## 1 Introduction  
Image alignment is a fundamental technique that has many applications for machine vision and image processing, including image retrieval, object recognition, pose estimation, industrial inspection, and target tracking. Because industrial inspections are usually conducted in well-controlled environments (e.g. the working distance between the lens and inspection objects is fixed), it is common practice to perform image alignment using constrained two-dimensional (2D) rigid transformation. This creates the possibility of establishing a fast image alignment algorithm that can be used for industrial inspection applications, and is the focus of this study. The research results are usually used in the quality control of manufacture automation. For example, two cases show the feasibility of this proposed research: (1) a robust vision system can inspect components fitted to several ranges of diesel engines at a critical stage and align in the engine assembly; (2) the quality assurance of surface mount devices (SMD) mounted on the printed circuit board (PCB). Image alignment techniques are generally divided into two major categories: a feature- or geometry-based method and an area- or intensity-based method [1]. The feature-based methods, of [PERSON] _et al._[2, 3] apply the directed Hausdorff distance to several algorithms to allow image alignment. [PERSON] _et al._[4] proposed a robust hierarchical Hausdorff distance to compare edge maps in a multi-level pyramidal structure. [PERSON] _et al._[5] also used the Hausdorff distance for image alignment in an inspection system for printed circuit boards. Peripherally, local descriptors are commonly used in several real-world applications, such as object recognition. [PERSON] [6] proposed a scale invariant feature transform (SIFT), which combines a scale invariant region detector and descriptor and uses the gradient distribution in detected regions. SIFT emphasizes local features at a particular point of interest that are invariant to scale and rotation. [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) compared the performance of descriptors that were computed for local interest regions. Even though SIFT is a useful method to describe the region of interest when the region has scale and rotates, SIFT-based matching fails when there are only a few feature points in the template image. The Zernike moments method Zernike (1958) is also used to pattern features for a pattern image and determines invariance to rotation, translation, and scale. To ensure computational efficiency, [PERSON] (2016) further proposed an accelerated version that uses a multi-resolution wavelet technique, projection, and Zernike moments. [PERSON] (2017) proposed a rotation-invariant template method that uses gradient information in the form of orientation codes. However, this method is time-consuming because the histogram must be computated. In order to increase the speed of the template-matching method that uses orientation codes, the novel rotation discriminative descriptor and an object search strategy were used [PERSON] (2018). The aforementioned invariant descriptors of points of interest in the image are resistant to partial occlusion and relatively insensitive to changes of viewpoint. The algorithms referred to are generally efficient in general applications, but they can result in unnecessary processing when the image alignment only involves rotation and limited scaling compensation, as with automatic optical inspection (AOI) applications. The area-based method is sometimes called the correlation-like or template-matching method and has been popular for decades because of its basic concepts. The small template is firstly applied to a large inspection image by sliding the template window on a pixel-by-pixel basis. The normalized cross correlation (NCC) is computed between the template and the inspection images. The maximum values or peaks for the computed correlation values indicate the matches between a template and sub-images in the inspection image.