四元数矩阵降维法：彩色图像去噪新策略

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本文是一篇发表在《多维系统与信号处理》(Multidimensional Systems and Signal Processing)国际期刊上的研究论文，其标题是"通过减少四元数矩阵奇异值分解对彩色图像进行去噪"。该研究由Shan Gai、Guowei Yang、Minghua Wan 和 Lei Wang 合作完成，发表日期为2013年，且论文的数字对象标识符(DOI)为10.1007/s11045-013-0268-x。主要知识点包括： 1. **彩色图像去噪方法**：论文探讨了利用四元数矩阵奇异值分解（Quaternion Matrix Singular Value Decomposition, QMSVD）技术来处理彩色图像噪声问题。QMSVD是一种高级数学工具，它能够在保持颜色空间信息的同时，有效地捕捉和去除图像中的冗余和噪声成分，从而提高图像质量。 2. **四元数在图像处理中的应用**：四元数是一种扩展复数，特别适用于处理旋转和方向信息。在图像处理中，它们能够提供一种更直观的方式来描述颜色和方向的变化，这对于色彩去噪和恢复图像细节具有重要意义。 3. **奇异值分解**：这是矩阵分析中的核心概念，通过分解矩阵为一组特征向量和相应的特征值，可以揭示数据的主要结构和模式。在彩色图像中，QMSVD能识别并消除那些对图像质量影响不大的低频噪声和冗余信息。 4. **论文贡献与限制**：文章可能介绍了作者们如何通过优化或简化四元数矩阵的奇异值分解过程，提高了算法的效率和去噪效果。同时，也可能讨论了这种方法在实际应用中的优势和可能存在的挑战，比如计算复杂度、对特定噪声类型的敏感性等。 5. **版权与引用政策**：文章强调了版权保护和学术诚信的重要性。读者在使用或自我存档论文时必须遵守 Springer Science + Business Media 的规定，确保在一定时间后（例如官方发布后12个月或之后）且给予原始出版物适当的承认，并链接到Springer网站的正式出版文章。这篇论文深入探讨了如何利用四元数矩阵奇异值分解这一技术来提升彩色图像的去噪效果，对于图像处理领域的研究人员和工程师来说，提供了有价值的新方法和理论支持。

Multidim Syst Sign Process

ably corrupted by noise during acquisition and transmission. In the last decades, a large

number of methods have been proposed for removing noise, thus leading to improvements

in denoising performance.

Several color-image-denoising techniques, which can remove noise by a point wise-

varying local mask, have been proposed in the spatial domain (Tschumperle and Deriche

2005). Takeda et al. (2007) constructed a local model of an image with a polynomial func-

tion. The ﬁltered image is obtained by modifying coefﬁcients that are calculated by weighted

least squares regression. Another denoising framework can remove noise by computing the

differences in spatial values among neighboring pixels (Coifman and Meer 2002; Buades et

al. 2006). The image-denoising algorithm based on partial differential equation was proposed

by Perona and Malik (1990).

The challenge faced by any image-denoising algorithm is the removal of noise while

producing a sharp image without information loss. Awate and Whitaker (2006), Baudes et

al. (2005) proposed a new bilateral ﬁlter based on patches. Non-local techniques obtain the

denoised image by minimizing the penalty term on the average weighted distance between

an image patch and all other patches (Kervrann and Boulanger 2006, 2008). Elad and

Aharon (2006) proposed kernel singular value decomposition (KSVD), which is a signif-

icant denoising method. A new denoising algorithm was proposed in Chatterjee and Milan-

far (2009) by combining dictionary-based methods with regression-based methods. Dabov

et al. (2007) proposed color-block matching and 3D ﬁltering (CBM3D), which can group

similar patches and perform ﬁltering in the transform domain. This dictionary-based algo-

rithm provides implicit modeling for natural images. Other color-image-denoising meth-

ods based on explicit models have been proposed in Liu et al. (2008), Muresan and Parks

(2003), Zhang et al. (2010). Then the patch-based locally optimal Wiener (PLOW) ﬁlter-

ing to exploit the patch redundancy of an image was proposed by Chatterjee and Milanfar

(2012).

Another class of image-denoising techniques is based on the transform domain. The noisy

image is ﬁrst transformed into the frequency domain. Smaller coefﬁcients usually correspond

to higher frequency components of the image signal, which are dominated by noise. There-

after, transformed coefﬁcientscan be processed by using the maximum a posteriori probability

estimation technique. Typical transform domain methods for image denoising include wavelet

(Rajpoot and Butt 2012; Vo et al. 2011; Hsia et al. 2012), Bayes least square–Gaussian scale

mixture (BLS–GSM) method (Portilla et al. 2003), and contourlet (Sheng and Zhen 2009;

Haji and Bui 2012). The state-of-the-art image-denoising method in the transform domain is

a high-order SVD (HOSVD) (Rajwade and Rangarajan 2013).

Existing denoising methods ﬁrst remove the noise in color images by separating the

image into R, G, and B channels before using the denoising algorithm on each channel.

The denoised image loses large amounts of information because of the inherent relation-

ships among the three components of the color image (i.e., R, G, and B). The proposed

algorithm in this paper can process the color image as a whole instead of separating it into

components. A good trade-off can be achieved between the denoising effect and information

preservation.

The rest of this paper is organized as follows. Section 2 reviews the theory of

reduced quaternion (RQ) matrix (RQM). Section 3 provides the details of the proposed

RQM singular value decomposition (RQMSVD). Section 4 presents the new color-image-

denoising method based on RQMSVD. Section 5 shows the experimental validation of

the proposed algorithm and compares the proposed algorithm with several recently pro-

posed denoising algorithms in terms of visual quality. Finally, the conclusion is given in

Sect. 6.

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