It's time for a break! We've been playing with random functions that look like TV white noise, our head is still spinning thinking about shaders, and our eyes are tired. Time to go out for a walk!
We feel the air on our skin, the sun in our face. The world is such a vivid and rich place. Colors, textures, sounds. While we walk we can't avoid noticing the surface of the roads, rocks, trees and clouds.
@ -23,17 +23,19 @@ We feel the air on our skin, the sun in our face. The world is such a vivid and
The unpredictability of these textures could be called "random," but they don't look like the random we were playing with before. The “real world” is such a rich and complex place! How can we approximate this variety computationally?
This was the question [Ken Perlin](https://mrl.nyu.edu/~perlin/) was trying to solve in the eary 1980s when he was commissioned to generate more realistic textures for the movie "Tron." In response to that, he came up with an elegant *Oscar winning* noise algorithm. (No biggie.)
After uncommenting it, notice how the transition between the peaks gets smooth. In some noise implementations you will find that programmers prefer to code their own cubic curves (like the following formula) instead of using the [```smoothstep()```](.../glossary/?search=smoothstep).
float u = f * f * (3.0 - 2.0 * f ); // custom cubic curve
@ -79,11 +81,12 @@ y = mix(rand(i), rand(i + 1.0), u); // using it in the interpolation
This *smooth randomness* is a game changer for graphical engineers or artists - it provides the ability to generate images and geometries with an organic feeling. Perlin's Noise Algorithm has been implemented over and over in different languages and dimensions to make mesmerizing pieces for all sorts of creative uses.
Now that we know how to do noise in 1D, it's time to move on to 2D. In 2D, instead of interpolating between two points of a line (```fract(x)``` and ```fract(x)+1.0```), we are going to interpolate between the four corners of the square area of a plane (```fract(st)```, ```fract(st)+vec2(1.,0.)```, ```fract(st)+vec2(0.,1.)``` and ```fract(st)+vec2(1.,1.)```).
Similarly, if we want to obtain 3D noise we need to interpolate between the eight corners of a cube. This technique is all about interpolating random values, which is why it's called **value noise**.
@ -132,9 +136,10 @@ Take a look at the following noise function.
We start by scaling the space by 5 (line 45) in order to see the interpolation between the squares of the grid. Then inside the noise function we subdivide the space into cells. We store the integer position of the cell along with the fractional positions inside the cell. We use the integer position to calculate the four corners' coordinates and obtain a random value for each one (lines 23-26). Finally, in line 35 we interpolate between the 4 random values of the corners using the fractional positions we stored before.
* Change the multiplier of line 45. Try to animate it.
@ -143,7 +148,7 @@ Now it's your turn. Try the following exercises:
* At what level of zoom does the noise start looking like random again?
* どのぐらいズームすると、ノイズがまたただのランダムに見え始めるでしょう？
* どのぐらいズームすると、ノイズがまたただのランダムに見え始めるでしょう。
* At what zoom level is the noise is imperceptible?
@ -159,33 +164,32 @@ Now it's your turn. Try the following exercises:
* Now that you've achieved some control over order and chaos, it's time to use that knowledge. Make a composition of rectangles, colors and noise that resembles some of the complexity of a [Mark Rothko](http://en.wikipedia.org/wiki/Mark_Rothko) painting.
Noise algorithms were originally designed to give a natural *je ne sais quoi* to digital textures. The 1D and 2D implementations we've seen so far were interpolations between random *values*, which is why they're called **Value Noise**, but there are more ways to obtain noise...
ノイズのアルゴリズムは本来、自然な *je ne sais quoi* （訳注：言葉では表せない質感、美しさ）をデジタルなテクスチャーに与えるために考え出されました。ここまで見てきた一次元と
ノイズのアルゴリズムは本来、自然な *je ne sais quoi* （訳注：言葉では表せない質感、美しさ）をデジタルなテクスチャーに与えるために考え出されました。ここまで見てきた一次元と二次元の実装はランダムな値の間の補間を行うもので、そのためバリューノイズと呼ばれていました。しかしノイズを作り出す方法はこれだけではありません。
[  ](../edit.html#11/2d-vnoise.frag)
As you discovered in the previous exercises, value noise tends to look "blocky." To diminish this blocky effect, in 1985 [Ken Perlin](https://mrl.nyu.edu/~perlin/) developed another implementation of the algorithm called **Gradient Noise**. Ken figured out how to interpolate random *gradients* instead of values. These gradients were the result of a 2D random function that returns directions (represented by a ```vec2```) instead of single values (```float```). Click on the following image to see the code and how it works.
Take a minute to look at these two examples by [Inigo Quilez](http://www.iquilezles.org/) and pay attention to the differences between [value noise](https://www.shadertoy.com/view/lsf3WH) and [gradient noise](https://www.shadertoy.com/view/XdXGW8).
Like a painter who understands how the pigments of their paints work, the more we know about noise implementations the better we will be able to use them. For example, if we use a two dimensional noise implementation to rotate the space where straight lines are rendered, we can produce the following swirly effect that looks like wood. Again you can click on the image to see what the code looks like.
@ -196,7 +200,7 @@ Like a painter who understands how the pigments of their paints work, the more w
Another way to get interesting patterns from noise is to treat it like a distance field and apply some of the tricks described in the [Shapes chapter](../07/).
@ -207,7 +211,7 @@ Another way to get interesting patterns from noise is to treat it like a distanc
A third way of using the noise function is to modulate a shape. This also requires some of the techniques we learned in the [chapter about shapes](../07/).
* What other generative pattern can you make? What about granite? marble? magma? water? Find three pictures of textures you are interested in and implement them algorithmically using noise.
* What about using noise for motion? Go back to the [Matrix chapter](../08/). Use the translation example that moves the "+" around, and apply some *random* and *noise* movements to it.
@ -237,11 +241,11 @@ For you to practice:
For Ken Perlin the success of his algorithm wasn't enough. He thought it could perform better. At Siggraph 2001 he presented the "simplex noise" in which he achieved the following improvements over the previous algorithm:
Ken Perlinにとって彼のアルゴリズムの成功は十分でありませんでした。彼はもっとパフォーマンスをあげられると考えたのです。2001年のSiggraphで発表したシンプレックスノイズでは以前のアルゴリズムに比べて下記に挙げるような改良を実現しました。
Ken Perlinにとって彼のアルゴリズムの成功は十分ではありませんでした。彼はもっとパフォーマンスをあげられると考えたのです。2001年のSiggraphで彼は、以前のアルゴリズムに比べて下記の改良を実現したシンプレックスノイズを発表しました。
* An algorithm with lower computational complexity and fewer multiplications.
* よりシンプルで乗算の回数が少ない済むアルゴリズム。
* よりシンプルで乗算の回数が少ない。
* A noise that scales to higher dimensions with less computational cost.
@ -253,8 +257,8 @@ Ken Perlinにとって彼のアルゴリズムの成功は十分でありませ
* A noise with well-defined and continuous gradients that can be computed quite cheaply.
* An algorithm that is easy to implement in hardware.
@ -262,42 +266,44 @@ Ken Perlinにとって彼のアルゴリズムの成功は十分でありませ
I know what you are thinking... "Who is this man?" Yes, his work is fantastic! But seriously, how did he improve the algorithm? Well, we saw how for two dimensions he was interpolating 4 points (corners of a square); so we can correctly guess that for [three (see an implementation here)](../edit.html#11/3d-noise.frag) and four dimensions we need to interpolate 8 and 16 points. Right? In other words for N dimensions you need to smoothly interpolate 2 to the N points (2^N). But Ken smartly noticed that although the obvious choice for a space-filling shape is a square, the simplest shape in 2D is the equilateral triangle. So he started by replacing the squared grid (we just learned how to use) for a simplex grid of equilateral triangles.
The simplex shape for N dimensions is a shape with N + 1 corners. In other words one fewer corner to compute in 2D, 4 fewer corners in 3D and 11 fewer corners in 4D! That's a huge improvement!
N次元に対して最も単純な形は N + 1 の頂点を持ちます。つまり二次元に対しては1頂点、三次元に対しては4頂点、四次元に対しては11頂点少ない計算で済むのです。これは大変な改善です。
N次元に対して最も単純な形は N + 1 の頂点を持ちます。つまり二次元に対しては1頂点、三次元に対しては4頂点、四次元に対しては11頂点少ない計算で済むのです。これは大幅な改善です。
In two dimensions the interpolation happens similarly to regular noise, by interpolating the values of the corners of a section. But in this case, by using a simplex grid, we only need to interpolate the sum of 3 corners.
How is the simplex grid made? In another brilliant and elegant move, the simplex grid can be obtained by subdividing the cells of a regular 4 cornered grid into two isosceles triangles and then skewing it until each triangle is equilateral.
Then, as [Stefan Gustavson describes in this paper](http://staffwww.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf): _"...by looking at the integer parts of the transformed coordinates (x,y) for the point we want to evaluate, we can quickly determine which cell of two simplices that contains the point. By also comparing the magnitudes of x and y, we can determine whether the point is in the upper or the lower simplex, and traverse the correct three corner points."_
In the following code you can uncomment line 44 to see how the grid is skewed, and then uncomment line 47 to see how a simplex grid can be constructed. Note how on line 22 we are subdividing the skewed square into two equilateral triangles just by detecting if ```x > y``` ("lower" triangle) or ```y > x``` ("upper" triangle).
Another improvement introduced by Perlin with **Simplex Noise**, is the replacement of the Cubic Hermite Curve ( _f(x) = 3x^2-2x^3_ , which is identical to the [```smoothstep()```](.../glossary/?search=smoothstep) function) for a Quintic Hermite Curve ( _f(x) = 6x^5-15x^4+10x^3_ ). This makes both ends of the curve more "flat" so each border gracefully stiches with the next one. In other words you get a more continuous transition between the cells. You can see this by uncommenting the second formula in the following graph example (or see the [two equations side by side here](https://www.desmos.com/calculator/2xvlk5xp8b)).
Ken Perlinがシンプレックスノイズで導入したもう1つの改善点は、3次エルミート曲線（f(x) = 3x^2-2x^3 これは [```smoothstep()```](.../glossary/?search=smoothstep)関数と同等です）を4次エルミート曲線（f(x) = 6x^5-15x^4+10x^3）で置き換えたことです。これにより曲線の両端がより平坦になり次の曲線により綺麗につながり、マス目間の変化がより滑らかになります。下記のグラフの2つ目の式のコメントを外すとこれを実際に見ることができます（もしくは[ここ](https://www.desmos.com/calculator/2xvlk5xp8b)で並べて比較することもできます）。
Ken Perlinがシンプレックスノイズで導入したもう1つの改善点は、3次エルミート曲線（f(x) = 3x^2-2x^3 これは [```smoothstep()```](.../glossary/?search=smoothstep)関数と同等です）を4次エルミート曲線（f(x) = 6x^5-15x^4+10x^3）で置き換えたことです。これによって曲線の両端がより平坦になり、隣の曲線に綺麗につながることで、マス目間の変化がより滑らかになります。下記のグラフの2つ目の式のコメントを外すと実際に見ることができます（もしくは[ここ](https://www.desmos.com/calculator/2xvlk5xp8b)で並べて比較することもできます）。
<divclass="simpleFunction"data="
@ -309,11 +315,11 @@ y = x*x*(3.0-2.0*x);
Note how the ends of the curve change. You can read more about this in [Ken's own words](http://mrl.nyu.edu/~perlin/paper445.pdf).
All these improvements result in an algorithmic masterpiece known as **Simplex Noise**. The following is a GLSL implementation of this algorithm made by Ian McEwan (and presented in [this paper](http://webstaff.itn.liu.se/~stegu/jgt2012/article.pdf)) which is overcomplicated for educational purposes, but you will be happy to click on it and see that it is less cryptic than you might expect.
[  ](../edit.html#11/2d-snoise-clear.frag)
@ -323,28 +329,28 @@ Well... enough technicalities, it's time for you to use this resource in your ow
* Contemplate how each noise implementation looks. Imagine them as a raw material, like a marble rock for a sculptor. What can you say about about the "feeling" that each one has? Squinch your eyes to trigger your imagination, like when you want to find shapes in a cloud. What do you see? What are you reminded of? What do you imagine each noise implementation could be made into? Following your guts and try to make it happen in code.
In the following chapters we will see some well known techniques to perfect your skills and get more out of your noise to design quality generative content with shaders. Until then enjoy some time outside contemplating nature and its intricate patterns. Your ability to observe needs equal (or probably more) dedication than your making skills. Go outside and enjoy the rest of the day!
kynd
9 years ago