In this preliminary study, we provide two methods for estimating the gradients of functions of real value. Both methods are built on derivative estimations that are calculated using the standard method or the Squire-Trapp method for any given direction. Gradients are computed as the average of derivatives in uniformly sampled directions. The first method uses a uniformly distributed set of axes that consists of orthogonal unit vectors that span the space. The second method only uses a uniformly distributed set of unit vectors. Both methods essentially minimize the error through an average of estimations to cancel error terms. Both methods are essentially a conceptual generalization of the method used to estimate normal fractal surfaces.

翻译：在此初步研究中，我們提出了兩種用於估計實數值函數梯度的方法。這兩種方法均基於對任意指定方向使用標準方法或Squire-Trapp方法計算得到的導數估計。梯度被計算為均勻採樣方向上導數的平均值。第一種方法使用一均勻分布的軸集，該軸集由張成空間的正交單位向量組成。第二種方法僅使用一均勻分布的單位向量集。兩種方法本質上通過對估計值進行平均來最小化誤差，從而抵消誤差項。這兩種方法實質上是對估計常規分形曲面所用方法的概念性推廣。