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math: shitloads of tweaks, optimisations, fixes and comments to the 

simplex noise code.
undefined
Sam Hocevar 10 years ago
parent
91451eaf70
commit
0af8c1fd6f
2 changed files with 264 additions and 126 deletions
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  1. +78
    -14
      demos/test/simplex.cpp
  2. +186
    -112
      src/lol/math/simplex_interpolator.h

+ 78
- 14
demos/test/simplex.cpp View File

@@ -17,44 +17,108 @@

using namespace lol;

/* Constants to tweak */
float const zoom = 0.03f;
int const octaves = 1;

int main(int argc, char **argv)
{
UNUSED(argc, argv);

float const zoom = 0.03f;
int const octaves = 10;

/* Create an image */
ivec2 const size(1280, 720);
Image img(size);
array2d<vec4> &data = img.Lock2D<PixelFormat::RGBA_F32>();

/* Fill image with simplex noise */
/* Declare plenty of allocators */
simplex_interpolator<2> s2;
simplex_interpolator<3> s3;
simplex_interpolator<4> s4;
simplex_interpolator<5> s5;
simplex_interpolator<6> s6;
simplex_interpolator<7> s7;
simplex_interpolator<8> s8;
simplex_interpolator<9> s9;
simplex_interpolator<10> s10;
simplex_interpolator<11> s11;
simplex_interpolator<12> s12;

/* Fill image with simplex noise */
for (int j = 0; j < size.y; ++j)
for (int i = 0; i < size.x; ++i)
{
int cell = j / (size.y / 2) * 3 + i / (size.x / 3);

float x = (float)i, y = (float)j;
float sum = 0.f;
int maxoctave = (j < size.y / 2) ? 1 : (1 << octaves);
float sum = 0.f, coeff = 0.f;
int const max_k = 1 << octaves;

for (int k = 1; k <= maxoctave; k *= 2)
for (int k = 1; k < max_k; k *= 2)
{
if (i < size.x / 3)
sum += 0.5f / k * s2.Interp(zoom * k * vec2(x, y));
else if (i < size.x * 2 / 3)
sum += 0.5f / k * s3.Interp(zoom * k * vec3(x, y, 0.f));
else
sum += 0.5f / k * s7.Interp(zoom * k * vec7(x, 0.f, 0.f, 0.f, y, 0.f, 0.f));
float t = 0.f;

switch (cell)
{
case 0:
t = s2.Interp(zoom * k * vec2(x, y));
break;
case 1:
t = s3.Interp(zoom * k * vec3(x, 1.f, y));
break;
case 2:
t = s4.Interp(zoom * k * vec4(x, 1.f, y, 2.f));
break;
case 3:
t = s6.Interp(zoom * k * vec6(x, 1.f, 2.f, y, 3.f, 4.f));
break;
case 4:
t = s8.Interp(zoom * k * vec8(x, 1.f, 2.f, 3.f, y, 4.f,
5.f, 6.f));
break;
case 5:
t = s12.Interp(zoom * k * vec12(x, 1.f, 2.f, 3.f, 4.f, 5.f,
y, 6.f, 7.f, 8.f, 9.f, 10.f));
break;
default:
break;
}

sum += 1.f / k * t;
coeff += 1.f / k;
}

float c = saturate(0.5f + sum);
float c = saturate(0.5f + 0.5f * sum / coeff);
data[i][j] = vec4(c, c, c, 1.f);
//data[i][j] = Color::HSVToRGB(vec4(c, 1.0f, 0.5f, 1.f));
}

#if 0
/* Mark simplex vertices */
vec2 diagonal = normalize(vec2(1.f));
vec2 dx = mat2::rotate(60.f) * diagonal;
vec2 dy = mat2::rotate(-60.f) * diagonal;
for (int j = -100; j < 100; ++j)
for (int i = -100; i < 100; ++i)
{
auto putpixel = [&](ivec2 p, vec4 c = vec4(1.f, 0.f, 1.f, 1.f))
{
if (p.x >= 0 && p.x < size.x - 1 && p.y >= 0 && p.y < size.y - 1)
data[p.x][p.y] = c;
};

ivec2 coord = ivec2(i / zoom * dx + j / zoom * dy);

vec2 g = s2.GetGradient((i + 0.1f) * dx + (j + 0.1f) * dy);
for (int t = 0; t < 40; ++t)
putpixel(coord + (ivec2)(g * (t / 2.f)), vec4(0.f, 1.f, 0.f, 1.f));

putpixel(coord);
putpixel(coord + ivec2(1, 0));
putpixel(coord + ivec2(0, 1));
putpixel(coord + ivec2(1, 1));
}
#endif

/* Save image */
img.Unlock2D(data);
img.Save("simplex.png");


+ 186
- 112
src/lol/math/simplex_interpolator.h View File

@@ -17,6 +17,27 @@
namespace lol
{

/*
* Simplex noise in dimension N
* ----------------------------
*
* The N-dimensional regular hypercube can be split into N! simplices that
* all have the main diagonal as a shared edge.
* - number of simplices: N!
* - number of vertices per simplex: N+1
* - number of edges: N(N+1)/2
* - minimum edge length: 1 (hypercube edges, e.g. [1,0,0,…,0])
* - maximum edge length: sqrt(N) (hypercube diagonal, i.e. [1,1,1,…,1])
*
* We skew the simplicial grid along the main diagonal by a factor f =
* sqrt(N+1), which means the diagonal of the initial parallelotope has
* length sqrt(N/(N+1)). The edges of that parallelotope have length
* sqrt(N/(N+1)), too. A formula for the maximum edge length was found
* empirically:
* - minimum edge length: sqrt(N/(N+1)) (parallelotope edges and diagonal)
* - maximum edge length: sqrt(floor((N+1)²/4)/(N+1))
*/

template<int N>
class simplex_interpolator
{
@@ -24,165 +45,218 @@ public:
simplex_interpolator(int seed = 0)
: m_seed(seed)
{
/* Matrix coordinate transformation to skew simplex referential is done
* by inverting the base matrix M which is written as follows:
*
* M = | a b b b … | M^(-1) = | c d d d … |
* | b a b b … | | d c d d … |
* | b b a b … | | d d c d … |
* | … | | … |
*
* where a, b, c, d are computed below ↴
*/

float b = (1.f - lol::sqrt(N + 1.f)) / lol::sqrt((float)N * N * N);
float a = b + lol::sqrt((N + 1.f) / N);

// determinant of matrix M
float det = a * (a + (N - 2) * b) - (b * b) * (N - 1);

float c = (a + (N - 2) * b) / det;
float d = -b / det;

#if 0
// Print some debug information
printf("Simplex Noise of Dimension %d\n", N);

long long int n = 1; for (int i = 1; i <= N; ++i) n *= i;
printf(" - each hypercube cell has %lld simplices "
"with %d vertices and %d edges\n", n, N + 1, N * (N + 1) / 2);

vec_t<float, N> diagonal(1.f);
printf(" - length of hypercube edges is 1, diagonal is %f\n",
length(diagonal));
printf(" - length of parallelotope edges and diagonal is %f\n",
length(unskew(diagonal)));

vec_t<float, N> vertices[N + 1];
vertices[0] = vec_t<float, N>(0.f);
for (int i = 0; i < N; ++i)
{
for (int j = 0; j < N; ++j)
vertices[i + 1] = vertices[i];
vertices[i + 1][i] += 1.f;
}
float minlength = 10.0f;
float maxlength = 0.0f;
for (int i = 0; i < N + 1; ++i)
for (int j = i + 1; j < N + 1; ++j)
{
m_base[i][j] = (i == j ? a : b);
m_base_inverse[i][j] = (i == j ? c : d);
float l = length(unskew(vertices[i] - vertices[j]));
minlength = min(minlength, l);
maxlength = max(maxlength, l);
}
}
printf(" - edge lengths between %f and %f\n", minlength, maxlength);
printf(" - predicted: %f and %f\n", sqrt((float)N/(N+1)),
sqrt((N+1)*(N+1)/4/(float)(N+1)));
printf("\n");
#endif
}

// Single interpolation
/* Single interpolation */
inline float Interp(vec_t<float, N> position) const
{
// Computing position in simplex referential
vec_t<float, N> simplex_position = m_base_inverse * position;

// Retrieving the closest floor point and decimals
// Retrieve the containing hypercube origin and associated decimals
vec_t<int, N> origin;
vec_t<float, N> pos;
get_origin(skew(position), origin, pos);

this->ExtractFloorDecimal(simplex_position, origin, pos);

vec_t<int, N> index_order = this->GetIndexOrder(pos);

// Resetting decimal point in regular orthonormal coordinates
pos = m_base * pos;
return get_noise(origin, pos);
}

return this->LastInterp(origin, pos, index_order);
/* Only for debug purposes: return the gradient vector */
inline vec_t<float, N> GetGradient(vec_t<float, N> position) const
{
vec_t<int, N> origin;
vec_t<float, N> pos;
get_origin(skew(position), origin, pos);
return get_gradient(origin);
}

protected:
inline float LastInterp(vec_t<int, N> origin,
vec_t<float, N> const & pos,
vec_t<int, N> const & index_order) const
inline float get_noise(vec_t<int, N> origin,
vec_t<float, N> const & pos) const
{
/* “corner” will traverse the simplex along its edges in
* orthonormal coordinates. */
vec_t<float, N> corner(0);
float result = 0;
/* For a given position [0…1]^N inside a regular N-hypercube, find
* the N-simplex which contains that position, and return a path
* along the hypercube edges from (0,0,…,0) to (1,1,…,1) which
* uniquely describes that simplex. */
vec_t<int, N> traversal_order;
for (int i = 0; i < N; ++i)
traversal_order[i] = i;

/* Naïve bubble sort — enough for now */
for (int i = 0; i < N; ++i)
for (int j = i + 1; j < N; ++j)
if (pos[traversal_order[i]] < pos[traversal_order[j]])
std::swap(traversal_order[i], traversal_order[j]);


/* Get the position in world coordinates, too */
vec_t<float, N> world_pos = unskew(pos);

/* “corner” will traverse the simplex along its edges in world
* coordinates. */
vec_t<float, N> world_corner(0.f);
float result = 0.f, coeff = 0.f;

for (int i = 0; i < N + 1; ++i)
{
vec_t<float, N> const delta = pos - corner;
vec_t<float, N> const &gradient = GetGradient(origin);

/* We use 4/3 because the maximum radius of influence for
* a given corner is sqrt(3/4). FIXME: check whether this
* holds for higher dimensions. */
float dist = 1.0f - 4.f / 3.f * sqlength(delta);
if (dist > 0)
#if 1
// FIXME: In “Noise Hardware” (2-17) Perlin uses 0.6 but Gustavson
// makes an exception for dimension 2 and uses 0.5 instead.
// I think m should actually increase with the dimension count.
float const m = N == 2 ? 0.5f : 0.6f;
float d = m - sqlength(world_pos - world_corner);
#else
// DEBUG: this is the linear contribution of each vertex
// in the skewed simplex. Unfortunately it creates artifacts.
float d = ((i == 0) ? 1.f : pos[traversal_order[i - 1]])
- ((i == N) ? 0.f : pos[traversal_order[i]]);
#endif

if (d > 0)
{
dist *= dist;
result += dist * dist * dot(gradient, delta);
vec_t<float, N> const &gradient = get_gradient(origin);

// Perlin uses 8d⁴ whereas Gustavson uses d⁴ and a final
// multiplication factor at the end.
//d = 8.f * d * d * d * d;
d = d * d * d * d;

//d = (3.f - 2.f * d) * d * d;
//d = ((6 * d - 15) * d + 10) * d * d * d;

result += d * dot(gradient, world_pos - world_corner);
coeff += d;
}

if (i < N)
{
corner += m_base[index_order[i]];
origin[index_order[i]] += 1;
vec_t<float, N> v(0.f);
v[traversal_order[i]] = 1.f;
world_corner += unskew(v);
origin[traversal_order[i]] += 1;
}
}

// FIXME: another paper uses the value 70 for dimension 2, 32 for
// dimension 3, and 27 for dimension 4; find where this comes from.
return result * 70.f / 16.f;
}

/* For a given position [0…1]^n inside a square/cube/hypercube etc.,
* find the simplex which contains that position, and return the path
* from (0,0,…,0) to (1,1,…,1) that describes that simplex. */
inline vec_t<int, N> GetIndexOrder(vec_t<float, N> const & pos) const
{
vec_t<int, N> result;
for (int i = 0; i < N; ++i)
result[i] = i;

/* Naïve bubble sort — enough for now */
for (int i = 0; i < N; ++i)
for (int j = i + 1; j < N; ++j)
if (pos[result[i]] < pos[result[j]])
std::swap(result[i], result[j]);

return result;
// FIXME: Gustavson uses the value 70 for dimension 2, 32 for
// dimension 3, and 27 for dimension 4; find where this comes from
// and maybe find a reasonable formula.
//return 70.f * result / coeff;
float const k = N == 2 ? 70 : N == 3 ? 32 : N == 4 ? 27 : 20;
return k * result;
}

inline vec_t<float, N> GetGradient(vec_t<int, N> origin) const
inline vec_t<float, N> get_gradient(vec_t<int, N> origin) const
{
/* Quick shuffle table:
* strings /dev/urandom | grep . -nm256 | sort -k2 -t: | sed 's|:.*|,|'
* Then just replace “256” with “0”. */
static int const shuffle[256] =
{
111, 14, 180, 186, 221, 114, 17, 79, 66, 46, 11, 81, 246, 200, 141,
172, 85, 244, 112, 92, 34, 106, 218, 205, 236, 7, 121, 115, 109,
131, 10, 96, 188, 148, 219, 107, 94, 182, 235, 163, 143, 213, 248,
202, 52, 154, 37, 241, 53, 129, 25, 60, 242, 38, 171, 63, 203, 255,
193, 6, 42, 209, 28, 176, 210, 159, 54, 144, 3, 71, 89, 116, 12,
237, 67, 216, 252, 178, 174, 164, 98, 234, 32, 26, 175, 24, 130,
128, 113, 99, 212, 62, 152, 75, 185, 73, 93, 31, 30, 151, 122, 173,
139, 91, 136, 162, 194, 208, 56, 101, 68, 69, 211, 44, 97, 55, 83,
33, 50, 119, 156, 149, 41, 157, 253, 247, 161, 47, 230, 166, 225,
204, 224, 13, 110, 123, 142, 64, 65, 155, 215, 120, 197, 140, 58,
77, 214, 126, 195, 179, 220, 232, 125, 147, 8, 39, 187, 27, 217,
100, 134, 199, 88, 206, 231, 250, 74, 2, 135, 9, 245, 118, 21, 243,
82, 183, 238, 87, 158, 61, 4, 177, 146, 153, 117, 249, 254, 233,
90, 222, 207, 48, 15, 18, 20, 239, 133, 0, 165, 138, 127, 169, 72,
1, 201, 145, 191, 192, 16, 49, 19, 95, 226, 228, 84, 181, 251, 36,
150, 22, 43, 70, 45, 105, 5, 189, 160, 196, 40, 59, 57, 190, 80,
104, 167, 78, 124, 103, 240, 184, 170, 137, 29, 23, 223, 108, 102,
86, 198, 227, 35, 229, 76, 168, 132, 51,
111, 14, 180, 186, 221, 114, 219, 79, 66, 46, 152, 81, 246, 200,
141, 172, 85, 244, 112, 92, 34, 106, 218, 205, 236, 7, 121, 115,
109, 131, 10, 96, 188, 148, 17, 107, 94, 182, 235, 163, 143, 63,
248, 202, 52, 154, 37, 241, 53, 129, 25, 159, 242, 38, 171, 213,
6, 203, 255, 193, 42, 209, 28, 176, 210, 60, 54, 144, 3, 71, 89,
116, 12, 237, 67, 216, 252, 178, 174, 164, 98, 234, 32, 26, 175,
24, 130, 128, 113, 99, 212, 62, 11, 75, 185, 73, 93, 31, 30, 44,
122, 173, 139, 91, 136, 162, 194, 41, 56, 101, 68, 69, 211, 151,
97, 55, 83, 33, 50, 119, 156, 149, 208, 157, 253, 247, 161, 133,
230, 166, 225, 204, 224, 13, 110, 123, 142, 64, 65, 155, 215, 9,
197, 140, 58, 77, 214, 126, 195, 179, 220, 232, 125, 147, 8, 39,
187, 27, 217, 100, 134, 199, 88, 206, 231, 250, 74, 2, 135, 120,
21, 245, 118, 243, 82, 183, 238, 150, 158, 61, 4, 177, 146, 153,
117, 249, 254, 233, 90, 222, 207, 48, 15, 18, 20, 16, 47, 0, 51,
165, 138, 127, 169, 72, 1, 201, 145, 191, 192, 239, 49, 19, 160,
226, 228, 84, 181, 251, 36, 87, 22, 43, 70, 45, 105, 5, 189, 95,
40, 196, 59, 57, 190, 80, 104, 167, 78, 124, 103, 240, 184, 170,
137, 29, 23, 223, 108, 102, 86, 198, 227, 35, 229, 76, 168, 132,
};

/* 16 random vectors; this should be enough for small dimensions */
auto v = [&]()
{
vec_t<float, N> ret;
for (int i = 0; i < N; ++i)
ret[i] = rand(-1.f, 1.f);
return normalize(ret);
};
/* Generate 2^(N+2) random vectors, but at least 2^5 (32) and not
* more than 2^20 (~ 1 million). */
int const gradient_count = 1 << min(max(N + 2, 5), 20);

static vec_t<float, N> const gradients[16] =
static auto build_gradients = [&]()
{
v(), v(), v(), v(), v(), v(), v(), v(),
v(), v(), v(), v(), v(), v(), v(), v(),
array<vec_t<float, N>> ret;
for (int k = 0; k < gradient_count; ++k)
{
vec_t<float, N> v;
for (int i = 0; i < N; ++i)
v[i] = rand(-1.f, 1.f);
ret << normalize(v);
}
return ret;
};

static array<vec_t<float, N>> const gradients = build_gradients();

int idx = m_seed;
for (int i = 0; i < N; ++i)
idx = shuffle[(idx + origin[i]) & 255];
idx ^= shuffle[(idx + origin[i]) & 255];

idx &= (gradient_count - 1);
#if 0
// DEBUG: only output a few gradients
if (idx > 2)
return vec_t<float, N>(0);
#endif
return gradients[idx];
}

return gradients[(idx ^ (idx >> 4)) & 15];
static inline vec_t<float, N> skew(vec_t<float, N> const &v)
{
/* Quoting Perlin in “Hardware Noise” (2-18):
* The “skew factor” f should be set to f = sqrt(N+1), so that
* the point (1,1,...1) is transformed to the point (f,f,...f). */
float const sum = dot(v, vec_t<float, N>(1));
float const f = sqrt(1.f + N);
return v + vec_t<float, N>(sum * (f - 1) / N);
}

static inline vec_t<float, N> unskew(vec_t<float, N> const &v)
{
float const sum = dot(v, vec_t<float, N>(1));
float const f = sqrt(1.f + N);
return v + vec_t<float, N>(sum * (1 / f - 1) / N);
}

/* For a given world position, extract grid coordinates (origin) and
* the corresponding delta position (pos). */
inline void ExtractFloorDecimal(vec_t<float, N> const & simplex_position,
vec_t<int, N> & origin,
vec_t<float, N> & pos) const
inline void get_origin(vec_t<float, N> const & simplex_position,
vec_t<int, N> & origin, vec_t<float, N> & pos) const
{
// Finding floor point index
for (int i = 0; i < N; ++i)
@@ -192,7 +266,7 @@ protected:
pos = simplex_position - (vec_t<float, N>)origin;
}

mat_t<float, N, N> m_base, m_base_inverse;
/* A user-provided random seed. Defaults to zero. */
int m_seed;
};



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