CN102003948A - High-precision optical stripe phase extraction method - Google Patents

Abstract

The invention discloses a high-precision optical stripe phase extraction method which comprises the following steps of: performing adaptive window Fourier transform on an optical stripe by using a formula; then, performing inverse Fourier transform on complete first-level frequency spectrum to obtain a first-level frequency spectrum light intensity component I1(x); and extracting the phase of each point of the optical stripe by using a formula. The phase extraction method of the invention has the characteristic that the telescopic factor a corresponding to each point in the optical stripe is adaptively changed. Compared with the traditional phase extraction technology, the invention has higher phase extraction precision.

Description

A high-precision optical fringe phase extraction method

technical field

The invention relates to the technical field of optical measurement, in particular to an optical fringe phase extraction method.

Background technique

Optical precision measurement using optical fringes (such as interference fringes, projected fringes) is a widely used technique. The automatic analysis of optical fringes is the key to this technology. Optical fringe phase extraction technology is a widely used automatic optical fringe analysis technology. The optical fringe phase extraction technology includes Fourier transform technology, phase shift technology and so on. Phase-shift technology requires multiple sinusoidally distributed fringes, which is generally not suitable for dynamic measurement, while Fourier transform technology only needs one fringe, which is especially suitable for dynamic measurement. In the Fourier transform technology, it is not strictly required that the optical fringes are sinusoidally distributed, but it is required to extract the first-order spectrum completely from the Fourier spectral space. If the first-order spectrum and the other-order spectrum are aliased, the Fourier transform technology cannot fully extract the first-order spectrum, which will lead to a large error in phase extraction. The reason for spectral aliasing is that the Fourier transform is a global transformation that cannot give the spatial (or time) domain information of the spectrum, and the high-level spectrum of a certain local area of the fringe may overlap with the low-level spectrum of another local area. . The windowed Fourier transform is proposed to make up for the poor locality of the Fourier transform. The windowed Fourier transform can separate the spectrum of different local areas to a certain extent, so as to improve the spectral aliasing problem of different local areas. Thereby improving the phase extraction precision. But the selection of the window scale is a key issue. Because of different window scales, the corresponding frequency resolution and spatial resolution are different. A wide window corresponds to high frequency resolution and low spatial resolution; conversely, a narrow window corresponds to low frequency resolution and high spatial resolution. In order to accurately extract the first-level spectrum, a higher frequency resolution should be ensured as much as possible; in order to separate the spectrum of different local areas so that they do not alias, a high spatial resolution is required. However, according to the Heisenberg uncertainty principle, the spatial resolution and the frequency resolution cannot reach the maximum value at the same time, so it is required to achieve a balance between the two: to improve the frequency resolution as much as possible without aliasing. It was once proposed that the window size is proportional to the local fringe period (or the reciprocal of the instantaneous frequency, which corresponds to the instantaneous frequency of the time-varying signal) [Weng Jiawen, Zhong Jingang, "Fourier transform of telescopic window in three-dimensional shape Application in Surface Measurement", Acta Optics Sinica, Vol.24, No.6, 725-729, 6/2004. Zheng Suzhen, Chen Wenjing, Su Xianyu, "Adaptive Window Fourier Transform 3D Surface Shape Detection Technology", Optoelectronic Engineering, Vol.32, No.9, 51-54, 2005. ], this window selection method, when the instantaneous frequency of the optical fringe is high, the window is narrow, resulting in low frequency resolution, which cannot guarantee the highest frequency resolution without aliasing, so it cannot Guaranteed to obtain higher phase extraction accuracy. We have also proposed to set the window size according to the instantaneous frequency gradient, which improves the above situation, but requires the use of an integral inequality and an empirical threshold to judge the size of the instantaneous frequency gradient. The empirical threshold brings great inconvenience to the actual operation [Jingang Zhong and Huiping Zeng, "Multiscale windowed Fourier transform for phase extraction of fringe patterns," Applied Optics, Vol.46, No.14, 2670-2675, 5/2007.].

After careful theoretical analysis, we proposed a new window setting method, which is relatively easy to implement in actual operation, and can improve the frequency resolution as much as possible without aliasing, and obtain high-precision phase extraction.

Contents of the invention

The purpose of the present invention is to propose a high-precision optical fringe phase extraction method.

对光学条纹进行自适应窗口傅里叶变换，其中I(x)为通过信号采集系统获得的光学条纹，x为光学条纹的坐标，

是窗口函数，a是伸缩因子，b是平移因子，f表示频率，j表示复数的虚部，exp(-j2πfx)＝e^-j2πfx；通过平移因子b控制窗口在光学条纹上逐点移动进行窗口傅里叶变换，每进行一次窗口傅里叶变换，就提取该次变换的一级频谱，将所有一级频谱相加，就得到光学条纹的完整一级频谱F(f₁)；再采用公式

对这完整一级频谱进行反傅里叶变换，利用公式

即可提取光学条纹各点的位相

其中Im[I₁(x)]为复函数I₁(x)的虚部，Re[I₁(x)]为I₁(x)的实部；其特征在于伸缩因子a的确定方法如下：首先提取光学条纹各点的瞬时频率f_inst(x)；根据瞬时频率f_inst(x)来确定光学条纹中的每一点x所对应的自适应窗口傅里叶变换的最大窗口宽度w_max(x)，最大窗口宽度w_max(x)按照以下规则来确定：在以x为中心的区间[x-Δx，x+Δx](Δx＞0)内的最大瞬时频率为

最小瞬时频率为

满足条件的Δx最大值为Δx_max，则w_max(x)＝2Δx_max；再根据最大窗口宽度w_max(x)确定伸缩因子a的大小。A high-precision optical fringe phase extraction method, using the formula

Perform adaptive window Fourier transform on the optical fringe, where I(x) is the optical fringe obtained by the signal acquisition system, x is the coordinate of the optical fringe,

Is the window function, a is the expansion factor, b is the translation factor, f represents the frequency, j represents the imaginary part of the complex number, exp(-j2πfx)=e ^-j2πfx ; the window is controlled by the translation factor b to move point by point on the optical fringe to carry out the window Fourier transform, each time a windowed Fourier transform is performed, the first-order spectrum of this transformation is extracted, and all the first-order spectra are added to obtain the complete first-order spectrum F(f ₁ ) of the optical fringe; then the formula

Perform an inverse Fourier transform on this complete first-order spectrum, using the formula

The phase of each point of the optical fringe can be extracted

Wherein Im[I ₁ (x)] is the imaginary part of the complex function I ₁ (x), Re[I ₁ (x)] is the real part of I ₁ (x); it is characterized in that the determination method of the expansion factor a is as follows: First extract the instantaneous frequency f _inst (x) of each point of the optical fringe; determine the maximum window width w _max (x) of the adaptive window Fourier transform corresponding to each point x in the optical fringe according to the instantaneous frequency f _inst (x) ), the maximum window width w _max (x) is determined according to the following rules: the maximum instantaneous frequency in the interval [x-Δx, x+Δx] (Δx>0) centered on x is

The minimum instantaneous frequency is

To meet the conditions The maximum value of Δx is Δx _max , then w _max (x)=2Δx _max ; then determine the scaling factor a according to the maximum window width w _max (x).

Further, the window function adopts the Gaussian function $G_a^b\left(x\right)=\frac{1}{a}g\left(\frac{x-b}{a}\right),$ $g\left(x\right)=\frac{1}{\sqrt{2\mathrm{\π}}}\exp (-\frac{x^2}{2}).$

Further, take the full width at 1/e ² of the maximum value of the Gaussian function as the width of the Gaussian window, and the scaling factor corresponding to each point x in the optical fringe

的Δx最大值

那么

In order to ensure that the first-order spectrum of the windowed Fourier transform does not overlap with the zero-order spectrum, it is best to set a minimum value of the maximum window width w _max (x). For Gaussian functions, the minimum value of w _max (x) is preferred use i.e. if the condition is met

Δx max

So

Further, the wavelet transform method is used to extract the instantaneous frequency of each point of the optical fringe. The wavelet transform method is a better method among the existing methods for extracting instantaneous frequency.

The theoretical basis of the present invention is as follows:

For any optical fringe, it can be expressed by the following formula:

为需要提取的位相。设I₁(x)为光学条纹的基傅里叶分量，有：Wherein, r(x) is a parameter representing contrast, A _n is a Fourier coefficient, and n is an integer,

is the phase to be extracted. Let I ₁ (x) be the base Fourier component of the optical fringe, we have:

Then the phase can be obtained by the following operation:

首先要获得I₁(x)。目前最常用的方法是先对(1)式表示的光学条纹进行傅里叶变换：Therefore, to extract the phase

First, I ₁ (x) must be obtained. At present, the most commonly used method is to perform Fourier transform on the optical fringe represented by formula (1):

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

Extract the first-order spectrum F(f ₁ ) from its Fourier spectrum, and then perform inverse Fourier transform on the first-order spectrum to get:

${\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; df</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The problem with this method is that if the primary spectrum F(f ₁ ) is aliased with other spectrums, the complete primary spectrum F(f ₁ ) cannot be extracted, thus limiting its application range.

The present invention adopts an adaptive window Fourier transform method, which can solve the above spectrum aliasing problem to a certain extent. The adaptive window Fourier transform is expressed as:

${\mathrm{<span\; class="notranslate">\; AWFT</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

是高斯函数，伸缩因子a(＞0)控制高斯函数的宽度，平移因子b在x轴上逐点移动，使高斯窗口在x轴上滑动。根据高斯函数的性质有：in,

is a Gaussian function, the scaling factor a (>0) controls the width of the Gaussian function, and the translation factor b moves point by point on the x-axis to make the Gaussian window slide on the x-axis. According to the properties of the Gaussian function:

${<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; db</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

Therefore the following relationship can be obtained:

${<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; AWFT</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; db</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdb</span>}$

$<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; db</span>}<span\; class="notranslate">\; \}</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fx</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}$

$<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>$

The above formula shows that the traditional Fourier transform spectrum can be obtained by superimposing the spectra of all window Fourier transforms. Therefore, the traditional Fourier transform primary spectrum can also be obtained by superimposing the primary spectrum of all window Fourier transforms:

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; AWFT</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; db</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Then according to (5) and (3) equations, the required phase can be obtained.

Then, how to determine the scaling factor a so that after each windowed Fourier transform, it is the key to ensure the separation of the first-order spectrum of the window Fourier transform from other spectrums, and at the same time, in order to accurately and completely extract the first-order spectrum , the frequency resolution of the windowed Fourier transform should be improved as much as possible. The present invention determines the size of expansion factor a according to this principle:

(a) As shown in Figure 1, in order to make the first-order spectrum in the window Fourier transform centered on x will not overlap with the second-order spectrum, the relationship must be satisfied:

(f ₁ ) _max &lt; (f ₂ ) _min . (10)

Since (f ₂ ) _min =2(f ₁ ) _min , formula (10) can be written as:

(f ₁ ) _max < 2(f ₁ ) _min . (11)

The above formula can also be expressed in the form of instantaneous frequency:

${<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; inst</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 2</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; mst</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

分别为以x为中心的区间[x-Δx，x+Δx](Δx＞0)内的最大瞬时频率、最小瞬时频率，满足条件(12)的Δx最大值为Δx_max，则自适应窗口傅里叶变换高斯窗口的宽度为：in

are respectively the maximum instantaneous frequency and the minimum instantaneous frequency in the interval [x-Δx, x+Δx] (Δx>0) centered on x, and the maximum value of Δx satisfying the condition (12) is Δx _max , then the adaptive window Fu The width of the Lie transform Gaussian window is:

w _max (x)=2Δx _max . (13)

Take the full width at 1/e ² of the maximum value of the Gaussian function as the Gaussian window, then:

w _max (x) = 4a, (14)

So,

$\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

(b) As shown in Figure 1, in order to make the first-order spectrum in the window Fourier transform centered on x will not overlap with the zero-order spectrum, the relationship must be satisfied:

f _b <(f ₁ ) _min , (16)

Expressed in the form of instantaneous frequency:

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; <</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; inst</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

For Gaussian functions, there are:

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&pi;a</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

So: $a>\frac{1}{\sqrt{2}\mathrm{\&pi;}{\left(f_{\mathrm{inst}}^x\right)}_{\min }},---\left(19\right)$

那么伸缩因子a的最小值必须满足条件(19)，才能保证一级频谱不会和零级频谱重叠。That is, the minimum instantaneous frequency within the interval [x-Δx _max , x+Δx _max ] centered at x is

Then the minimum value of the scaling factor a must satisfy the condition (19), so as to ensure that the first-order spectrum will not overlap with the zero-order spectrum.

Combining formulas (15) and (19), we have:

According to formula (14), the above formula can also be written as follows:

Because the present invention adopts the self-adaptive expansion factor a, the first-order spectrum after the window Fourier transform of each point in the image is not mixed with the zero-order spectrum and the second-order spectrum to the greatest extent, so it is different from the existing Fourier Compared with phase extraction techniques such as phase extraction techniques such as Fourier transformation with fixed window, wavelet transformation, etc., the present invention has higher phase extraction precision.

Description of drawings

Figure 1 is a schematic diagram of a windowed Fourier transform spectrum.

Figure 2 is a one-dimensional optical fringe diagram.

Fig. 3 is a graph of instantaneous frequency distribution of optical fringes.

Figure 4 is a distribution diagram of the maximum window width.

Fig. 5 is a distribution diagram of Gaussian function scaling factor.

Fig. 6 is a phase diagram obtained by the phase extraction method of the present invention.

Fig. 7 is an optical fringe diagram of the structured light projection of the embodiment.

FIG. 8 is a phase diagram of optical fringe extraction of structured light projection by the phase extraction method of the present invention.

Fig. 9 is the contour of lips reconstructed from the phase map extracted by the phase extraction method of the present invention.

Fig. 10 is a phase diagram of the extraction of structured light projection optical fringes by the traditional Fourier transform phase extraction method.

Fig. 11 is the lip contour reconstructed from the phase map extracted by the traditional Fourier transform phase extraction method.

Fig. 12 is a phase diagram of extraction of optical fringes of structured light projection by wavelet transform phase extraction method.

Figure 13 is the lip contour reconstructed from the phase map extracted by wavelet transform phase extraction method.

Detailed ways

The present invention will be described in further detail below in conjunction with the accompanying drawings.

Figure 2 shows the one-dimensional optical fringe pattern that requires phase extraction. The optical fringe obtained by the signal acquisition device can be expressed as a discretized form I(m), where m is an integer greater than or equal to zero, that is, 0≤m≤M . Applying the method of the present invention to carry out phase extraction to the optical fringes shown in Figure 2, the specific steps are as follows:

(1) Firstly, the instantaneous frequency distribution f _inst (m) of the optical fringe is obtained by using the existing technology, such as the wavelet transform method, and the unit of the frequency is Hz, as shown in FIG. 3 .

的Δ最大值Δ_max。其中Δ为大于或等于零的整数；

为区间[m-Δ，m+Δ]内瞬时频率的最大值，表示为

为区间[m-Δ，m+Δ]内瞬时频率的最小值，表示为

得到以点m为中心的高斯窗口函数的最大窗口宽度：(2) Determine that the condition is satisfied within the interval [m-Δ, m+Δ] centered on point m

The maximum value of Δ Δ _max . Where Δ is an integer greater than or equal to zero;

is the maximum value of the instantaneous frequency in the interval [m-Δ, m+Δ], expressed as

is the minimum value of the instantaneous frequency in the interval [m-Δ, m+Δ], expressed as

Get the maximum window width of the Gaussian window function centered at point m:

如图4所示。

As shown in Figure 4.

(3) according to the relation w _max (m)=4a (m) of Gaussian window width and expansion factor a, determine point by point the size of the expansion factor a (m) of the Gaussian function centered on point m:

As shown in Figure 5.

(4) Make the Gaussian window move point by point on the optical fringe to perform discrete adaptive window Fourier transform:

${\mathrm{<span\; class="notranslate">\; AWFT</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="j"\; filenames="HSA00000282968400011.png,HSA00000282968400031.png"\; state="\{\{state\}\}">j</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;fm</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

k为大于或等于零的整数且0≤k≤M，这里的k与上述的b的意义相同。每进行一次窗口傅里叶变换，提取其一级频谱将所有的一级频谱相加即得到全局傅里叶变换的一级频谱

in,

k is an integer greater than or equal to zero and 0≤k≤M, where k has the same meaning as b above. Every time a windowed Fourier transform is performed, its primary spectrum is extracted Add all the first-order spectrum to get the first-order spectrum of the global Fourier transform

(5) Perform inverse Fourier transform on F(f ₁ ) to get The phase is then extracted by the following formula: As shown in Figure 6.

Example

The method of the invention is used to extract the phase of the optical fringes obtained in the structured light projection profilometry, thereby realizing the surface profile measurement. FIG. 7 is an optical fringe diagram of structured light projection, which is an optical fringe diagram obtained by projecting structured light on a human face. Fig. 8 is a phase diagram extracted from the optical fringe pattern of structured light projection in the area in the middle box of Fig. 7 by the phase extraction method of the present invention. Fig. 9 is the lip contour reconstructed based on Fig. 8 . Fig. 10 is a phase diagram extracted by the traditional Fourier transform phase extraction method for the optical fringe pattern extracted from the structured light projection of the area in the middle box of Fig. 7 . Fig. 11 is the lip contour reconstructed based on Fig. 10 . Fig. 12 is the phase diagram extracted from the optical fringe pattern of the structured light projection in the middle box of Fig. 7 by the wavelet transform phase extraction method. Fig. 13 is the lip contour reconstructed based on Fig. 12 . Judging from the reconstructed lip contour effects in Fig. 9, Fig. 11 and Fig. 13, the effect in Fig. 9 is obviously better than that in Fig. 11 and Fig. 13. It can be seen that the phase extraction method of the present invention has higher phase extraction accuracy.

Claims (5)

1. A high-precision optical stripe phase extraction method adopts a formula

Performing an adaptive windowed Fourier transform on the optical fringes, wherein I (x) is the optical fringes obtained by the signal acquisition system, x is the coordinates of the optical fringes,

is a window function, a is a scaling factor, b is a translation factorSub, f denotes frequency, j denotes imaginary part of complex number; controlling the window to move point by point on the optical stripe by a translation factor b to perform window Fourier transform, extracting the primary frequency spectrum of the transform every time the window Fourier transform is performed, and adding all the primary frequency spectrums to obtain a complete primary frequency spectrum F (F) of the optical stripe₁) (ii) a Then using the formulaPerforming inverse Fourier transform on the complete primary spectrum by using formula

The phase of each point of the optical stripe can be extracted

Wherein Im [ I₁(x)]Is a complex function I₁(x) Imaginary part of, Re [ I ]₁(x)]Is I₁(x) The real part of (a); the method for determining the scaling factor a is characterized by comprising the following steps: firstly, extracting the instantaneous frequency f of each point of the optical stripe_inst(x) (ii) a According to instantaneous frequency f_inst(x) To determine the maximum window width w of the adaptive window Fourier transform corresponding to each point x in the optical fringe_max(x) Maximum window width w_max(x) Determined according to the following rules: in the interval [ x- Δ x, x + Δ x ] centered at x](Δ x > 0) a maximum instantaneous frequency of

Minimum instantaneous frequency of

Satisfies the conditions

Δ x of (a) is maximum Δ x_maxThen w is_max(x)＝2Δx_max(ii) a Then according to the maximum window width w_max(x) The magnitude of the scaling factor a is determined.

2. The method according to claim 1, wherein the phase extraction method comprises: the window function adopts a Gaussian function

3. The method according to claim 1 or 2, wherein the phase extraction method comprises: the scale factor corresponding to each point x in the optical stripe

4. The method according to claim 3, wherein the phase extraction method comprises: determining a maximum window width w_max(x) When, if

Then

5. The method according to claim 1, wherein the phase extraction method comprises: and extracting the instantaneous frequency of each point of the optical stripe by adopting a wavelet transform method.

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