Measure Object Shape

Measure Object Shape adds the shape values of CellProfiler’s MeasureObjectSizeShape that objects don’t carry anyway: the convex area and CellProfiler’s solidity, the Euler number, and the radii. Area, perimeter, Feret diameters, axes and the other object metrics are always measured and need no step.

An object with a hole: the convex area counts the pixels whose centre lies in the convex hull (dashed); the Euler number is 1 piece minus 1 hole = 0; the radius of a pixel is its distance to the nearest pixel outside the object

It is a step of its own because the radii need a distance transform of every object, too slow to pay for in every analysis.

Parameters

Parameter	Description
ID	The step’s ID, unique in the project. Its values’ names in the results start with it (e.g. `shape_01`).
Objects	The objects measured.

Measurements

Six values per object:

Measurement	Unit	Description
Convex area	px²	The number of pixels whose centre lies in (or on) the object’s convex hull.
Solidity (convex area)	0-1	Area / convex area.
Euler number	-	The number of pieces minus the number of holes; 1 for a solid object.
Max radius	px	The largest distance of a pixel to the outside: about the radius of the largest disk that fits in.
Mean radius	px	The mean distance of the pixels to the outside.
Median radius	px	The median distance of the pixels to the outside.

How it works

Convex area and solidity

The convex hull is taken around the midpoints of the pixels’ edges (the points half a pixel left, right, above and below each pixel’s centre), as scikit-image’s convex_hull_image, which CellProfiler uses. The Convex area is the number of pixels whose centre lies inside or on this hull. It is a pixel count, never smaller than the area.

\text{Solidity} = \frac{A}{A_{\text{convex}}}

1 means convex; bays, notches and holes lower it. This solidity matches CellProfiler’s. It differs slightly from the Solidity every object carries, which divides by the area of the hull polygon around the pixel corners. For small objects the two can differ noticeably.

Euler number

The number of connected pieces minus the number of holes, with pixels connected through their 8 neighbours (so holes through 4). An object is usually one piece, so 1 means no hole, 0 one hole, -1 two holes. It is computed from the 2 × 2 pixel patterns of the mask (Gray’s bit quads), as scikit-image’s euler_number:

E = \frac{Q_1 - Q_3 - 2\,Q_D}{4}

with $Q_1$ the 2 × 2 windows with one object pixel, $Q_3$ those with three, and $Q_D$ those with two diagonal ones.

Radii

The radius of a pixel is its exact Euclidean distance (from pixel centre to pixel centre) to the nearest pixel that is not part of the object. A pixel on the outline therefore has radius 1. Holes count as outside. The image border does not: the object is measured as if it were surrounded by background. Max, Mean and Median radius are taken over all the object’s pixels; with an even number of pixels the median is the mean of the two middle values.

The Max radius is about the radius of the largest disk that fits into the object, so it gives half the thickness of elongated objects independent of their length.