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In this paper, the 1-D mapping is studied for estimating a general projective transformation between images. First, a ray model of image patches is proposed to convert a complex projective deformation of a patch in the Cartesian space into two simple rigid shifts of radial lines in the log-polar/inverse-polar space. To capture these shifts via direct search, two types of radial image line matching, viz. 1-D log-polar mapping and 1-D inverse-polar mapping, are formally defined. Also, the unique features of these shifts are analyzed. Then, a novel two-step mathematical framework is devised for transformation estimation. In the first step, the 1-D log-polar mapping is applied to estimating four affine parameters via two alternative methods for simulated/real images. In the second step, the 1-D inverse-polar mapping is applied to estimating two projective parameters. By a direct search combined with the two-step line matching, two translational parameters are also determined. Finally, the performance of the framework is evaluated first in estimating a variety of projective transformations by comparing the estimates with their ground truths and between the two proposed methods, and second in registering a wide range of challenging image pairs by comparing with a state-of-the-art feature-based method.