Perlin Noise

Gradient Interface

We begin by introducing a new partial class asset for Noise, this time named Noise.Gradient. In it we declare the IGradient interface.

using Unity.Mathematics;

using static Unity.Mathematics.math;

public static partial class Noise {

	public interface IGradient {}
}

The purpose of a gradient is to evaluate a function, with a hash and a relative coordinate as parameters. Define a vectorized Evaluate method signature for this, initially for 1D noise, which means that besides the hash it needs to have parameter for the X coordinates.

	public interface IGradient {
		float4 Evaluate (SmallXXHash4 hash, float4 x);
	}

Now we can add a Value struct type in the same partial class that implements IGradient for 1D Value noise. It simply ignores the coordinates and returns the A floats of the hash, like our lattice noise currently does.

	public struct Value : IGradient {

		public float4 Evaluate (SmallXXHash4 hash, float4 x) => hash.Floats01A;
	}

To also support 2D and 3D noise add Evaluate variants with two and three coordinate parameters to IGradient.

	public interface IGradient {
		float4 Evaluate (SmallXXHash4 hash, float4 x);

		float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y);

		float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y, float4 z);
	}

Value must implement these methods as well, again by simply returning the hash value and ignoring the coordinates. The unused parameters will not affect performance because Burst eliminates all method invocation overhead and unused values.

	public struct Value : IGradient {

		public float4 Evaluate (SmallXXHash4 hash, float4 x) => hash.Floats01A;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) => hash.Floats01A;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y, float4 z) =>
			hash.Floats01A;
	}

Gradient Input

To provide the relative coordinates for gradients we'll add a vectorized float field named g to LatticeSpan4 in Noise.Lattice.

	struct LatticeSpan4 {
		public int4 p0, p1;
		public float4 g;
		public float4 t;
	}

We find the relative coordinates by subtracting p0 from the coordinates in GetLatticeSpan4.

		span.p0 = (int4)points;
		span.p1 = span.p0 + 1;
		span.g = coordinates - span.p0;

Now we can adjust Lattice1D.GetNoise4 so it invokes Evaluate on Value instead of directly getting a value from the hash. Pass the relevant hash and the relative coordinates as arguments to Evaluate.

	public struct Lattice1D : INoise {

		public float4 GetNoise4(float4x3 positions, SmallXXHash4 hash) {
			LatticeSpan4 x = GetLatticeSpan4(positions.c0);

			var g = default(Value);
			return lerp(
				g.Evaluate(hash.Eat(x.p0), x.g), g.Evaluate(hash.Eat(x.p1), x.g), x.t
			) * 2f - 1f;
		}
	}

At this point we still get the same result as before for 1D Value noise and the generated assembly code for the job is also exactly the same. But to get an idea of how gradient noise works we can temporarily change Value.Evaluate so it becomes the simple gradient function `f(x)=x`.

		public float4 Evaluate (SmallXXHash4 hash, float4 x) => x;

The resulting pattern is a sequence of linear 1D ramps that go from −1 to 1 in between lattice points. There is no apparent blending because we use the exact same gradient on both sides of each span, relative to p0 on both ends. To turn it into proper gradient noise the second gradient must be based on coordinates relative to p1. So replace the single g field in LatticeSpan4 with g0 and add a new g1 field.

	struct LatticeSpan4 {
		public int4 p0, p1;
		public float4 g0, g1;
		public float4 t;
	}

Because the second lattice point of a span sits one unit further along that dimension, we find g1 in GetLatticeSpan4 by subtracting one from g0.

		span.p0 = (int4)points;
		span.p1 = span.p0 + 1;
		span.g0 = coordinates - span.p0;
		span.g1 = span.g0 - 1f;

Adjust Lattice1D.GetNoise4 so it uses the correct relative coordinates: p0 together with g0 and p1 together with g1.

			return lerp(
				g.Evaluate(hash.Eat(x.p0), x.g0), g.Evaluate(hash.Eat(x.p1), x.g1), x.t
			) * 2f - 1f;

If we had used linear interpolation the resulting pattern would be flat because the gradients would have canceled each other. But because we use smooth C2-continuous interpolation each gradient is dominant near its lattice point and we get a wave pattern.

Note that the gradients are zero at their lattice points and that the interpolated pattern is also zero in the middle of a span, where the gradients cancel each other.

The pattern that we get is wholly negative because we apply a range adjustment afterwards. This adjustment is specific to Value noise, so let's restore the noise and move the adjustment to the Value methods.

	public struct Value : IGradient {

		public float4 Evaluate (SmallXXHash4 hash, float4 x) => hash.Floats01A * 2f - 1f;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) =>
			hash.Floats01A * 2f - 1f;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y, float4 z) =>
			hash.Floats01A * 2f - 1f;
	}

Then remove it from Lattice1D.GetNoise4.

			return lerp(
				g.Evaluate(hash.Eat(x.p0), x.g0), g.Evaluate(hash.Eat(x.p1), x.g1), x.t
			);
			//) * 2f - 1f;

Generic Lattice

Now that we have a gradient version of Value noise we can make Lattice1D generic by giving it an IGradient struct type parameter, like we did for Noise.Job.

	public struct Lattice1D<G> : INoise where G : struct, IGradient {

		public float4 GetNoise4(float4x3 positions, SmallXXHash4 hash) {
			LatticeSpan4 x = GetLatticeSpan4(positions.c0);

			var g = default(G);
			return lerp(
				g.Evaluate(hash.Eat(x.p0), x.g0), g.Evaluate(hash.Eat(x.p1), x.g1), x.t
			);
		}
	}

Adjustment Lattice2D in the same way, this time using the 2D variant of IGradient.Evaluate. Make sure to use the correct relative coordinates for all lattice points.

	public struct Lattice2D<G> : INoise where G: struct, IGradient {

		public float4 GetNoise4 (float4x3 positions, SmallXXHash4 hash) {
			…

			var g = default(G);
			return lerp(
				lerp(
					g.Evaluate(h0.Eat(z.p0), x.g0, z.g0),
					g.Evaluate(h0.Eat(z.p1), x.g0, z.g1),
					z.t
				),
				lerp(
					g.Evaluate(h1.Eat(z.p0), x.g1, z.g0),
					g.Evaluate(h1.Eat(z.p1), x.g1, z.g1),
					z.t
				),
				x.t
			);
			//) * 2f - 1f;
		}
	}

And update Lattice3D as well.

	public struct Lattice3D<G> : INoise where G : struct, IGradient {

		public float4 GetNoise4 (float4x3 positions, SmallXXHash4 hash) {
			…

			var g = default(G);
			return lerp(
				lerp(
					lerp(
						g.Evaluate(h00.Eat(z.p0), x.g0, y.g0, z.g0),
						g.Evaluate(h00.Eat(z.p1), x.g0, y.g0, z.g1),
						z.t
					),
					lerp(
						g.Evaluate(h01.Eat(z.p0), x.g0, y.g1, z.g0),
						g.Evaluate(h01.Eat(z.p1), x.g0, y.g1, z.g1),
						z.t
					),
					y.t
				),
				lerp(
					lerp(
						g.Evaluate(h10.Eat(z.p0), x.g1, y.g0, z.g0),
						g.Evaluate(h10.Eat(z.p1), x.g1, y.g0, z.g1),
						z.t
					),
					lerp(
						g.Evaluate(h11.Eat(z.p0), x.g1, y.g1, z.g0),
						g.Evaluate(h11.Eat(z.p1), x.g1, y.g1, z.g1),
						z.t
					),
					y.t
				),
				x.t
			);
			//) * 2f - 1f;
		}
	}

We now have to explicitly declare that we are using the Value noise versions of lattice noise jobs in NoiseVisualization.

	static ScheduleDelegate[] noiseJobs = {
		Job<Lattice1D<Value>>.ScheduleParallel,
		Job<Lattice2D<Value>>.ScheduleParallel,
		Job<Lattice3D<Value>>.ScheduleParallel
	};

Everything works the same as before, but it is now possible to add other gradient noise types with only a little bit of extra code.

Second Gradient Noise Type

To implement Perlin noise, add a Perlin struct type to Noise.Gradient that implements IGradient, just like Value. Intially set all evaluations to zero.

	public struct Perlin : IGradient {

		public float4 Evaluate (SmallXXHash4 hash, float4 x) => 0f;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) => 0f;

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y, float4 z) => 0f;
	}

Then turn the noise jobs array in NoiseVisualization into a two-dimensional array. Elements of such arrays have two index components, so the array type changes from ScheduleDelegate[] to ScheduleDelegate[,]. Then wrap the existing initialization in a second set of curly braces and insert a new set for the Perlin noise jobs before the Value noise set.

	static ScheduleDelegate[,] noiseJobs = {
		{
			Job<Lattice1D<Perlin>>.ScheduleParallel,
			Job<Lattice2D<Perlin>>.ScheduleParallel,
			Job<Lattice3D<Perlin>>.ScheduleParallel
		},
		{
			Job<Lattice1D<Value>>.ScheduleParallel,
			Job<Lattice2D<Value>>.ScheduleParallel,
			Job<Lattice3D<Value>>.ScheduleParallel
		}
	};

To make switching between the noise types possible add an enum configuration field for Perlin and Value noise and use it as the first array index argument in UpdateVisualization.

	public enum NoiseType { Perlin€, Value€ }

	[SerializeField]
	NoiseType type;
	
	…
	
	protected override void UpdateVisualization (
		NativeArray<float3x4> positions, int resolution, JobHandle handle
	) {
		noiseJobs[(int)type, dimensions - 1](
			positions, noise, seed, domain, resolution, handle
		).Complete();
		noiseBuffer.SetData(noise);
	}

1D Gradients

Ken Perlin never made a proper 1D noise variant, because it isn't very useful, but we do because it makes understanding higher dimensions easier. We already tested 1D gradient noise earlier, with the fixed gradient function `f(x)=x`. The idea of Perlin noise is that the gradients of lattice points can be different. In the case of 1D the most obvious other gradient would simple be the negative version of what we already used: `f(x)=-x`. We can use the first bit of the hash to determine whether we select the positive or negative version.

	public struct Perlin : IGradient {
		
		public float4 Evaluate (SmallXXHash4 hash, float4 x) =>
			select(-x, x, ((uint4)hash & 1) == 0);
		
		…
	}

This leads to four possible gradient interpolations per lattice span: positive-positive, negative-negative, positive-negative, and negative-negative. As positive and negative mirror each other there are only two unique cases: same and different gradients.

The noise reaches a maximum amplitude of 0.5 halfway between lattice points with opposite gradients. This is the case because both are 0.5 at that point and we end up with their average. Ideally the noise has a maximum amplitude of 1, which we can achieve by doubling the gradients.

		public float4 Evaluate (SmallXXHash4 hash, float4 x) =>
			2f * select(-x, x, ((uint4)hash & 1) == 0);

Variable Gradients

I name the noise that we get at this point binary Perlin noise, because its gradients can only have two states. Thus the noise consists of sequences of gradients that all point in the same direction, except when there is a sign flip. The flips show up as maximum-amplitude waves while the rest of the pattern consists of small identical small waves. This looks very rigid, so let's try a different approach.

Instead of performing a binary selection, use the hash to scale the relative coordinates by a factor in the −1–1 range. We still need to double that to achieve a maximum amplitude of 1.

		public float4 Evaluate (SmallXXHash4 hash, float4 x) =>
			2f * (hash.Floats01A * 2f - 1f) * x;

This looks more interesting, as we get a large variety of gradients with different amplitudes, both positive and negative. This does make it far less likely that the maximum amplitude is reached, so the average amplitude of the noise is reduced.

We can also combine the variable and the binary approach, using the first bit to select the sign and scaling by float A.

		public float4 Evaluate (SmallXXHash4 hash, float4 x) =>
			2f * hash.Floats01A * select(-x, x, ((uint4)hash & 1) == 0);

But that would include the first bit in both choices, which introduces a dependency. To keep the choices independent use the ninth bit to determine the sign instead.

			2f * hash.Floats01A * select(-x, x, ((uint4)hash & 1 << 8) == 0);

The advantage of this approach is that we can introduce a minimum amplitude, regardless of the gradient direction. This way we can prevent degenerate areas from appearing, where the scale ends up close to zero for multiple successive lattice points, which would produce a flat region. The simplest approach is to set the minimum amplitude to 1 and add the hash float to that. The result can be considered a mix of the binary and variable approaches, guaranteeing a minimum wave amplitude but adding some variety on top of that.

		public float4 Evaluate (SmallXXHash4 hash, float4 x) =>
			(1f + hash.Floats01A) * select(-x, x, ((uint4)hash & 1 << 8) == 0);

2D Gradients

Moving on to 2D Perlin noise, we again start with binary gradients in a single dimension, based on the first hash bit, and see how that looks.

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) =>
			select(-x, x, ((uint4)hash & 1) == 0);

The result is an interpolation of different binary 1D noise bands in one dimension along the other dimension. Let's again replace this with a an approach based on float A to make the gradients more varied.

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) =>
			(hash.Floats01A * 2f - 1f) * x;

In the case of 2D noise we are not limited to axis-aligned gradients, the gradient vector can rotate a full 360°. To generate such a vector we can use an approach similar to how we generate an octahedron-sphere shape, but reduced to two dimensions.

We start with a line along X, from −1 to 1. Then we make Y equal to 0.5 minus the absolute of X, like we defined Z when creating the first half of an octahedron, but now in one less dimension. This creates a wedge that can be considered an opened square. Then we subtract X plus 0.5 floored from X to shift the negative portions of the square so it is closed.

The result is a square with its corners on the X and Y axes at distance 0.5 from the origin. The gradient vectors thus point from the origin to somewhere on the edge of this square.

To evaluate the gradient in 2D we sum its X and Y components. Finally, we double the result so the axis-aligned vectors end up with a length of 1.

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y) {
			float4 gx = hash.Floats01A * 2f - 1f;
			float4 gy = 0.5f - abs(gx);
			gx -= floor(gx + 0.5f);
			return (gx * x + gy * y) * 2f;
		}

If we want to turn the square distribution into a proper circle we can do so by normalizing the gradient vectors via division by their length. Note that this doesn't produce a uniform circular distribution of gradients, because they're uniformly distributed along the square instead. They're more concentrated near the square's corners, just like the point on the octahedron sphere are more concentrated near the corners of the octahedron. However this isn't a problem because the distribution is symmetrical and varied enough.

			return (gx * x + gy * y) * rsqrt(gx * gx + gy * gy);

However, this doesn't make much difference so we can omit that part to make the noise code run faster. Normalization does make the diagonal gradients stronger, but the noise itself still needs to be normalized, which will also take care of that.

			return (gx * x + gy * y) * 2f;

Normalized 2D Noise

To normalize the noise we need to determine its current maximum amplitude. If a lattice square has four gradients that all point to its center then its maximum value will be reached in the middle. As that will be the average of four gradients that are equal, we only need to calculate the value of a single gradient at that point. A perfectly diagonal gradient is `f(x,y)=(x+y)/2` so the amplitude at the middle would be `f(0.5,0.5)=1/2`. However, this is not the maximum amplitude of the entire noise. There is another gradient configuration that has a greater amplitude.

The maximum in a single dimension is 0.5, in the middle of a lattice span when both gradients point straight toward each other along that axis. This is unnormalized 1D binary Perlin noise. If the two gradients of the other span of a lattice square both point straight at the opposite span, then during interpolation along the second dimension we might end up with a value that exceeds 0.5 somewhere.

The maximum lies at the halfway point in the first dimension and somewhere along the interpolation in the second dimension. So this is effectively a smooth interpolation between an axis-aligned gradient and 0.5. Thus we have to find the maximum of the function `xs(1-x)+0.5s(x)` with `s(t)=6t^5-15t^4+10t^3` with 0–1 for the domain of `x`.

The expanded form of that function is `6x(1-x)^5-15x(1-x)^4+10x(1-x)^3+3x^5-7.5x^4+5x^3`, which simplifies to `-6x^6+18x^5-17.5x^4+5x^3+x`.

To find the maximum we have to take the derivative of that function and solve for zero, because where the rate of change is zero the function has reached a maximum or minimum. Solving that equation is possible but nontrivial. To get an idea of what the function looks like we can visualize it with Desmos. By selecting the graph line we can also immediately learn that the maximum is 0.5353 at X coordinate 0.6509. So it is indeed higher than 0.5.

We can get a more precise answer by asking Wolfram|Alpha to maximize the function. It can even give us an exact answer but it is quite complex, so we make do with the approximation. 0.53528 will do, one digit more precise than what Desmos showed. Divide the gradient by that value to arrive at normalized 2D Perlin noise.

			return (gx * x + gy * y) * (2f / 0.53528f);

3D Gradients

To create 3D Perlin noise we have to do the same thing that we did for the 2D version, expanded to include the third dimension. For this we can use the same octahedron-based approach that we used to generate the octahedron sphere. In this case we need two random values in the −1–1 range. We'll use floats A and D for this. D is preferred over B or C because a single bit shift is faster than both a shift and a mask operation.

		public float4 Evaluate (SmallXXHash4 hash, float4 x, float4 y, float4 z) {
			float4 gx = hash.Floats01A * 2f - 1f, gy = hash.Floats01D * 2f - 1f;
			float4 gz = 1f - abs(gx) - abs(gy);
			float4 offset = max(-gz, 0f);
			gx += select(-offset, offset, gx < 0f);
			gy += select(-offset, offset, gy < 0f);
			return (gx * x + gy * y + gz * z);
		}

Finding the maximum amplitude works the same as for 2D. It is found while interpolating along the third dimension with a gradient that points along that axis, towards a 2D lattice rect that has achieved its maximum value, which is 0.53528. So we have to maximize the same function as before, except with the constant 0.5 replaced with 0.53528: `xs(1-x)+0.53528s(x)`, which can be expanded to `6x(1-x)^5-15x(1-x)^4+10x(1-x)^3+ 0.53528(6x^5-15x^4+10x^3)`. Feeding that to Desmos yields 0.5629 at X coordinate 0.6732.

And Wolfram|Alpha tell us it is 0.56290 at 0.67321.

return (gx * x + gy * y + gz * z) * (1f / 0.56290f);

The next tutorial is Noise Variants.