對於影像放大技術而言，如何解決在邊緣、輪廓上的高頻訊號部分發生的假影以及模糊的現象是影像放大時的一個重要課題，在局部自相似性的影像放大方法中觀察到了，在以較小的放大倍率放大的圖像中，小的區塊會有自相似性，因此可以被使用來還原放大後所失去的高頻訊號，因為高頻訊號中主要包含了邊緣以及輪廓部分的資訊，對此我們提出了基於梯度搜尋的方法，此方法可以利用高頻訊號的特徵來擷取相似性高的區塊。首先我們觀察了在局部自相似性方法中所找到的範例區塊的分布，並且藉由觀測結果提出一個基於分布比例的雙邊濾波器來使的邊緣能加銳利，與區域自相似方法比較，此方法能夠節省大量的區塊搜尋時間並達到相似的結果。另一方面，對於影像放大中的高頻訊號重建，我們提出了基於1-D 以及 2-D梯度的搜尋，2-D梯度的搜尋中考慮了八個不同方向的推導，使得在邊緣部分的變化可以良好的保存，藉由此方法搜尋到的的區塊對於原本的圖像會有高度的相似性，此外，我們更提出了一個淘汰機制來避免搜尋到錯誤的範例區塊，這個機制能結果能達到更好的視覺效果。相較於其他方法，重建後的高頻訊號保存了更好的邊緣以及輪廓的資訊。對比於其他近年來的方法，我們所提出的方法中有較高的PSNR以及SSIM數值。

For image up-scaling, blurs and ringing at the edges or contours of the high fre-
quency components in the super resolution image is one of the major problems.
The local-self-example image up-scaling method observe that small patches are
similar to themselves upon small scaling factors so they use small patch search to
reconstruct the high frequency components. Since the high frequency components
contain mainly the contour and edge information, we propose a gradient based
searching approach that utilizes the features of the high frequency components
to extract similar patches. In this paper we observe the distribution of example
patches' distribution at rst and based on the result we proposed an distribution
based bilateral lter to sharp the edge region and this approach can have less
computation time but have similar result since the search process is removed. On
the other side, we present another 1-D and 2-D based gradient search for recon-
structing the high frequency components in the super resolution image. The 2-D
gradient based search takes eight-direction derivations into consideration so that
most edge variations are well preserved, the found patches have high structure
similarity between the original image and the up-scaling image. Besides, we also
proposed a selection mechanism to avoid the mismatching patch error and achieve
better performance. Compared with other recent methods, the reconstructed high
frequency components of our proposed methods maintain more edge and contour
details. our proposed approaches have better PSNR and SSIM values than other
approach.

[1] J. A. Parker, R. V. Kenyon, and D. E. Troxel, "Comparison of interpolation
methods for image resampling," IEEE Transactions on Medical Imaging, pp.
31-39, 1983.
[2] H. Hou and H. C. Andrews, "Cubic splines for image interpolation and digital
ltering," IEEE Transaction on Acoustics, Speech and Signal Processing, pp.
508-517, 1978.
[3] Y. W. Tai, WS Tong, and CK Tang, "Perceptually-inspired and edge-directed
color image super-resolution" IEEE Conference on Computer Vision and Pat-
tern Recognition, vol. 2, pp. 1948-1955,2006.
[4] R. Fattal, "Image upsampling via imposed edge statistics," ACM Transaction
on Graphics, vol.26, no.3, pp. 56-65, Article 95, July 2007.
[5] G. Freeman and R. Fattal, "Image and video upscaling from local self-
examples," ACM Transaction on Graphics, vol.30, no.2, pp. 12:1-12:11, Apr
2011.
[6] W. T. Freeman, T. R. Jones, and E. C. Pasztor, "Example-based super-
resolution," IEEE Computer Graphics and Applications, vol. 22, no.2, pp.
56-65, Mar/Apr 2002.
49[7] C. Damkat, "Single image super-resolution using self-examples and texture
synthesis" Signal, Image and Video Processing, pp. 343-352, 2011.
[8] D. Glasner, S. Bagon, and M. Irani, "Super-resolution from a single image,"
IEEE Conference on Computer Vision, pp. 349-356, 2009.
[9] J. Sun, Z. Xu and H. Y. Shum, "Image super-resolution using gradient prole
prior," IEEE Conference on Computer Vision and Pattern Recognition, 2008.
[10] Q. Shan, Z. Li, J. Jia, and C. Tang, Fast image/video upsampling," ACM
Transaction on Graphics, vol. 27, no. 5, pp. 17, 2008.
[11] L. Hong, Y. Wan, and A. Jain, Fingerprint image enhancement: algorithm
and performance evaluation," IEEE Transaction on Pattern Analysis and
Machine Intelligence, pp. 777-789, 1998.
[12] J. Yang, Z. Lin, and S. Cohen,Fast image super-resolution based on in-
place example regression," IEEE Conference on Computer Vision and Pattern
Recognition , pp. 1059-1066, 2013.
[13] J. Allebach, and P. W. Wong,"Edge-directed interpolation," IEEE Interna-
tional Conference on Image Processing, pp. 707-710, 1996.
[14] X. Li, and M. T. Orchard,"New Edge-Directed Interpolation," IEEE Trans-
actions on Image Processing, pp. 1521-1527, 2001.
[15] D. Su, and P. Willis, "Image Interpolation by PixelLevel DataDependent
Triangulation" Computer Graphics Forum 23, Vol. 23. No. 2, pp. 189-201,
2004.
50[16] Z. Lin, and H. Y. Shum,"Fundamental limits of reconstruction-based super-
resolution algorithms under local translation," IEEE Pattern Analysis and
Machine Intelligence, pp. 83-97, 2004.
[17] J. Yang, and J. Wright,"Image Super-Resolution as Sparse Representation
of Raw Image Patches,"IEEE Conference on Computer Vision and Pattern
Recognition , pp. 1-8, 2008.
[18] J. Yang, and J. Wright,"Image Super-Resolution Via Sparse Representation,"
IEEE Transactions on Image Processing, pp. 2861-2873, 2010.
[19] K. I. Kim, and Y. Kwon,"Example-Based Learning for Single-Image Super-
Resolution," IEEE Pattern Recognition, pp. 456-465, 2008.
[20] N. Suetake, M. Sakano, and E. Uchino, "Image super-resolution based on
local self-similarity" Optical review, pp. 26-30, 2008.
[21] B. S. Morse, and D. Schwartzwald,"Image Magnication Using Level-Set Re-
construction," IEEE Computer Society Conference on Computer Vision and
Pattern Recognition, pp. I333-I340, 2001.
[22] D. Zhang, and X. Wu,"An Edge-Guided Image Interpolation Algorithm via
Directional Filtering and Data Fusion" IEEE Transactions on Image Process-
ing, pp. 2226-2238, 2006.
[23] S. Dai, M. Han, W. Xu, and Y Wu,"Soft Edge Smoothness Prior for Al-
pha Channel Super Resolution" IEEE Conference on Computer Vision and
Pattern Recognition, pp. 1-8, 2007.
51[24] M. Li, and T. Q. Nguyen,"Markov Random Field Model-Based Edge-Directed
Image Interpolation" IEEE Conference on Computer Vision and Pattern
Recognition, pp. 1121-1128, 2008.
[25] Z. Lin, and H. Y. Shum,"Fundamental Limits of Reconstruction-Based Su-
per resolution Algorithms under Local Translation" IEEE Transactions on
Pattern Analysis and Machine Intelligence, pp. 83-97, 2004.
[26] W. T. Freeman, EC. Pasztor, OT. Carmichael,"Learning Low-Level Vision"
IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 25-47,
2000.
[27] S. Baker, and T. Kanade, "Limits on super-resolution and how to break
them" IEEE Transactions on Pattern Analysis and Machine Intelligence, pp.
1167-1183, 2002.
[28] M. F. Tappen, B. C. Russell, and W. T. Freeman,"Exploiting the Sparse
Derivative Prior for Super-Resolution and Image Demosaicing" IEEE Work-
shop on Statistical and Computational Theories of Vision, 2003.
[29] H. Chang, D. Y. Yeung, and Y. Xiong, "Super-Resolution Through Neighbor
Embedding" IEEE Computer Conference on Computer Vision and Pattern
Recognition, 2004.
[30] Q. Wang, X. Tang, and H. Shum, "Patch Based Blind Image Super Res-
olution" IEEE International Conference on Computer Vision, pp. 709-716,
2005.
52[31] Z. Wang, A. C. Bovik, and H. R. Sheikh,"Image Quality Assessment: From
Error Visibility to Structural Similarity" IEEE Transactions on Image Pro-
cessing, pp. 600-612, 2004.
[32] S. Borman, and R. L. Stevenson, "Super-resolution from image sequences-
a review" IEEE IEEE Computer Society on Circuits and Systems, Midwest
Symposium, pp. 374, 1998.
[33] M. Elad and A. Feuer, "Super-resolution reconstruction of image sequences"
IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 21,
no. 9, pp. 817-834, 1999.
[34] S. C. Park, M. K. Park and M. G. Kang, "Super-resolution image reconstruc-
tion: a technical overview" IEEE Signal Processing Magazine, pp. 21-36,
2003.
[35] S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, "Fast and Robust
Multiframe Super Resolution" IEEE Transactions on Image Processing, vol.
13, no. 10, pp. 1327-1344, 2004.
[36] R. Xiong, J. Xu, and F. Wu, "A lifting-based wavelet transform supporting
non-dyadic spatial scalability" IEEE Transactions on Image Processing, pp.
1861-1864, 2006.
[37] S. Pollock, and I. L. Cascio, "Non-dyadic wavelet analysis" Optimisation,
Econometric and Financial Analysis, pp. 167-203, 2007.
[38] L. Wang, S. Xiang, G. Meng, H. Y. Wu, and C. Pan, "Edge-Directed Single-
Image Super-Resolution via Adaptive GradientMagnitude Self-Interpolation"
53IEEE transactions on circuits and systems for video technology, pp. 1289-
1299, 2013.
[39] K. I. Kim, and Y. Kwon, "Single-image super-resolution using sparse regres-
sion and natural image prior" IEEE Transactions on Pattern Analysis and
Machine Intelligence, pp. 1127-1133, 2010.
[40] K. Zhang, X. Gao, D. Tao, and X. Li , "Single Image Super-Resolution With
Non-Local Means and Steering Kernel Regression" IEEE Transactions on
Image Processing, pp. 4544-4556, 2012.
[41] S. Villena, M. Vega, S. D. Babacan, R. Molina and A. K. Katsaggelos,
"Bayesian combination of sparse and non-sparse priors in image super res-
olution" Digital Signal Processing, pp. 530-541, 2013.
[42] X. Li, X. Gao, Y. Hu, D. Tao, and B. Ning, "A multi-frame image super-
resolution method," Signal Process., vol. 90, no. 2, pp. 405-414, 2010.