Establishing mathematical models is a ubiquitous and effective method to understand the objective world. Due to the complex physiological structures and dynamic behaviors, the mathematical representation of the human face is an especially challenging task. In this paper, an explicit function called GmFace is proposed for face image representation in the form of a multi-Gaussian function. The model utilizes the advantages of two-dimensional Gaussian function which provides a symmetric bell surface with a controllable shape. The GmNet is then designed using Gaussian functions as neurons, with parameters that correspond to each of the parameters of GmFace in order to transform the problem of GmFace parameter solving into a network optimization problem of GmNet. Furthermore, using GmFace, several face image transformation operations can be realized mathematically through simple parameter computation. Experimental results demonstrate that GmFace has a superior representation ability for face images compared to convolutional autoencoder (CAE), principal component analysis (PCA) and discrete cosine transform (DCT) method.

建立数学模型是了解客观世界的一种普遍有效的方法。由于复杂的生理结构和动态行为，人脸的数学表示是一项特别具有挑战性的任务。在本文中，提出了一种称为GmFace的显式函数，用于以多高斯函数的形式表示人脸图像。该模型利用了二维高斯函数的优势，该函数提供了形状可控的对称钟形表面。然后使用高斯函数作为神经元设计GmNet，其参数与GmFace的每个参数相对应，以便将GmFace参数求解的问题转化为GmNet的网络优化问题。此外，使用GmFace，通过简单的参数计算，可以在数学上实现多种面部图像转换操作。实验结果表明，与卷积自动编码器（CAE），主成分分析（PCA）和离散余弦变换（DCT）方法相比，GmFace具有更好的人脸图像表示能力。