# Find Feature Shapes¶

## Group (Subgroup)¶

Statistics (Morphological)

## Description¶

This Filter calculates the second-order moments of each Feature in order to determine the principal axis lengths, principal axis directions, aspect ratios and moment invariant Omega3s. The principal axis lengths are those of a "best-fit" ellipsoid. The algorithm for determining the moments and these values is as follows:

1. For each Cell, determine the x, y and z distance to the centroid of the Feature that owns the Cell
2. For each Cell, calculate Ixx, Iyy, Izz, Ixy, Ixz and Iyz using the x, y and z distances determined in step 1.
3. Sum the individual Ixx, Iyy, Izz, Ixy, Ixz and Iyz values for all Cells belonging to the same Feature
4. Find the eigenvalues and eigenvectors of the 3x3 symmetric matrix defined by the 6 values calculated in step 3 for each Feature
5. Use the relationship of principal moments to the principal axis lengths for an ellipsoid, which can be found in [4], to determine the Semi-Axis Lengths
6. Calculate the Aspect Ratios from the Semi-Axis Lengths found in step 5.
7. Determine the Euler angles required to represent the principal axis directions in the sample reference frame and store them as the Feature's Axis Euler Angles.
8. Calculate the moment variant Omega3 as definied in [2] and is discussed further in [1] and [3]

None

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## Required Objects¶

Kind Default Name Type Component Dimensions Description
Cell Attribute Array FeatureIds int32_t (1) Specifies to which Feature each Cell belongs
Feature Attribute Array Centroids float (3) X, Y, Z coordinates of Feature center of mass
Attribute Matrix CellFeatureData Cell Feature N/A Feature Attribute Matrix of the selected Feature Ids

## Created Objects¶

Kind Default Name Type Component Dimensions Description
Feature Attribute Array AspectRatios float (2) Ratio of semi-axis lengths (b/a and c/a) for best-fit ellipsoid to Feature
Feature Attribute Array AxisEulerAngles float (3) Euler angles (in radians) necessary to rotate the sample reference frame to the reference frame of the Feature, where the prinicpal axes of the best-fit ellipsoid are (X, Y, Z)
Feature Attribute Array SemiAxisLengths float (3) Semi-axis lengths (a, b, c) for best-fit ellipsoid to Feature
Feature Attribute Array Omega3s float (1) 3rd invariant of the second-order moment matrix for the Feature, does not assume a shape type (i.e., ellipsoid)
Feature Attribute Array Volumes float (1) The volume of each Feature

## References ##¶

[1] Representation and Reconstruction of Three-dimensional Microstructures in Ni-based Superalloys, AFOSR FA9550-07-1-0179 Final Report, 20 Dec 2010.

[2] On the use of moment invariants for the automated classifcation of 3-D particle shapes, J. MacSleyne, J.P. Simmons and M. De Graef, Modeling and Simulations in Materials Science and Engineering, 16, 045008 (2008).

[3] n-Dimensional Moment Invariants and Conceptual Mathematical Theory of Recognition n-Dimensional Solids, Alexander G. Mamistvalov, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 8, AUGUST 1998, p. 819-831.

[4] M. Groeber, M. Uchic, D. Dimiduk, and S. Ghosh. A Framework for Automated Analysis and Simulation of 3D Polycrystalline Microstructures, Part 1: Statistical Characterization Acta Materialia, 56 (2008), 1257-1273.

## Example Pipelines¶

• (01) SmallIN100 Morphological Statistics
• (06) SmallIN100 Synthetic

Please see the description file distributed with this Plugin