok, this is going to be a long explanation!!!
A. Inner Product Computation
Let us start by dividing up 2D space ( for now!!) into square-shaped cells. At each vertex we randomly place a unit gradient vector labelled \(r_{i}\) that points in a random direction. See the figure below:
We wish to calculate the Perlin Noise amount at any point of interest inside the cell.
- We draw difference vectors to the point from each of the 4 vertices.
- We compute the vector dot product with each of the \(r_{i}\) and the above difference vectors. ( 4 dot products )
- These are shown in the text print at the side of the figure.
Dot Products are Scalars with Polarity
Note how the 4 dot products change as you move the mouse/touchpad. This changes the 4 gradient vectors and hence the scalar dot products change in amplitude and polarity.
In a typical Perlin Noise implementation, the gradient vectors are fixed after an initial setup. So each gradient vector generates a range of dot-product values as the point of interest moves within the cell.
B. Interpolation of Dot Product values
With the 4 scalar dot products, we are now ready to compute the Perlin Noise value at the point of interest. There are several ways of doing this:
- Simply take the average
- Take a weighted average, with fixed weights.
- Use a wieghting/interpolating function: The closer a point of interest is to one or other of the cell vertices, the higher is the contribution of the corresponding dot-product.
The third approach is the one embedded within (all?) Perlin Noise implementations. The interpolating function is: \[ f(t) = 6t^5-15t^4+10t^3 \] and looks like this:
Interpolation Function \(f(t)\) has smooth ends
Both \(\frac{df(t)}{dt}\) and \(\frac{d^2f(t)}{dt}\) are continuous at the ends of the range of the function (t = 0 and t = 1).
\[ \begin{array}{lcl}f'(t) & = & \ \frac{d}{dt}[6*t^5 - 15*t^4 + 10*t^3]\\ & = & 30 * (t^4 - 2 * t^3 + t^2)\\ & = & 0 \ \text{@ t = 0 and t = 1} \end{array} \]
\[ \begin{array}{lcl}f''(t) & = & \ \frac{d^2}{dt^2}[6*t^5 - 15*t^4 + 10*t^3]\\ & = &60 * (2 * t^3 - 3 * t^2 + t)\\ & = & 0 \ \text{@ t = 0 and t = 1} \end{array} \] This ensures that there are not sudden changes in the noise function near about the vertices.
D. Fractal Overlay and Combining
Now that we have one grid full of a layer of noise generated by weighted dot-products, we can appreciate one more thing: we can overlay the space with several layers of such noise values. Why would this be a good idea?
This multiple layer overlay creates a very natural-looking fractal-ness in the resulting noise function. Most natural looking shapes like landscapes, mountains, vegetables, flames…all have this self-similar structure where when one zooms in, the magnified function looks pretty much like the un-zoomed version!!
So how we create and merge overlays? We create several more-closely-spaced grids, and generate noise in the same way. These layers of noise-s are scaled by a factor (Usually \(\frac{1}{2^n}\)), where \(n\) is the “order”of the layer. Each new finely-spaced layer generates similar-looking noise functions, which are combined with smaller and smaller weights to achieve that final polished fractal look of Perlin Noise.
We will explore this fractality with code.
| Package | Version | Citation |
|---|---|---|
| ambient | 1.0.2 | @ambient |
| mosaicCalc | 0.6.4 | @mosaicCalc |
| plot3D | 1.4.1 | @plot3D |