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Geodesic Active Fields—A Geometric Framework for Image Registration

机译:测地活动场—图像配准的几何框架

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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 [1]. 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, -n-nto 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.
机译:在本文中,我们提出了一种称为测地线活动场的新型几何框架,用于常规图像配准。在图像配准中,人们寻找最能将一幅图像映射到另一幅图像的基础变形场。这是经典的不适定逆问题,通常可以通过添加正则项来解决。在这里,我们提出了配准项和正则项之间的乘法耦合,结果证明等效于将变形场嵌入加权最小表面问题中。然后,通过最小化流将变形场驱动到对应于配准问题的解的谐波图。该提议的配准方法在图像分割中与众所周知的测地线活动轮廓模型具有相似的相似性,其中分割项(边缘检测器函数)也通过乘法与正则项(长度函数)耦合。实际上,我们提出的几何模型实际上是对Caselles-Kimmel-Sapiro [1]引入的曲线和曲面的加权长度问题的矢量场的精确数学概括。变形场的能量是通过从标准配准模型中借用适当图像距离加权的Polyakov能量来测量的。我们研究了三种不同的加权函数,单峰图像的平方误差和近似绝对误差,以及多峰图像的局部联合熵。与针对特定应用量身定制的专业最新技术相比,我们的几何框架涉及许多重要方面。首先,我们用于配准的一般公式适用于任何可参数化,光滑和可区分的表面,包括非平坦和多尺度图像。在后一种情况下,多尺度图像同时在所有尺度上被配准,并且空间和尺度之间的关系本质上被考虑在内。其次,就我们所知,此方法是文献中引入的第一个重新参数化不变配准方法。第三,配准项(即局部图像差异)和正则项之间的乘法耦合自然会导致正则化强度的数据相关调整。最后,通过在变形场上选择度量,可以在经典高斯和更有趣的各向异性,类似电视的正则化之间自由插值。

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