臺灣博碩士論文加值系統

此論文主要為討論局部多項式近似法又稱局部多項式迴歸，於影像處理之應用。局部多項式近似法主要運用於訊號回復。
第二章將介紹局部多項式近似法觀測訊號之模型及權重視窗大小的選擇。並解釋如何運用多項式近似法來合成和恢復訊號。權重視窗的頻寬在訊號近似準確度中扮演很重要的角色。為了求出理想的視窗寬度，我們運用統計學上方法—信賴區間。
對於相對上比較複雜的影像訊號，使用具方向性的視窗能保留更多影像的細節。第四章中介紹的非等方向性的LPA-ICI kernel即是具方向性的視窗的一種。
第五章將介紹局部多項式近似法應用於時頻分析。
局部多項式近似法於影像處理之應用則留到論文的最後一章，包括去雜訊、去模糊、及彩色濾鏡陣列。

The main idea of this thesis is to discuss local polynomial approximation (LPA) employed in image processing. It is also called local polynomial regression. LPA is used to reconstruct signals.
In Chapter 2, the observation model of LPA is introduced, and so is the basis weighting window. How it is applied to signal synthesis and reconstruction is also explained here. The bandwidth of weighting window plays an important role in the accuracy of signal estimation. In order to find the ideal window size, we employ a statistical technique, intersection of confidence intervals (ICI).
For relatively complex image signals, using directional windows can reserve more details of the image. The anisotropic LPA-ICI kernel in Chapter 4 is one of the directional windows.
The time-frequency transform using LPA is briefly introduced in Chapter 5.
Applications of Local Polynomial Approximation technique in image processing are in the last Chapter, including denoising, deblurring, and color filter array (CFA).

口試委員會審定書 #
誌謝 i
中文摘要 ii
ABSTRACT iii
CONTENTS iv
LIST OF FIGURES vii
LIST OF TABLES ix
Chapter 1 Introduction 1
1.1 Local Polynomial Approximation 1
1.2 Adaptive Scale Selection 2
Chapter 2 Local Polynomial Approximation 5
2.1 Basis of Local Polynomial Approximation 5
2.1.1 Idea of Local Polynomial Approximation 5
2.1.2 Weighting Window 9
2.1.3 Estimate Calculation 12
2.2 Kernel LPA Estimates 16
2.2.1 Estimation of Signal 16
2.2.2 Estimation of Derivative 17
2.2.3 Reproducing Polynomial Kernels 18
2.3 Nonparametric Regression 23
2.3.1 Regression function 23
2.3.2 Nadaraya-Watson estimate 24
2.3.3 LPA estimate 24
2.4 Shift-Invariant Kernels 25
Chapter 3 Adaptive Scale Selection 30
3.1 Discrete LPA Accuracy 30
3.1.1 Estimation Errors 30
3.1.2 Variance 31
3.1.3 Bias 31
3.2 Scale 34
3.2.1 Variant Scale 34
3.2.2 Invariant Scale 39
3.3 Intersection of Confidence Intervals 41
3.3.1 LPA-ICI Algorithm 41
3.3.2 Complexity and Convergence 48
3.3.3 Threshold Adjustment 49
3.4 Example: 1-D De-noising 49
Chapter 4 Anisotropic LPA-ICI 53
4.1 Directional LPA 53
4.1.1 Polynomials 53
4.1.2 Directional Windowing and Rotation 54
4.1.3 Calculations 55
4.1.4 Shift-invariant Kernels 56
4.2 Accuracy Analysis 58
4.2.1 Polynomial 58
4.2.2 Basic Estimates 61
4.2.3 Rotated Estimates 65
4.3 Adaptive Scale 70
4.3.1 Scale Setting 70
4.3.2 Estimates and Estimates Fusion 70
4.3.3 Complexity 71
Chapter 5 Local Polynomial Fourier Transform 72
5.1 Time-Frequency Transform 72
5.2 Local Polynomial Time-Frequency Transform 73
Chapter 6 Applications in Image Processing 75
6.1 Image Denoising 75
6.1.1 Example I: Cameraman 256x256 75
6.1.2 Example II: Lena 512x512 77
6.1.3 Example: III: Different Anisotropism 78
6.2 Image Deblurring 82
6.2.1 Example I: Cameraman 256x256 with BSNR 25dB 82
6.2.2 Example II: Camera man 256x256 with BSNR 40dB 84
6.3 Color Filter Array 87
6.3.1 Example I: Kodak 01 89
6.3.2 Example II: Kodak 05 91
6.3.3 Example III: Kodak 07 93
Chapter 7 Conclusions and Future Work 95
REFERENCE 96

[1]M. Unser, A. Aldroubi, M. Eden, “Polynomial Spline Signal Approximations: Filter Design and Asymptotic Equivalence with Shannon’s Sampling Theorem,” IEEE Trans. Information Theory, vol. 38, NO. 1, pp95-103, Jan. 1992.
[2]V. Katkovnik, “On Adaptive Local Polynomial Approximation with Varying Bandwidth, ” IEEE, pp2321-2324, 1998.
[3]V. Katkovnik, “A New Method for Varying Bandwidth Selection, ” IEEE Trans. Signal Processing, vol. 47, NO. 9, Sep. 1999.
[4]S. C. Chan and Z. Zhang, “Multi-Resolution Analysis of Non-Uniform Data with Jump Discontinuities and Impulse Noise Using Robust Local Polynomial Regression, ” IEEE ICASSP, II-pp769-772, 2004.
[5]S. C. Chan and Z. Zhang, “Robust Local Polynomial Regression with Adaptive Bandwidth, ” IEEE ISCAS, III-pp333-336, 2004.
[6]Nadaraya, E.A., “On estimating regression,” Theory Prob. Appl., vol. 9, pp141-142, 1964.
[7]V. Katkovnik, K. Egiazarian, J. Astola, “ Application of the ICI principle to window size adaptive median filtering, ” Signal Processing 83, pp251-257, 2003.
[8]D. Barash, “A Fundamental Relationship between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Equation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, NO. 6, pp844-847, June 2002.
[9]Katkovnik, V., A. Foi, K. Egiazarian, and J. Astola, “Directional varying scale approximations for anisotropic signal processing”, Process., EUSIPCO 2004, pp. 101-104, 2004.
[10]Foi, A., and V. Katkovnik, “From local polynomial approximation to pointwise shape-adaptive transforms: an evolutionary nonparametric regression perspective”, Proc. Int. TICSP Workshop SMMSP 2006, Florence, September 2006.
[11]V. Katkovnik, “A new form of the Fourier transform for time-varying frequency estimation, ” Signal Processing, pp187-200, 1995.
[12]Y. Wei and G. Bi, “Efficient Algorithm for Modified Local Polynomial Time Frequency Transform, ” IEEE ICASSP, 2006.
[13]Igor Djurović, “Robust Adaptive Local Polynomial Fourier Transform,” IEEE Signal Processing Letters, vol. 11, NO. 2, pp201-204, Feb. 2004.
[14]L. Stanković, “Order Adaptive Local Polynomial FT Based Interference Rejection in Spread Spectrum Communication Systems,” IEEE Tran. Instrumentation and Measurement, vol. 54, NO. 6, pp2156-2162, Dec. 2005.
[15]S. Djukanović, M. Daković, L. Stanković, “Local Polynomial Fourier Transform Receiver for Nonstationary Interference Excision in DSSS Communications,” IEEE Trans. Signal Processing, vol. 56, NO. 4, pp1627-1636, April 2008.
[16]V. Katkovnik, “Wavelets and Local Polynomial Approximation,” TFTS, pp237-240, 1996.
[17]Katkovnik, V., K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule”, J. of Math. Imaging and Vision, vol. 16, no. 3, pp. 223-235, 2002.
[18]R. Charnigo, J. Sun, R. Muzic Jr., “A Semi-Local Paradigm for Wavelet Denoising, ” IEEE Trans. Image Processing, pp1-12, 2006.
[19]H. Takeda, S. Farsiu, P. Milanfar, “Kernel Regression for Image Processing and Reconstruction, ” IEEE Trans. Image Processing, vol. 16, NO. 2, pp349-366, Feb. 2007.
[20]H. Takeda, S. Farsiu, P. Milanfar, “ Deblurring Using Regularized Locally Adaptive Kernel Regression,” IEEE Trans. Image Processing, vol. 17, NO. 4, pp550-563, April 2008.
[21]Paliy, D., V. Katkovnik, R. Bilcu, S. Alenius, K. Egiazarian, “Spatially Adaptive Color Filter Array Interpolation for Noiseless and Noisy Data”, International Journal of Imaging Systems and Technology (IJISP), Special Issue on Applied Color Image Processing, vol. 17, iss. 3, pp. 105-122, October 2007.
[22]V. Katkovnik, J. Astola, K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process., vol. 17, no. 6, pp. 833-846, June 2008.
[23]Paliy D., M. Trimeche, V. Katkovnik, S. Alenius, “Demosaicing of Noisy Data: Spatially Adaptive Approach”, Proc. of SPIE-IS&T Electronic Imaging 2007, Computational Imaging IV, Vol. 6497, 649720, San Jose, CA, January 2007.
[24]Paliy D., R. Bilcu, V. Katkovnik, M. Vehvilainen, “Color Filter Array Interpolation Based on Spatial Adaptivity”, Proc. of SPIE-IS&T Electronic Imaging 2007, Computational Imaging IV, Vol. 6497, 649707, San Jose, CA, January 2007.
[25]Paliy, D., V. Katkovnik, R. Bilcu, S. Alenius, K. Egiazarian, “Spatially Adaptive Color Filter Array Interpolation for Noiseless and Noisy Data”, International Journal of Imaging Systems and Technology (IJISP), Special Issue on Applied Color Image Processing, vol. 17, iss. 3, pp. 105-122, October 2007.
[26]Paliy, D., A.Foi, R. Bilcu, V. Katkovnik, “Denoising and Interpolation of Noisy Bayer Data with Adaptive Cross-Color Filters”, SPIE-IS&T Electronic Imaging, Visual Communications and Image Processing 2008, vol. 6822, San Jose, CA, January 2008.