The purpose of this paper is to analyze a nonlinear elasticity model previously introduced by the authors for comparing two images, regarded as bounded open subsets of $\R^n$ together with associated vector-valued intensity maps. Optimal transformations between the images are sought as minimisers of an integral functional among orientation-preserving homeomorphisms. The existence of minimisers is proved under natural coercivity and polyconvexity conditions, assuming only that the intensity functions are bounded measurable. Variants of the existence theorem are also proved, first under the constraint that finite sets of landmark points in the two images are mapped one to the other, and second when one image is to be compared to an unknown part of another. The question is studied as to whether for images related by a linear mapping the unique minimizer is given by that linear mapping. For a natural class of functional integrands an example is given guaranteeing that this property holds for pairs of images in which the second is a scaling of the first by a constant factor. However for the property to hold for arbitrary pairs of linearly related images it is shown that the integrand has to depend on the gradient of the transformation as a convex function of its determinant alone. This suggests a new model in which the integrand depends also on second derivatives of the transformation, and an example is given for which both existence of minimizers is assured and the above property holds for all pairs of linearly related images.

翻译：本文旨在分析作者先前提出的用于比较两幅图像的非线性弹性模型，其中图像被视为$\R^n$中的有界开子集及其关联的向量值强度映射。图像间的最优变换通过在保向同胚中寻找积分泛函的极小化子来实现。在仅假设强度函数为有界可测的条件下，通过自然的强制性条件和多凸性条件证明了极小化子的存在性。本文还证明了存在性定理的若干变体：首先在两幅图像中有限个地标点必须相互映射的约束下，其次当一幅图像需与另一幅图像的未知部分进行比较时。本文研究了当图像通过线性映射相关联时，该线性映射是否为唯一极小化子的问题。针对一类自然的泛函被积函数，给出了一个示例，保证该性质在第二幅图像为第一幅图像常数倍缩放的情况下成立。然而，要使该性质对任意线性相关的图像对均成立，研究表明被积函数必须仅依赖于变换梯度的行列式之凸函数。这启发了新的模型，其中被积函数还依赖于变换的二阶导数，并给出了一个示例，既保证了极小化子的存在性，又使上述性质对所有线性相关的图像对均成立。