Diffeomorphic registration has become a powerful approach for seeking a
smooth and invertible spatial transformation between two coordinate systems
which have been measured via the template and reference images. While the
pointwise volume-preserving constraint is effective for some problems, it is
too stringent for many other problems especially when the local deformations
are relatively large, because it may lead to a poor large-deformation for
enforcing local matching.In this paper, we propose a novel bi-variant
diffeomorphic image registration model with the soft constraint of Jacobian
equation, which allows local deformations to shrink and grow in a flexible
range.The Jacobian determinant of the transformation is explicitly controlled
by optimizing the relaxation function. To prevent deformation folding and
enhance the smoothness of deformation, we not only impose a positivity
constraint in optimizing the relaxation function, but also employ a regularizer
to ensure the smoothness of the relaxation function.Furthermore, the positivity
constraint ensures that is as close to one as possible, which helps to obtain a
volume-preserving transformation on average.We further analyze the existence of
the minimizer for the variational model and propose a penalty splitting method
with a multilevel strategy to solve this model. Numerical experiments show that
the proposed algorithm is convergent, and the positivity constraint can control
the range of relative volume and not compromise registration accuracy.
Moreover, the proposed model produces diffeomorphic maps for large deformation,
and achieves better performance compared to the several existing registration
models.

本文提出了一种具有雅可比方程软约束的新型双变量微分同胚图像配准模型，该模型允许局部变形在一个灵活的范围内收缩和扩大，通过优化松弛函数来明确控制变换的雅可比行列式，并采用正则化器来确保松弛函数的平滑性。数值实验表明，所提出的算法收敛性强，正性约束能控制相对体积的范围而不影响配准准确性。此外，相对于其他几种现有的配准模型，该模型在大变形下产生了微分同胚映射并取得了更好的性能。

一种用于松弛雅可比行列式约束的差函一致性变分模型

A Bi-variant Variational Model for Diffeomorphic Image Registration with  Relaxed Jacobian Determinant Constraints

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem:
given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for
assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and
$H\in\R_+^{k\times n}$ such that $V=WH$. Exact, non trivial, nonnegative
factorizations do not always exist, hence it is interesting to pose the
approximate NMF problem. The criterion which is commonly employed is
I-divergence between nonnegative matrices. The problem becomes that of finding,
for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An
iterative algorithm, EM like, for the construction of the best pair $(W, H)$
has been proposed in the literature. In this paper we interpret the algorithm
as an alternating minimization procedure \`a la Csisz\'ar-Tusn\'ady and
investigate some of its stability properties. NMF is widespreading as a data
analysis method in applications for which the positivity constraint is
relevant. There are other data analysis methods which impose some form of
nonnegativity: we discuss here the connections between NMF and Archetypal
Analysis.

本文研究非负矩阵因式分解（NMF）问题，提出了基于 I-divergence 的非负矩阵近似分解方法，使用类 EM 算法进行迭代，并探讨了其稳定性及与原型分析（Archetypal Analysis）的联系。