Mathematical Problems in Engineering

Volume 2015, Article ID 852016, 10 pages

http://dx.doi.org/10.1155/2015/852016

## Phase Error Caused by Speed Mismatch Analysis in the Line-Scan Defect Detection by Using Fourier Transform Technique

School of Mechatronic Engineering, China University of Mining & Technology, 1 Daxue Road, Xuzhou, Jiangsu 221116, China

Received 8 April 2015; Revised 18 June 2015; Accepted 23 June 2015

Academic Editor: Oleg V. Gendelman

Copyright © 2015 Eryi Hu and Yuan Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The phase error caused by the speed mismatch issue is researched in the line-scan images capturing 3D profile measurement. The experimental system is constructed by a line-scan CCD camera, an object moving device, a digital fringe pattern projector, and a personal computer. In the experiment procedure, the detected object is moving relative to the image capturing system by using a motorized translation stage in a stable velocity. The digital fringe pattern is projected onto the detected object, and then the deformed patterns are captured and recorded in the computer. The object surface profile can be calculated by the Fourier transform profilometry. However, the moving speed mismatch error will still exist in most of the engineering application occasion even after an image system calibration. When the moving speed of the detected object is faster than the expected value, the captured image will be compressed in the moving direction of the detected object. In order to overcome this kind of measurement error, an image recovering algorithm is proposed to reconstruct the original compressed image. Thus, the phase values can be extracted much more accurately by the reconstructed images. And then, the phase error distribution caused by the speed mismatch is analyzed by the simulation and experimental methods.

#### 1. Introduction

It is known that the Fourier transform profilometry was widely used in the machine vision, industry monitoring, production surface inspection, and so forth [1–5]. By using this type of phase extracting algorithm, the phase distribution of the deformed fringe patterns can be obtained with only one frame of image [6, 7], so that the projection grating phase measurement method can be applied in the dynamic or moving object surface profile inspection [8]. Furthermore, the line-scan or the time delay and integration (TDI) CCD camera is used to capture the surface image in the moving object inspection [9, 10]. In particular, the applications of the TDI camera have been reported for the dynamic inspection of rotating objects [11, 12], and the Fourier transform profilometry is used to obtain the surface profile information. The error analysis about the projection grating Fourier transform profilometry has been reported [8, 13], from which it is found that the main reasons of the phase extraction error are the nonlinear response of the CCD, the random noise, the quantization of grey levels, spatial carrier frequency, calibration error, and so forth.

However, compared with the TDI CCD camera image capturing system, there is no time delay and integration procedure of the charges in the line-scan CCD image sensors [12], so that the image capture speed of the line-scan CCD camera will be much faster than the TDI CCD, and the price of the line-scan camera will be much lower than the TDI CCD. Similarly, the phase error will also exist when the line-scan and the moving speed of the detected object are mismatch with each other. As there is an obvious difference of the operation principle between the proposed two kinds of CCD camera, the measurement error of the line-scan CCD camera caused by the speed mismatch will be discussed in depth in this paper. A line-scan 3D surface profile measurement system is constructed to obtain the surface profile of the moving object. The object surface height values are calculated by the Fourier transform profilometry. The simulation and experimental methods are applied to analyze the phase error distribution of the detected 3D model. In order to decline the phase error into the lower level, an image recovering algorithm will be applied to reconstruct the distorted images.

#### 2. The Principle of the Line-Scan Fourier Transform Profilometry

##### 2.1. The Experimental System

The configuration of the line-scan Fourier transform profilometry experimental setup is founded on the conventional projection grating systems. As shown in Figure 1, the experimental system is constructed by an image line-scan CCD camera, an object moving device, a fringe pattern digital projector, a speed coder, and a personal computer. Only one frame of parallel fringe pattern with cosine function modulated intensity is projected onto the moving object plane at an incidence angle. The line-scan CCD camera is put upon the moving device to obtain the deformed fringe patterns modulated by the surface profile of the detected object. The optical axis of the camera is normal to the reference plane and the line-scan direction is perpendicular to the moving direction. Furthermore, the projected fringe direction is parallel to the moving direction of the object. The detected object is put on the motorized translation stage. When the detected object is moving in a stable velocity with the stage, the fringe pattern is captured by the line-scan CCD camera line by line. The speed coder is used to detect the moving speed of the object, and the pulse is sent back to the computer for the synchronization between the object moving and the line-scan. Furthermore, the image capturing system must be calibrated to get an original setting about the trigger of the line-scan CCD camera. However, the moving speed mismatch error will still exist in most of the engineering applications even after an image system calibration. Finally, the deformed images are recorded in a personal computer for the next calculation. The object surface height can be obtained after extracting the fringe deformation between the reference and the detected surface grating. In this study, the Fourier transform method, which is shown in the following, is used to evaluate the fringe deformation. When the speed mismatch error is unavoidable, the phase error of the deformed fringe pattern will be introduced in the measurement.