The fast recognition and classification of objects with rectangular shapes are an essential component in the automatic detection of printed circuit board (PCB) defects

such as through automatic optical inspection or automatic X-ray inspection. The image obtained generally covers only a certain part of a PCB that contains rectangular shapes. The amount

positions

orientations

and exact shapes are almost undetermined. Most traditional algorithms for rectangular region detection are based on the Hough transform or on its derivations. In these approaches

each potential region must be voted on in a 2D or 3D domain. This process increases time complexity and lowers computation speed. By contrast

shape angle theory is an effective method for detecting rectangular regions in contour inspection. Shape angle theory can avoid the inaccuracy and computational inefficiency induced by Hough transform. The accuracy and time complexity of shape angle calculation depend on the efficiency of the tangent calculation at discrete points on the contour. This calculation is currently performed using either Vialard's algorithm or its derived methods. These procedures can be complicated and inaccurate when applied to rectangles with relatively smooth edges.A new method based on Fourier fitting has been proposed in this paper to solve this problem. A polar coordinate system transform is first applied to the discrete points. Then

Fourier series are used to fit envelopes and to calculate the derivative of the envelope function to obtain the tangent. The integrated chip FUSION 878A is derived from a gray-scale image to verify the accuracy and reliability of this method. The average computing time of the proposed method is 1.5775 s. By contrast

the traditional approach reports an average computing time of 156.155 s. The proposed method avoids iterations in handling rectangle contours. Therefore

time complexity is significantly reduced by two orders of magnitude. Moreover

the proposed method generated a precise outcome. The improved approach is applied to rectangle region detection problems. Experimental results indicate the acceptable accuracy and reliability of this algorithm. The following conclusions are drawn: 1) the proposed method exhibits little time complexity when calculating the tangent vector of the rectangle contour. The findings of both experiments and theoretical analysis suggest that specifically

time complexity is reduced from (n) to (n). 2) We examine the effects of breadth-to-length ratio and rectangle dimensions on shape angle. Understanding these effects can provide reference information for template selection and data matching in industrial applications. 3) This approach improves on the shape angle method and can effectively classify rectangles in images by implementing rectangle detection based on shape angle theory.