Signal Processing Laboratory, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
IEEE Trans Image Process. 2011 May;20(5):1300-12. doi: 10.1109/TIP.2010.2093904. Epub 2010 Nov 18.
In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including nonflat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Second, this method is, to the best of our knowledge, the first reparametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.
在本文中，我们提出了一种新的几何框架，称为测地主动场，用于一般的图像配准。在图像配准中，人们寻找最佳的变形场，将一幅图像映射到另一幅图像上。这是一个经典的不适定逆问题，通常通过添加正则化项来解决。在这里，我们提出了一种注册项和正则化项之间的乘法耦合，这被证明等价于将变形场嵌入到加权最小曲面问题中。然后，变形场通过一个最小化流驱动，朝着对应于注册问题解的调和映射移动。这种用于注册的方法与图像分割中著名的测地主动轮廓模型非常相似，其中分割项（边缘检测函数）与正则化项（长度函数）通过乘法耦合。实际上，我们提出的几何模型实际上是 Caselles-Kimmel-Sapiro 引入的曲线和曲面加权长度问题的向量场的精确数学推广。变形场的能量是通过用适当的图像距离加权的 Polyakov 能量来测量的，这是从标准注册模型借来的。我们研究了三种不同的加权函数，用于单模态图像的平方误差和近似绝对误差，以及用于多模态图像的局部联合熵。与专门为特定应用定制的专门的最先进方法相比，我们的几何框架有重要的贡献。首先，我们的注册通用公式适用于任何可参数化、平滑和可微的曲面，包括非平面和多尺度图像。在后一种情况下，多尺度图像在所有尺度上同时注册，并且空间和尺度之间的关系内在地被考虑到。其次，据我们所知，这种方法是文献中首次引入的重参数化不变注册方法。第三，注册项（即局部图像差异）和正则化项之间的乘法耦合自然导致正则化强度的自适应调整。最后，通过选择变形场的度量，我们可以在经典的高斯正则化和更有趣的各向异性、TV 样正则化之间自由插值。
