【摘要】 得益于时频分析和多分辨特性，小波变换轮廓术（Wavelet Transform Profilometry,WTP）能从单帧条纹中高精度提取携带被测物面高度信息的相位图。小波变换轮廓术的相位解调难易程度和精度依赖于小波的选取。复小波可以直接用于条纹相位解调。一维高斯函数的n阶导数满足小波的容许条件，且任意边带的频谱分布具有平滑、非对称的特点。它们作为一维实小波基函数，结合2D复小波通用框架，可以用于构造方向性好、平滑且适合条纹分析的二维复小波，有效提高小波变换轮廓术的相位计算精度。通过理论分析和数字计算，详细说明了用一维高斯函数一阶、二阶和三阶导数分别构造的2D复小波空-频域特性，并将其应用于低信噪比和低对比度条纹分析以重建物面三维面形。以PMP结果作为参考值，平面测量时，所构造的三个2D高斯复小波三维面形重建误差的STD均小于0.07；非均匀反射物面测量时，采用构造小波的2D WTP解调存在低信噪条纹时，重建结果优于复墨西哥帽小波和传统Fan小波。在非暗场区域，重建误差的STD均小于0.05，有效提高了测量精度。研究工作丰富了2D WTP的小波基函数库，拓展了2D WTP在光学测量和条纹分析中的应用前景。更多还原

【Abstract】 Benefiting from the time-frequency analysis and multi-resolution characteristics of wavelet transform,wavelet transform profilometry(WTP)can accurately extract the phase map carrying the height information of the measured surface from a single frame of fringe pattern. The difficulty and accuracy of phase demodulation in wavelet transform profilometry depend on the choice of wavelet. Complex wavelets can be directly used for fringe phase demodulation. The nth-order derivatives of the one-dimensional Gaussian function satisfy the admissibility condition of wavelets,and the spectral distribution of each side-band exhibits smooth and asymmetric characteristics. These 1D functions,as 1D real wavelet bases,combined with the 2D complex wavelet framework,can be used to construct two-dimensional complex wavelets with good directionality and smoothness,suitable for fringe analysis,effectively improving the accuracy of phase calculation in wavelet transform profilometry. The spatial-frequency characteristics of 2D complex wavelets,constructed from the first,second,and third derivatives of a one-dimensional Gaussian function,are thoroughly examined through theoretical analysis and numerical calculations. These wavelets are then applied to the analysis of low signal-to-noise ratio and low-contrast fringes to reconstruct the 3D surface profile of the object. Using Phase Measuring Profilometry(PMP)results as a reference,the standard deviation(STD)of 3D surface reconstruction errors,based on the three 2D complex Gaussian wavelets developed in this study,remains consistently below 0.07 in planar measurements. For non-uniform reflective surfaces with low signal-to-noise ratio fringes,the 2D WTP demodulation method,employing the proposed wavelets,outperforms both the complex Mexican hat wavelet and the traditional Fan wavelet in terms of reconstruction accuracy. In non-dark-field regions,the reconstruction error STD is maintained below 0.05,effectively enhancing measurement precision. This research not only enriches the wavelet basis function library for 2D WTP but also expands its potential applications in optical measurement and fringe pattern analysis.更多还原