Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, CH-8092 Zurich, Switzerland.
IEEE Trans Image Process. 2011 Jun;20(6):1543-57. doi: 10.1109/TIP.2010.2102044. Epub 2010 Dec 23.
This paper addresses the problem of image alignment based on random measurements. Image alignment consists of estimating the relative transformation between a query image and a reference image. We consider the specific problem where the query image is provided in compressed form in terms of linear measurements captured by a vision sensor. We cast the alignment problem as a manifold distance minimization problem in the linear subspace defined by the measurements. The transformation manifold that represents synthesis of shift, rotation, and isotropic scaling of the reference image can be given in closed form when the reference pattern is sparsely represented over a parametric dictionary. We show that the objective function can then be decomposed as the difference of two convex functions (DC) in the particular case where the dictionary is built on Gaussian functions. Thus, the optimization problem becomes a DC program, which in turn can be solved globally by a cutting plane method. The quality of the solution is typically affected by the number of random measurements and the condition number of the manifold that describes the transformations of the reference image. We show that the curvature, which is closely related to the condition number, remains bounded in our image alignment problem, which means that the relative transformation between two images can be determined optimally in a reduced subspace.
本文针对基于随机测量的图像配准问题。图像配准包括估计查询图像和参考图像之间的相对变换。我们考虑了这样一个具体问题，即查询图像以视觉传感器获取的线性测量的压缩形式提供。我们将配准问题建模为测量定义的线性子空间中的流形距离最小化问题。当参考模式在参数字典上稀疏表示时，可以给出表示参考图像的位移、旋转和各向同性缩放综合的变换流形的封闭形式。我们表明，在字典构建在高斯函数上的特殊情况下，目标函数可以分解为两个凸函数的差（DC）。因此，优化问题成为一个 DC 程序，它可以通过割平面法全局求解。解的质量通常受随机测量的数量和描述参考图像变换的流形的条件数的影响。我们表明，在我们的图像配准问题中，曲率（与条件数密切相关）保持有界，这意味着可以在降低的子空间中最优地确定两幅图像之间的相对变换。
