Setting up the studio for #root phenotyping with @RhizoVision @JohnInnesCentre only a few hundreds photos 📸 ahead! good luck Acacia & @Looking_at_soil

Here's a neat trick from integral geometry - a beautiful topic that bridges geometry, probability and statistics. Let's say you have some curve with a random shape, possibly even self-intersecting. How can you measure its length? This isn't just a parlor trick - it has many practical applications. For example, the curve could be a strand of DNA or a twisted length of wire. You can think of the curve as a collection of tiny segments of course. You can then measure each segment and add up the results. OK, but you can go further and take the segments to be so small that they are almost like points. You can then add up the (red) "points". In practice, this is not altogether easier. So we need a more convenient mechanism for doing this. One way to do it is to drop lines that intersect with the shape and count the number of intersections. You can do this with a mesh or a grid too. The curve's length is the sum total of intersections n(ρ,θ) of all lines (in polar coords) with the curve (counting multiplicities). This is the beautiful Crofton formula: Length = 1/2 ∫∫ n(ρ,θ) dρ dθ The 1/2 is there because oriented lines are a double cover of un-oriented lines If this looks a little like the Radon Transform from tomography (CT scans and x-rays) it's because it is. The Radon transform can be viewed as a measure-theoretic generalization of Crofton's formula! Crofton's formula has higher dimensional analogs too that also allow for computing surface areas, not just arc lengths. It even generalizes to Riemannian surfaces; where the integral is over a natural measure on the space of geodesics. The classic text on the topic is Luis A. Santaló 'Integral Geometry and Geometric Probability' There's also a terrific set of introductory slides that are great as a starting point to learn about this beautiful subject. math.utah.edu/~treiberg/IntG…

Happy #WorldBiodiversityDay! Remember that soils contain a lot of biodiversity. They're probably the most biodiversity-dense ecosystem. And this biodiversity works for us.