Making Noise Predictably

Meeting Ken Perlin

Abstract

Can you make the same random noise again?

Figure 1: TRON the Movie

What noise based outcomes will we see today?

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, fire, marble, terrain, mountains, and clouds.

Inspiration

Arch

Cat Fur Up Close

Figure 2: Perlin Noise based Textures

What is Perlin Noise?

Ok, this is going to be a long explanation!!!

A. Inner Product Computation

Let us take some inspiration from the Mahabharata first: Here is an evocative picture of an event from the great epic, where Bheeshma is lying on a bed of arrows, and the arrows are piercing his body in random directions. The arrows are like the gradient vectors in Perlin Noise, and the point of interest is like Bheeshma’s body. The dot product of the gradient vectors with the difference vectors to the point of interest gives us the noise value at that point.:

Figure 3: Bheeshma On a Bed of Arrows

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.

NoteDot 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 weighting/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:

\[ f(t) = 6t^5-15t^4+10t^3 \tag{1}\]

and looks like this:

NoteInterpolation Function \(f(t)\) has smooth ends

Both \(\frac{df(t)}{dt}\) and \(\frac{d^2f(t)}{dt^2}\) 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 overlaid on the first one, and generate noise in the same way. These layers of noise-s are scaled by a factor (Usually \(\Large{\frac{1}{2^n}}\)), 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. For now, here is Ken Perlin’s own explanation from 1999:

“The outline of my algorithm to create noise is very simple. Given an input point P, look at each of the surrounding grid points. In two dimensions there will be four surrounding grid points; in three dimensions there will be eight. In n dimensions, a point will have 2n surrounding grid points.
For each surrounding grid point Q, choose a pseudo-random gradient vector G. It is very important that for any particular grid point you always choose the same gradient vector.
Compute the inner product G . (P-Q). This will give the value at P of the linear function with gradient G which is zero at grid point Q.
Now you have 2n of these values. Interpolate between them down to your point, using an S-shaped cross-fade curve (eg: 3t2-2t3) to weight the interpolant in each dimension. This step will require computing n S-curves, followed by 2n-1 linear interpolations.”

— Ken Perlin

Creating Textures and Waveforms with Perlin Noise

Using p5.js

Here is a landscape generated using Perlin Noise:

Using R

The ambient package allows us to create a variety of noise patterns, including Perlin Noise. The commands are: gen_perlin() and noise_perlin(), whose arguments are:

• dim: The dimensions (height, width, (and depth)) of the noise to be generated. The length determines the dimensionality of the noise.
• frequency: Determines the granularity of the features in the noise.
• interpolator:How should values between sampled points be calculated? Either ‘linear’, ‘hermite’, or ‘quintic’ (default), ranging from lowest to highest quality.
• fractal: The fractal type to use. Either ‘none’, ‘fbm’ (default), ‘billow’, or ‘rigid-multi’. It is suggested that you experiment with the different types to get a feel for how they behaves.
• octaves: The number of noise layers used to create the fractal noise. Ignored if fractal = ‘none’. Defaults to 3.
• lacunarity: The frequency multiplier between successive noise layers when building fractal noise. Ignored if fractal = ‘none’. Defaults to 2.
• gain: The relative strength between successive noise layers when building fractal noise. Ignored if fractal = ‘none’. Defaults to 0.5.
• pertubation: The perturbation to use. Either ‘none’ (default), ‘normal’, or ‘fractal’. Defines the displacement (warping) of the noise, with ‘normal’ giving a smooth warping and ‘fractal’ giving a more erratic warping.
• pertubation_amplitude: The maximal perturbation distance from the origin. Ignored if pertubation = ‘none’. Defaults to 1.
• x, y, z: Coordinates to get noise value from
• seed: The seed to use for the noise. If NULL a random seed will be used:

noise2 <- noise_perlin(
  dim = c(400, 400), # height/width
  frequency = 0.01, # Lower = less granular, more organic
  interpolator = "quintic", #' linear', 'hermite', or 'quintic'
  fractal = "fbm", # Try "billow" , "rigid-multi"
  octaves = 5,
  lacunarity = 2,
  gain = 0.8, # Default = 0.5 giving 1/2^n scaling
  pertubation = "none", # Note the incorrect spelling
  pertubation_amplitude = 1 # Note the incorrect spelling
)
## generates a matrix
noise2 %>% as_tibble()
# Plot the matrix
plot(as.raster(normalise(noise2)))

V1	V2	V3	V4	V5	V6	V7	V8	V9	V10	V11	V12	V13	V14	V15	V16	V17	V18	V19	V20	V21	V22	V23	V24	V25	V26	V27	V28	V29	V30	V31	V32	V33	V34	V35	V36	V37	V38	V39	V40	V41	V42	V43	V44	V45	V46	V47	V48	V49	V50	V51	V52	V53	V54	V55	V56	V57	V58	V59	V60	V61	V62	V63	V64	V65	V66	V67	V68	V69	V70	V71	V72	V73	V74	V75	V76	V77	V78	V79	V80	V81	V82	V83	V84	V85	V86	V87	V88	V89	V90	V91	V92	V93	V94	V95	V96	V97	V98	V99	V100	V101	V102	V103	V104	V105	V106	V107	V108	V109	V110	V111	V112	V113	V114	V115	V116	V117	V118	V119	V120	V121	V122	V123	V124	V125	V126	V127	V128	V129	V130	V131	V132	V133	V134	V135	V136	V137	V138	V139	V140	V141	V142	V143	V144	V145	V146	V147	V148	V149	V150	V151	V152	V153	V154	V155	V156	V157	V158	V159	V160	V161	V162	V163	V164	V165	V166	V167	V168	V169	V170	V171	V172	V173	V174	V175	V176	V177	V178	V179	V180	V181	V182	V183	V184	V185	V186	V187	V188	V189	V190	V191	V192	V193	V194	V195	V196	V197	V198	V199	V200	V201	V202	V203	V204	V205	V206	V207	V208	V209	V210	V211	V212	V213	V214	V215	V216	V217	V218	V219	V220	V221	V222	V223	V224	V225	V226	V227	V228	V229	V230	V231	V232	V233	V234	V235	V236	V237	V238	V239	V240	V241	V242	V243	V244	V245	V246	V247	V248	V249	V250	V251	V252	V253	V254	V255	V256	V257	V258	V259	V260	V261	V262	V263	V264	V265	V266	V267	V268	V269	V270	V271	V272	V273	V274	V275	V276	V277	V278	V279	V280	V281	V282	V283	V284	V285	V286	V287	V288	V289	V290	V291	V292	V293	V294	V295	V296	V297	V298	V299	V300	V301	V302	V303	V304	V305	V306	V307	V308	V309	V310	V311	V312	V313	V314	V315	V316	V317	V318	V319	V320	V321	V322	V323	V324	V325	V326	V327	V328	V329	V330	V331	V332	V333	V334	V335	V336	V337	V338	V339	V340	V341	V342	V343	V344	V345	V346	V347	V348	V349	V350	V351	V352	V353	V354	V355	V356	V357	V358	V359	V360	V361	V362	V363	V364	V365	V366	V367	V368	V369	V370	V371	V372	V373	V374	V375	V376	V377	V378	V379	V380	V381	V382	V383	V384	V385	V386	V387	V388	V389	V390	V391	V392	V393	V394	V395	V396	V397	V398	V399	V400
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# Using the generator
grid <- long_grid(seq(1, 10, length.out = 1000), seq(1, 10, length.out = 1000))
grid$noise <- gen_perlin(grid$x, grid$y,
  octaves = 5,
  frequency = 1.2
)
plot(grid, noise)
##
grid %>%
  gf_point(y ~ x,
    colour = ~noise,
    size = 0.01, show.legend = F
  ) %>%
  gf_refine(
    scale_color_gradient(
      low = "orangered",
      high = "black"
    ),
    coord_fixed()
  ) %>%
  gf_theme(theme_void())

It seems the ambient package cannot generate 1D-Perlin noise, so we cannot generate say time waveforms based on this idea. p5.js of course can do 1D.

There is also another new package called [opensimplex2] that can generate simplex noise, that is said to be computationally less intense compared to Perlin Noise. We will discuss that in the next section. TBD.

Videos

Of course there are videos by Dan Shiffman on the topic of Perlin Noise:

1. Perlin Noise in p5.js. https://www.youtube.com/watch?v=Qf4dIN99e2w&list=PLRqwX-V7Uu6ZV4yEcW3uDwOgGXKUUsPOM&index=3&pp=iAQB
2. Perlin Noise in 2D. https://www.youtube.com/watch?v=ikwNrFvnL3g&list=PLRqwX-V7Uu6ZV4yEcW3uDwOgGXKUUsPOM&index=6&pp=iAQB
3. Perlin Noise in Detail. https://www.youtube.com/watch?v=D1BBj2VaBl4&list=PLRqwX-V7Uu6ZV4yEcW3uDwOgGXKUUsPOM&index=7&pp=iAQB
4. Graphing Perlin Noise. https://www.youtube.com/watch?v=y7sgcFhk6ZM&list=PLRqwX-V7Uu6ZV4yEcW3uDwOgGXKUUsPOM&index=5&pp=iAQB

Wait, But Why?

• Generating random waveforms and textures is an important part of Creative Computational projects.
• These ideas are used in landscape, texture and sound generation.
• “Normal” random noise is too jagged to have the natural look that we would like
• Perlin Noise creates both a smoothness and a fine-grain random structure in an arbitrary number of dimensions.
• This ends up looking more “organic” and “natural”. Think cats.

References

1. Ken Perlin.(1999). Making Noise. https://web.archive.org/web/20151221035155/http://www.noisemachine.com/talk1/index.html. Based on a talk presented at GDCHardCore on Dec 9, 1999.
2. https://www.scratchapixel.com/lessons/procedural-generation-virtual-worlds/perlin-noise-part-2/perlin-noise.html
3. https://www.khanacademy.org/computing/computer-programming/programming-natural-simulations/programming-noise/a/perlin-noise
4. https://adrianb.io/2014/08/09/perlinnoise.html
5. https://www.arendpeter.com/Perlin_Noise.html
6. https://mzucker.github.io/html/perlin-noise-math-faq.html
7. https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/
8. Gorilla Sun Blog. Perlin Noise. https://www.gorillasun.de/tag/perlin-noise/
9. The ambient package in R: https://ambient.data-imaginist.com/

Textbooks

1. Patricio Gonzalez Vivo & Jen Lowe. The Book of Shaders. https://thebookofshaders.com/11/

Resources

1. https://opengameart.org/content/700-noise-textures
2. https://github.com/sighack/perlin-noise-fields

Other Tools

1. Strudel
2. Pure Data
3. Envion https://www.peamarte.it/env/envion_v3.6.html

R Package Citations

Package	Version	Citation
ambient	1.0.3	Pedersen and Peck (2025)

Pedersen, Thomas Lin, and Jordan Peck. 2025. ambient: A Generator of Multidimensional Noise. https://doi.org/10.32614/CRAN.package.ambient.