The Math Before the Model

Every forester or arborist we meet asks the same thing within about ten minutes of the conversation: How accurate is your data compared to ground truth?

Very fair question, and it has a straight answer if one is interested in the final numbers: in our validation work against field calipers we found every stem in the plot, held position accuracy to 0.45 metres RMSE, and saw no systematic bias in DBH.

But there's something much deeper in the answer, that as a company who put science in the forefront, we can't ignore.

When the tape isn't ground truth

Laser scan and a caliper aren't really comparable. If you put a tape around a stem, then scan the same stem with a laser, the numbers won't match. Neither instrument is broken. How big an irregular object measures depends on the scale you look at it from, and bark is about as irregular as surfaces get. It's the same reason a coastline gets longer the smaller your ruler is.

So the tape isn't ground truth either, it's a measurement with a scale baked into it, and the industry has spent a century agreeing to treat it as the standard. That agreement works, and we're not suggesting anyone to abandon it. It just depends on everyone using the same tool. The moment you change tools, you need to know what changed and why, or the new number doesn't help you.

Forty years on the problem

Bernhard Stoinski has been working on that problem for about 40 years.

He's a mathematician by training, in data analysis, modelling and logic, and his early research went after a fairly stubborn question: how do you write down professional judgement in a form a computer can actually use? He started with rule-based systems and moved to fuzzy databases, where a query comes back with degrees of similarity rather than a yes or a no. That sounds abstract until you've stood in a stand with someone experienced, they don't tell you a stem is sound or unsound, they tell you it looks alright, but come back and check it after a wet winter. You can't store that in a binary system.

From there Bernhard moved on to formalising decisions, and presented the framework to the German Mathematical Society in Karlsruhe. Trees were the first place he applied it, in risk management, and they've been his subject ever since. Volume calculation, standardising inspection, working out how to fold biological and structural and legal requirements into one assessment. He published through the nineties and two thousands in AFZ-Der Wald, spoke at the FLL road safety conferences and the SVK arboriculture seminars in Hannover, and taught on the European Tree Technician programme. That last one counts for a lot, he spent years in rooms with people who put their hands on trees for a living.

Then laser scanning arrived, and with it a harder problem than measurement: a raw scan of a stand is a few hundred million points that don't know what they belong to. Before you can measure anything you have to work out which points are ground and which are vegetation, where one crown ends and the next one starts, what's stem and what's branch. Get that wrong and everything after it is wrong too. A suppressed stem you failed to separate out isn't a rounding error, it's a tree missing from your stand table.

Which is where his early work turns out to have been useful preparation. Segmentation boundaries aren't crisp, there's no exact point where a stem becomes a branch, and no true line between two crowns that have grown into each other. Ask two experienced arborists to draw it and you'll get two answers, both of them defensible. Most systems deal with this by forcing every point into a hard yes or no and then living with the consequences. Bernhard was building systems for graded membership in the 1990s, long before anyone was pointing lasers at forests. The maths he was doing then fits the point cloud now, and he runs that work here.

He's still publishing. With Steffen Rust, Professor of Arboriculture at the University of Applied Sciences and Arts in Göttingen and one of the developers of stress wave tomography, he worked out how to identify species straight from structural features in the point cloud, hitting 96 per cent accuracy across four species. Other research groups now cite that paper when they benchmark their own. The most recent work, with Rust, Andreas Detter and others, uses a laser-based pulling test to measure how a stem inclines, bends and stiffens along its length. That's not a record of where the wood is. It's a record of how the tree handles load.

In total he's published dozens of papers and conference contributions, spread across fuzzy logic and expert systems, mathematical logic, arboricultural practice and standards, and more recently LiDAR and machine learning applied to trees.

Insight, not numbers

Which brings it back round to the question we started with: The honest answer isn't a single number, a number on its own just asks you to trust us. It's that we can show you where our figure differs from your caliper, by how much, and why. Scale, geometry, instrument. You can argue with that, and you can take it to a buyer who disputes your cruise.

Richard Hamming, who spent his career building the numerical methods the rest of computing was built on, put it in one line in 1962:

"The purpose of computing is insight, not numbers."

That's the distinction underneath all of this, and it isn't maths against machine learning. Machine learning is maths, we use it heavily and the results are extraordinary. But a model's structure comes out of the data, and the parameters inside it don't correspond to anything you can point at. It gives you an answer and a confidence score, what it can't give you is an account of itself, so when it's wrong it's wrong quietly and there's no thread to pull. A derivation works the other way round. The structure is stated first, every step is open, and anyone who disagrees can go and check it. On a timber sale that is the difference between a figure that holds up when someone leans on it and one that doesn't.

So the maths isn't sitting under the machine learning as a formality. It's what turns an output into something you can put your name to, it's also what makes the thing hold together at scale, across different sensors, different forests and different ground conditions, rather than working beautifully on the stand it was tuned for and falling apart on the next one.

There's a pattern here that's older than any of this: in 1912 Einstein had the physics of general relativity and no mathematics to write it down in. He went to his friend Marcel Grossmann, who went through the literature and came back with the answer: the problem had already been solved. Riemann on curved manifolds, Christoffel, Ricci and Levi-Civita on the calculus that went with them. Work done decades earlier, for its own sake, by people with no idea what it would eventually be for. Without it, no general relativity.

The maths is usually already there, it just tends to be finished long before anyone needs it. When Bernhard writes about his own career, he starts with the question that's run underneath all of it, through every field he's moved through:

How can observed and experienced reality be represented in a form that can be described mathematically, analysed logically, and made comprehensible and reproducible?

He was building self-learning systems on that principle in 1996, in a world with no laser scanners, no point clouds and no forestry application in sight. Thirty years on, it turns out to be the right maths for the problem we're solving. We're extremely happy to have such a strong foundation leading our scientific team.

Selected work

• Rust, S., Göcke, L., Liebisch, J., Coelho-Duarte, A. P., Sergio, A., Detter, A., Stoinski, B. A Novel Laser-Based Tree-Pulling Test Method to Measure Stem Inclination, Bending, and Spatially Resolved Structural Stiffness. Forests.
• Rust, S., Stoinski, B. (2024). Enhancing Tree Species Identification in Forestry and Urban Forests through Light Detection and Ranging Point Cloud Structural Features and Machine Learning. Forests, 15:188. doi:10.3390/f15010188
• Rust, S., Stoinski, B. Using Artificial Intelligence to Assist Tree Risk Assessment. Arboriculture & Urban Forestry, 48(2), 138–146.
• Rust, S., Stoinski, B. Enhancing Urban Tree Risk Assessment through LiDAR and Machine Learning. 23rd International Nondestructive Testing and Evaluation of Wood Symposium, Campinas.
• Stoinski, B. Extension of Category Theory using a PL0 Calculus Functor to Form Propositional Morphisms in Multi-Agent Systems. Logic Colloquium 2023, ASL European Summer Meeting, Milano.
• Stoinski, B. Standardisierung der Baumkontrolle. AFZ-Der Wald, 4, 46–49.
• Stoinski, B. The Use of Neuro-Fuzzy Databases as Self-Learning Expert Systems in Theory and Practical Application. EUFIT96, Fourth European Congress on Intelligent Techniques and Soft Computing, Aachen.