2D Perlin Noise

Question

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);
}

Answer 1

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.

Answer 2

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.

Answer 3

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)
                )
            )
    );
}