图像恢复中的凸能量泛函正则化模型综述

目的凸能量泛函正则化模型(EFRM)的综述论文在国内外还少有报道,为使即将进入该领域的研究者全面了解发展现状,结合图像恢复,对该领域国内外研究现状进行综述。方法在参考大量文献的基础上,从凸EFRM的起因、组成、处理和发展趋势等方面加以总结和比较。首先,给定反问题,无法获得可行解,解决此问题的有效方法是建立EFRM。其次,从能量泛函的组成,分析拟合项和正则项的适用条件,给出引起图像模糊的5种点扩散函数,阐述权重的重要性及确定方法。再次,将能量泛函的拟合项和正则项分为整体处理、单独处理,分析空域、变换域和混合域正则化模型求解算法,评述模型和算法的优缺点。最后,指出图像恢复EFRM的发展趋势及存在的问题。结果一般说来,无法直接求解由拟合项、正则项和权重组成的原始凸EFRM,然而,通过转化模型、对偶模型和原始—对偶模型,利用数值代数、矩阵论和优化理论对转化模型进行整体处理、分裂处理,可以设计出高效、快速求解算法。结论图像恢复中的EFRM研究虽然取得了很多有意义的理论与应用成果,但随着大规模数据处理问题的不断涌现,建立准确的数学模型,设计高效快速的求解算法以及分析算法的收敛性等理论问题有待进一步深入研究。

Survey on convex energy functional regularization model of image restoration

Objective Local and international survey research on the convex energy functional regularization model(EFRM) are rare. To obtain comprehensive understanding of potential research, research development is surveyed in the field. Method Based on numerous references, we summarize and compare convex EFRM from four aspects, namely, original, composition, treatment, and development. First, given the inverse problem, obtaining feasible solutions is impossible. The effective method of solving the problem is to establish EFRM. Second, after establishing the energy functional model, the application conditions of the fitting and regularization terms are analyzed. Five types of point spread functions that can make image blur are provided. The significance and weight determination method are presented. Third, the fitting and regularization terms of the energy functional model are sorted by entire and separate treatments, and the computational algorithms of the regularization model are analyzed in space, transform, and hybrid domains. The advantages and disadvantages of the models and algorithms are determined. Finally, the development trends and existence problems of EFRM of image restoration are highlighted. Result Generally, directly solving primal convex EFRM is impossible; however, with transformation model, dual model, and primal-dual model of the primal model and by taking advantage of numerical analysis, matrix theory, and optimization theory, designing effective and rapid algorithms to completely and separately solve the transformation model becomes possible. Conclusion Meaningful theories and applicable results of EFRM of image restoration have been obtained. However, with the continuing appearance of large-scale data processing problems, several theoretical problems, such as establishment of accurate mathematical models, design of effective and fast solving algorithms, and analysis of algorithm convergence properties, require further research.