图像配准中方向场正则化模型的适定性和收敛性

Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration

图像配准是图像处理的一个重要方面.方向场正则化模型是现有配准方法中效果相对突出的模型.然而它仍然无法正确对齐所有感兴趣的区域.因此,本文从理论角度研究方向场正则化模型,希望寻找模型设计可能存在的问题.由于模型中有一个直接变量和一个由直接变量通过常微初值问题确定的间接变量,所以它是数学上的一类新颖的正则化模型.方向场正则化模型定义为min_(v){α||v||_(H)^(2)+ρ(T(y^(v)(τ)),S)},其中T和S分别是模板图像和参考图像,y^(v)(τ):x■y^(v)(τ;0,x)是由初值问题dy/ds=v(s,y),y(0)=x的解y^(v)(s;0,x)定义的变换,ρ是相似性泛函,α>0是正则化参数,H是希尔伯特空间.本文首先证明了方向场正则化模型有稳定解,然后证明了其收敛性.结合y^(v)(τ)与v的收敛关系和正则化问题的经典理论可得上述结论.然而,在现有理论下,ρ,S和T需满足较强的条件.本文通过充分利用y^(v)(τ)的性质,提出了关于ρ,S和T的相对弱的条件.此外,我们还验证了配准常用的3个相似性泛函都满足所提条件.

Image registration is fundamental to image processing.The vector field regularization model performs relatively well among a large number of registration methods.However,it still can’t correspond to all interested regions across images correctly.Therefore,we hope to study the theory of the vector field regularization model to see whether there are some problems with the design of the model.Moreover,as there are two unknowns which are related by an initial value problem in the regularization model,it is novel in mathematics.The vector field regularization model takes the form minv{α||v||_(H)^(2)+ρ(T(y^(v)(τ)),S)},where T is a template image,S is a reference image,y^(v)(τ):x■y^(v)(τ;0,x)is a transformation determined by the solution y^(v)(s;0,x)of the initial value problem dy/ds=v(s,y),y(0)=x,ρis a similarity functional,α>0 is a regularization parameter and H is a Hilbert space.In this paper,we firstly show the vector field regularization model has stable solutions and then demonstrate its convergence.The above results can be obtained by the standard arguments of regularized problems together with the convergence relation of y^(v)(τ)and v.However,the requirements forρ,S and T are relatively strong under the existing regularization theory.We give relatively weak conditions for p,S and T by taking full advantage of the good properties of y^(v)(τ).In addition,we verify that three commonly used similarity functionals in image registration satisfy the given conditions.