3D Perlin杂分析导Perlin
我目前正在实施一个3D培林利用Shader Model 4（的DirectX 10 HLSL）噪音凹凸贴图。产生噪声本身不是一个很大的问题（还有吨教程和codeS左右），但我还没有发现有3D Perlin杂分析衍生物。
I am currently implementing a 3D Perlin noise bump mapping using Shader Model 4 (DirectX 10 HLSL). Generating the noise itself is not a big problem (there are tons of tutorials and codes around) but what I have not found are analytical derivatives of 3D Perlin noise.
的唯一网站走的是衍生品在内的伊尼戈Quilez的网站和相关的 GameDev.net讨论。的问题是，在所述第一连杆的噪声是基于值，而不是基于梯度（这是我的要求），在第二连杆，还有只有2D梯度噪声衍生物
The only sites taking the derivatives into account are Ińigo Quilez's site and a related GameDev.net discussion. The problem is that in the first link the noise is value based, not gradient based (which is a requirement for me), in the second link, there's only 2D gradient noise derivative.
请注意，我不是在寻找数值衍生工具的要求要产生4相邻的噪声样本，这就是太多的开销。
Note that I'm not looking for numerical derivatives as those require 4 neighboring noise samples to be generated and that's way too much overhead.
有没有人计算，这些衍生品？有没有正在使用他们一个参考实现？
Has anyone calculated these derivatives? Is there a reference implementation that is using them?
推荐答案
我也无法找到今天在网络上的解决方案，所以我试图得到它。
I also could not found a solution on the web today, so I tried to derive it.
三维Perlin杂的首先的符号定义。
Firstly the notations of a 3D Perlin noise is defined.
假设3D Perlin的噪声由三线性插值计算为
Assume the 3D Perlin noise is computed by the trilinear interpolation as
n = Lerp(
Lerp(
Lerp(dot000, dot100, u),
Lerp(dot010, dot110, u),
v),
Lerp(
Lerp(dot001, dot101, u),
Lerp(dot011, dot111, u),
v),
w)
其中， U ， v ，是W 是插因素进行分数坐标的五次多项式（即提高Perlin杂）：
where u, v, w are the interpolation factors by the quintic polynomial of fraction coordinates (i.e., improved Perlin noise):
x0 = frac(x)
y0 = frac(y)
z0 = frac(z)
x1 = x0 - 1
y1 = y0 - 1
z1 = z0 - 1
u = x0 * x0 * x0 * (x0 * (6 * x0 - 15) + 10)
v = y0 * y0 * y0 * (y0 * (6 * y0 - 15) + 10)
w = z0 * z0 * z0 * (z0 * (6 * z0 - 15) + 10)
和点___ 是梯度矢量（GX ___，gy___，GZ ___） s的格点的点积和分数坐标：
and dot___s are dot products of the gradient vectors (gx___, gy___, gz___)s at lattice points and the fraction coordinates:
dot000 = gx000 * x0 + gy000 * y0 + gz000 * z0
dot100 = gx100 * x1 + gy100 * y0 + gz100 * z0
dot010 = gx010 * x0 + gy010 * y1 + gz010 * z0
dot110 = gx110 * x1 + gy110 * y1 + gz110 * z0
dot001 = gx001 * x0 + gy001 * y0 + gz001 * z1
dot101 = gx101 * x1 + gy101 * y0 + gz101 * z1
dot011 = gx011 * x0 + gy011 * y1 + gz011 * z1
dot111 = gx111 * x1 + gy111 * y1 + gz111 * z1
计算的衍生品
首先， U ， v 和是W
u' = 30 * x0 * x0 * (x0 - 1) * (x0 - 1)
v' = 30 * y0 * y0 * (y0 - 1) * (y0 - 1)
w' = 30 * z0 * z0 * (z0 - 1) * (z0 - 1)
通过扩大 N 与线性插值（A，B，T）= A +（B - A）* T ，
n = dot000
+ u(dot100 - dot000)
+ v(dot010 - dot000)
+ w(dot001 - dot000)
+ uv(dot110 - dot010 - dot100 + dot000)
+ uw(dot101 - dot001 - dot100 + dot000)
+ vw(dot011 - dot001 - dot010 + dot000)
+ uvw(dot111 - dot011 - dot101 + dot001 - dot110 + dot010 + dot100 - dot000)
再取偏导数 N ，
nx = gx000
+ u' (dot100 - dot000)
+ u (gx100 - gx000)
+ v (gx010 - gx000)
+ w (gx001 - gx000)
+ u'v (dot110 - dot010 - dot100 + dot000)
+ uv (gx110 - gx010 - gx100 + gx000)
+ u'w (dot101 - dot001 - dot100 + dot000)
+ uw (gx101 - gx001 - gx100 - gx000)
+ vw (gx011 - gx001 - gx010 + gx000)
+ u'vw(dot111 - dot011 - dot101 + dot001 - dot110 + dot010 + dot100 - dot000)
+ uvw (gx111 - gx011 - gx101 + gx001 - gx110 + gx010 + gx100 - gx000)
ny = gy000
+ u (gy100 - gy000)
+ v' (dot010 - dot000)
+ v (gy010 - gy000)
+ w (gy001 - gy000)
+ uv' (dot110 - dot010 - dot100 + dot000)
+ uv (gy110 - gy010 - gy100 + gy000)
+ uw (gy101 - gy001 - gy100 + gy000)
+ v'w (dot011 - dot001 - dot010 + dot000)
+ vw (gy011 - gy001 - gy010 + gy000)
+ uv'w(dot111 - dot011 - dot101 + dot001 - dot110 + dot010 + dot100 - dot000)
+ uvw (gy111 - gy011 - gy101 + gy001 - gy110 + gy010 + gy100 - gy000)
nz = gz000
+ u (gz100 - gz000)
+ v (gz010 - gz000)
+ w' (dot001 - dot000)
+ w (gz001 - gz000)
+ uv (gz110 - gz010 - gz100 + gz000)
+ uw' (dot101 - dot001 - dot100 + dot000)
+ uw (gz101 - gz001 - gz100 + gz000)
+ vw' (dot011 - dot001 - dot010 + dot000)
+ vw (gz011 - gz001 - gz010 + gz000)
+ uvw'(dot111 - dot011 - dot101 + dot001 - dot110 + dot010 + dot100 - dot000)
+ uvw (gz111 - gz011 - gz101 + gz001 - gz110 + gz010 + gz100 - gz000)
然后（NX，NY，NZ）是噪声功能梯度向量（偏导数）。
Then (nx, ny, nz) is the gradient vector (partial derivatives) of the noise function.
一些常见子前pression可以分解出来，如果编译器不能处理它。例如：
Some common sub-expression can be factored out, if the compiler cannot handle it. For example:
uv = u * v
vw = v * w
uw = u * w
uvw = uv * w
在扩展 N 被多次重复使用的系数。它们可以通过计算：
The coefficients in the expanded n are reused multiple times. They can be computed by:
k0 = dot100 - dot000
k1 = dot010 - dot000
k2 = dot001 - dot000
k3 = dot110 - dot010 - k0
k4 = dot101 - dot001 - k0
k5 = dot011 - dot001 - k1
k6 = (dot111 - dot011) - (dot101 - dot001) - k3
另外，衍生品也有类似的系数，
Also the derivatives has similar coefficients,
gxk0 = gx100 - gx000
gxk1 = gx010 - gx000
...
的计算 N 可以使用的扩展形式 K0 ... K6 和
The computation of n can uses the expanded form with k0, ... k6 as well.
该解决方案已被证实对中心差分法。
This solution has been verified against central difference method.
虽然这种解决方案看起来笨拙，我的实验（仅CPU，SSE）表明，该解决方案计算这些衍生品只招约 50％的额外时间可以计算单个3D Perlin杂样。
Although this solution looks clumsy, my experiment (CPU only, SSE) showed that, computing these derivatives by this solution only incurs about 50% extra time to computing a single 3D Perlin noise sample.
有限差分将至少需要300％额外的时间（做额外的3个样本）或600％（做6个样品的中央差）。
Finite difference will at least need 300% extra time (doing extra 3 samples) or 600% (doing 6 samples for central difference).
因此，该解决方案是在性能更好，并且也应该更加数值稳定
Therefore, this solution is better in performance, and should also be more numerically stable.
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