From a 2D Shape to a String Structure Using the Symmetry Set 1st edition by Arjan Kuijper, Ole Fogh Olsen, Peter Giblin, Philip Bille, Mads Nielsen ISBN 3540219835 9783540219835

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ISBN 10:       3540219835
ISBN 13: 978-3540219835
Author:        Arjan Kuijper, Ole Fogh Olsen, Peter Giblin, Philip Bille, Mads Nielsen 
 

Many attempts have been made to represent families of 2D shapes in a simpler way. These approaches lead to so-called structures as the Symmetry Set (<span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="SS”>SS) and a subset of it, the Medial Axis (<span id="MathJax-Element-2-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="MA”>MA).

In this paper a novel method to represent the <span id="MathJax-Element-3-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="SS”>SS as a string is presented. This structure is related to so-called arc-annotated sequences, and allows faster and simpler query algorithms for comparison and database applications than graph structures, used to represent the <span id="MathJax-Element-4-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="MA”>MA.

Example shapes are shown and their data structures derived. They show the stability and robustness of the <span id="MathJax-Element-5-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="SS”>SS and its string representation.

From a 2D Shape to a String Structure Using the Symmetry Set 1st Table of contents:

• Introduction

  • 1.1 Background and Motivation
  • 1.2 Symmetry and Its Importance in Shape Analysis
  • 1.3 Challenges in Shape Representation and Symmetry Detection
  • 1.4 From Shape to String: The Proposed Transformation
  • 1.5 Contributions and Objectives of the Paper
  • 1.6 Structure of the Paper

• Related Work

  • 2.1 Shape Representation Techniques in Computational Geometry
  • 2.2 Symmetry Detection in 2D Shapes
  • 2.3 String-Based Representations in Shape Analysis
  • 2.4 Applications of Symmetry in Pattern Recognition and Computer Vision
  • 2.5 Limitations of Existing Methods

• Mathematical Background

  • 3.1 Definitions and Concepts of Symmetry in 2D Shapes
  • 3.2 The Symmetry Set: Properties and Computation
  • 3.3 Types of Symmetry: Reflective, Rotational, Translational
  • 3.4 Geometric Transformations and Their Relation to Symmetry
  • 3.5 Mathematical Foundations for Shape to String Mapping

• Symmetry Set Computation

  • 4.1 Overview of Symmetry Set Extraction
  • 4.2 Algorithms for Computing the Symmetry Set of a 2D Shape
  • 4.3 Handling Complex Shapes with Multiple Symmetries
  • 4.4 Robustness and Stability of Symmetry Detection
  • 4.5 Case Studies: Symmetry Sets of Simple and Complex Shapes

• From Shape to String Structure

  • 5.1 The Concept of String Representation of Shapes
  • 5.2 Encoding Symmetries as Strings: A Mapping Approach
  • 5.3 Structural Properties of the String Representation
  • 5.4 Transformation Rules: How Shape Features Map to Strings
  • 5.5 Examples and Illustrations of Shape to String Conversion

• Applications of the String Structure

  • 6.1 Shape Classification and Recognition Using String Structures
  • 6.2 Symmetry-Based Matching and Comparison of Shapes
  • 6.3 Efficient Shape Retrieval Systems
  • 6.4 Robotics and Path Planning Using Symmetry Information
  • 6.5 Medical Imaging: Shape Representation and Diagnosis

• Algorithm Implementation

  • 7.1 Preprocessing: Shape Representation and Symmetry Detection
  • 7.2 Symmetry Set Computation Algorithm
  • 7.3 String Construction from Symmetry Data
  • 7.4 Computational Complexity and Optimization Considerations
  • 7.5 Software Implementation and Tools Used

• Experimental Setup and Evaluation

  • 8.1 Datasets and Test Shapes for Evaluation
  • 8.2 Performance Metrics: Accuracy, Efficiency, and Robustness
  • 8.3 Benchmarking Against Traditional Shape Representation Methods
  • 8.4 Results: Symmetry Set Detection and String Conversion
  • 8.5 Evaluating the Effectiveness of String Structures in Applications

• Results and Discussion

  • 9.1 Visual Examples of Shape to String Transformation
  • 9.2 Impact of Symmetry Set on Shape Analysis Performance
  • 9.3 Comparison with Other Shape Representation Techniques
  • 9.4 Robustness to Shape Variations and Noise
  • 9.5 Insights into the Practical Use of String Structures

• Challenges and Future Directions

  • 10.1 Handling Non-Simple and Irregular Shapes
  • 10.2 Extending the Method to 3D Shapes and Higher Dimensions
  • 10.3 Real-Time Computation and Scalability for Large Datasets
  • 10.4 Integration with Machine Learning for Automated Shape Recognition
  • 10.5 Future Research in Symmetry and String-Based Shape Representation

• Conclusion

  • 11.1 Summary of Key Contributions and Findings
  • 11.2 Practical Implications of the Proposed Method
  • 11.3 Limitations and Future Work in Shape to String Transformation
  • 11.4 Closing Remarks

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