摘  要：图像处理问题，在八九十年代，主要是通过数学建模的方式，使用微分的思想，最后再通过小波分析来对问题，进行解决。具体的思想成熟是在20世纪初，在之前数学模型的基础上，通过变分的方式建模，使用变分偏微分方程，之后用数值的方法对目标函数实现最优化。用数学方法语言来表达图像分解问题即是：将一个完整图像分为结构分量、纹理分量、噪音分量。对面图像分解技术模型的不断发展，本文重点研究基于MCA（形态成分分析）的图像分解技术。

具体内容分为一下几点：

   分析形态成分分析在图像分解问题中的发展状况，对形态成分分析的定义有一定的了解，明白用形态成分分析的方式处理图像分解问题有什么优点。

   稀疏表示是处理图像分解问题的理论基础，根据不同的情况范围，通过曲波变换、离散余弦，由多样信号进行处理和使用，从多尺度的角度为基础并以此展开拓展与开发。且文中的字典设计和优化算法，也是通过稀疏表示里的学习算法和稀疏编码得到。

   引入MCA算法，使用偏微分和变分的数学方法，为图像分解问题的数学理论公式提供支撑。并使用公式，表现出这类问题中解的存在性、唯一性和规则性，将非凸函数和非线性组合的通过字典转换为凸函数和线性组合来解决问题。

该论文包括对稀疏表示的定义，和在此基础上演变出的盲源分离的介绍，在盲源分离的理论基础上，弄清MCA（形态成分分析）的本质，接着通过数学偏微分和变分的方法对其模型求解，再由优化算法得到最终解。

最后的实验结果通过编写程序，在计算机上得以实现图像分解，验证出该算法的有效性。

关键词： 字典  盲源分离  图像的分解技术  稀疏表示   形态成分分析

Image Decomposition Based on Morphological Analysis

Abstract：Image processing problem, in the eighties and nineties, mainly through mathematical modeling, the use of differential thinking, and finally through the wavelet analysis to solve the problem. The concrete thought is mature in the early 20th century, on the basis of the previous mathematical model, through the variational way modeling, the use of variational partial differential equations, followed by numerical methods to optimize the objective function. Using the mathematical method to express the image decomposition problem is: a complete image is pided into structural components, texture components, noise components. In this paper, we focus on the image decomposition technology based on MCA (morphological component analysis), which is based on the continuous development of the image decomposition technology.

Specific content is pided into a few points:

   Analyze the development of morphological analysis in morphological decomposition, and find out how to deal with the problem of image decomposition by means of morphological analysis.

   Sparse representation is the theoretical basis for the problem of image decomposition. According to the different situation range, the discrete cosine is processed and used by various waveforms, and is based on multi-scale and expanded and developed The And the text of the dictionary design and optimization algorithm, but also through the sparse representation of the learning algorithm and sparse coding to get.

   Introduce the MCA algorithm, using the partial differential and variational mathematical methods to support the mathematical theory of image decomposition. And use the formula to show the existence, uniqueness and regularity of the solution in this kind of problem. The nonconvex function and the nonlinear combination are transformed into convex function and linear combination to solve the problem.

This paper includes the definition of sparse representation and the introduction of blind source separation on the basis of this. On the basis of the theory of blind source separation, the essence of MCA (morphological analysis) is clarified, And then the final solution is obtained by the optimization algorithm.