IEEE Trans Pattern Anal Mach Intell. 2017 Apr;39(4):803-817. doi: 10.1109/TPAMI.2016.2560816. Epub 2016 Apr 29.
This paper presents a novel image descriptor to effectively characterize the local, high-order image statistics. Our work is inspired by the Diffusion Tensor Imaging and the structure tensor method (or covariance descriptor), and motivated by popular distribution-based descriptors such as SIFT and HoG. Our idea is to associate one pixel with a multivariate Gaussian distribution estimated in the neighborhood. The challenge lies in that the space of Gaussians is not a linear space but a Riemannian manifold. We show, for the first time to our knowledge, that the space of Gaussians can be equipped with a Lie group structure by defining a multiplication operation on this manifold, and that it is isomorphic to a subgroup of the upper triangular matrix group. Furthermore, we propose methods to embed this matrix group in the linear space, which enables us to handle Gaussians with Euclidean operations rather than complicated Riemannian operations. The resulting descriptor, called Local Log-Euclidean Multivariate Gaussian (LEMG) descriptor, works well with low-dimensional and high-dimensional raw features. Moreover, our descriptor is a continuous function of features without quantization, which can model the first- and second-order statistics. Extensive experiments were conducted to evaluate thoroughly LEMG, and the results showed that LEMG is very competitive with state-of-the-art descriptors in image classification.
本文提出了一种新颖的图像描述符，能够有效地描述局部的、高阶的图像统计信息。我们的工作受到扩散张量成像和结构张量方法（或协方差描述符）的启发，并受到基于分布的流行描述符（如 SIFT 和 HoG）的启发。我们的想法是将一个像素与在邻域中估计的多元高斯分布相关联。挑战在于，高斯空间不是线性空间，而是黎曼流形。我们首次证明，通过在这个流形上定义乘法运算，可以为高斯空间配备李群结构，并且它与上三角矩阵组的一个子群同构。此外，我们提出了在线性空间中嵌入这个矩阵群的方法，这使得我们可以用欧几里得运算而不是复杂的黎曼运算来处理高斯。由此得到的描述符称为局部对数欧式多变量高斯（LEMG）描述符，它可以很好地处理低维和高维原始特征。此外，我们的描述符是特征的连续函数，没有量化，可以对一阶和二阶统计进行建模。我们进行了广泛的实验来彻底评估 LEMG，结果表明，LEMG 在图像分类方面与最先进的描述符非常有竞争力。
