影像超解析技術是一個非常重要的方法，它能將低解析度影像還原成清晰的高解析度影像。本論文提出一個以非線性降維為基礎的超解析法，此超解析法使用的降維方法為two-directional two-dimensional locality preserving projection ((2D)2-LPP)。(2D)2-LPP為本論文提出之新的降維方法，其採用流形學習(Manifold learning)，利用內部流形結構把高維空間投影到低維空間，並且不失去重要的資料。(2D)2-LPP可解決流形學習法中非樣本資料的問題(out-of-sample problem)，改進新資料降維的速度。此外，(2D)2-LPP針對二維影像降維，能比PCA、LPP等方法保留更多資料。實驗資料從AR和FERET資料庫選取出適當的人臉。實驗結果表示本論文提出之方法比PCA、2D-PCA、(2D)2-PCA以及2D-LPP超解析的方法有更佳的PSNR。

Super-resolution is an important method to reconstruct high-resolution images from low-resolution images. In this paper, a nonlinear dimension reduction algorithm based on two-directional two-dimensional locality preserving projection ((2D)2-LPP) is proposed for face image super-resolution. The (2D)2-LPP is a new manifold learning method proposed by this paper. It detects the intrinsic manifold structure of high space and preserves the structure in low space by projection. The projection approach in the (2D)2-LPP resolves the out-of-sample problem in embedding-based manifold learning methods, and improves the speed in reducing the dimension of a new sample data. Moreover, the (2D)2-LPP preserves more accurate manifold structure by directly operating on 2D images rather than operating flattened 1D vector as PCA and LPP do. Extensive experiments are conducted on the AR and FERET databases. Experimental results show that the proposed method performs better than PCA,2D-PCA,(2D)2-PCA, and 2D-LPP super-resolution methods in PSNR.

Chapter 1 Introduction ...........................................1
1.1 Background..........................................................1
1.2 Motivation..........................................................3
1.3 Brief introduction of dimension reduction...........................................................4
1.3.1 PCA, 2D-PCA ,and (2D)2-PCA.................................................................5
1.3.2 LPP and 2D-LPP.................................................................8
1.4 The organization of this paper..............................................................10
Chapter 2 The Framework of Dimension Reduction-based Super-resolution...............11
2.1 Super-resolution by dimension reduction..........................................................11
2.2 PCA-based super-resolution.........................................................16
2.3 2D-PCA and 2D-LPP based super-resolution......................................................17
Chapter 3 (2D)2 Locality Preserving Projection for Super-resolution...........................19
3.1 Learning the super-resolution manifold by (2D)2-LPP........................................20
3.2 Dimension expansion for super-resolving images...............................................23
3.3 Algorithms for learning and super-resolution......................................................25
Chapter 4 Experimental Results............................................................27
4.1 Super-resolution results............................................................28
4.2 Cumulative percentage of variance...........................................................30
4.3 Regularization parameter λ..................................................................31
4.4 Training Number.............................................................32
4.5 Noise Robustness.........................................................34
4.6 Compare a low-resolution image and a feature to a high-resolution image.........35
Chapter 5 Conclusions and Future Work...............................................................37
References.........................................................38

[1] Kim, S. P., Bose, N. K., and Valenzuela, H. M., "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Transactions Acoustic, Speech, Signal Processing, Vol. 38, Issue 6, pp. 1013-1027, 1990.
[2] Gunturk, B. K., Batur, A. U., Altunbasak, Y., Hayes, M. H., and Mersereau, R. M., "Eigenface-domain super-resolution for face recognition," IEEE Transactions on Image Processing, Vol. 12, No.5, pp. 597-606, 2003.
[3] Baker, S. and Kanade, T., "Limits on super-resolution and how to break them," IEEE Transactions Pattern Analysis and Machine Intelligence, Vol. 24, No. 1, pp. 1167-1183, 2002.
[4] Lindley, C. A., Practical image processing in C, Wiley-Interscience, USA, pp. 23-25, 1994.
[5] Valenzuela, H. M., "A two-dimensional recursive model for bilinear systems with applications to image reconstruction," IEEE Transactions on Circuits and System, Vol. 37, No. 3, pp. 354-363, 1990.
[6] Keys, R. G., "Cubic convolution interpolation for digital image processing," IEEE Transactions Acoustic, Speech and Signal Processing, Vol. 29, Issue 6, pp. 1153-1160, 1981.
[7] Clark, J. J., Palmer, M. R., and Laurence, P. D., "A transformation method for the reconstruction of functions from nonuniformly spaced samples," IEEE Transactions Acoustic, Speech and Signal Processing, Vol. 33, Issue 5, pp. 1151-1165, 1985.
[8] Park, S. C., Park, M. K., and Kang, M. G., "Super-resolution image reconstruction: a technical overview," IEEE Signal Processing Magazine, Vol. 20, Issue 3, pp. 21-36, 2003.
[9] Stark, H. and Oskoui, P., "High resolution image recovery from image-plane arrays, using convex projections," J. Opt. Soc. Am. A, Vol. 6, No. 11, pp. 1715-1726, 1989.
[10] Irani, M. and Peleg, S., "Improving resolution by image registration," CVGIP: Graphical Models and Image Processing, Vol. 53, Issue 3, pp. 231-239, 1991.
[11] Tipping, M. E. and Bishop, C. M., "Bayesian image super-resolution," Proceedings, International Neural Information Processing Systems Conference, Vancouver, Canada, pp. 1279-1286, 2002.
[12] Kanemura, A., Maeda, S., and Ishii, S., "Edge-preserving bayesian image superresolution based on compound markov random fields," Proceedings, International Artificial Neural Networks Conference, San Sebastian, Spain, pp. 611-620, 2007.
[13] Zhuang, Y., Zhang, J., and Wu, F., " Hallucinating faces: LPH super-resolution and neighbor reconstruction for residue compensation," Pattern Recognition, Vol. 40, No. 11, pp. 3178-3194, 2007.
[14] Capel, D. and Zisserman, A., "Super-resolution from multiple views using learnt image models," Proceedings, International Computer Vision and Pattern Recognition Conference, Hawaii, USA, pp. 627-634, 2001.
[15] Chakrabarti, A., Rajagopalan, A. N., and Chellappa, R., "Super-resolution of face images using kernel PCA-based prior," IEEE Transactions on Multimedia, Vol. 38, Issue 4, pp. 888-892, 2007.
[16] Chang, H., Yeung, D., and Xiong, Y., "Super-resolution through neighbor embedding," Proceedings, International Computer Vision and Pattern Recognition Conference, Washington, USA, pp. 275-282, 2004.
[17] Jain, A. K., Flynn, P., and Ross, A. A., Handbook of Biometrics, Springer-Verlag, USA, pp. 62-64, 2007.
[18] Wang, Y. K. and Huang, C. R., "Face image super-resolution using two-dimensional locality preserving projection," Proceedings, International Intelligent Information Hiding and Multimedia Signal Processing Conference, Kyoto, Japna, 2009 (Accepted).
[19] Kumar, B. G. V. and Aravind, R., "A 2D model for face superresolution," Proceedings, International Pattern Recognition Conference, Florida, USA, pp.1-4, 2008.
[20] Chen, S., Zhao, H., Kong, M., and Luo, B., "2D-LPP: a two-dimensional extension of locality preserving projects," Neurocomputing, Vol. 70, Issue 4-6, pp. 912-921, 2007.
[21] Niu, B., Yang, Q., Shiu, S. C. K., and Pal, S. K., "Two-dimensional Laplacianfaces method for face recognition," Pattern Recognition, Vol. 41, Issue 10, pp. 3237-3243, 2008.
[22] Turk, M. A. and Pentland, A. P., "Face recognition using eigenfaces," Proceedings, International Computer Vision and Pattern Recognition Conference, Maui, USA, pp. 586-591, 1991.
[23] Yang, J., Zhang, D., Frangi, A. F., and Yang, J., "Two-dimensional PCA: a new approach to appearance-based face representation and recognition," IEEE Transactions Pattern Analysis and Machine Intelligence, Vol. 26, Issue 1, pp. 131-137, 2004.
[24] Zhang, D. and Zhou, Z., "(2D)2PCA: Two-directional two-dimensional PCA for efficient face representation and recognition," Neurocomputing, Vol. 69, Issue 1-3, pp. 224-231, 2005.
[25] Roweis, S. T. and Saul, L. K., "Nonlinear dimensionality reduction by locally linear embedding," Science, Vol. 290, No. 5500, pp. 2323-2326, 2000.
[26] Tenenbaum, J. B., Silva, V. D., and Langford, J. C., "A global geometric framework for nonlinear dimeensionality reduction," Science, Vol. 290, No. 11, pp. 2319-2323, 2000.
[27] Belkin, M. and Niyogi, P., "Laplacian eigenmaps and spectral techniques for embedding and clustering," Proceedings, International Neural Information Processing System Conference, Cambridge, USA, pp. 585-591, 2002.
[28] Donoho, D. L. and Grimes, C., "Hessian eigenmaps: new locally linear embedding techniques for high dimensional data," Proceedings, International National Academy of Sciences Conference, Washington, USA, pp. 5591-5596, 2003.
[29] Bengio, Y., Paiement, J. F., and Vincent, P. “Out-of-sample extensions for LLE, ISOMAP, MDS, eigenmaps and spectral clustering,” Proceedings, International Neural Information Processing Systems Conference, Vancouver, Canada, pp. 177-184, 2003.
[30] He, X. and Niyogi, P., "Locality preserving projections," Proceedings, International Neural Information Processing Systems Conference, Vancouver, Canada, pp. 585-591, 2003.