Procedural noise functions are fundamental tools in computer graphics used for synthesizing virtual geometry and texture patterns. Ideally, a procedural noise function should be compact, aperiodic, parameterized, and randomly accessible. Traditional lattice noise functions such as Perlin noise, however, exhibit periodicity due to the axial correlation induced while hashing the lattice vertices to the gradients. In this paper, we introduce a parameterized lattice noise called prime gradient noise (PGN) that minimizes discernible periodicity in the noise while enhancing the algorithmic efficiency. PGN utilizes prime gradients, a set of random unit vectors constructed from subsets of prime numbers plotted in polar coordinate system. To map axial indices of lattice vertices to prime gradients, PGN employs Szudzik pairing, a bijection F: ℕ² → ℕ. Compositions of Szudzik pairing functions are used in higher dimensions. At the core of PGN is the ability to parameterize noise generation though prime sequence offsetting which facilitates the creation of fractal noise with varying levels of heterogeneity ranging from homogeneous to hybrid multifractals. A comparative spectral analysis of the proposed noise with other noises including lattice noises show that PGN significantly reduces axial correlation and hence, periodicity in the noise texture. We demonstrate the utility of the proposed noise function with several examples in procedural modeling, parameterized pattern synthesis, and solid texturing.

程序噪声函数是计算机图形学中用于合成虚拟几何图形和纹理图案的基本工具。理想情况下，过程噪声函数应该是紧凑的，非周期性的，参数化的并且可以随机访问。但是，传统的晶格噪声函数（例如Perlin噪声）由于将晶格顶点哈希化为梯度而引起的轴向相关性而表现出周期性。在本文中，我们介绍了一种称为素梯度噪声（PGN）的参数化点阵噪声，该参数可最大程度地减少噪声中可辨别的周期性，同时提高算法效率。PGN利用素数梯度，即从极坐标系中绘制的素数子集构造的一组随机单位向量。为了将晶格顶点的轴向索引映射到素梯度，PGN使用了Szudzik配对，即双射˚F：ℕ ² →ℕ。Szudzik配对函数的成分用于更高的维度。PGN的核心是通过主要序列偏移对噪声生成进行参数化的能力，这有助于创建具有从同质到混合多重分形的不同异质性水平的分形噪声。对建议的噪声与其他噪声（包括晶格噪声）进行的频谱比较分析表明，PGN显着降低了轴向相关性，从而降低了噪声纹理的周期性。我们通过程序建模，参数化模式合成和实体纹理中的几个示例演示了所提出的噪声函数的实用性。