# Shape Metrics Formulas

Shape metrics are extracted from binary images obtained from the segmentation of the 3D volumes.
Two types of primary shape metrics are extracted : 3D shape metrics, extracted from the segmented 3D volume, and 2D shape metrics, extracted from 2D projections of the segmented 3D volume.
Secondary shape metrics are also computed using the primary shape metrics.
For this experiment, the shape metrics are extracted and computed for two channels: actin and nucleus.

## General notation

L = Axis of interest.
ROI = Region Of Interest in image, the foreground object. It defines the voxels/pixels over which features are extracted.
N = Number of voxels/pixels in the ROI.
X dimension = dimension along X axis of the voxel/pixel in the image.
Y dimension = dimension along Y axis of the voxel/pixel in the image.
Z dimension = dimension along Z axis of the voxel in the image.

Each shape metric computation is associated with one formula. For metrics related to axis, the computation can yield up to six values per channel:

1. Three measures denoted with X, Y, Z and obtained from the image stack without orientation.
2. Three measures denoted with L1, L2, L3 and obtained from the oriented cell and correspond to gyration moments.

## Gyration tensors

To calculate the gyration tensors, Sij, the position vectors of each cell were centered about its mass (CM) to measure the deviation of the cell from its center and to neglect effects due to cell orientation: The principal moments (eigenvalues), L1, L2, and L3, of the gyration tensor, S, were calculated, where L1 < L2 < L3. The square roots of the eigenvalues (L10.5, L20.5, L30.5) are the characteristic semi-axis lengths (radii) of the ellipsoid that describes the shape of the cell, while the eigenvectors of S describe the cell’s original orientation.

# Primary metrics: 3D Shape metrics

These shape metrics are computed on the binary mask of the 3D image.

## Depth along axis

Caliper length along the axis of interest, computed projecting each foreground voxel on the axis of interest in physical dimension. The depth is the length of the segment formed between the minimum and the maximum of the projections.

Notation: L-Depth

Six values are computed over the six axis of interest.

## Square root of gyration moments

Computed from the gyration tensors, square root of gyration moments.

Notation: Sqrt(L)

Three values are computed over the three gyration moments.

## Surface Area

Surface area of the ROI, computed using the Marching Cube algorithm affecting local surface area to each cube configuration. The surface area of the object is the sum of all the computed local surface areas. For more details, please refer to the initial paper  .

Notation: Surface Area

## Center of mass

Coordinates of the center of mass of the ROI. Notation: Center of mass (x,y,z)

## Volume

Volume of the foreground object is the sum of the volumes of each voxel part of the ROI.

Notation: Volume

# Primary metrics: 2D Shape metrics

These shape metrics are computed on max projections of the segmented 3D image.

## Perimeter

Perimeter is computed from the max projection of the foreground object along the axis of interest. The computation is done using the Marching Square algorithm affecting local perimeter to each square configuration.
The perimeter of the projected object is the sum of all the computed local perimeters. For more details, please refer to the initial paper  .

Notation: L-Perimeter

Six values are computed over the six axis of interest.

## Area

Area is computed from the max projection of the foreground object along the axis of interest. The area is the sum of the areas of all the projected ROI pixels. Notation: L-Area

Six values are computed over the six axis of interest.

## Bounding Box Aspect Ratio

Bounding box aspect ratio is computed from the max projection of the foreground object along the axis of interest. The bounding box of the ROI is computed to determine the aspect ratio. Notation: L-Aspect Ratio

Six values are computed over the six axis of interest.

# Secondary metrics

These metrics are computed using the primary shape metrics extracted from the 3D segmented image and its projections.

## Ratio (L1,L3)

Ratio of shortest gyration moment to longest gyration moment. Notation: sqrt(L1)/sqrt(L3)

## Ratio (L2,L3)

Ratio of middle gyration moment to longest gyration moment. Notation: sqrt(L2)/sqrt(L3)

## Ratio (L1,L2)

Ratio of shortest gyration moment to middle gyration moment. Notation: sqrt(L1)/sqrt(L2)

## Ratio (L3,L1)

Ratio of longest gyration moment to shortest gyration moment. Notation: sqrt(L3)/sqrt(L1)

## Ratio (L3,L2)

Ratio of longest gyration moment to middle gyration moment. Notation: sqrt(L3)/sqrt(L2)

## Ratio (L2,L1)

Ratio of middle gyration moment to shortest gyration moment. Notation: sqrt(L2)/sqrt(L1) ## Asphericity Notation: Asphericity

## Acylindricity Notation: Acylindricity

## Anisotropy Notation: Anisotropy

## Nucleus to Actin Surface Area Ratio

Ratio of nucleus to actin surface area.

Notation: Nucleus to Actin Surface Area

## Nucleus to Actin Volume Ratio

Ratio of nucleus to actin volume.

Notation: Nucleus to Actin Volume

## Nucleus to Actin L1-Depth Ratio

Ratio of nucleus to actin L1-Depth.

Notation: Nucleus to Actin L1-Depth

## Nucleus to Actin Distance

Distance from center of mass of nucleus to center of mass of actin using Euclidian distance.

Notation: Nucleus to Actin Distance

# References

 ^ Lindblad, Joaquim, "Surface area estimation of digitized 3D objects using weighted local configurations", Image and Vision Computing , Volume 23, Issue 2, pp.111-122, Feb. 2005
https://www.sciencedirect.com/science/article/pii/S0262885604001507