傳統的圖片被數位與顯影，並以矩形網格的方式呈現，然而這樣的方式也延伸出例如網格的連續性或像素間距等議題。因此，傳統的圖片經由不明確的限制條件所產生的矩形網格來解析較低的影像處理到較佳的解析度。其數學形態圖像處理概念常以二進制、灰度和模糊規律六邊形等方法，以研究低分辨率圖像和多足步行機器人等特定主題。其研究提出六邊形採樣的圖像方法，其方法包括網格的形成和圖像處理等考量。其中，圖像處理包括二進制、灰度、不同尺寸與形狀、定向模糊結構元素、雜訊除去和邊緣檢測的一連串應用模糊形態學運算等等，以便使低分辨率圖像邊緣檢測來探討多足步行機器人的行走等策略。邊緣檢測之目的是使多足機器人避開站立於邊緣、缺口或障礙等場所，其低計算能力對於即時影像應用的效果非常顯著。然而除了行走策略考量之外，當損壞一條腿時，則機器人需要一些替代策略來完成任務。因此，本研究提出了一種可移動的滑動腿設計來解決這個問題。其故障腿可以分離或其他腿可以經由協調，來獲得更好的替代步態配置的指令，使故障腿滑動到更好的位置。基於腿部序列，步幅，縱向穩定性和效率等考量，以評估替代步態的進行。最後提出故障腿的處理建議表，以顯示不同步態序列漸進的效率。這些表格可以提供替代的受傷步態，以提供故障腿資訊的處理選項，以便給多足機器人腿部嚴重故障時的一些後續處理策略。因此，經由此建議表和程序以便使多足機器人可以克服任何故障事件，來保持穩定和效率。所以此研究目的是運用性能評價，來證明整合二進制、灰度、六邊形網格與模糊形態學等圖像處理方法，以針對低分辨率圖像和多足機器人行走策略之邊緣檢測的探討，並呈現更加穩健的應用結果。

Traditionally images are digitized, processed and displayed in a rectangular grid. But rectangular grid has many inherent ambiguities such as continuity, inter-pixel distance, etc. These ambiguities restrict rectangular grid to obtain better results in low resolution image processing. This study considers a particular topic of mathematical morphology image processing based on binary, grayscale and fuzzy discipline for hexagonally sampled low resolution images and multi-legged robot walking strategies. The proposed research presents a methodology for hexagonally sampled images that consist of processing, and display of processed images on hexagonal grid. Image processing includes binary, grayscale and fuzzy morphological operations with different sizes, shapes and directional fuzzy structuring elements with an application of noise removal and edge detection. Edge detection method for low resolution images are very significant for multi-legged robot walking strategy with low computation power in real time applications. Edge detection method can aid multi-legged robot to avoid standing zone edges, gaps/obstacles etc. Moreover, if a leg is damaged, then robot needs some alternative strategies to complete its mission. Thus, this study proposes a removable leg and sliding leg approach to solve this problem. A fault leg can be detaches and other legs can be slide to better position by the command of operator to get better alternative gait configuration. Based on leg sequence, stride length, longitudinal stability and efficiency, alternative gaits are evaluated. This study recommends tables for different gait sequence with progressive efficiency. These tables can provide options for alternative gait and information about certain damaged leg. Moreover, a procedure for a multi‐legged robot to complete its mission after serious leg failure is included. By taking the recommended tables and procedure, the multi-legged robot can overcome any fault event and maintain stability and efficiency. In addition, performance evaluation conducted to demonstrate that hexagonal grid structure coupled with binary, grayscale and fuzzy morphological image processing is more robust than the rectangular counterpart in many applications including edge detection for low resolution images, multi-legged robot walking strategy etc.

Acknowledgement + ii
摘要 + iii
Abstract + iv
Contents + vi
List of Figures + ix
List of Tables + xiv
Chapter 1 Introduction + 1
1.1 Image definition + 1
1.2 Edge Model Definition + 3
1.3 Why Do We Need Edge Detection + 4
1.4 Hexagonal Image & Mathematical Morphology + 5
1.5 Previous Research Work + 9
1.6 Motivation and Goals + 15
Chapter 2 Binary Morphology for Hexagonal Image + 17
2.1 Resampling (Rectangular to Hexagonal) + 18
2.2 Hexagonal Structuring Elements + 21
2.3 Hexagonal Binary Morphological Operator + 22
2.4 Noise Removal and Edge Detection + 22
2.5 Performance Evaluation + 26
Chapter 3 Grayscale Morphology for Hexagonal Image + 28
3.1 Hexagonal Structuring Elements + 28
3.2 Hexagonal Grayscale Morphological Operator+ 29
3.3 Edge Detection (Morphological Gradient) + 31
3.4 Edge Enhancement (Top-Hat Transformation) + 33
3.5 Performance Evaluation + 35
Chapter 4 Fuzzy Morphology for Hexagonal Image + 38
4.1 Methodology + 40
4.1.1 Image Resampling + 40
4.1.2 Image Fuzzification + 40
4.1.3 Fuzzy Morphology + 43
4.1.4 Noise Removal and Edge Detection + 45
4.2 Performance Evaluation + 48
Chapter 5 Better Walking Strategies for Multi-legged Robots + 53
5.1 Inverse Kinematics of Multi-legged Robots + 54
5.2 An Image based Walking strategy + 56
5.2.1 Construction of Discontinuous Terrain + 57
5.2.2 Simplified Forward Gait + 61
5.2.3 Rotational Gait (Around the Center of Gravity) + 63
5.2.4 Maximal Angle of Rotation + 63
5.2.5 Rotation around Any Point + 65
5.2.6 The Algorithm for Gait Selection + 67
5.2.7 Gait Strategy + 68
5.2.8 Gait Selection for Forward Walking + 70
5.2.9 Strategy for Walking + 71
5.3 Walking Strategy with Damaged Leg + 76
5.3.1 The Use of Removable Sliding Legs+ 76
5.3.2 Axial Stability for an Adjusted Gait+ 78
5.3.3 Alternative Gait Configuration+ 79
5.3.4 Comparison of Alternative Gait + 83
5.3.5 Fixed Position Adjustment (FP) + 85
5.3.6 Non Fixed Position Adjustment (NFP) + 86
5.3.7 Sliding gait in Circular Multi-legged Robot + 88
5.3.8 Procedure to resume its task after leg failure + 90
5.4 Comparison with Previous Research Work + 92
Chapter 6 Conclusion and Future Work + 98
6.1 Discussion + 98
6.2 Contributions + 99
6.3 Possible Future Work + 100
References + 101

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