Voronoi Noise  



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Voronoi Noise produces very structured random patterns which can be given different characters by manipulating the distance calculations done between random points. Three dimensions are used to generate noise, so animation is possible (using time as the third dimension). This image shows frames taken at z=0 to z=1 in steps of 0.05; and looks like stars. 
There are a lot of possible variations from the two enums which select the shape:
enum DistanceMethod {Length, Length2, Manhattan, Chebychev, Quadratic, Minkowski4, Minkowski5}; enum VoronoiType {First, Second, Third, Fourth, Difference21, Difference32, Crackle};The algorithm comes up with four numbers to choose between and VoronoiType lets you specify which one, or which popular combination. The following table shows what the combinations are doing:
VoronoiType Difference21 Second  Third Difference32 Third  Second Crackle Max(1.0, 10*Get(...Difference21);
The DistanceMethod alters both the look and the time taken. Each DistanceMethod will be called 27 times per pixel:This is the code used to generate the images:
DistanceMethod Length sqrt(x*x+y*y+z*z); Euclidean (shortest line). Square Root may be slow for some applications. Length2 (x*x+y*y+z*z) The length squared. Saves the slow Square Root for some calculations. Manhattan The length of the distance in axial directions (as if travelling around a city). Chebychev The length of the longest Axial journey. Quadratic (x*x+y*y+z*z+x*y+x*z+y*z) The sum of everything multiplied by everything else! Minkowski4 Same as Minkowski(4); pow(x*x*x*x+y*y*y*y+z*z*z*z,0.25); Minkowski5 Same as Minkowski(0.5);
general Minkowski is very slow:
double Minkowski(p) {return pow(pow(fabs(x), p) + pow(fabs(y), p) + pow(fabs(z), M_E), 1.0/p);}for(WORD x=GetWidth(); x; ) { // Loops are best comparing the counter with zero, so this counts down. double u=double(x)/Width; // Calculate the parameter (ratio in the range [0,1]) indicating how far through the loop is. for(DWORD y=GetHeight(); y;) { double v=double(y)/GetHeight(); // Calculate the parameter (ratio in the range [0,1]) indicating how far through the loop is. BYTE B=BYTE(Round(255*Clamped(0.0, Voronoi::Get(x/10.0,y/10.0, 0.5, Voronoi::Difference21, Voronoi::Quadratic), 1.0))); // Turn the parameter into a Greyscale Level. SetPixel(x,y, RGB(B,B,B)); } }GetWidth(), GetHeight() and SetPixel are members of CPixelBlock which was used as a base class.
Round and Clamped are declared in Global.h.The following images show the patterns available. Be aware that the Voronoi::Get method returns values that may be outside the [0,1] range, so the Clamping results in a lot of white on some images, and that the the negatives of the images (reverse black and white) are as important as the versions here:
The DistanceMethod 'Crackle' is supposed to be white, but to see what lies behind the white, the following images have been autobalanced rather than clamped:
To generate these images, the image was saved using DWORD greyscale values: SetPixel(x,y, Round(10000000*Voronoi::Get(...)) The minumum and maximum values were stored and in a second pass, each DWORD was Mapped to the usual [0,255] range to autobalance the greyscale levels:void SetLevels(DWORD Minimum=0, DWORD Maximum=255) { // A Compander (Compressor/Expander) DWORD MinLevel=ULONG_MAX, MaxLevel=0; for(DWORD* ptr=&Bits[Width*Height1]; ptr>=Bits; ptr) { if(MinLevel>*ptr) MinLevel=*ptr; if(MaxLevel<*ptr) MaxLevel=*ptr; } for(DWORD* ptr=&Bits[Width*Height1]; ptr>=Bits; ptr) { *ptr=Interpolate(Parameterize(*ptr, MinLevel,MaxLevel), Minimum,Maximum); } GreyScale=Maximum; }Interpolate and Parameterize are declared in Global.h. There is also a page devoted to Interpolation.Another Dimension
Adding a third dimension allows noise over a three dimensional surface, or 2D animation with time as the third dimension. To create a looping animation, the following code uses z in the Interval [0,1] and uses that to blend between two adjacent noise function cells:
for(int Frame=20; Frame; Frame) { double z=Frame*1/20.0; for(WORD x=GetWidth(); x;) { for(WORD y=GetHeight(); y;) { double d=(1z)*Voronoi::Get(x/10.0,y/10.0, z , Voronoi::First, Voronoi::Length2) +z*Voronoi::Get(x/10.0,y/10.0, z1, Voronoi::First, Voronoi::Length2); SetLevel(x,y, Round(255*Clamped(0.0, 1d, 1.0))); } } CString S; S.Format("Frame%02u.bmp",Frame); Save(S); }GetWidth(), GetHeight(), SetLevel() and Save are members of CPixelBlock which was used as a base class.
Round and Clamped are declared in Global.h.
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