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    <p>I also could not found a solution on the web today, so I tried to derive it.</p> <p>Firstly the notations of a 3D Perlin noise is defined.</p> <h1>Notation</h1> <p>Assume the 3D Perlin noise is computed by the trilinear interpolation as</p> <pre><code>n = Lerp( Lerp( Lerp(dot000, dot100, u), Lerp(dot010, dot110, u), v), Lerp( Lerp(dot001, dot101, u), Lerp(dot011, dot111, u), v), w) </code></pre> <p>where <code>u</code>, <code>v</code>, <code>w</code> are the interpolation factors by the quintic polynomial of fraction coordinates (i.e., improved Perlin noise):</p> <pre><code>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) </code></pre> <p>and <code>dot___</code>s are dot products of the gradient vectors <code>(gx___, gy___, gz___)</code>s at lattice points and the fraction coordinates:</p> <pre><code>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 </code></pre> <h1>Computing the derivatives</h1> <p>First, compute derivatives of <code>u</code>, <code>v</code> and <code>w</code></p> <pre><code>u' = 30 * x0 * x0 * (x0 - 1) * (x0 - 1) v' = 30 * y0 * y0 * (y0 - 1) * (y0 - 1) w' = 30 * z0 * z0 * (z0 - 1) * (z0 - 1) </code></pre> <p>By expanding <code>n</code> with <code>Lerp(a, b, t) = a + (b - a) * t</code>,</p> <pre><code>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) </code></pre> <p>Then take partial derivatives of <code>n</code>,</p> <pre><code>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) </code></pre> <p>,</p> <pre><code>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) </code></pre> <p>,</p> <pre><code>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) </code></pre> <p>Then <code>(nx, ny, nz)</code> is the gradient vector (partial derivatives) of the noise function.</p> <h1>Optimization</h1> <p>Some common sub-expression can be factored out, if the compiler cannot handle it. For example:</p> <pre><code>uv = u * v vw = v * w uw = u * w uvw = uv * w </code></pre> <p>The coefficients in the expanded <code>n</code> are reused multiple times. They can be computed by:</p> <pre><code>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 </code></pre> <p>Also the derivatives has similar coefficients,</p> <pre><code>gxk0 = gx100 - gx000 gxk1 = gx010 - gx000 ... </code></pre> <p>The computation of <code>n</code> can uses the expanded form with <code>k0</code>, ... <code>k6</code> as well.</p> <h1>Final words</h1> <p>This solution has been verified against central difference method.</p> <p>Although this solution looks clumsy, my experiment (CPU only, SSE) showed that, computing these derivatives by this solution only incurs about <strong>50% extra time</strong> to computing a single 3D Perlin noise sample. </p> <p>Finite difference will at least need 300% extra time (doing extra 3 samples) or 600% (doing 6 samples for central difference).</p> <p>Therefore, this solution is better in performance, and should also be more numerically stable.</p>
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