### Simplex Noise for OSL part 2: turbulence

In this article a small update on my previous implementation of Simplex Noise for OSL: an extra feature to add turbulence in the form of fractional brownian motion (fBM) to the noise

## Fractional Brownian Motion

In a previous article I showed how to implement 2, 3 and 4 dimensional simplex noise in OSL. In this article I show how this shader can be augmented with fractional brownian motion. Examples for 1 (= unperturbed noise), 2 and 3 octaves are shown here:

The code shown below isn't much altered from the previous version except for a loop around the core noise generation to add and scale additional octaves of noise. The shader node it creates gets extra input sockets for octaves, lacunarity and H which have the same meaning as in the regular Perlin implementation of fBM enhanced noise.

`/** A speed-improved simplex noise algorithm for 2D, 3D and 4D in OSL.** Based on example Java code by Stefan Gustavson (stegu@itn.liu.se).* Optimisations by Peter Eastman (peastman@drizzle.stanford.edu).* Better rank ordering method by Stefan Gustavson in 2012.** This could be speeded up even further, but it's useful as it is.** OSL port Michel Anders (varkenvarken) 2013-02-04* original comment is is left in place, OSL specific comments* are preceded by MJA** This code was placed in the public domain by its original author,* Stefan Gustavson. You may use it as you see fit, but* attribution is appreciated.** 2013-02-09 (varkenvarken) added fBM turbulence (Octaves) */int fastfloor(float x) { int xi = (int)x; return x < xi ? xi-1 : xi;}// MJA it is safe to overload functions in OSL // so here's some extra dot() versionsfloat dot(vector g, float x, float y) { return g[0]*x + g[1]*y; }float dot(vector g, float x, float y, float z) { return g[0]*x + g[1]*y + g[2]*z; }float dot(vector g, float t, float x, float y, float z, float w) { return g[0]*x + g[1]*y + g[2]*z + t*w; }shader simplexnoise( point Pos = P, float t = 0, float Scale = 1, int Octaves = 1, float H = 1, float lacunarity=2, int Dim = 2, output float fac = 0){ vector grad3[12] = {vector(1,1,0),vector(-1,1,0),      vector(1,-1,0),vector(-1,-1,0),      vector(1,0,1),vector(-1,0,1),      vector(1,0,-1),vector(-1,0,-1),      vector(0,1,1),vector(0,-1,1),      vector(0,1,-1),vector(0,-1,-1)}; // MJA I couldn't get OSL to compile an array of structs so // I separated a vec4 into a vector and a float vector grad4v[32]= {vector(0,1,1) ,vector(0,1,1)  ,      vector(0,1,-1) ,vector(0,1,-1),      vector(0,-1,1),vector(0,-1,1) ,      vector(0,-1,-1),vector(0,-1,-1),      vector(1,0,1) ,vector(1,0,1)  ,      vector(1,0,-1) ,vector(1,0,-1),      vector(-1,0,1),vector(-1,0,1) ,      vector(-1,0,-1),vector(-1,0,-1),      vector(1,1,0) ,vector(1,1,0)  ,      vector(1,-1,0) ,vector(1,-1,0),      vector(-1,1,0),vector(-1,1,0) ,      vector(-1,-1,0),vector(-1,-1,0),      vector(1,1,1) ,vector(1,1,-1) ,      vector(1,-1,1) ,vector(1,-1,-1),      vector(-1,1,1),vector(-1,1,-1),      vector(-1,-1,1),vector(-1,-1,-1)}; float grad4t[32]= { 1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,      1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,      0,0,0,0,0,0,0,0};       int perm[512] = {151,160,137,91,90,15, 131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23, 190, 6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33, 88,237,149,56,87,174,20,125,136,171,168, 68,175,74,165,71,134,139,48,27,166, 77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244, 102,143,54, 65,25,63,161, 1,216,80,73,209,76,132,187,208, 89,18,169,200,196, 135,130,116,188,159,86,164,100,109,198,173,186, 3,64,52,217,226,250,124,123, 5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42, 223,183,170,213,119,248,152, 2,44,154,163, 70,221,153,101,155,167, 43,172,9, 129,22,39,253, 19,98,108,110,79,113,224,232,178,185, 112,104,218,246,97,228, 251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107, 49,192,214, 31,181,199,106,157,184, 84,204,176,115,121,50,45,127, 4,150,254, 138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180, 151,160,137,91,90,15, 131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23, 190, 6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33, 88,237,149,56,87,174,20,125,136,171,168, 68,175,74,165,71,134,139,48,27,166, 77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244, 102,143,54, 65,25,63,161, 1,216,80,73,209,76,132,187,208, 89,18,169,200,196, 135,130,116,188,159,86,164,100,109,198,173,186, 3,64,52,217,226,250,124,123, 5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42, 223,183,170,213,119,248,152, 2,44,154,163, 70,221,153,101,155,167, 43,172,9, 129,22,39,253, 19,98,108,110,79,113,224,232,178,185, 112,104,218,246,97,228, 251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107, 49,192,214, 31,181,199,106,157,184, 84,204,176,115,121,50,45,127, 4,150,254, 138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180}; // MJA precomputing this table instead of calculating it as done in the  // original code saves 30% running time. int permMod12[512] = { 7, 4, 5, 7, 6, 3, 11, 1, 9, 11, 0, 5, 2, 5, 7, 9, 8,  0, 7, 6, 9, 10, 8, 3, 1, 0, 9, 10, 11, 10, 6, 4, 7, 0, 6, 3, 0, 2, 5, 2, 10,  0, 3, 11, 9, 11, 11, 8, 9, 9, 9, 4, 9, 5, 8, 3, 6, 8, 5, 4, 3, 0, 8, 7, 2, 9,  11, 2, 7, 0, 3, 10, 5, 2, 2, 3, 11, 3, 1, 2, 0, 7, 1, 2, 4, 9, 8, 5, 7, 10,  5, 4, 4, 6, 11, 6, 5, 1, 3, 5, 1, 0, 8, 1, 5, 4, 0, 7, 4, 5, 6, 1, 8, 4, 3,  10, 8, 8, 3, 2, 8, 4, 1, 6, 5, 6, 3, 4, 4, 1, 10, 10, 4, 3, 5, 10, 2, 3, 10,  6, 3, 10, 1, 8, 3, 2, 11, 11, 11, 4, 10, 5, 2, 9, 4, 6, 7, 3, 2, 9, 11, 8, 8,  2, 8, 10, 7, 10, 5, 9, 5, 11, 11, 7, 4, 9, 9, 10, 3, 1, 7, 2, 0, 2, 7, 5, 8,  4, 10, 5, 4, 8, 2, 6, 1, 0, 11, 10, 2, 1, 10, 6, 0, 0, 11, 11, 6, 1, 9, 3, 1,  7, 9, 2, 11, 11, 1, 0, 10, 7, 1, 7, 10, 1, 4, 0, 0, 8, 7, 1, 2, 9, 7, 4, 6, 2,  6, 8, 1, 9, 6, 6, 7, 5, 0, 0, 3, 9, 8, 3, 6, 6, 11, 1, 0, 0, 7, 4, 5, 7, 6, 3,  11, 1, 9, 11, 0, 5, 2, 5, 7, 9, 8, 0, 7, 6, 9, 10, 8, 3, 1, 0, 9, 10, 11, 10,  6, 4, 7, 0, 6, 3, 0, 2, 5, 2, 10, 0, 3, 11, 9, 11, 11, 8, 9, 9, 9, 4, 9, 5, 8,  3, 6, 8, 5, 4, 3, 0, 8, 7, 2, 9, 11, 2, 7, 0, 3, 10, 5, 2, 2, 3, 11, 3, 1, 2,  0, 7, 1, 2, 4, 9, 8, 5, 7, 10, 5, 4, 4, 6, 11, 6, 5, 1, 3, 5, 1, 0, 8, 1, 5, 4,  0, 7, 4, 5, 6, 1, 8, 4, 3, 10, 8, 8, 3, 2, 8, 4, 1, 6, 5, 6, 3, 4, 4, 1, 10, 10,  4, 3, 5, 10, 2, 3, 10, 6, 3, 10, 1, 8, 3, 2, 11, 11, 11, 4, 10, 5, 2, 9, 4, 6, 7,  3, 2, 9, 11, 8, 8, 2, 8, 10, 7, 10, 5, 9, 5, 11, 11, 7, 4, 9, 9, 10, 3, 1, 7, 2,  0, 2, 7, 5, 8, 4, 10, 5, 4, 8, 2, 6, 1, 0, 11, 10, 2, 1, 10, 6, 0, 0, 11, 11, 6,  1, 9, 3, 1, 7, 9, 2, 11, 11, 1, 0, 10, 7, 1, 7, 10, 1, 4, 0, 0, 8, 7, 1, 2, 9, 7,  4, 6, 2, 6, 8, 1, 9, 6, 6, 7, 5, 0, 0, 3, 9, 8, 3, 6, 6, 11, 1, 0, 0};  // Skewing and unskewing factors for 2, 3, and 4 dimensions float F2 = 0.5*(sqrt(3.0)-1.0); float G2 = (3.0-sqrt(3.0))/6.0; float F3 = 1.0/3.0; float G3 = 1.0/6.0; float F4 = (sqrt(5.0)-1.0)/4.0; float G4 = (5.0-sqrt(5.0))/20.0; float pwr = 1.0; float pwHL = pow(lacunarity,-H);    if(Dim == 2){  // 2D simplex noise  fac = 0;  float xin=Pos[0]*Scale, yin=Pos[1]*Scale;  for(int p=0; p   float n0, n1, n2; // Noise contributions from the three corners   // Skew the input space to determine which simplex cell we're in   float s = (xin+yin)*F2; // Hairy factor for 2D   int i = fastfloor(xin+s);   int j = fastfloor(yin+s);   float t = (i+j)*G2;   float X0 = i-t; // Unskew the cell origin back to (x,y) space   float Y0 = j-t;   float x0 = xin-X0; // The x,y distances from the cell origin   float y0 = yin-Y0;   // For the 2D case, the simplex shape is an equilateral triangle.   // Determine which simplex we are in.   int i1, j1; // Offsets for second (middle) corner of simplex in (i,j) coords   if(x0>y0) {i1=1; j1=0;} // lower triangle, XY order: (0,0)->(1,0)->(1,1)   else {i1=0; j1=1;}      // upper triangle, YX order: (0,0)->(0,1)->(1,1)   // A step of (1,0) in (i,j) means a step of (1-c,-c) in (x,y), and   // a step of (0,1) in (i,j) means a step of (-c,1-c) in (x,y), where   // c = (3-sqrt(3))/6   float x1 = x0 - i1 + G2; // Offsets for middle corner in (x,y) unskewed coords   float y1 = y0 - j1 + G2;   float x2 = x0 - 1.0 + 2.0 * G2; // Offsets for last corner in (x,y) unskewed coords   float y2 = y0 - 1.0 + 2.0 * G2;   // Work out the hashed gradient indices of the three simplex corners   int ii = i & 255;   int jj = j & 255;   int gi0 = permMod12[ii+perm[jj]];   int gi1 = permMod12[ii+i1+perm[jj+j1]];   int gi2 = permMod12[ii+1+perm[jj+1]];   // Calculate the contribution from the three corners   float t0 = 0.5 - x0*x0-y0*y0;   if(t0 < 0) n0 = 0.0;   else {   t0 *= t0;   n0 = t0 * t0 * dot(grad3[gi0], x0, y0);  // (x,y) of grad3 used for 2D gradient   }   float t1 = 0.5 - x1*x1-y1*y1;   if(t1 < 0) n1 = 0.0;   else {   t1 *= t1;   n1 = t1 * t1 * dot(grad3[gi1], x1, y1);   }   float t2 = 0.5 - x2*x2-y2*y2;   if(t2 < 0) n2 = 0.0;   else {   t2 *= t2;   n2 = t2 * t2 * dot(grad3[gi2], x2, y2);   }   // Add contributions from each corner to get the final noise value.   // The result is scaled to return values in the interval [-1,1].   fac += (70.0 * (n0 + n1 + n2)) * pwr;   pwr *= pwHL;   xin *= lacunarity;   yin *= lacunarity;  } } else if(Dim == 3){  // 3D simplex noise  fac = 0 ;  float xin=Pos[0]*Scale, yin=Pos[1]*Scale, zin=Pos[2]*Scale;  for(int p=0; p   float n0, n1, n2, n3; // Noise contributions from the four corners   // Skew the input space to determine which simplex cell we're in   float s = (xin+yin+zin)*F3; // Very nice and simple skew factor for 3D   int i = fastfloor(xin+s);   int j = fastfloor(yin+s);   int k = fastfloor(zin+s);   float t = (i+j+k)*G3;   float X0 = i-t; // Unskew the cell origin back to (x,y,z) space   float Y0 = j-t;   float Z0 = k-t;   float x0 = xin-X0; // The x,y,z distances from the cell origin   float y0 = yin-Y0;   float z0 = zin-Z0;   // For the 3D case, the simplex shape is a slightly irregular tetrahedron.   // Determine which simplex we are in.   int i1, j1, k1; // Offsets for second corner of simplex in (i,j,k) coords   int i2, j2, k2; // Offsets for third corner of simplex in (i,j,k) coords   if(x0>=y0) {   if(y0>=z0)    { i1=1; j1=0; k1=0; i2=1; j2=1; k2=0; } // X Y Z order    else if(x0>=z0) { i1=1; j1=0; k1=0; i2=1; j2=0; k2=1; } // X Z Y order    else { i1=0; j1=0; k1=1; i2=1; j2=0; k2=1; } // Z X Y order   }   else { // x0 < y0   if(y0 < z0) { i1=0; j1=0; k1=1; i2=0; j2=1; k2=1; } // Z Y X order   else if(x0 < z0) { i1=0; j1=1; k1=0; i2=0; j2=1; k2=1; } // Y Z X order   else { i1=0; j1=1; k1=0; i2=1; j2=1; k2=0; } // Y X Z order   }   // A step of (1,0,0) in (i,j,k) means a step of (1-c,-c,-c) in (x,y,z),   // a step of (0,1,0) in (i,j,k) means a step of (-c,1-c,-c) in (x,y,z), and   // a step of (0,0,1) in (i,j,k) means a step of (-c,-c,1-c) in (x,y,z), where   // c = 1/6.   float x1 = x0 - i1 + G3; // Offsets for second corner in (x,y,z) coords   float y1 = y0 - j1 + G3;   float z1 = z0 - k1 + G3;   float x2 = x0 - i2 + 2.0*G3; // Offsets for third corner in (x,y,z) coords   float y2 = y0 - j2 + 2.0*G3;   float z2 = z0 - k2 + 2.0*G3;   float x3 = x0 - 1.0 + 3.0*G3; // Offsets for last corner in (x,y,z) coords   float y3 = y0 - 1.0 + 3.0*G3;   float z3 = z0 - 1.0 + 3.0*G3;   // Work out the hashed gradient indices of the four simplex corners   int ii = i & 255;   int jj = j & 255;   int kk = k & 255;   int gi0 = permMod12[ii+perm[jj+perm[kk]]];   int gi1 = permMod12[ii+i1+perm[jj+j1+perm[kk+k1]]];   int gi2 = permMod12[ii+i2+perm[jj+j2+perm[kk+k2]]];   int gi3 = permMod12[ii+1+perm[jj+1+perm[kk+1]]];   // Calculate the contribution from the four corners   float t0 = 0.6 - x0*x0 - y0*y0 - z0*z0;   if(t0 < 0) n0 = 0.0;   else {   t0 *= t0;   n0 = t0 * t0 * dot(grad3[gi0], x0, y0, z0);   }   float t1 = 0.6 - x1*x1 - y1*y1 - z1*z1;   if(t1 < 0) n1 = 0.0;   else {   t1 *= t1;   n1 = t1 * t1 * dot(grad3[gi1], x1, y1, z1);   }   float t2 = 0.6 - x2*x2 - y2*y2 - z2*z2;   if(t2 < 0) n2 = 0.0;   else {   t2 *= t2;   n2 = t2 * t2 * dot(grad3[gi2], x2, y2, z2);   }   float t3 = 0.6 - x3*x3 - y3*y3 - z3*z3;   if(t3 < 0) n3 = 0.0;   else {   t3 *= t3;   n3 = t3 * t3 * dot(grad3[gi3], x3, y3, z3);   }   // Add contributions from each corner to get the final noise value.   // The result is scaled to stay just inside [-1,1]   fac += (32.0*(n0 + n1 + n2 + n3))*pwr;   pwr *= pwHL;   xin *= lacunarity;   yin *= lacunarity;   zin *= lacunarity;  } } else if ( Dim == 4 ) {  // 4D simplex noise, better simplex rank ordering method 2012-03-09  fac = 0;  float x=Pos[0]*Scale, y=Pos[1]*Scale, z=Pos[3]*Scale, w=t*Scale;  for(int p=0; p   float n0, n1, n2, n3, n4; // Noise contributions from the five corners   // Skew the (x,y,z,w) space to determine which cell of 24 simplices we're in   float s = (x + y + z + w) * F4; // Factor for 4D skewing   int i = fastfloor(x + s);   int j = fastfloor(y + s);   int k = fastfloor(z + s);   int l = fastfloor(w + s);   float t = (i + j + k + l) * G4; // Factor for 4D unskewing   float X0 = i - t; // Unskew the cell origin back to (x,y,z,w) space   float Y0 = j - t;   float Z0 = k - t;   float W0 = l - t;   float x0 = x - X0;  // The x,y,z,w distances from the cell origin   float y0 = y - Y0;   float z0 = z - Z0;   float w0 = w - W0;   // For the 4D case, the simplex is a 4D shape I won't even try to describe.   // To find out which of the 24 possible simplices we're in, we need to   // determine the magnitude ordering of x0, y0, z0 and w0.   // Six pair-wise comparisons are performed between each possible pair   // of the four coordinates, and the results are used to rank the numbers.   int rankx = 0;   int ranky = 0;   int rankz = 0;   int rankw = 0;   if(x0 > y0) rankx++; else ranky++;   if(x0 > z0) rankx++; else rankz++;   if(x0 > w0) rankx++; else rankw++;   if(y0 > z0) ranky++; else rankz++;   if(y0 > w0) ranky++; else rankw++;   if(z0 > w0) rankz++; else rankw++;   int i1, j1, k1, l1; // The integer offsets for the second simplex corner   int i2, j2, k2, l2; // The integer offsets for the third simplex corner   int i3, j3, k3, l3; // The integer offsets for the fourth simplex corner   // simplex[c] is a 4-vector with the numbers 0, 1, 2 and 3 in some order.   // Many values of c will never occur, since e.g. x>y>z>w makes x < z, y < w and x < w   // impossible. Only the 24 indices which have non-zero entries make any sense.   // We use a thresholding to set the coordinates in turn from the largest magnitude.   // Rank 3 denotes the largest coordinate.   i1 = rankx >= 3 ? 1 : 0;   j1 = ranky >= 3 ? 1 : 0;   k1 = rankz >= 3 ? 1 : 0;   l1 = rankw >= 3 ? 1 : 0;   // Rank 2 denotes the second largest coordinate.   i2 = rankx >= 2 ? 1 : 0;   j2 = ranky >= 2 ? 1 : 0;   k2 = rankz >= 2 ? 1 : 0;   l2 = rankw >= 2 ? 1 : 0;   // Rank 1 denotes the second smallest coordinate.   i3 = rankx >= 1 ? 1 : 0;   j3 = ranky >= 1 ? 1 : 0;   k3 = rankz >= 1 ? 1 : 0;   l3 = rankw >= 1 ? 1 : 0;   // The fifth corner has all coordinate offsets = 1, so no need to compute that.   float x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w) coords   float y1 = y0 - j1 + G4;   float z1 = z0 - k1 + G4;   float w1 = w0 - l1 + G4;   float x2 = x0 - i2 + 2.0*G4; // Offsets for third corner in (x,y,z,w) coords   float y2 = y0 - j2 + 2.0*G4;   float z2 = z0 - k2 + 2.0*G4;   float w2 = w0 - l2 + 2.0*G4;   float x3 = x0 - i3 + 3.0*G4; // Offsets for fourth corner in (x,y,z,w) coords   float y3 = y0 - j3 + 3.0*G4;   float z3 = z0 - k3 + 3.0*G4;   float w3 = w0 - l3 + 3.0*G4;   float x4 = x0 - 1.0 + 4.0*G4; // Offsets for last corner in (x,y,z,w) coords   float y4 = y0 - 1.0 + 4.0*G4;   float z4 = z0 - 1.0 + 4.0*G4;   float w4 = w0 - 1.0 + 4.0*G4;   // Work out the hashed gradient indices of the five simplex corners   int ii = i & 255;   int jj = j & 255;   int kk = k & 255;   int ll = l & 255;   int gi0 = perm[ii+perm[jj+perm[kk+perm[ll]]]] % 32;   int gi1 = perm[ii+i1+perm[jj+j1+perm[kk+k1+perm[ll+l1]]]] % 32;   int gi2 = perm[ii+i2+perm[jj+j2+perm[kk+k2+perm[ll+l2]]]] % 32;   int gi3 = perm[ii+i3+perm[jj+j3+perm[kk+k3+perm[ll+l3]]]] % 32;   int gi4 = perm[ii+1+perm[jj+1+perm[kk+1+perm[ll+1]]]] % 32;   // Calculate the contribution from the five corners   // MJA because I couldn't get OSL to compile an array of structs   // the 4-vectors for the gradients are split into an array with   // regular vectors (x,y,z) in grad4v and an array of floats   // (for the w component) in grad4t   float t0 = 0.6 - x0*x0 - y0*y0 - z0*z0 - w0*w0;   if(t0 < 0) n0 = 0.0;   else {   t0 *= t0;   n0 = t0 * t0 * dot(grad4v[gi0], grad4t[gi0], x0, y0, z0, w0);   }   float t1 = 0.6 - x1*x1 - y1*y1 - z1*z1 - w1*w1;   if(t1 < 0) n1 = 0.0;   else {   t1 *= t1;   n1 = t1 * t1 * dot(grad4v[gi1], grad4t[gi1], x1, y1, z1, w1);   }   float t2 = 0.6 - x2*x2 - y2*y2 - z2*z2 - w2*w2;   if(t2 < 0) n2 = 0.0;   else {   t2 *= t2;   n2 = t2 * t2 * dot(grad4v[gi2], grad4t[gi2], x2, y2, z2, w2);   }   float t3 = 0.6 - x3*x3 - y3*y3 - z3*z3 - w3*w3;   if(t3 < 0) n3 = 0.0;   else {   t3 *= t3;   n3 = t3 * t3 * dot(grad4v[gi3], grad4t[gi3], x3, y3, z3, w3);   }   float t4 = 0.6 - x4*x4 - y4*y4 - z4*z4 - w4*w4;   if(t4 < 0) n4 = 0.0;   else {   t4 *= t4;   n4 = t4 * t4 * dot(grad4v[gi4], grad4t[gi4], x4, y4, z4, w4);   }   // Sum up and scale the result to cover the range [-1,1]   fac += (27.0 * (n0 + n1 + n2 + n3 + n4))*pwr;   pwr *= pwHL;   x   *= lacunarity;   y   *= lacunarity;   z   *= lacunarity;   w   *= lacunarity;  } }}`

## Example node setup

The sample images were created with the following node setup: