小波在光學系統上之應用

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姓名

許金益(Jin-Yi Sheu) 查詢紙本館藏

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光電科學與工程學系

論文名稱

小波在光學系統上之應用
(The Applications of Optical System by Wavelet Transformation Method)

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摘要(中)

一般光學系統的賽德像差可以用澤尼克（Zernike）多項式展開，因為我們所測量到的波面均會出現雜訊，所以利用傳統的最小平方差方法來求得澤尼克係數會產生所謂的數值不穩定（誤差），在此我們提出一個新方法可以解決以上所提到的問題。首先將干涉實驗所得之條紋算出其波面，再將此波面作小波轉換，利用這組新的數據來求賽德像差的大小。從模擬的結果可得知此種新方法所得到的數據比傳統的方法要精確很多，而且所需要的計算時間大約少了十多倍。
另外我們也利用了離散小波轉換來提高影像的對比度。首先我們從螢光顯微鏡取出兩張不同波長的影像，利用影像融合的方法使圖像的對比度提高，藉此可以分辨生物晶片上的不同圖像。
最後用疊紋影像及小波轉換的方法來做微米距離的量測。此方法是藉由光柵及CCD相機之像素的重疊產生影像，然後利用小波轉換來計算兩疊紋間之寬度，並藉由此寬度算出光柵所移動之微小距離，這種測量方法所需之實驗設備簡單又經濟，而且所得到的實驗結果有很高的準確度。

摘要(英)

Seidel aberration coefficients can be expressed by Zernike coefficients. The least-squares matrix inversion method of determining the Zernike coefficients from a sampled wave front with measurement noise has been found numerically unstable. We present a new method to estimate the Seidel aberration coefficients by using a two-dimensional discrete wavelet transform and a technique (wavelet transform) for determining the spherical aberration and defocusing of a rotationally symmetric optical system. Compared with the least-squares matrix inversion method, their performance are more stable under input of Gaussian white noise and we obtain not only the more accurate Seidel aberration coefficients but also speed the computation. The simulated wave fronts are fitted, and results are shown for spherical aberration, coma, astigmatism, and defocus.
Furthermore, We introduce a contrast and aberration correct image fusion method with the discrete wavelet transform to identify the micro-array biochip. The image fusion method is applied to fuse two images from different microscopes. The results show that the fused image can get better analysis of the details at each original micro spot biochip. Finally, A new approach based on the moiré theory and wavelet transform is proposed for measuring the micro-range distance between a charge-couple-device (CCD) camera and a two-dimensional reference grating. The micro-range distance is determined by measuring the pitch of the moiré pattern image, which is digitized by a CCD camera. A one-dimensional wavelet transform algorithm is applied to estimate the pitch of the moiré pattern. Experimental results prove that this technique is very efficient and highly accuracy, this method evaluates the micro-range distance with a suitable filter (suitable dilation factor ) to obtain a unique value of the average pitch of the moiré image. It is therefore immune to the noise and able to estimate the micro-range distance accurately.

關鍵字(中)

★ 小波轉換
★ 賽德像差
★ 影像融合

關鍵字(英)

★ wavelet transform
★ seidel aberration
★ image fusi

論文目次

Contents
Abstract
Contents i
Table Captions ii
Figure Captions iii
Glossary of Notation vi
1. Introduction 1
1.1 The history of wavelet transform 1
1.2 The studied motivation and studied purpose 2
1.3 The applications of wavelet transform 3
1.3.1 Analysis of Seidel aberration of optical system 3
1.3.2 Image fusion 4
1.3.3 Micro-range measurements 5
References 6
2. Theory 10
2.1 The window Fourier Transform 10
2.2 Wavelet Transform 12
2.3 Discrete Wavelet Transform 15
2.4 Multiresolution analysis 18
2.5 Two-dimensional wavelet decomposition algorithm 20
References 23
3. Analysis of Seidel aberration by use of discrete wavelet transform 24
3.1 Seidel aberration coefficients computed with the Zernike polynomials 25
3.2 Seidel aberration coefficients computed by the discrete wavelet Transform 28
3.3 Computer simulation 31
3.4 Conclusion 38
References 39
4. Analysis of Wave-Aberration by Use of the Wavelet Transform 41
4.1 Computed aberration coefficients by the least-squares method 42
4.2 Computed aberration coefficients by the wavelet transform 45
4.3 Computer simulation 49
4.4 Conclusion 55
References 56
5. The new image fusion method applied in two wavelengths detection of Biochip spot 58
5.1 Correct the aberration by software 59
5.2 Image fusion 60
5.3 Experiment 64
5.4 Result 66
5.5 Conclusion 67
References 68
6. Analysis of CCD Moiré Pattern to Micro-range Measurements Using the Wavelet Transform 69
6.1 Background 70
6.2 Moiré pattern and image processing 73
6.3 Experiment result and discussion 77
6.4 Conclusion 79
References 81
7. Summary and future work 84
Table Captions
Table 3.1 Zernike polynomial up to fourth degree 27
Table 3.2 Results of computer simulation 33
Table 3.3 Results of computer simulation 33
Table 3.4 Results of computer simulation 34
Table 4.1 Seidel polynomials in Cartersian Coordinate 42
Table 4.2 Results of computer simulation 51
Table 4.3 SNRs of the different algorithms 55
Table 5.1 Experimental results 67
Table 6.1 Summary of the experimental results 80
Figure Captions
Fig. 2.1 Time-frequency localization windows for the Gabor transform. 12
Fig. 2.2 Time-frequency localization windows for the wavelet transform. 15
Fig. 2.3 The schematic diagram of wavelet transform. (a) the decomposition process. (b) the reconstructed process. 18
Fig. 2.4 Schematic diagram of the two-dimension wavelet decomposition. 22
Fig. 3.1 Contour of the test wave front estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LS method. 35
Fig. 3.2 Contour of the test wave front estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LS method. 36
Fig. 3.3 Contour of the test wave front estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LS method. 37
Fig. 3.4 Simulated cure (dot line) and cure fitting with added noise by DWT method (solid line) and by LS method (dashed line) of the test wave-fronts:(a) , (b) , (c) , (d) , (e) , and (f) 38
Fig. 4.1 (a) Mexican-hat wavelet and (b) its Fourier spectrum . 47
Fig. 4.2 Spectra of noisy signal (solid line) and Mexican-hat wavelet (dashed line) for dilation factor (a) =3.2, (b) =1.4, (c) =0.7,and (d) =0.3. 49
Fig. 4.3 The (a) contour of the original, (b) reconstructed by the WT method, and (c) by the LS method in a unit square exit pupil. 53
Fig. 4.4 Wave fronts are derived using the WT method (solid line), the LS method (dashed line), and the true wave function (dot line) of the axial case ( ). 54
Fig. 4.5 The comparison of SNRs under input noise. 54
Fig. 5.1 excitation: BP 510-560. beamsplitter: FT 580 emission: LP 590. 65
Fig. 5.2 excitation: BP 450-459. beamsplitter: FT 510 emission: LP 520. 65
Fig. 5.3 test image of blue filter. 66
Fig. 5.4 test image of green filter. 66
Fig. 5.5 Fusion result of Fig. 5.3 and Fig. 5.4 using DWT method. 67
Fig. 6.1 (a) Mexican-hat wavelet and (b) its Fourier spectrum . 74
Fig. 6.2 The experimental set up of the optical system. 74
Fig. 6.3 (a) The original one-dimensional data, (b) the estimated data of Fig. 3(a) by the WT method, (c) the estimated data of Fig. 3(b) by a threshold, and (d) the derivation data of Fig. 3(c). 77
Fig. 6.4 The tested results. 80
Fig. 6.5 Spectrum of noisy signal (solid line) and Mexican-hat wavelet (dashed line) for dilation factor (a) =0.42, (b) =0.9, (c) =1.8, and (d) =2.7. 81
Glossary of Notation
1. Real numbers
2. Integers
3. Continuous time signal
4. Norm
5. Finite energy functions
6. Scaling function
7. Wavelet function
8. Hierarchy level
9. Scaling function coefficients
10. Wavelet coefficients
11. Orthogonal projection of a function onto the space
12. Scaling function space at resolution
13. Wavelet space at resolution
14. Inner product of and
15. Fourier transform
16. Short-time windowed Fourier transform
17. Wavelet transform
18. Window width in the time domain
19. Window width in the frequency domain
20. Direct sum of two vector space

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指導教授

張榮森(Rong-Seng Chang)

審核日期

2002-7-11

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