论文标题
纯四元基质矩阵的低等级纯四元素近似
Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices
论文作者
论文摘要
Quaternion矩阵成功地用于许多彩色图像处理应用中。特别是，纯季基矩阵可用于表示颜色图像的红色，绿色和蓝色通道。可以使用Quaternion奇异值分解获得纯Quaternion矩阵的低级别近似值。但是，从所得的低级别近似矩阵可能不是纯的四元中，即低级别矩阵包含对颜色图像表示无用的真实组件，因此这种近似值不是最佳的。本文的主要贡献是找到一个最佳的排名 - $ r $ pure Quaternion矩阵的近似值（纯Quaternion矩阵（颜色图像）。我们的想法是在低级别的四元素矩阵歧管上使用投影，以及对具有零实际分量的季节矩阵的投影，并开发交替的投影算法以找到如此最佳的低率纯Quaternion矩阵近似值。投影算法的收敛可以通过证明低级别的四元基质歧管和零实际组件季节矩阵歧义具有非平凡的相交点来确定。提出了关于合成纯四个矩阵和颜色图像的数值示例，以说明投影算法可以找到纯Quaternion矩阵或彩色图像的最佳低级纯季节近似。
Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a non-trivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.
