影像放大是一種用來增加影像解析度的重要方法，早期的影像放大方法採用插值法例如雙立方內插法來放大影像，但由於在影像放大過程失去高頻細節造成放大後的影像在邊緣區域有模糊化及震盪效應情況的產生。近年來所提出的方法像是區域自成像超解析度能達到非常良好的影像放大效果，但伴隨著高複雜度因為這些方法需要對整張影像作高頻訊號的回復。
在本篇論文，我們提出一種利用邊緣圖來實作影像放大的方法。藉由預測放大後影像的邊緣區域，我們回復所預測邊緣區域的高頻訊號來提高影像邊緣銳利度及減少影像邊緣的震盪效應。我們提出採用三次雲形線插值法來放大邊緣曲線的方式達成放大邊緣圖的目的。此外，如果邊緣曲線放大在不經過調整的情況以三次雲形線插值法放大會產生曲線鋸齒現象，我們也提出一種簡單的平滑函式去改善這種曲線鋸齒現象以及維持影像本身的邊緣輪廓。
相比於其他影像放大方法採用回復整張放大影像的高頻訊號，我們藉由只回復邊緣部分的高頻訊號來降低百分之九十的運算時間。實驗結果顯示我們所提出的方法能達到與區域自成像超解析度方法近似的結果。

Image up-scaling is an important technique to increase the resolution of an image. While earlier interpolation based approaches such as the bilinear and the bicubic method cause blurring and ringing artifacts in edge regions of the up-scaled image due to the loss of high frequency details. Recent approaches such as the local-self example super resolution can achieve very promising up-scaling results while their computation cost are high because they recover high frequency components of the whole image.
In this paper, we proposed an image up-scaling method via an up-scaled edge map. By predicting edge regions of the up-scaled image, we recover high frequency components of edge regions of the up-scaled image to improve the sharpness and reduce ringing artifacts. We propose an edge curve scaling method with cubic spline interpolation to up-scale an edge map. If an edge curve is directly applied to the cubic spline interpolation function for edge curve up-scaling , the edge curve scaling results have zigzag artifacts. We also propose a simple smoothing function to avoid the zigzag problems and maintain the contour shape of images.
Our methods can reduce execution time by 90% because we only perform high frequency components recovery on edge regions while other methods adopt to recover the high frequency components of every points in the up-scaled image. Experimental results show that we can achieve similar performances with the local self example super resolution method.

Contents
1 Introduction 1
1.1 Background of Image Up-sampling . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and Problem Description . . . . . . . . . . . . . . . . . 4
1.3 Goal and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Related Works 9
2.1 Interpolation Based Methods . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Gradient Analytic Methods . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Patch Based Super-Resolution . . . . . . . . . . . . . . . . . . . . . 12
3 Proposed Up-Scaling Method 16
3.1 Scalable Edge Map based Image Up-Scaling Method . . . . . . . . . 16
4 Scalable Edge Map 20
4.1 Scalable Edge Map Block Diagram . . . . . . . . . . . . . . . . . . 20
4.2 1D Piecewise Curve Extraction . . . . . . . . . . . . . . . . . . . . 22
4.3 2D Piecewise Curve Extraction . . . . . . . . . . . . . . . . . . . . 26
4.4 Smoothing Function for Extracted Piecewise Curves . . . . . . . . . 27
4.5 Curve Fitting Functions Calculation . . . . . . . . . . . . . . . . . . 29
4.6 Edge Map Up-Scaling Result . . . . . . . . . . . . . . . . . . . . . . 31
5 High Frequency Component Recovery 34
5.1 Patch Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 High Frequency Component Reconstruction . . . . . . . . . . . . . 36
5.3 Ringing Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 Computation Complexity Analysis . . . . . . . . . . . . . . . . . . 38
6 Experimental Results 41
7 Conclusion and Future Work 51
7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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