楊初平,劉建斌,譚穗妍,翁嘉文
(華南農(nóng)業(yè)大學(xué)物理系,廣州510642)
應(yīng)用頻率積分相位解調(diào)測(cè)量徑向畸變
楊初平,劉建斌,譚穗妍,翁嘉文
(華南農(nóng)業(yè)大學(xué)物理系,廣州510642)
為了測(cè)量光學(xué)成像系統(tǒng)的徑向畸變,采用載頻條紋模板,應(yīng)用瞬時(shí)頻率積分法提取因徑向畸變而產(chǎn)生的徑向調(diào)制相位;推導(dǎo)了條紋徑向調(diào)制相位與瞬時(shí)頻率的關(guān)系式,并導(dǎo)出徑向調(diào)制相位和徑向畸變位移關(guān)系;采用小波頻率估計(jì)提取畸變條紋徑向瞬時(shí)頻率,并對(duì)其進(jìn)行積分獲得畸變條紋的徑向調(diào)制相位;應(yīng)用徑向調(diào)制相位和立方卷積插值算法對(duì)畸變圖像進(jìn)行了校正,得出了詳細(xì)的理論分析和實(shí)驗(yàn)結(jié)果。結(jié)果表明,上述方法是可行的。
測(cè)量與計(jì)量;畸變測(cè)量;載頻條紋;相位解調(diào);徑向畸變
畸變普遍存在于光學(xué)成像系統(tǒng)中。徑向畸變測(cè)量就是獲得徑向畸變位移函數(shù)即畸變像點(diǎn)相對(duì)于無畸變像點(diǎn)的位移,校正畸變圖像。通過設(shè)定特征點(diǎn)、提取特征點(diǎn)的畸變位移的標(biāo)定模板[1-7]是目前測(cè)量徑向畸變的主要方法。如果把這些特征點(diǎn)的灰度按正弦函數(shù)分布排列,則測(cè)量因徑向畸變導(dǎo)致的特征點(diǎn)的位移將轉(zhuǎn)化成測(cè)量灰度條紋徑向調(diào)制相位。本文中以縱向正弦載頻灰度條紋作為模板,把徑向位置畸變轉(zhuǎn)化為徑向調(diào)制相位,采用小波頻率估計(jì)提取畸變條紋徑向瞬時(shí)頻率并進(jìn)行積分運(yùn)算得到徑向調(diào)制相位。與目前提取條紋調(diào)制相位都需要進(jìn)行相位解包[8-14]不同,采用瞬時(shí)頻率積分相位解調(diào)方法提取變形條紋調(diào)制相位方法沒有產(chǎn)生包裹相位,因而無需經(jīng)相位解包過程,便可直接得到調(diào)制相位。然后換算為徑向位置畸變,實(shí)現(xiàn)對(duì)圖像的校正。
畸變圖像建立如圖1所示的坐標(biāo)系。徑向畸變相對(duì)于圖像中心點(diǎn)O具有旋轉(zhuǎn)對(duì)稱性,在同一圓周上的點(diǎn)具有相等的位置畸變。設(shè)(x0,y0),(x,y),(xd,yd)分別表示圖像中心點(diǎn)、理想像點(diǎn)和對(duì)應(yīng)的畸
Fig.1 Diagram illustrating radial distortion of fringe patterns
式中,徑向畸變位移函數(shù)T(r)是描述畸變像點(diǎn)相對(duì)于無畸變像點(diǎn)的位移,表示為:
通過實(shí)驗(yàn)得到Δr,代入(1)式并結(jié)合灰度插值得到校正圖像。
2.1 徑向調(diào)制相位與徑向瞬時(shí)頻率
以條形正弦灰度模板進(jìn)行成像,采集的畸變條紋和無畸變條紋的關(guān)系如圖1所示,其中徑向OAB經(jīng)過圖像中心O點(diǎn)。采集的無畸變條紋像的灰度可表示如下:
式中,A1(x)是背景灰度,B1(x)是正弦灰度的幅值,相位分布是(2πf0x+φ),f0是頻率,φ是初始相位。
一方面,由于畸變,采集的條形模板灰度不再是正弦分布,可表示為:
式中,相位分布是[2πf0x+φ(x)+φ],不再是周期性分布,而是出現(xiàn)沿著徑向發(fā)生變化的調(diào)制相位函數(shù)φ(x)?;儣l紋像與無畸變條紋像比較,因徑向畸變產(chǎn)生的相位差即徑向調(diào)制相位函數(shù)為:
另一方面,徑向畸變表現(xiàn)為畸變像點(diǎn)相對(duì)于無畸變像點(diǎn)的位置移動(dòng),如圖1所示,徑向畸變使B點(diǎn)移到A點(diǎn),產(chǎn)生的徑向畸變位移函數(shù)表示為Δr(x)。則徑向畸變相位函數(shù)與徑向畸變位移函數(shù)的關(guān)系為:
因此,只要得到徑向調(diào)制相位就可以計(jì)算徑向畸變位移分布。
畸變條形模板灰度分布不再是固定的頻率f0,而是隨著徑向發(fā)生變化?;儣l形徑向瞬時(shí)頻率定義為:
把f0移到等號(hào)左邊,兩邊積分得:φ(x)-φ(0)=在畸變像的中心點(diǎn),畸變?yōu)?,則瞬時(shí)頻率f(0)=f0,而且φ(0)=0。因此徑向調(diào)制相位函數(shù)為:
因此,若得到瞬時(shí)頻率分布f(x),則可以計(jì)算徑向各個(gè)畸變點(diǎn)的徑向調(diào)制相位。在以往的條紋調(diào)制相位解調(diào)過程中,由于采用反正切函數(shù)計(jì)算條紋每個(gè)點(diǎn)的調(diào)制相位,大于2π的調(diào)制相位將被折斷到(-π,π)范圍內(nèi),為了獲得真實(shí)調(diào)制相位,需要對(duì)解調(diào)相位進(jìn)行解包處理。采用瞬時(shí)頻率積分不會(huì)出現(xiàn)相位折斷,因而無需相位解包過程就能夠直接計(jì)算真實(shí)的調(diào)制相位。
2.2 頻率估計(jì)
條紋上各點(diǎn)瞬時(shí)頻率的計(jì)算可以采用連續(xù)小波變換進(jìn)行估計(jì)。對(duì)1維函數(shù)f(x),其連續(xù)小波變換系數(shù)Wf(a,b)定義為[11-13]:
母函數(shù)ψ(x)采用Gabor小波函數(shù)[12]:
式中,ψa,b(x)為母函數(shù)經(jīng)過尺度因子a伸縮和平移因子b平移后得到的小波序列,ψa,b*(x)是ψa,b(x)的共軛函數(shù)。對(duì)實(shí)信號(hào)f(x)的各個(gè)b點(diǎn)采用不同a值進(jìn)行連續(xù)小波變換,得到一系列的小波系數(shù)Wf(a,b)。小波系數(shù)的幅值A(chǔ)(a,b)和相位φ(a,b)可以通過Wf(a,b)的實(shí)部Re[Wf(a,b)]和虛部Im[Wf(a,b)]分別表示為[12]:
對(duì)信號(hào)各個(gè)位置b,通過檢測(cè)不同a值的小波變換系數(shù)幅值A(chǔ)(a,b),其中最大值所對(duì)應(yīng)的尺度因子a就是最佳尺度,記為a(b),信號(hào)在b點(diǎn)的瞬時(shí)頻率可以估計(jì)為f(b)=1/a(b)。把變形條紋I2(x,y)過中心點(diǎn)O沿OAB方向的灰度分布作為f(x),按照上述過程可以提取畸變條紋徑向OAB的瞬時(shí)頻率分布f(x)=1/a(x),經(jīng)過瞬時(shí)頻率積分得到徑向調(diào)制相位函數(shù)。
調(diào)整攝像機(jī)光軸垂直參考平面,然后采集圖像。圖2是成像系統(tǒng)采集的變形條紋(640pixel× 480pixel),中心點(diǎn)O即零畸變點(diǎn)位于(320,240)。圖3是OAB方向的瞬時(shí)頻率分布,以O(shè)為中心呈左右對(duì)稱。圖4是對(duì)O點(diǎn)右側(cè)瞬時(shí)頻率積分得到徑向調(diào)制相位的原始曲線和多項(xiàng)式擬合曲線。按照(4)式把圖4中的徑向調(diào)制相位轉(zhuǎn)化為徑向畸變位移,采用立方卷積插值法[15]對(duì)畸變圖像(見圖5)進(jìn)行校正,校正圖像如圖6所示。
Fig.2 Distorted fringe pattern
Fig.3 Local frequency along OAB
Fig.4 Modulated phase along OAB
Fig.5 Distorted image
Fig.6 Calibrated image
為了測(cè)量成像系統(tǒng)的徑向畸變,以正弦灰度條紋模板作為成像對(duì)象,把徑向位置畸變轉(zhuǎn)化為變形條紋徑向調(diào)制相位;該方法只需要采集1幅畸變條紋圖像,該變形條紋不僅含有徑向畸變信息,而且在畸變條紋中心點(diǎn)含有無畸變條紋像信息,采用小波頻率估計(jì)提取徑向頻率并進(jìn)行瞬時(shí)頻率積分計(jì)算徑向調(diào)制相位,實(shí)現(xiàn)畸變校正。
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M easurement of radial distortion based on frequency integration phase demodulation
YANG Chuping,LIU Jianbin,TAN Suiyan,WENG Jiawen
(Department of Physics,South China Agricultural University,Guangzhou 510642,China)
Tomeasure the radial distortion in optical imaging systems,adopting the straight sinusoidal carrier-fringe pattern,the radial modulated phase resulting from radial distortion was extracted by means of instantaneous frequency integration.The formula between the instantaneous frequency and the radial modulated phase was deduced,and the conversion formula between the radial modulated phase and the radial distortion displacement was obtained.The instantaneous frequency of the distorted fringe pattern in the radial direction was calculated by using frequency estimation of Gabor wavelet transform,and integrated to obtain the radialmodulated phase.The radialmodulated phase and the cubic convolution interpolation algorithm were used to calibrate the distorted image.Experimental results demonstrate that the method is available.
measurement and metrology;distortion measurement;carrier frequency fringe pattern;phase demodulation;radial distortion
O438
A
10.7510/jgjs.issn.1001-3806.2014.03.026
1001-3806(2014)03-0402-04
國(guó)家自然科學(xué)基金資助項(xiàng)目(61307011);廣東省自然科學(xué)基金資助項(xiàng)目(9151064201000035)
楊初平(1970-),男,副教授,研究方向?yàn)楣庑畔⑻幚怼?/p>
E-mail:yangchp@scau.edu.cn
2013-07-09;
2013-10-08