Introduction

• Random gradient vector lookup.
• S-curve evaluation.
• Grid-point dot product evaluation.
• Interpolation across all dimensions.

I developed the following "dynamic noise" algorithm in March 2004. In this context, the word "dynamic" refers to the inherently mutable nature of this noise function. In other words, it moves. While rereading my senior design project from the year before (a hardware-description-language implementation of improved noise with minor novelties), I noticed I had obliquely mentioned that the noise algorithm could likely be made dynamic. With a bit of tinkering, this algorithm emerged. I do not know whether something similar has arisen any place in the field, but the function has proven useful, so it seemed time to make certain it was passed on. (All other attempts to do so have seemed unusually thwarted by the universe.) Okay, enough about myself.

The general enhancement of this algorithm is noise movement without need of slicing through a higher dimension, so much less computation is required. The general drawbacks are slightly augmented memory usage as compared to traditional noise due to scaling of the gradient vectors, and use of an iterative timestepping approach that makes backtracking to a previous noise state or changing flow velocity more difficult than in the traditional approach of slicing through a higher dimension.

Method

We should begin by describing a 2D implementation of improved noise. Perlin provides code for the 3D case accompanying his 2002 brief, but leaves other cases out. We can use our own sensibilities to create a 2D case. The main difference is the set of possible gradient direction vectors. We will place them within the set {[1,2], [1,-2], [-1,2], [-1,-2], [2,1], [2,-1], [-2,1], [-2,-1]}. These vectors do not align with either the grid lines or the grid diagonals, as Perlin suggests they should not.

Generating dynamic noise is very similar to making improved noise, with one primary difference: all 3-bit gradient direction vectors are scaled by an unsigned value. This produces a slightly less dense noise function, but a more controllable one. For the 2D case, we will scale the gradient vectors with a 3-bit unsigned value. The 3-bit [0..7] range simply emerged from guesswork and testing. It seems to be the smallest bit-width that produces mostly decent results, at least with a constant change factor in scale. The change in gradient vector scale per time step is stored in a 1-bit value, representing +1 or -1 change. When changing, the gradient vectors just scale up and down, up and down: 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, ... The gradient vector scale change is made based on a coin toss: true then change, else leave it alone. I believe I had also tried other nonlinear methods like bit-shifting scale and changing the scale sinusoidally to get higher-order continuity over time, but they did not work out so well.

All this yields the following values in the gradient LUT for the 2D case:

• 3 bits for scale, indicating a value in range [0..7].
• 3 bits for direction, indicating [1,2], [1,-2], [-1,2], [-1,-2], [2,1], [2,-1], [-2,1], or [-2,-1].
• 1 bit for scale delta, indicating +1 or -1.

The magic is that, when gradient vectors are scaled by zero, we can swap around the direction vectors without any noticeable effect. Then, when they are rescaled, the noise function takes on the new form determined by the gradient direction vectors. As gradient vectors change scale, new vectors are always reaching zero scale, and so gradient vector swapping occurs every time step. If you are familiar with the improved noise function, this may sound pretty simple. And it is!

This approach also implies the constraint that gradient direction vectors must be stored in writable memory, of course, not constants.

The 1D case is trivial, and the 3D case can be extrapolated from the Perlin improved noise code and an understanding of the dynamic noise approach. Higher dimensions are not presented.

Economy & Conclusion