在過去二十年，小波轉換在視訊處理、電腦視覺及資料壓縮等領域已變成了一種標準的處理技術。這些應用所處理的往往都是取樣後的訊號，因此離散小波轉換已成為一廣泛被採用的計算工具。在電腦視覺的應用上，光流計算是一重要的課題。本論文探討如何由小波方法，從一連串的影像中精確的找出運動物體的光流資訊並將之應用於影像切割、影像追蹤等方面。

本論文實現不同的尺度函數的小波光流估測。我們比較D4、 D6、CDF 6/2(6)、CDF 9/7(7)、CDF 9/7(9)尺度函數在小波方法上的表現。我們計算2個合成影像Translating Tree and Yosemitec與4個實際影像Rubik cube、Hamburg Taxi、Coastguard、Football的光流。對合成影像，Daubechies 4小波基底沒利用降解析的角度平均誤差分別為19.85o、34.23o並需要130.6、156.8秒的計算時間;實際影像分別需要144.4、182.9、238.1、181.5秒的計算時間。Daubechies 6小波基底沒利用降解析的角度平均誤差分別為0.701o、5.136o並需要153.5、200.0、163.0、199.9、256.3、197.8秒的計算時間。Daubechies 6/2(6) 小波基底沒利用降解析的角度平均誤差分別為0.633o、5.369o並需要149.7、202.1、163.6、198.8、255.3、197.5秒的計算時間。Daubechies 9/7(7) 小波基底沒利用降解析的角度平均誤差分別為0.644o、5.073o並需要180.9、249.9、180.1、238.7、308.0、233.4秒的計算時間。Daubechies 9/7(9) 小波基底沒利用降解析的角度平均誤差分別為0.661o、4.475o並需要265.4、318.5、209.0、253.1、315.7、250.5秒的計算時間。然後比較垂直與偶垂直基底時，我們可以知道偶垂直基底有比垂直基底有更好的表現。且因我們利用了降解析與原有的小波方法合併，大大減少了原有的計算時間。

Over the past twenty years, the wavelet transform has become a standard technique in many fields such as signal processing, computer vision and data compression. In these applications, the signals to be processed are usually sampled, so the discrete wavelet transform (DWT) is used extensively. Optical flow calculation is an essential problem in computer vision. This thesis investigates how to obtain accurate optical flow of moving objects from image sequences by using wavelets.

This thesis reports the computation time and accuracy of optical flow calculation using different scaling functions. We compare the performance of the Daubechies D4, D6, CDF , CDF and CDF scaling functions employed in the wavelet-based optical flow estimation. We estimated the optical flow from two kinds of synthetic sequences: Translating Tree and Yosemitec and four kinds of real sequences: Rubik’s cube, Hamburg Taxi, Coastguard and Football. The average errors and computation time of the D4 are 19.85o, 34.23o degrees and 130.6, 156.8, 144.4, 182.9, 238.1 and 181.5 seconds without resolution reduction. The average errors and computation time of the D6 are 0.701o, 5.136o degrees and 153.5, 200.0, 163.0, 199.9, 256.3 and 197.8 seconds without resolution reduction. The average errors and computation time of the CDF are 0.633o, 5.369o degrees and 149.7, 202.1, 163.6, 198.8, 255.3 and 197.5 seconds without resolution reduction. The average errors and computation time of the CDF are 0.644o, 5.073o degrees and 180.9, 249.9, 180.1, 238.7, 308.0 and 233.4 seconds without resolution reduction. The average errors and computation time of the CDF are 0.661o, 4.475o degrees and 265.4, 318.5, 209.0, 253.1, 315.7 and 250.5 seconds without resolution reduction. When compared with the orthogonal bases the biorthogonal bases, have better performance. Finally, we reduce the computation time by combining the wavelet-based method and resolution reduction.

[1] E. H. Adelson and J. R. Bergen, ”Spatiotemporal Energy Models for the Perception of Motion,” Journal of Optical Society of America, A: Optics and Image Science, Vol. 2, Page(s): 284 – 299, 1985.
[2] C. P. Bernard, “Discrete Wavelet Analysis for Fast Optic Flow Computation,” Feb. 26, 1999.
[3] J. L. Barron, D. J. Fleet and S. S. Beauchemin, "Performance of Optical Flow Techniques," Internation Journal of Computer Vision 12, Page(s): 43 - 77, 1994.
[4] P. J. Burt, C. Yen and X. Xu, “Multi-resolution Flow through Motion Analysis,” Proceeding of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Page(s): 246 – 252, 1983.
[5] C. H. Chen, Y. C. Chiou, M. K. Wu and Y. F. Li, “An Efficient Video Object Segmentation Algorithm Using Change Detection and Background Updating Technique,” International Conference on Systems and Signals, 2005.
[6] A. Cohen, I. Daubechies and J. C. Feauveau, “Biorthogonal Bases of Compactly Supported Wavelets,” Communications on Pure and Applied Mathematics, Vol. 45, Page(s): 485 – 560, 1992.
[7] L. F. Chen, J. C. Lin and H. Y. M. Liao, “Wavelet-Based Optical Flow Estimation,” Pattern Recognition, 2000. Proceedings. 15th International Conference, Vol. 3, Page(s): 1056 – 1059, Sept. 2000.
[8] L. F. Chen, H. Y. M. Liao and J. C. Lin, ”Wavelet-Based Optical Flow Estimation,” Circuits and Systems for Video Technology, IEEE Transactions, Vol. 12, No. 1, Page(s): 1 – 12, Jan. 2002.
[9] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Philadelphia: Society for Industrial and Applied Mathematics, 1992.
[10] I. Daubechies, “Orthogonal Bases of Compactly Supported Wavelets,” Society for Industrial and Applied Mathematics, Vol. 24, No. 2, Page(s): 499 – 519, Mar. 1993.
[11] A. Graps, “An Introduction to Wavelets,” IEEE Computational Science and Engineering, Vol. 2, No. 2, Page(s): 50-61, Jun. 1995.
[12] R. C. Gonzalez and R. E. Woods., Digital Image Processing, 2nd ed., NJ: Prentice Hall, 2002.
[13] B. K. P. Horn and B. G. Schunck, “Determining Optical Flow,” Artificial Intelligence, Vol. 17, Page(s): 185 - 203, 1981.
[14] A. Latto, H. L. Resnikoff and E. Tenenbaum, Aware, Inc. “The Evaluation of Connection Coefficients of Compactly Supported Wavelets,” Aug. 1999.
[15] S. H. Lai and B. C. Vemuri, “Robust and Efficient Algorithm for Optical Flow Computation,” Proceeding of IEEE International Conference on Computer Vision, Page(s): 455 – 460, 1995.
[16] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed., NJ: Academic Press, 1999.
[17] W. Sweldens,” Wavelets: What Next ?,” Proceedings of the IEEE, Vol. 84, No. 4, Page(s): 680 - 685, Apr. 1996.
[18] G. Strang and T. Nguyen, Wavelets and Filter Banks, New York: Wellesley- Cambridge Press, 1997.
[19] A. M. Tekalp, Digital Video Processing, NJ: Prentice Hall, 1995.
[20] M. Vetterli, ”Filter Banks Allowing Perfect Reconstruction,” Signal Processing, Vol. 10, No. 3, Page(s): 219 - 244, Apr. 1986.
[21] P. P. Vaidyanathan and Z. Doganata, ”The Role of Lossless Systems in Modern Digital Signal Processing: A tutorial,” Education, IEEE Transactions, Vol. 32, No. 3, Page(s): 181-197, Aug. 1989.
[22] F. Y. M. Wan, Introduction to the Calculus of Variations and Its Application. London, U.K.: Chapman & Hall, 1995.
[23] Y. T. Wu, T. Kanade, J. Cohn and C. C. Li, “Optical Flow Estimation Using Wavelet Motion Model,” Proceeding of IEEE International Conference on Computer Vision, Page(s): 992 – 998, Jan. 1998.