I am trying to replicate the outcome of this link using linear convolution in spatial-domain.
Images are first converted to 2d double arrays and then convolved. Image and kernel are of the same size. The image is padded before convolution and cropped accordingly after the convolution.
As compared to the FFT-based convolution, the output is weird and incorrect.
How can I solve the issue?
Note that I obtained the following image output from Matlab which matches my C# FFT output:
.
Update-1: Following #Ben Voigt's comment, I changed the Rescale() function to replace 255.0 with 1 and thus the output is improved substantially. But, still, the output doesn't match the FFT output (which is the correct one).
.
Update-2: Following #Cris Luengo's comment, I have padded the image by stitching and then performed spatial convolution. The outcome has been as follows:
So, the output is worse than the previous one. But, this has a similarity with the 2nd output of the linked answer which means a circular convolution is not the solution.
.
Update-3: I have used the Sum() function proposed by #Cris Luengo's answer. The result is a more improved version of **Update-1**:
But, it is still not 100% similar to the FFT version.
.
Update-4: Following #Cris Luengo's comment, I have subtracted the two outcomes to see the difference:
,
1. spatial minus frequency domain
2. frequency minus spatial domain
Looks like, the difference is substantial which means, spatial convolution is not being done correctly.
.
Source Code:
(Notify me if you need more source code to see.)
public static double[,] LinearConvolutionSpatial(double[,] image, double[,] mask)
{
int maskWidth = mask.GetLength(0);
int maskHeight = mask.GetLength(1);
double[,] paddedImage = ImagePadder.Pad(image, maskWidth);
double[,] conv = Convolution.ConvolutionSpatial(paddedImage, mask);
int cropSize = (maskWidth/2);
double[,] cropped = ImageCropper.Crop(conv, cropSize);
return conv;
}
static double[,] ConvolutionSpatial(double[,] paddedImage1, double[,] mask1)
{
int imageWidth = paddedImage1.GetLength(0);
int imageHeight = paddedImage1.GetLength(1);
int maskWidth = mask1.GetLength(0);
int maskHeight = mask1.GetLength(1);
int convWidth = imageWidth - ((maskWidth / 2) * 2);
int convHeight = imageHeight - ((maskHeight / 2) * 2);
double[,] convolve = new double[convWidth, convHeight];
for (int y = 0; y < convHeight; y++)
{
for (int x = 0; x < convWidth; x++)
{
int startX = x;
int startY = y;
convolve[x, y] = Sum(paddedImage1, mask1, startX, startY);
}
}
Rescale(convolve);
return convolve;
}
static double Sum(double[,] paddedImage1, double[,] mask1, int startX, int startY)
{
double sum = 0;
int maskWidth = mask1.GetLength(0);
int maskHeight = mask1.GetLength(1);
for (int y = startY; y < (startY + maskHeight); y++)
{
for (int x = startX; x < (startX + maskWidth); x++)
{
double img = paddedImage1[x, y];
double msk = mask1[x - startX, y - startY];
sum = sum + (img * msk);
}
}
return sum;
}
static void Rescale(double[,] convolve)
{
int imageWidth = convolve.GetLength(0);
int imageHeight = convolve.GetLength(1);
double maxAmp = 0.0;
for (int j = 0; j < imageHeight; j++)
{
for (int i = 0; i < imageWidth; i++)
{
maxAmp = Math.Max(maxAmp, convolve[i, j]);
}
}
double scale = 1.0 / maxAmp;
for (int j = 0; j < imageHeight; j++)
{
for (int i = 0; i < imageWidth; i++)
{
double d = convolve[i, j] * scale;
convolve[i, j] = d;
}
}
}
public static Bitmap ConvolveInFrequencyDomain(Bitmap image1, Bitmap kernel1)
{
Bitmap outcome = null;
Bitmap image = (Bitmap)image1.Clone();
Bitmap kernel = (Bitmap)kernel1.Clone();
//linear convolution: sum.
//circular convolution: max
uint paddedWidth = Tools.ToNextPow2((uint)(image.Width + kernel.Width));
uint paddedHeight = Tools.ToNextPow2((uint)(image.Height + kernel.Height));
Bitmap paddedImage = ImagePadder.Pad(image, (int)paddedWidth, (int)paddedHeight);
Bitmap paddedKernel = ImagePadder.Pad(kernel, (int)paddedWidth, (int)paddedHeight);
Complex[,] cpxImage = ImageDataConverter.ToComplex(paddedImage);
Complex[,] cpxKernel = ImageDataConverter.ToComplex(paddedKernel);
// call the complex function
Complex[,] convolve = Convolve(cpxImage, cpxKernel);
outcome = ImageDataConverter.ToBitmap(convolve);
outcome = ImageCropper.Crop(outcome, (kernel.Width/2)+1);
return outcome;
}
Your current output looks more like the auto-correlation function than the convolution of Lena with herself. I think the issue might be in your Sum function.
If you look at the definition of the convolution sum, you'll see that the kernel (or the image, doesn't matter) is mirrored:
sum_m( f[n-m] g[m] )
For the one function, m appears with a plus sign, and for the other it appears with a minus sign.
You'll need to modify your Sum function to read the mask1 image in the right order:
static double Sum(double[,] paddedImage1, double[,] mask1, int startX, int startY)
{
double sum = 0;
int maskWidth = mask1.GetLength(0);
int maskHeight = mask1.GetLength(1);
for (int y = startY; y < (startY + maskHeight); y++)
{
for (int x = startX; x < (startX + maskWidth); x++)
{
double img = paddedImage1[x, y];
double msk = mask1[maskWidth - x + startX - 1, maskHeight - y + startY - 1];
sum = sum + (img * msk);
}
}
return sum;
}
The other option is to pass a mirrored version of mask1 to this function.
I have found the solution from this link. The main clue was to introduce an offset and a factor.
factor is the sum of all values in the kernel.
offset is an arbitrary value to fix the output further.
.
#Cris Luengo's answer also raised a valid point.
.
The following source code is supplied in the given link:
private void SafeImageConvolution(Bitmap image, ConvMatrix fmat)
{
//Avoid division by 0
if (fmat.Factor == 0)
return;
Bitmap srcImage = (Bitmap)image.Clone();
int x, y, filterx, filtery;
int s = fmat.Size / 2;
int r, g, b;
Color tempPix;
for (y = s; y < srcImage.Height - s; y++)
{
for (x = s; x < srcImage.Width - s; x++)
{
r = g = b = 0;
// Convolution
for (filtery = 0; filtery < fmat.Size; filtery++)
{
for (filterx = 0; filterx < fmat.Size; filterx++)
{
tempPix = srcImage.GetPixel(x + filterx - s, y + filtery - s);
r += fmat.Matrix[filtery, filterx] * tempPix.R;
g += fmat.Matrix[filtery, filterx] * tempPix.G;
b += fmat.Matrix[filtery, filterx] * tempPix.B;
}
}
r = Math.Min(Math.Max((r / fmat.Factor) + fmat.Offset, 0), 255);
g = Math.Min(Math.Max((g / fmat.Factor) + fmat.Offset, 0), 255);
b = Math.Min(Math.Max((b / fmat.Factor) + fmat.Offset, 0), 255);
image.SetPixel(x, y, Color.FromArgb(r, g, b));
}
}
}
I try to make a meteor like this video,
but I can only get like this:
This is my simplex noise:
public class PerlinNoise{
int B = 256;
int[] m_perm = new int[B+B];
Texture2D m_permTex;
public int octava;
public float frequencia, amplitud;
public PerlinNoise(int seed, float frec, float amp, int oct)
{
octava = oct;
amplitud = amp;
frequencia = frec;
UnityEngine.Random.seed = seed;
int i, j, k;
for (i = 0 ; i < B ; i++)
{
m_perm[i] = i;
}
while (--i != 0)
{
k = m_perm[i];
j = UnityEngine.Random.Range(0, B);
m_perm[i] = m_perm[j];
m_perm[j] = k;
}
for (i = 0 ; i < B; i++)
{
m_perm[B + i] = m_perm[i];
}
}
float FADE(float t) { return t * t * t * ( t * ( t * 6.0f - 15.0f ) + 10.0f ); }
float LERP(float t, float a, float b) { return (a) + (t)*((b)-(a)); }
float GRAD1(int hash, float x )
{
int h = hash & 15;
float grad = 1.0f + (h & 7);
if ((h&8) != 0) grad = -grad;
return ( grad * x );
}
float GRAD2(int hash, float x, float y)
{
int h = hash & 7;
float u = h<4 ? x : y;
float v = h<4 ? y : x;
return (((h&1) != 0)? -u : u) + (((h&2) != 0) ? -2.0f*v : 2.0f*v);
}
float GRAD3(int hash, float x, float y , float z)
{
int h = hash & 15;
float u = h<8 ? x : y;
float v = (h<4) ? y : (h==12 || h==14) ? x : z;
return (((h&1) != 0)? -u : u) + (((h&2) != 0)? -v : v);
}
float Noise1D( float x )
{
//returns a noise value between -0.5 and 0.5
int ix0, ix1;
float fx0, fx1;
float s, n0, n1;
ix0 = (int)Mathf.Floor(x); // Integer part of x
fx0 = x - ix0; // Fractional part of x
fx1 = fx0 - 1.0f;
ix1 = ( ix0+1 ) & 0xff;
ix0 = ix0 & 0xff; // Wrap to 0..255
s = FADE(fx0);
n0 = GRAD1(m_perm[ix0], fx0);
n1 = GRAD1(m_perm[ix1], fx1);
return 0.188f * LERP( s, n0, n1);
}
public float Noise2D( float x, float y )
{
int ix0, iy0, ix1, iy1;
float fx0, fy0, fx1, fy1, s, t, nx0, nx1, n0, n1;
ix0 = (int)Mathf.Floor(x);
iy0 = (int)Mathf.Floor(y);
fx0 = x - ix0;
fy0 = y - iy0;
fx1 = fx0 - 1.0f;
fy1 = fy0 - 1.0f;
ix1 = (ix0 + 1) & 0xff; // Wrap to 0..255
iy1 = (iy0 + 1) & 0xff;
ix0 = ix0 & 0xff;
iy0 = iy0 & 0xff;
t = FADE( fy0 );
s = FADE( fx0 );
nx0 = GRAD2(m_perm[ix0 + m_perm[iy0]], fx0, fy0);
nx1 = GRAD2(m_perm[ix0 + m_perm[iy1]], fx0, fy1);
n0 = LERP( t, nx0, nx1 );
nx0 = GRAD2(m_perm[ix1 + m_perm[iy0]], fx1, fy0);
nx1 = GRAD2(m_perm[ix1 + m_perm[iy1]], fx1, fy1);
n1 = LERP(t, nx0, nx1);
return 0.507f * LERP( s, n0, n1 );
}
float Noise3D( float x, float y, float z )
{
//returns a noise value between -1.5 and 1.5
int ix0, iy0, ix1, iy1, iz0, iz1;
float fx0, fy0, fz0, fx1, fy1, fz1;
float s, t, r;
float nxy0, nxy1, nx0, nx1, n0, n1;
ix0 = (int)Mathf.Floor( x ); // Integer part of x
iy0 = (int)Mathf.Floor( y ); // Integer part of y
iz0 = (int)Mathf.Floor( z ); // Integer part of z
fx0 = x - ix0; // Fractional part of x
fy0 = y - iy0; // Fractional part of y
fz0 = z - iz0; // Fractional part of z
fx1 = fx0 - 1.0f;
fy1 = fy0 - 1.0f;
fz1 = fz0 - 1.0f;
ix1 = ( ix0 + 1 ) & 0xff; // Wrap to 0..255
iy1 = ( iy0 + 1 ) & 0xff;
iz1 = ( iz0 + 1 ) & 0xff;
ix0 = ix0 & 0xff;
iy0 = iy0 & 0xff;
iz0 = iz0 & 0xff;
r = FADE( fz0 );
t = FADE( fy0 );
s = FADE( fx0 );
nxy0 = GRAD3(m_perm[ix0 + m_perm[iy0 + m_perm[iz0]]], fx0, fy0, fz0);
nxy1 = GRAD3(m_perm[ix0 + m_perm[iy0 + m_perm[iz1]]], fx0, fy0, fz1);
nx0 = LERP( r, nxy0, nxy1 );
nxy0 = GRAD3(m_perm[ix0 + m_perm[iy1 + m_perm[iz0]]], fx0, fy1, fz0);
nxy1 = GRAD3(m_perm[ix0 + m_perm[iy1 + m_perm[iz1]]], fx0, fy1, fz1);
nx1 = LERP( r, nxy0, nxy1 );
n0 = LERP( t, nx0, nx1 );
nxy0 = GRAD3(m_perm[ix1 + m_perm[iy0 + m_perm[iz0]]], fx1, fy0, fz0);
nxy1 = GRAD3(m_perm[ix1 + m_perm[iy0 + m_perm[iz1]]], fx1, fy0, fz1);
nx0 = LERP( r, nxy0, nxy1 );
nxy0 = GRAD3(m_perm[ix1 + m_perm[iy1 + m_perm[iz0]]], fx1, fy1, fz0);
nxy1 = GRAD3(m_perm[ix1 + m_perm[iy1 + m_perm[iz1]]], fx1, fy1, fz1);
nx1 = LERP( r, nxy0, nxy1 );
n1 = LERP( t, nx0, nx1 );
return 0.936f * LERP( s, n0, n1 );
}
public float FractalNoise1D(float x, int octNum, float frq, float amp)
{
float gain = 1.0f;
float sum = 0.0f;
for(int i = 0; i < octNum; i++)
{
sum += Noise1D(x*gain/frq) * amp/gain;
gain *= 2.0f;
}
return sum;
}
public float FractalNoise2D(float x, float y, int octNum, float frq, float amp)
{
float gain = 1.0f;
float sum = 0.0f;
for(int i = 0; i < octNum; i++)
{
sum += Noise2D(x*gain/frq, y*gain/frq) * amp/gain;
gain *= 2.0f;
}
return sum;
}
public int Noise2D(Vector3Int v3) {
return Mathf.RoundToInt(FractalNoise2D(v3.x,v3.z,octava,frequencia,amplitud));
}
public int Noise2D(int x, int z)
{
return Mathf.RoundToInt(FractalNoise2D(x, z, octava, frequencia, amplitud));
}
public float FractalNoise3D(float x, float y, float z, int octNum, float frq, float amp)
{
float gain = 1.0f;
float sum = 0.0f;
for(int i = 0; i < octNum; i++)
{
sum += Noise3D(x*gain/frq, y*gain/frq, z*gain/frq) * amp/gain;
gain *= 2.0f;
}
return sum;
}
public void LoadPermTableIntoTexture()
{
m_permTex = new Texture2D(256, 1, TextureFormat.Alpha8, false);
m_permTex.filterMode = FilterMode.Point;
m_permTex.wrapMode = TextureWrapMode.Clamp;
for(int i = 0; i < 256; i++)
{
float v = (float)m_perm[i] / 255.0f;
m_permTex.SetPixel(i, 0, new Color(0,0,0,v));
}
m_permTex.Apply();
}
}
This is my implementation:
public void Resimulate() {
PerlinNoise per = new PerlinNoise(seed, frec, amp, octa);
for (int x = 0; x < TamX; x++)
{
for (int y = 0; y < TamY; y++)
{
for (int z = 0; z < TamZ; z++)
{
if (per.FractalNoise3D(x, y, z, octa, frec, amp)>0)
{
lista.Add(new Bloque(new Vector3Int(x, y, z)));
}
}
}
}
}
I found information on various pages, blogs and here but found nothing that worked for me. If anyone has information I would greatly appreciate your help.
Thank you for helping me.
First, it's important to note that the code you have is not Simplex noise! It's the older "Perlin noise" algorithm that tends to exhibit visually significant directional artifacts.
I would suggest avoiding both Perlin noise and Simplex noise altogether. Perlin because of the directional artifacts, and Simplex because you run into patent issues with the 3D implementation.
There's an algorithm I've designed recently to get around both of those issues, that's starting to be adopted by a fair number of game developers, called OpenSimplex noise. Code: https://gist.github.com/KdotJPG/b1270127455a94ac5d19
(EDIT: Here's a C# port: https://gist.github.com/omgwtfgames/601497972e4e30fd9c5f)
Do something like this:
const double S = 24; //Sparsity of noise features
public void Resimulate() {
//To get fractal noise, ideally you want multiple instances of noise so you don't get odd behavior near (0,0,0)
OpenSimplexNoise n1 = new OpenSimplexNoise(seed);
OpenSimplexNoise n2 = new OpenSimplexNoise(seed+1);
OpenSimplexNoise n3 = new OpenSimplexNoise(seed+2);
for (int x = 0; x < TamX; x++) {
for (int y = 0; y < Tamy; z++) {
for (int z = 0; z < TamZ; z++) {
double n = (n1.eval(x / S, y / S, z / S)
+ n2.eval(x / S / 2, y / S / 2, z / S / 2) * .5
+ n3.eval(x / S / 4, y / S / 4, z / S / 4) * .25)
/ (1 + .5 + .25);
double dx = (TamX / 2.0 - x);
double dy = (TamY / 2.0 - y);
double dz = (TamZ / 2.0 - z);
double d = dx*dx + dy*dy + dz*dz;
//d is now your squared distance from the center.
//n is your fractal noise value
//you want to combine d and n to form some threshold that
//determines whether or not there is a block.
//threshold = d / C1 + n + C2 > 0 where C1 and C2 are constants you choose
}
}
}
}
EDIT: Here's a visual difference between Perlin and OpenSimplex:
Left is noise(x, y, 0) grayscale
Next is noise(x, y, 0) > 0 ? white : black
Next is |noise(x, y, 0)| > 0.1 ? white : black
Next is noise(x, y, 0.5) grayscale
I have fully mastered the art of Perlin Noise in 3D, and now I'm trying to use my same implementation for a 2D algorithm.
The problem seems to be in picking my gradient directions. In 3D I use 16 gradients in evenly distributed directions and this works great.
In 2D I figured I'd use 8 gradients. up, down, left, right, and the four diagonal directions.
Here is what I get:
The general look of the noise is always correct, but the edges of the squares don't quite match up.
I have also tried using other gradients or fewer gradients but get similar results.
Here in another example you can see that the edges do match up sometimes and the results are fine in that area -
When I don't use gradients and instead just interpolate between a value picked randomly at each of the 4 corners I get the right results, which is what makes me think it is the gradient part that is messing it up.
Here is my code:
//8 different gradient directions
private Point[] grads = new Point[] {
new Point(0, 1), new Point(1, 1), new Point(1, 0), new Point(1, -1),
new Point(0, -1), new Point(-1, -1), new Point(-1, 0), new Point(-1, 1),};
//takes the dot product of a gradient and (x, y)
private float dot2D(int i, float x, float y)
{
return
grads[i].X * x + grads[i].Y * y;
}
public float Noise2D(float x, float y)
{
int
ix = (int)(x),
iy = (int)(y);
x = x - ix;
y = y - iy;
float
fx = fade(x),
fy = fade(y);
ix &= 255;
iy &= 255;
// here is where i get the index to look up in the list of
// different gradients.
// hashTable is my array of 0-255 in random order
int
g00 = hashTable[ix + hashTable[iy ]],
g10 = hashTable[ix + 1 + hashTable[iy ]],
g01 = hashTable[ix + hashTable[iy + 1]],
g11 = hashTable[ix + 1 + hashTable[iy + 1]];
// this takes the dot product to find the values to interpolate between
float
n00 = dot2D(g00 & 7, x, y),
n10 = dot2D(g10 & 7, x, y),
n01 = dot2D(g01 & 7, x, y),
n11 = dot2D(g11 & 7, x, y);
// lerp() is just normal linear interpolation
float
y1 = lerp(fx, n00, n10),
y2 = lerp(fx, n01, n11);
return
lerp(fy, y1, y2);
}
I'm in a bit of a rush, but this might be helpful. I adapted Perlin's reference implementation to C#. For 2D, just use the 3D Noise() function with a fixed z parameter. (public static float Noise(float x, float y, float z) towards the end of the class.)
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Microsoft.Xna.Framework;
using System.Diagnostics;
namespace GoEngine.Content.Entities
{
public class NoiseMaker
{
/// adapted from http://cs.nyu.edu/~perlin/noise/
// JAVA REFERENCE IMPLEMENTATION OF IMPROVED NOISE - COPYRIGHT 2002 KEN PERLIN.
private static int[] p = new int[512];
private static int[] permutation = { 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
};
static NoiseMaker()
{
CalculateP();
}
private static int _octaves;
private static int _halfLength = 256;
public static void SetOctaves(int octaves)
{
_octaves = octaves;
var len = (int)Math.Pow(2, octaves);
permutation = new int[len];
Reseed();
}
private static void CalculateP()
{
p = new int[permutation.Length * 2];
_halfLength = permutation.Length;
for (int i = 0; i < permutation.Length; i++)
p[permutation.Length + i] = p[i] = permutation[i];
}
public static void Reseed()
{
var random = new Random();
var perm = Enumerable.Range(0, permutation.Length).ToArray();
for (var i = 0; i < perm.Length; i++)
{
var swapIndex = random.Next(perm.Length);
var t = perm[i];
perm[i] = perm[swapIndex];
perm[swapIndex] = t;
}
permutation = perm;
CalculateP();
}
public static float Noise(Vector3 position, int octaves, ref float min, ref float max)
{
return Noise(position.X, position.Y, position.Z, octaves, ref min, ref max);
}
public static float Noise(float x, float y, float z, int octaves, ref float min, ref float max)
{
var perlin = 0f;
var octave = 1;
for (var i = 0; i < octaves; i++)
{
var noise = Noise(x * octave, y * octave, z * octave);
perlin += noise / octave;
octave *= 2;
}
perlin = Math.Abs((float)Math.Pow(perlin,2));
max = Math.Max(perlin, max);
min = Math.Min(perlin, min);
//perlin = 1f - 2 * perlin;
return perlin;
}
public static float Noise(float x, float y, float z)
{
int X = (int)Math.Floor(x) % _halfLength;
int Y = (int)Math.Floor(y) % _halfLength;
int Z = (int)Math.Floor(z) % _halfLength;
if (X < 0)
X += _halfLength;
if (Y < 0)
Y += _halfLength;
if (Z < 0)
Z += _halfLength;
x -= (int)Math.Floor(x);
y -= (int)Math.Floor(y);
z -= (int)Math.Floor(z);
var u = Fade(x);
var v = Fade(y);
var w = Fade(z);
int A = p[X] + Y, AA = p[A] + Z, AB = p[A + 1] + Z, // HASH COORDINATES OF
B = p[X + 1] + Y, BA = p[B] + Z, BB = p[B + 1] + Z; // THE 8 CUBE CORNERS,
return MathHelper.Lerp(
MathHelper.Lerp(
MathHelper.Lerp(
Grad(p[AA], x, y, z) // AND ADD
,
Grad(p[BA], x - 1, y, z) // BLENDED
,
u
)
,
MathHelper.Lerp(
Grad(p[AB], x, y - 1, z) // RESULTS
,
Grad(p[BB], x - 1, y - 1, z)
,
u
)
,
v
)
,
MathHelper.Lerp(
MathHelper.Lerp(
Grad(p[AA + 1], x, y, z - 1) // CORNERS
,
Grad(p[BA + 1], x - 1, y, z - 1) // OF CUBE
,
u
)
,
MathHelper.Lerp(
Grad(p[AB + 1], x, y - 1, z - 1)
,
Grad(p[BB + 1], x - 1, y - 1, z - 1)
,
u
)
,
v
)
,
w
);
}
static float Fade(float t) { return t * t * t * (t * (t * 6 - 15) + 10); }
static float Grad(int hash, float x, float y, float z)
{
int h = hash & 15; // CONVERT LO 4 BITS OF HASH CODE
float u = h < 8 ? x : y, // INTO 12 GRADIENT DIRECTIONS.
v = h < 4 ? y : h == 12 || h == 14 ? x : z;
return ((h & 1) == 0 ? u : -u) + ((h & 2) == 0 ? v : -v);
}
}
}
Update
Okay, I managed to create a working 2D version. Here's the class:
/// implements improved Perlin noise in 2D.
/// Transcribed from http://www.siafoo.net/snippet/144?nolinenos#perlin2003
/// </summary>
public static class Noise2d
{
private static Random _random = new Random();
private static int[] _permutation;
private static Vector2[] _gradients;
static Noise2d()
{
CalculatePermutation(out _permutation);
CalculateGradients(out _gradients);
}
private static void CalculatePermutation(out int[] p)
{
p = Enumerable.Range(0, 256).ToArray();
/// shuffle the array
for (var i = 0; i < p.Length; i++)
{
var source = _random.Next(p.Length);
var t = p[i];
p[i] = p[source];
p[source] = t;
}
}
/// <summary>
/// generate a new permutation.
/// </summary>
public static void Reseed()
{
CalculatePermutation(out _permutation);
}
private static void CalculateGradients(out Vector2[] grad)
{
grad = new Vector2[256];
for (var i = 0; i < grad.Length; i++)
{
Vector2 gradient;
do
{
gradient = new Vector2((float)(_random.NextDouble() * 2 - 1), (float)(_random.NextDouble() * 2 - 1));
}
while (gradient.LengthSquared() >= 1);
gradient.Normalize();
grad[i] = gradient;
}
}
private static float Drop(float t)
{
t = Math.Abs(t);
return 1f - t * t * t * (t * (t * 6 - 15) + 10);
}
private static float Q(float u, float v)
{
return Drop(u) * Drop(v);
}
public static float Noise(float x, float y)
{
var cell = new Vector2((float)Math.Floor(x), (float)Math.Floor(y));
var total = 0f;
var corners = new[] { new Vector2(0, 0), new Vector2(0, 1), new Vector2(1, 0), new Vector2(1, 1) };
foreach (var n in corners)
{
var ij = cell + n;
var uv = new Vector2(x - ij.X, y - ij.Y);
var index = _permutation[(int)ij.X % _permutation.Length];
index = _permutation[(index + (int)ij.Y) % _permutation.Length];
var grad = _gradients[index % _gradients.Length];
total += Q(uv.X, uv.Y) * Vector2.Dot(grad, uv);
}
return Math.Max(Math.Min(total, 1f), -1f);
}
}
Call it like this:
private void GenerateNoiseMap(int width, int height, ref Texture2D noiseTexture, int octaves)
{
var data = new float[width * height];
/// track min and max noise value. Used to normalize the result to the 0 to 1.0 range.
var min = float.MaxValue;
var max = float.MinValue;
/// rebuild the permutation table to get a different noise pattern.
/// Leave this out if you want to play with changing the number of octaves while
/// maintaining the same overall pattern.
Noise2d.Reseed();
var frequency = 0.5f;
var amplitude = 1f;
var persistence = 0.25f;
for (var octave = 0; octave < octaves; octave++)
{
/// parallel loop - easy and fast.
Parallel.For(0
, width * height
, (offset) =>
{
var i = offset % width;
var j = offset / width;
var noise = Noise2d.Noise(i*frequency*1f/width, j*frequency*1f/height);
noise = data[j * width + i] += noise * amplitude;
min = Math.Min(min, noise);
max = Math.Max(max, noise);
}
);
frequency *= 2;
amplitude /= 2;
}
if (noiseTexture != null && (noiseTexture.Width != width || noiseTexture.Height != height))
{
noiseTexture.Dispose();
noiseTexture = null;
}
if (noiseTexture==null)
{
noiseTexture = new Texture2D(Device, width, height, false, SurfaceFormat.Color);
}
var colors = data.Select(
(f) =>
{
var norm = (f - min) / (max - min);
return new Color(norm, norm, norm, 1);
}
).ToArray();
noiseTexture.SetData(colors);
}
Note that I've used a couple of XNA structures (Vector2 and Texture2D), but it should be pretty clear what they do.
If you want higher frequency (more "noisy") content with fewer octaves, increase the initial frequency value that used in the octave loop.
This implementation uses "improved" Perlin noise, which should be a bit faster than the standard version. You might also have a look at Simplex noise, which is quite a bit faster at higher dimensions.
I had to change this:
n00 = dot2D(g00 & 7, x, y),
n10 = dot2D(g10 & 7, x, y),
n01 = dot2D(g01 & 7, x, y),
n11 = dot2D(g11 & 7, x, y);
to this:
n00 = dot2D(g00 & 7, x , y ),
n10 = dot2D(g10 & 7, x - 1, y ),
n01 = dot2D(g01 & 7, x , y - 1),
n11 = dot2D(g11 & 7, x - 1, y - 1);
Basically just subtracting 1 from the x and y where needed.
If you plug in a zero value for z into your 3D equation and simply follow the math through, removing terms, you'll see that you end up with a simpler equation in the end.
Your implementation looks kind of different to the one I'm using though.
Here's a comparison of a 3D and 2D function I'm using (in JavaScript):
noise3d: function(x, y, z)
{
// Find unit cube that contains point.
var X = Math.floor(x) & 255,
Y = Math.floor(y) & 255,
Z = Math.floor(z) & 255;
// Find relative x,y,z of point in cube.
x -= Math.floor(x);
y -= Math.floor(y);
z -= Math.floor(z);
// Compute fade curves for each of x,y,z.
var u = fade(x),
v = fade(y),
w = fade(z);
// Hash coordinates of the corners.
var A = p[X ] + Y, AA = p[A] + Z, AB = p[A + 1] + Z,
B = p[X + 1] + Y, BA = p[B] + Z, BB = p[B + 1] + Z;
// Add blended results from 8 corners of cube.
return scale(
lerp(
w,
lerp(
v,
lerp(
u,
grad(p[AA], x, y, z),
grad(p[BA], x - 1, y, z)
),
lerp(
u,
grad(p[AB], x, y - 1, z),
grad(p[BB], x - 1, y - 1, z)
)
),
lerp(
v,
lerp(
u,
grad(p[AA + 1], x, y, z - 1),
grad(p[BA + 1], x - 1, y, z - 1)
),
lerp(
u,
grad(p[AB + 1], x, y - 1, z - 1),
grad(p[BB + 1], x - 1, y - 1, z - 1)
)
)
)
);
}
The 2D version involves fewer computations.
noise2d: function(x, y)
{
// Find unit square that contains point.
var X = Math.floor(x) & 255,
Y = Math.floor(y) & 255;
// Find relative x,y of point in square.
x -= Math.floor(x);
y -= Math.floor(y);
// Compute fade curves for each of x,y.
var u = fade(x),
v = fade(y);
// Hash coordinates of the corners.
var A = p[X ] + Y, AA = p[A], AB = p[A + 1],
B = p[X + 1] + Y, BA = p[B], BB = p[B + 1];
// Add blended results from the corners.
return scale(
lerp(
v,
lerp(
u,
grad(p[AA], x, y, 0),
grad(p[BA], x - 1, y, 0)
),
lerp(
u,
grad(p[AB], x, y - 1, 0),
grad(p[BB], x - 1, y - 1, 0)
)
)
);
}