## 1 Introduction
The method also requires an approximate initial guess of the transformation parameters, then solves the parameters by iteratively optimizing a non-linear objective function that describes the visual discrepancy between the template image and the re-aligned image. The accuracy of the method depends on the initial guess and the transformation scale when the initial parameters are far from the true value or the transformation is too large to approximate to the homography or affine model, the method would fail into a local minimum. Such adverse conditions are common in the 2D-3D image alignment applications such as face alignment [12; 13], scene/object alignment [14; 15; 16; 17; 18], and volumetric medical image alignment [19; 20]. To avoid the influence of initial parameters, face alignment methods use landmark points along the face contour in both 2D and 3D dimensions to build the objective function [12]. By facilitating the landmarks as the shape prior, [PERSON] et al. [13] applied a cascaded coupled-regressor to update the 2D landmarks and the 3D face shape iteratively, achieving the state-of-art accuracy in current 2D-3D face alignment methods. For general 3D objects, shape priors like face landmarks are unavailable, therefore the researchers designed the objective functions using the image features like edge [14], intensity texture [15], surface normal [16], or object structure [21]. By combining the feature tracker and the robust estimator, the image-feature-based objective functions are effective to solve the small and temporal continuous non-linear rigid transformations. However, when the tracker is lost, those methods would fail because of re-initialization errors. Among those image features, the strong gradient feature proposed by [PERSON] et al. in [14] and the descriptor filed proposed by [PERSON] et al. in [15] have already built a numerical connection between the 2D image and the 3D model of one 3D object, which are able to address the alignment problem if a transformation prior is provided. Besides the global objective functions, discrete transformation models have also been studied to tackle the non-linear transformations. A classic method is the mutual information based image registration (MI) [18], which calculates the mutual information of the image partitions by histogramming their joint intensities or using a kernel estimator for the joint density. According to the deep analysis of MI presented by [PERSON], [PERSON] et al. [19], the performance of MI is highly related to the partition  
Figure 2: The obvious visible registration error when assuming the model’s origin coincides with the marker’s origin. In this example, a \(4\times 4\) identity matrix is set as the offset matrix. (**a**) The origin coincidence assumption. (**b**) The visible registration error. strategy of the target object and the configuration of the histogram bin, which are experimentally chosen according to its application object. [PERSON] et al. [20] also used a discrete model to address the global non-linear transformation between an original 3D object and its broken fragments. The method assumes that each fragment complies an affine model and solves the parameters by a polynomial equation, which greatly increases the computation cost. Another way to solve the non-linear transformation is to use regression models trained by deep neural networks [16; 17; 22]. However, both the universality and accuracy of the learning-based method are limited by the quantity and quality of the training data, which is hard to be generalized on objects out of the training dataset. To sum up, given an approximate parameter prior, the image-feature-based objective function is a relatively feasible method to solve the non-linear 2D-3D image alignment for an industrial product. Aiming at eliminating the three types of errors, this work integrates both the marker pose prior and the gradient feature to calculate an approximate prior of the offset matrix, then optimize the prior parameters using the dense descriptor to achieve the minimal alignment error.