Atomic Pattern Indexing via Triangulation Mesh Analysis
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Solution Overview
Problem
Current methods for indexing atomic patterns in crystal structures are complex, computationally expensive, and prone to errors due to their reliance on geometric phase analysis and nearest neighbor search algorithms, which fail to accurately handle curved directions and irregularities in crystal structures.
Innovation Solution
A method involving image processing to acquire and triangulate points representing atomic patterns, determining direction vectors through statistical analysis of triangle properties, and iteratively indexing points using a Delaunay triangulation algorithm and geometric transformation to align with orthonormal vectors, reducing algorithmic cost and improving accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If geometric phase analysis (GPA) procedure is used to determine direction vectors, then the indexing can be performed, but the implementation becomes complex and does not necessarily give good results for all types of crystal structure
Solution Approach 1:
The patent extracts the essential geometric information (direction vectors A and B) from the crystal structure image and separates the indexing process from the complex GPA procedure. By using a simplified geometric approach that directly identifies lattice points and determines direction vectors through basic geometric relationships, the method eliminates the need for complex Fourier transform-based GPA while maintaining indexing capability.
Solution Approach 2:
Instead of using GPA to first determine direction vectors and then perform indexing, the patent inverts the approach by directly identifying lattice points through geometric relationships and deriving direction vectors from these points. This reversal simplifies the overall process and improves adaptability to different crystal structures.
2Reliability
If nearest neighbor search algorithm is used for indexing, then the indexing can be performed, but the algorithmic cost becomes O(n2) which is computationally expensive
Solution Approach 1:
The patent segments the indexing process into two stages: first identifying lattice points using geometric relationships, then determining direction vectors from these points. This segmentation avoids the need for exhaustive nearest neighbor searches across all points, reducing computational complexity from O(n2) to a more efficient algorithm that processes points in a structured manner.
Solution Approach 2:
The patent performs preliminary identification of lattice points and determination of direction vectors before performing the actual indexing of all points. By pre-establishing the geometric framework (lattice points and direction vectors), the subsequent indexing operation becomes computationally efficient rather than requiring expensive nearest neighbor searches for each point.
3Ease of manufacture
If comparing actual structure to a regular mesh is used for indexing, then the indexing can be performed, but indexing errors occur when main direction is curved under deformation or when crystal has irregularities
Solution Approach 1:
The patent employs a dynamic approach by identifying lattice points and direction vectors directly from the actual crystal structure image rather than comparing to a fixed regular mesh. This allows the method to adapt to curved directions and irregularities in the crystal structure, maintaining accuracy even under deformation conditions where a static mesh comparison would fail.
Solution Approach 2:
The patent changes the fundamental parameter from a fixed regular mesh to dynamically identified lattice points and direction vectors. By allowing these parameters to be derived from the actual structure rather than imposed from a regular grid, the method can accommodate variations in crystal structure including curvature and irregularities, improving reliability while maintaining process simplicity.
Data Source
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AI summary
This process consists of: acquiring (104) an image; processing (108) the image to obtain a file of localized points (Pn); applying (110) a triangulation algorithm to the points in the file to obtain a triangular mesh, a point being connected to a neighboring point by a side of a triangle; determining (120) crystal direction vectors from the triangles in the mesh; and indexing (140) each point, by iterations from a point chosen as the origin, selecting a point indexed in the previous iteration as the central point, then checking whether a point neighboring the central point corresponds to a translation of the central point along one of the direction vectors, respectively along one of their opposites, and, if so, incrementing, respectively decrementing, the index of the central point along the principal direction associated with the translation vector to index the neighboring point from the indices of the central point.