Shape is what is left after removing position, rotation and size.

Removing position, rotation and size

On a plate, each seed has a position, an orientation and a size. None of this is shape. If the seed rolls to one side, if the plate turns or if the photo is taken closer, the shape stays the same. So the three can be removed, one at a time, to see what is left.

1 · On the plate 2 · Without position 3 · Without rotation 4 · Without size See the four steps

The center used is the centroid, the balance point of the seed. The rotation lays flat the major axis of the inertia ellipse, the direction in which length is measured. Size is removed by dividing everything by the length. After the three steps the 24 seeds become a bundle, and what still separates one from another is shape alone: one plumper, another more curved, another with a finer tip.

Size in pixels misleads

The orchid set in the kit has 38 crops of tetrazolium plates, with 1,614 seeds outlined in the set's label and marked as viable (red embryo) or non-viable. Setting aside those that touch the edge of the crop leaves 1,407: 775 viable and 632 non-viable. When the length of each one is measured in pixels, the viable seeds look much larger, with a median of 144 px against 90 px.

But the crops are not at the same scale. The median length per crop ranges from 69 to 189 px, and the classes are not spread evenly among them: some crops hold almost only non-viable seeds, which look small in the image. With no ruler in the photo, there is no way to know how much of this is larger seeds and how much is different magnification. When each length is divided by the median of its crop, the difference nearly vanishes (right-hand plot). In the 16 crops with at least five seeds of each class, the viable seed comes out, at the median, only 5% longer than the non-viable seed from the same crop.

The mean shape of each class

With position, rotation and size removed, the seeds can be overlaid and averaged. Each aligned outline becomes a mask, the masks are summed, and the curve where half of the seeds cover the point is the mean shape. It is the same idea as the mean seed shape and the shape space of the 145 silhouettes in Learn, now with the orchids separated by class.

The mean viable seed is slightly more elongated than the non-viable one: a length-to-width ratio of 4.2 against 3.8. The difference exists, but it is small next to the variation within each class, which the 25% and 75% curves show. Shape alone does not decide viability. The tetrazolium test exists precisely because the decision lies in the color of the embryo, not in the outline.

Three numbers for shape

A whole shape does not fit in a spreadsheet column. A shape descriptor summarizes the outline in a number that does not change with position, rotation or size. Three of them are enough to start, and each one sees something different.

Elongation is the length divided by the width. Here it comes from the rotating caliper around the seed, the Feret diameters: the largest opening over the smallest. It equals 1 for a circle and is close to 4 for an orchid seed.

Circularity compares the area with the perimeter, \(4\pi A/P^2\). It equals 1 for a circle and for no other figure, and it drops when the shape elongates or when the edge becomes indented. An ellipse that is twice as long as it is wide gives about 0.84.

Solidity is the area of the seed divided by the area of its convex hull, the rubber band stretched around it: \(A/A_{\text{fecho}}\). It equals 1 for a seed with no concavity and drops when the seed is curved, when it has a notch, or when two touching seeds become a single object.

It is worth looking at the row sorted by each of the three descriptors. Elongation and circularity move together, because a long seed has a large perimeter for the area it covers. Solidity takes another path: a seed can be short and low in solidity (curved like a comma) or long and fully solid (straight like a spindle). So none of the three replaces the others.

In SeedCounter

Go deeper

To see it in motion: the rotating caliper around the seed, the seed drawn with circles and the post A seed is a sum of circles, with the video.

Further reading

1. Kendall D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society 16(2), 81–121. doi:10.1112/blms/16.2.81
2. Dryden I. L., Mardia K. V. (2016). Statistical Shape Analysis, with Applications in R. 2nd ed. Wiley.
3. Tanabata T., Shibaya T., Hori K., Ebana K., Yano M. (2012). SmartGrain: high-throughput phenotyping software for measuring seed shape through image analysis. Plant Physiology 160(4), 1871–1880. doi:10.1104/pp.112.205120
4. Koklu M., Ozkan I. A. (2020). Multiclass classification of dry beans using computer vision and machine learning techniques. Computers and Electronics in Agriculture 174, 105507. doi:10.1016/j.compag.2020.105507

Grain phenotyping programs measure these same numbers: SmartGrain measures length, width, area, perimeter and circularity [3], and the dry bean dataset of Koklu and Ozkan has 16 dimension and shape descriptors for 13,611 grains [4].