Six numbers for a shape, and the convention behind each one.

Area: counting pixels or integrating the outline

In a segmented image, the seed is a set of pixels. The simplest area comes from counting them, \(A_{\text{px}} = N\), in px², which becomes mm² when multiplied by the square of the pixel side. The other area comes from the outline. If the edge is a polygon with vertices \((x_i, y_i)\), \(i = 0, \dots, n-1\), closed with \((x_n, y_n) = (x_0, y_0)\), the area follows from the shoelace formula:

The two areas do not give the same number, and the difference has an exact formula when the polygon passes through the centers of the edge pixels, such as the 8-neighbor trace that follows the edge. The vertices become points of an integer lattice, and Pick's theorem [6] applies.

The polygon through the centers loses half a pixel per edge pixel. On a digital ellipse of 221 × 64 px this removes 1.9% of the count; on one of 55 × 16 px, 7.9%. Neither area is wrong. The count has no bias when a pixel is included by its center, and the polygon measures the region half a pixel inward. What matters is knowing which of the two is in the spreadsheet, and not mixing them in the same calculation (this is what goes wrong in circularity, further below).

Perimeter: the length depends on the ruler

The perimeter is the most treacherous of the six measurements. A real edge is irregular at many scales, and the measured length grows as the ruler shrinks, which is Mandelbrot's coastline of Britain problem [5]. In an image the ruler is the pixel, and even a perfectly smooth edge becomes a staircase.

Freeman [1] encoded the edge as a chain of steps between neighboring pixels, each step in one of 8 directions, 45° apart. The natural length of the chain counts 1 for each horizontal or vertical step and \(\sqrt2\) for each diagonal:

This length overestimates, and we can work out by how much. Take a straight stretch of edge, of length 1, at angle \(\theta \in [0, \pi/4]\) to the horizontal. It advances \(\cos\theta\) horizontally and \(\sin\theta\) vertically. The shortest path on the 8-neighbor grid takes \(\sin\theta\) diagonal steps and \(\cos\theta - \sin\theta\) straight ones, and measures

It equals 1 at 0° and at 45°, and reaches \(\sqrt{4 - 2\sqrt2} \approx 1{,}0824\) at 22.5°: up to 8.2% more only because of the direction of the stretch.

The calculation was checked on real data. On 1,904 rice grains from a published dataset [10][11], the ratio between the perimeters from the two rulers, SeedCounter's and the dataset's, came to 1.054052, within 0.07% of the theoretical value: one of the two follows the 8-neighbor chain and the other corrects for the bias. The correction gets the mean right but not each stretch: with the factor 0.948, a stretch at 22.5° still comes out 2.6% too long and a horizontal stretch comes out 5.2% too short. Estimators that look at several steps at once do better [3][4].

Circularity and the isoperimetric inequality

Some values to keep at hand: the square gives \(\pi/4 \approx 0{,}785\); the ellipse whose length is twice its width, 0.84; the regular polygon with \(n\) sides, \(C_n = \pi / (n\tan(\pi/n))\), which gives 0.9986 with 48 sides. The orchid seed in the kit sits near 0.4.

In an image, however, circularity easily exceeds 1. Inequality (8) holds for a curve and the area of that same curve. When the area comes from one convention and the perimeter from another, it guarantees nothing. The table measures the same digital disk in four ways.

Circularity of digital disks, mean of 40 disks with the center drawn at random inside the pixel; in orange, values above 1. Our own simulation, rerun with the page's script.

The column for the polygon through the centers never exceeds 1, because there area and perimeter belong to the same curve. It stays below 1 and, for the larger disks, near \(1/1{,}0548^2 \approx 0{,}899\): the chain bias, squared. The 4-neighbor pixel edge tends to \(\pi^2/16 \approx 0{,}617\), because it measures the disk as a staircase of length \(8r\). And combining the pixel count, which extends to the outer edge, with a corrected perimeter, which passes through the centers, gives small circles with circularity near 1.4. Circularity can be compared only between measurements made with the same convention and at the same resolution.

Feret diameters and rotating calipers

The Feret diameter in a given direction is the opening of a caliper with its jaws perpendicular to that direction. With \(u_\theta = (\cos\theta, \sin\theta)\),

Since \(\langle p, u\rangle\) is linear in \(p\), the maximum and the minimum over the seed are the same as over its convex hull, and in a convex polygon they fall on vertices. The hull is therefore enough, and Andrew's monotone chain computes it in \(O(n\log n)\) [8].

The rotating calipers of Shamos and Toussaint use this: two supporting lines advance around the hull in one direction only and find both diameters in linear time once the hull is known [9]. The rotating caliper in Learn shows the same thing in motion. Feret is the most stable descriptor on the page: it depends only on the hull, and the hull ignores concavities and inward undulations.

Aspect ratio: two conventions

Length over width looks like a single calculation. There are two in use. The first is the caliper ratio, \(R_F = F_{\max}/F_{\min}\). The second comes from the inertia ellipse. The second-order central moments of the region,

form the covariance matrix of the region. With eigenvalues \(\lambda_1 \ge \lambda_2\), the uniform ellipse with the same moments has semi-axes \(a = 2\sqrt{\lambda_1}\) and \(b = 2\sqrt{\lambda_2}\), in the direction of the eigenvectors, and

For a polygon, the moments follow from Green's theorem, as the area does. The centroid, for example, is \(\bar x = \frac{1}{6A}\sum_i (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i)\), and there are formulas of the same kind for \(\mu_{20}\), \(\mu_{11}\) and \(\mu_{02}\). This is how the track's mean seed shape aligned the seeds.

In the orchid seeds of the kit the median of \(R_F/R_E\) is 1.01, but in 32% of them the two ratios differ by more than 10%, and which one is larger depends on the shape. In curved seeds (solidity below 0.8), the Feret ratio comes out 9% lower at the median, because the minimum width spans the whole bend, not just the grain. In the nearly convex, angular label polygons (solidity of 0.95 or more), it comes out 7% higher, for the same reason as the square. Eccentricity carries the same information as \(R_E\), but squeezed near 1 for long seeds: ratio 3 gives 0.943, ratio 4 gives 0.968, ratio 5 gives 0.980. To compare elongated seeds, the ratio discriminates better than eccentricity.

Solidity and the convex hull

The hull \(\operatorname{conv}\Omega\) is the smallest convex set that contains the seed, the rubber band stretched around it. \(S = 1\) when the seed is convex, and it drops with any concavity: curvature, notch, break, two touching seeds. It is the natural descriptor for questioning an outline, because a cluster almost always has lower solidity than an isolated seed of the same species.

Like the area, the hull has two conventions in use: the convex polygon through the pixel centers, or the set of pixels that falls inside it. The second is larger, and the solidity changes along with it. It is one of the two conventions that explain the divergences in rice, just below.

What each number ignores

The ISO 9276-6 standard gives rules and nomenclature for describing particle shape, with almost all of the definitions in two dimensions, precisely because the measurement usually comes from image analysis [12]. The table summarizes what each of the six ignores and what gets in its way.

All six are invariant to translation and rotation in the continuous plane. On the grid, not exactly: rotating the seed in the photo does not change the pixel count, apart from the edge, but it does change the chain perimeter by up to 8.2% on a straight stretch, by equation (5). The three without units do not need the ruler, and that is why the Shape track could compare crops with different magnifications.

With real seeds

The 1,407 orchid seeds in the kit that do not touch the edge of the crop, measured on the label polygon:

Real data Mean ± standard deviation and median by class. Descriptors without units only: the crops do not share the same magnification, and what is measured in pixels mixes seed and zoom (see Size is misleading).

The viable class comes out slightly more elongated and slightly less circular, as the track's mean seed shape already showed. Solidity calls for caution, because it measures the label as much as the seed. Polygons with 4 to 8 vertices can hardly represent a concavity, and those of 13 to 20 already record curves and tips:

Real data Medians by range of the number of vertices of the label polygon. Labels with 60 vertices or more follow the edge pixels, as a staircase, and carry the chain bias with them.

Within the same vertex range, the two classes are similar in solidity. Since the viable seeds were outlined with more vertices (median 12 versus 9), part of the solidity difference between the classes in the first table comes from the outline and not from the seed. A descriptor measures the outline it receives.

Instrument: the polygon that loses vertices

The instrument runs the whole path of a measurement. A reference shape, the outline of a real seed from the kit smoothed to a curve, or an exact figure, is drawn on a pixel grid. The edge is followed with 8 neighbors, and the trace is reduced to \(n\) vertices taken at equal intervals along it. At each stage the calculations are redone. Roughness adds undulations 3 to 12 px long to the edge, like those a real segmentation leaves.

Three things to look for. Without roughness and with many vertices, the trace perimeter stays near 5% above the reference, the Freeman factor of equation (6), and the trace circularity falls by 10 to 15%: the perimeter goes up and the area of the polygon through the centers goes down, by equation (3). With roughness, the polygon with few vertices cuts across the undulations, shortens the perimeter and inflates the circularity. And no \(n\) eliminates the error for every shape: the point where the curve crosses 1 changes with the shape, with the resolution and with the roughness. The area hardly moves with any of this; the perimeter, and with it the circularity, moves with everything.

In SeedCounter

Where it fails

References

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