颜色图像滤波的四元数扩散方法
Quaternion Diffusion for Color Image Filtering
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摘要: 扩散方程方法是图像处理的基本方法之一，对于图像的去噪、增强、边缘提取等能取得很好的效果。传统的颜色图像扩散方程很少考虑颜色图像通道对间的直接相互作用，本文提出了基于四元数扩散的新的颜色图像扩散方法，很好的处理了颜色图像通道间的作用。四元数分析是将复数推广到四维的结果，不满足乘法的交换率。虽然讲四元数用于图像处理已有了一些研究，但都是研究四元数傅立叶分析、四元数小波分析或将简化的四元数偏微分方程用于灰度图像处理，而本文第一次将四元数扩散用于颜色图像处理，这样的好处包括： 1 ）可以避免一般非线性扩散的阶梯效应； 2 ）四元数扩散不需要对图像导数进行高斯卷积等归整化操作； 3 ）更为省时。 在理论上，本文给出了对于线性四元数扩散方程的解，并对解进行了分析。以前对于图像的四元数分析都是使用四元数的相位形式，但这样无法对于四元数扩散方程进行分析。本文 首先对四元数的定义和各种运算性质进行了简述，然后通过引入四元数指数形式的乘运算并证明了几个引理，克服了四元数运算的乘法不可交换性，完成了线性四元数扩散方程的求解。通过对线性四元数扩散方程的近似解进行研究，证明了当方程的参数在一定范围内可以具有对于颜色图像导数的归整化。文中还分析了线性和非线性四元数方程中各个参数对于扩散结果的影响。文章还用专门的部分分析了四元数扩散与其它颜色扩散的关系，如四元数扩散与最近提出的广义空间彩色扩散的关系，及近似情况下四元数扩散与最近提出的复彩色扩散一定意义下的一致性。 在算法和实验上，本文给出了线性和非线性四元数扩散方程的形式、计算方法和应用示例。在非线性四元数扩散方程部分，首先从理论上分析然后用实验验证了非线性四元数扩散可以避免阶梯效应的能力，然后分析并用实验验证了这种方法对于亮度变化场景的边缘提取可以取得很好的效果。本文还给出了适合于四元数扩散方程的稳定的计算策略。为了表明四元数扩散对于颜色扩散的有效性，本文使用归一化平均平方误差和平均彩色误差作为评价指标，实验结果表明对于不同噪声方差和不同实验图像，线性和非线性四元数扩散的综合指标（还包含所用时间）都分别优于最近提出的线性和非线性广义空间彩色扩散方法。Abstract: How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the direct interactions between color channel pairs. In this paper, a new method of color diffusion considering these effectsis presented, which is based on quaternion diffusion (QD) equation. In addition to showing the solution to linear QD and its analysis, one form of nonlinear QD is discussed. Compared with other color diffusion techniques, considering the interactions between channel pairs, QD has the following advantages: 1) staircasing effect is avoided; 2) as diffusion tensor, the image derivative is regularized without requiring additional convolution; 3) less time is needed. Experimental results demonstrate the effectiveness of linear and nonlinear QD applied to natural color images for denoising by both visual and quantitative evaluations.
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