Wang Yulong, Kou Kit Ian, Zou Cuiming, Tang Yuan Yan
IEEE Trans Image Process. 2021;30:3637-3649. doi: 10.1109/TIP.2021.3064193. Epub 2021 Mar 17.
Sparse representation has achieved great success across various fields including signal processing, machine learning and computer vision. However, most existing sparse representation methods are confined to the real valued data. This largely limit their applicability to the quaternion valued data, which has been widely used in numerous applications such as color image processing. Another critical issue is that their performance may be severely hampered due to the data noise or outliers in practice. To tackle the problems above, in this work we propose a robust quaternion valued sparse representation (RQVSR) method in a fully quaternion valued setting. To handle the quaternion noises, we first define a new robust estimator referred as quaternion Welsch estimator to measure the quaternion residual error. Compared to the conventional quaternion mean square error, it can largely suppress the impact of large data corruption and outliers. To implement RQVSR, we have overcome the difficulties raised by the noncommutativity of quaternion multiplication and developed an effective algorithm by leveraging the half-quadratic theory and the alternating direction method of multipliers framework. The experimental results show the effectiveness and robustness of the proposed method for quaternion sparse signal recovery and color image reconstruction.
稀疏表示在包括信号处理、机器学习和计算机视觉在内的各个领域都取得了巨大成功。然而，大多数现有的稀疏表示方法都局限于实值数据。这在很大程度上限制了它们对四元数取值数据的适用性，而四元数取值数据已广泛应用于诸如彩色图像处理等众多应用中。另一个关键问题是，在实际应用中，由于数据噪声或离群值，它们的性能可能会受到严重影响。为了解决上述问题，在这项工作中，我们在全四元数取值的环境下提出了一种鲁棒的四元数取值稀疏表示（RQVSR）方法。为了处理四元数噪声，我们首先定义了一种新的鲁棒估计器，称为四元数韦尔奇估计器，以测量四元数残差误差。与传统的四元数均方误差相比，它可以在很大程度上抑制大数据损坏和离群值的影响。为了实现RQVSR，我们克服了四元数乘法不可交换性带来的困难，并利用半二次理论和乘子交替方向法框架开发了一种有效的算法。实验结果表明了该方法在四元数稀疏信号恢复和彩色图像重建方面的有效性和鲁棒性。
