現在二維影像在生活中已經是非常容易取得，所以這些影像常用來做許 多分析，包含相當常見之應用如影像比對、影像分割等等，而透過整合這些 應用亦能創造出非常有趣的題目。在本篇論文中，我們針對二維影像的一些 特徵探討潛在的應用，包含多重涵義的圖地反轉影像和立體摺紙設計。多重 涵義的圖地反轉影像在心理學上指的是一種視覺感知現象，且多以二值化的 影像呈現，根據觀察者所注視的位置可將影像分為兩個區域，而此兩個相鄰 的區域分別代表圖與地，圖與地的感知現象會根據觀察著看的位置而改變。 立體摺紙設計是一個有趣的藝術形式，把要呈現的影像以一張帶有切割與折 線的紙來呈現。當打開這個摺紙設計，影像將會以立體的結構呈現且隨著開 合角度改變。
在第一部份，我們提出了一個系統用來產生多重涵義的圖地影像，演算 法主要分成三個步驟，首先經過片段輪廓匹配找出兩個物件最佳的片段輪 廓，再根據幾何特性來平衡視覺主體的競爭將兩個片段輪廓透過形變結合在 一起，接著找出最佳的剪裁窗口和圖像二值化來最大化兩個物件的輪廓，使 得反轉影像的效果呈現最佳，最後我們透過大量產生多重涵義的圖地的影像 來論證我們方法的有效性。
在第二部分，我們提出了一個立體摺紙設計方法使用二維圖像作為輸入 影像，由於缺乏三維的幾何資訊，我們進行新的理論分析以確保設計的摺疊 性與穩定性，基於圖的表示法，我們進一步提出一個最佳化演算法來最佳化 拓樸結構與幾何形狀，在我們的系統中，我們也允許使用者來指定設計上的 折線位置，最後我們對於產生的實例來進行評估且與其他方法比較來顯示我 們方法的有效性。

Thanks to the high availability of 2D images, images can be analyzed in many ways,including quite common applications such as image matching and image segmentation.The integration of these applications can also create very interesting topics.In this work, we explore the potential applications based on the characteristics of 2D images, including ambiguous figure-ground images and paper pop-up design.Ambiguous figure-ground images, mostly represented by binary images, are fascinating as they present viewers a visual phenomena of perceiving multiple interpretations from a single image. Paper pop-up is a fascinating artistic form which involves only a piece of paper with cuts and folds. Interesting geometric structure `pops up' when the paper is opened.
In the first part, we investigate the theory behind this ambiguous perception and present an automatic algorithm to generate such images.We model the problem as a binary image composition using two object contours and approach it through a three-stage pipeline.Thealgorithm first performs a partial shape matching to find a good partial contour matching between objects. Then we combine matched contours into a compound contour using an adaptive contour deformation, followed by computing an optimal cropping window and image binarization for the compound contour that maximize the object contours.We have tested our system using a wide range of input objects and generated a large number of convincing examples with or without user guidance.The efficiency of our system and quality of results are verified through an extensive experimental study.
In the second part, we present the first paper pop-up design framework that takes 2D images instead of 3D models as input.Due to the lack of 3D geometry information, we perform novel theoretic analysis to ensure the foldability and stability of the resultant design.Based on a novel graph representation of the paper pop-up plan, we further propose a practical optimization algorithm via mixed integer programming that jointly optimizes the topology and geometry of the 2D plan.We also allow the user to interactively explore the design space and specify constraints for creative design.We evaluate our framework on a variety of examples. The experiments and comparisons exhibit both the efficiency and efficacy of our framework.

致􅍯謝i
中文摘􃗓要ii
Abstract iii
Contents v
List of Figures vii
1 Introduction . . . . . . . . . . . . . . . . 1
1.1 Generating Ambiguous Figure-Ground Images . . . . . . . . . . . . . . . . 1
1.2 Image-based Paper Pop-up Design . . . . . . . . . . . . . . . . . . . . . . 4
2 Related Works 7
2.1 Perceptual Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Computational Arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Shape Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Paper Craft Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Computational Paper Pop-ups . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Photo Pop-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Algorithm Part I 12
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Partial Shape Matching . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1.1 Content-Aware Shape Matching Metric . . . . . . . . . . 14
3.2.1.2 Initial Sub-Matching Contours . . . . . . . . . . . . . . . 16
3.2.2 Sub-Matching Contours Clustering . . . . . . . . . . . . . . . . . . 17
3.2.2.1 Optimal Partial Contour Matching . . . . . . . . . . . . . 17
3.2.3 Adaptive Contour Deformation . . . . . . . . . . . . . . . . . . . . 18
3.2.4 Image Cropping and Binarization . . . . . . . . . . . . . . . . . . . 20
3.2.5 User Controls and Application . . . . . . . . . . . . . . . . . . . . 22
4 Algorithm Part II 23
4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1.1 Pop-up Plans . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1.2 Foldability . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2.1 Initialize boundary fold lines . . . . . . . . . . . . . . . . 30
4.2.2.2 Initialize inner fold lines . . . . . . . . . . . . . . . . . . . 31
4.2.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.3.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.3.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.3.3 MIP formulation . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.4 Ensuring Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Results and Discussion 38
5.1 Generating Ambiguous Figure-Ground Images . . . . . . . . . . . . . . . . 38
5.1.1 Timing Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.2 Shape Matching Metric Evaluation . . . . . . . . . . . . . . . . . . 40
5.1.3 Performance of Partial Shape Matching . . . . . . . . . . . . . . . 40
5.1.4 Ranking Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.5 Figure-Ground Ambiguity Test . . . . . . . . . . . . . . . . . . . . 42
5.1.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Image-based Paper Pop-up Design . . . . . . . . . . . . . . . . . . . . . . 44
6 Conclusion and Future Work . . . . . . . . . . . . . . . . 47
6.1 Generating Ambiguous Figure-Ground Images . . . . . . . . . . . . . . . . 47
6.2 Image-based Paper Pop-up Design . . . . . . . . . . . . . . . . . . . . . . 48
A The Detailed Timings of the Results . . . . . . . . . . . . . . . . . . . . . . 49
Bibliography . . . . . . . . . . . . . . . . . . . . . . 50

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