对应用于自适应光学的角膜像差重建算法的效能分析
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摘要
目的:分别使用构造正则方程、奇异值分解、Householder变换三种解法解决自适应光学中角膜像差重建中线性模型的最小二乘解问题,比较重建精度、耗时以及可靠性,为应用于实时矫正眼球像差的自适应光学系统中波前重建最优算法的选择提供参考及理论依据。方法:使用Pentacam角膜地形图仪获取20眼正常眼的角膜地形图前表面高度数据。数据范围为中央6mm直径的瞳孔区,根据Pentacam提供的参数生成最佳拟合球面(best fit sphere,BFS),二者之差即为模拟的角膜前表面的像差数据。数据采样率为100μm(仪器设置),编制采样程序对其分别使用100、300、500tμm的分辨率进行采样,对每个波前像差数据矩阵计算梯度,求得x方向及y方向的一阶偏导,此即为波前像差的斜率数据(即为模拟夏克-哈特曼传感器采样)。获得波前斜率数据后使用zernike多项式(分别使用1~130个模式项数)重建波前数据,对于上述三种采样率,都分别使用构造正则方程法、奇异值分解与Householder变换三种算法获得重建波面的模式系数。根据获得的模式系数获得重建波面,使用其与原始波面的差值的残差均方根(root mean square,RMS)考察重建精度,同时使用MATLAB函数tic、tio考察重建耗时,并计算线性模型的条件数、观察模式系数矩阵解的合理性以考察算法的可靠性。结果:重建精度:三种采样率时,Householder变换与奇异值分解的表现是一致的,而构造正则方程法则在分辨率为500μm时表现较差,主要为高阶(zernike模式数K>88)时RMS值明显的不稳定(表现为无规律的大幅度的波动)。重建耗时:三种采样率时,随着阶数的增加,奇异值分解的耗时增加明显高于Householder变换与构造正则方程法,而另二者的差异性表现得并不直观。算法可靠性:采样率越高,使用的zernike多项式的阶数越低,结果越可靠。同时构造正则方程法在采样率较低时相比较于另二者的计算稳定性明显较差。结论:Householder变换在精确以及高效两方面优于另两者,且当采样率较高时,三种解法的可靠性几乎相同。而当采样率较低时,Householder变换仍有相对较稳定的表现,此结果为应用于眼科医学的自适应光学系统中波前重建最优算法的选择提供了参考及理论依据。
AIM:To investigate the accuracy,time-consuming and reliability of three Zernike algorithms in reconstruction of the corneal aberration in adaptive optics.METHODS:Elevation data of corneal front surface collected on 20 normal eyes over a 6mm pupil were converted into the simulated original wave-front data by subtracting the best fitting sphere,which was then resampled at resolutions of 100,300,and 500μm.Differences in elevation between adjacent pixels were used to generate simulated wave-front slope data,which were used to reconstruct wave-front by three algorithms:the regularized solution,the singular value decomposition,and Householder transform separately.The number of Zernike modes was from 1 to 130 separately in each reconstruction procedure.Each new wave-front map generated was directly compared to the originally sampled wave-front and the residual root-mean-square(RMS) error between the original and reconstructed map was recorded,also we investigate the time-consuming and reliability of the solution by calculating the condition numbers of the linear model and observing the mode coefficient matrix.RESULTS:Householder transformation performed as well as the singular value decomposition by three sampling rates in reconstruction accuracy,while the regularized solution showed unacceptable results when the number Zernike modes used higher than 88 by the resolution was 500μm.With the modes number increased,the time that the singular value decomposition consumed increased more obviously than the time that the Householder transformation and the regularized solution consumed,and the difference between the latter two didn't show obviously.The higher the sampling rate was,also the lower the Zernike exponent number was,and the more reliable the result was,and the instability of regularized solution is more serious than the other two at the low sampling rate.CONCLUSION:Householder transformation is superior the other two in accuracy as well as the highly effectiveness,and the reliability of three algorithms was almost identical at high sampling rate,while the Householder transformation still showed relatively stable performance at low sampling rate,which provides the reference and the theory basis of choice to the optimal algorithm which is applied in the adaptive optics system of real-time correction eyeball's aberration wave-front reconstruction.
AIM:To investigate the accuracy,time-consuming and reliability of three Zernike algorithms in reconstruction of the corneal aberration in adaptive optics.METHODS:Elevation data of corneal front surface collected on 20 normal eyes over a 6mm pupil were converted into the simulated original wave-front data by subtracting the best fitting sphere,which was then resampled at resolutions of 100,300,and 500μm.Differences in elevation between adjacent pixels were used to generate simulated wave-front slope data,which were used to reconstruct wave-front by three algorithms:the regularized solution,the singular value decomposition,and Householder transform separately.The number of Zernike modes was from 1 to 130 separately in each reconstruction procedure.Each new wave-front map generated was directly compared to the originally sampled wave-front and the residual root-mean-square(RMS) error between the original and reconstructed map was recorded,also we investigate the time-consuming and reliability of the solution by calculating the condition numbers of the linear model and observing the mode coefficient matrix.RESULTS:Householder transformation performed as well as the singular value decomposition by three sampling rates in reconstruction accuracy,while the regularized solution showed unacceptable results when the number Zernike modes used higher than 88 by the resolution was 500μm.With the modes number increased,the time that the singular value decomposition consumed increased more obviously than the time that the Householder transformation and the regularized solution consumed,and the difference between the latter two didn't show obviously.The higher the sampling rate was,also the lower the Zernike exponent number was,and the more reliable the result was,and the instability of regularized solution is more serious than the other two at the low sampling rate.CONCLUSION:Householder transformation is superior the other two in accuracy as well as the highly effectiveness,and the reliability of three algorithms was almost identical at high sampling rate,while the Householder transformation still showed relatively stable performance at low sampling rate,which provides the reference and the theory basis of choice to the optimal algorithm which is applied in the adaptive optics system of real-time correction eyeball's aberration wave-front reconstruction.
引文
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3 Wahl DJ,Jian Y,Bonora S,et al.Wavefront sensorless adaptive optics fluorescence biomicroscope for in vivo retinal imaging in mice.Biomed Opt Express 2015;7(1):1-12
4 Zhou X,Bedggood P,Bui B,et al.Contrast-based sensor-less adaptive optics for retinal imaging.Biomed Opt Express 2015;6(9):3577-3595
5 Ai D,Yang J,Fan J,et al.Adaptive Tensor-Based Principal Component Analysis for Low-Dose CT Image Denoising.PLoS One 2015;10(5):e0126914
6 Bonora S,Jian Y,Zhang P,et al.Wavefront correction and highresolution in vivo OCT imaging with an objective integrated multi-actuator adaptive lens.Opt Express 2015;23(17):21931-21941
7 Zawadzki RJ,Zhang P,Zam A,et al.Adaptive-optics SLO imaging combined with widefield OCT and SLO enables precise 3D localization of fluorescent cells in the mouse retina.Biomed Opt Express 2015;6(6):2191-2210
8 Jin P,Li X.Correction of image drift and distortion in a scanning electron microscopy.J Microsc 2015;260(3):268-280
9 Jahnke M,Wirbelauer C,Pham DT.Influence of age on optical aberrations of the human eye.Ophthalmologe 2006;103(7):596-604
10 Wang JM,Liu CL,Luo YN,et al.Statistical virtual eye model based on wavefront aberration.Int J Ophthalmol 2012;5(5):620-624
11 Porter J,Guirao A,Cox IG,et al.Monochromatic aberrations of the human eye in a large population.J Opt Soc Am A 2001;18(8):1793-1803
12 Smolek MK,Klyce SD.Zernike polynomial fitting fails to represent all visually significant corneal aberrations.Invest Ophthalmol Vis Sci 2003;44(11):4676-4681
13 Klyce SD,Karon MD,Smolek MK.Advantages and disadvantages of the Zernike expansion for representing wave aberration of the normal and aberrated eye.J Refract Surg 2004;20(5):537-541
14 Dai GM.Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms.J Refract Surg 2006;22(9):943-948
15 Yoon G,Pantanelli S,MacRae S.Comparison of Zernike and Fourier wavefront reconstruction algorithms in representing corneal aberration of normal and abnormal eyes.J Refract Surg 2008;24(6);582-590
16 Cao ZL,Liao WH,Shen JX.A new algorithm for human eye's wavefront aberration fitting with Zernike polynomial.Optics and Precision Engineering 2006;14(2):308-314
17 Gao WW,Shen JX,Li BM,et al.Comparative Study on Algorithms for Wave-Front AberrationReconstruction of Human Eye.Guang Pu Xue Yu Guang Pu Fen Xi 2010;30(8):2232-2235
18 Wang P,Menon R.Computational spectroscopy via singularvalue decomposition and regularization.Opt Express 2014;22(18):21541-21550
19 Huang G,Chen J,Benesty J.Direction-of-arrival estimation of passive acoustic sources in reverberant environments based on the Householder transformation.J Acoust Soc Am 2015;138(5):3053-3060
20 Kushida N.Correction;Condition Number Estimation of Preconditioned Matrices.PLoS One 2015;10(6):e0130920
21 Chen C,Chen X,Gao M,et al.Contextual influence on the tilt after-effect in foveal and para-foveal vision.Neurosci Bull 2015;31(3):307-316
2 Bonora S,Zawadzki RJ.Wavefront sensorless modal deformable mirror correction in adaptive optics;optical coherence tomography.Opt Lett2013;38(22):4801-4804
3 Wahl DJ,Jian Y,Bonora S,et al.Wavefront sensorless adaptive optics fluorescence biomicroscope for in vivo retinal imaging in mice.Biomed Opt Express 2015;7(1):1-12
4 Zhou X,Bedggood P,Bui B,et al.Contrast-based sensor-less adaptive optics for retinal imaging.Biomed Opt Express 2015;6(9):3577-3595
5 Ai D,Yang J,Fan J,et al.Adaptive Tensor-Based Principal Component Analysis for Low-Dose CT Image Denoising.PLoS One 2015;10(5):e0126914
6 Bonora S,Jian Y,Zhang P,et al.Wavefront correction and highresolution in vivo OCT imaging with an objective integrated multi-actuator adaptive lens.Opt Express 2015;23(17):21931-21941
7 Zawadzki RJ,Zhang P,Zam A,et al.Adaptive-optics SLO imaging combined with widefield OCT and SLO enables precise 3D localization of fluorescent cells in the mouse retina.Biomed Opt Express 2015;6(6):2191-2210
8 Jin P,Li X.Correction of image drift and distortion in a scanning electron microscopy.J Microsc 2015;260(3):268-280
9 Jahnke M,Wirbelauer C,Pham DT.Influence of age on optical aberrations of the human eye.Ophthalmologe 2006;103(7):596-604
10 Wang JM,Liu CL,Luo YN,et al.Statistical virtual eye model based on wavefront aberration.Int J Ophthalmol 2012;5(5):620-624
11 Porter J,Guirao A,Cox IG,et al.Monochromatic aberrations of the human eye in a large population.J Opt Soc Am A 2001;18(8):1793-1803
12 Smolek MK,Klyce SD.Zernike polynomial fitting fails to represent all visually significant corneal aberrations.Invest Ophthalmol Vis Sci 2003;44(11):4676-4681
13 Klyce SD,Karon MD,Smolek MK.Advantages and disadvantages of the Zernike expansion for representing wave aberration of the normal and aberrated eye.J Refract Surg 2004;20(5):537-541
14 Dai GM.Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms.J Refract Surg 2006;22(9):943-948
15 Yoon G,Pantanelli S,MacRae S.Comparison of Zernike and Fourier wavefront reconstruction algorithms in representing corneal aberration of normal and abnormal eyes.J Refract Surg 2008;24(6);582-590
16 Cao ZL,Liao WH,Shen JX.A new algorithm for human eye's wavefront aberration fitting with Zernike polynomial.Optics and Precision Engineering 2006;14(2):308-314
17 Gao WW,Shen JX,Li BM,et al.Comparative Study on Algorithms for Wave-Front AberrationReconstruction of Human Eye.Guang Pu Xue Yu Guang Pu Fen Xi 2010;30(8):2232-2235
18 Wang P,Menon R.Computational spectroscopy via singularvalue decomposition and regularization.Opt Express 2014;22(18):21541-21550
19 Huang G,Chen J,Benesty J.Direction-of-arrival estimation of passive acoustic sources in reverberant environments based on the Householder transformation.J Acoust Soc Am 2015;138(5):3053-3060
20 Kushida N.Correction;Condition Number Estimation of Preconditioned Matrices.PLoS One 2015;10(6):e0130920
21 Chen C,Chen X,Gao M,et al.Contextual influence on the tilt after-effect in foveal and para-foveal vision.Neurosci Bull 2015;31(3):307-316