lm3_mp_conversion_factor__sqrt_fit

One-parameter model predicting a herbarium sheet image's pixel-per-centimeter conversion factor from its size in megapixels. Default mp_conversion_factor model in LeafMachine3 >= 3.0.0.

$${\mathrm{c}\mathrm{f}}_{\mathrm{p}\mathrm{x}\mathrm{/}\mathrm{c}\mathrm{m}}=k\sqrt{\mathrm{M}\mathrm{P}},k=27.112084,\mathrm{M}\mathrm{P}=\frac{w\cdot h}{{10}^6}\backslash mathrm\{cf\}\_\{\backslash mathrm\{px/cm\}\}\; \backslash ;=\backslash ;\; k\backslash ,\backslash sqrt\{\backslash mathrm\{MP\}\},\; \backslash qquad\; k\; =\; 27.112084,\; \backslash qquad\; \backslash mathrm\{MP\}\; =\; \backslash frac\{w\; \backslash cdot\; h\}\{10^\{6\}\}$$

LM3 metadata

Field	Value
LM3 action	mp_conversion_factor
LM3_default_model	true
LM3_version	3.0.0 (minimum supported; later releases remain compatible)
Model key	sqrt_fit
Form	$\mathrm{c}\mathrm{f}=k\sqrt{\mathrm{M}\mathrm{P}}\backslash mathrm\{cf\}\; =\; k\backslash sqrt\{\backslash mathrm\{MP\}\}$, $k=27.112084k\; =\; 27.112084$ (in code: cf_px_per_cm = k * sqrt(megapixels))
Input	megapixels of the image, $\mathrm{M}\mathrm{P}=w\cdot h\mathrm{/}{10}^6\backslash mathrm\{MP\}\; =\; w\; \backslash cdot\; h\; /\; 10^\{6\}$, in the frame being measured
Output	pixels per cm in that same frame
Parameters	1
Fitted	2026-08-12

The form is exact under uniform resize. Scaling both sides of an image by $ss$ scales its megapixels by $s^{2}s^\{2\}$, so

$$k\sqrt{s^2\mathrm{M}\mathrm{P}}=s\cdot k\sqrt{\mathrm{M}\mathrm{P}}k\backslash ,\backslash sqrt\{s^\{2\}\backslash ,\backslash mathrm\{MP\}\}\; \backslash ;=\backslash ;\; s\; \backslash cdot\; k\backslash ,\backslash sqrt\{\backslash mathrm\{MP\}\}$$

and the prediction on the resized image is the original prediction times the resize factor. The model can therefore be evaluated in any frame.

Files

Folder	File	Use
json/	model.json	LM3 runtime artifact (k, fit metrics, ranges)
fit/	fit_mp_cf.py, fit_data.csv	reproduces model.json with numpy
docs/	fit_708_sheets_light.png, fit_708_sheets_dark.png	the fit figure, one per theme

SHA-256 of every file is in manifest.json.

Training data

708 human ruler measurements on 549 herbarium sheets from 48 herbaria (US 318, NY 106, O 45, BR 39, NCU 22, others 178). Ground truth is the manually measured length of 1 cm on each sheet's ruler (cm_1_avg) from the LeafMachine2 ruler quality-control set; image dimensions come from the same set; rows kept where 20 < cf < 400. Megapixels 14.6 to 36.2, conversion factor 88.8 to 172.5 px/cm. All rows are used for the fit; there is no train/val/test split. The table ships as fit/fit_data.csv (filename, herbarium code, width, height, megapixels, cf); specimen images are not distributed.

Fit

Least squares through the origin:

$$k=\frac{\sum _i\sqrt{{\mathrm{M}\mathrm{P}}_i}{\mathrm{c}\mathrm{f}}_i}{\sum _i{\mathrm{M}\mathrm{P}}_i}k\; \backslash ;=\backslash ;\; \backslash frac\{\backslash sum\_i\; \backslash sqrt\{\backslash mathrm\{MP\}\_i\}\backslash ;\backslash mathrm\{cf\}\_i\}\{\backslash sum\_i\; \backslash mathrm\{MP\}\_i\}$$

Unconstrained checks on the same rows agree with the form: fitting $\mathrm{c}\mathrm{f}=a{\mathrm{M}\mathrm{P}}^b\backslash mathrm\{cf\}\; =\; a\backslash ,\backslash mathrm\{MP\}^\{b\}$ gives $b=0.4904b\; =\; 0.4904$ (theory 0.5), and fitting $\mathrm{c}\mathrm{f}=k\sqrt{\mathrm{M}\mathrm{P}}+c\backslash mathrm\{cf\}\; =\; k\backslash sqrt\{\backslash mathrm\{MP\}\}\; +\; c$ gives $c=0.64c\; =\; 0.64$ px/cm (theory 0).

Evaluation

Data	n	RMSE (px/cm)	R²	Mean abs. error	p95	Max
fit rows (in-sample)	708	4.53	0.905	2.3%	7.5%	32.2%
20 independently hand-labeled sheets, each resized across 1 to 38.6 MP	480			3.7%	10.9%	13.0%

The second row's error is the same inside and outside the fitted 14.6 to 36.2 MP range. A two-parameter linear fit on the same 708 rows reaches RMSE 4.37 inside the range and is wrong by up to 193% outside it.

Figure

The 708 measurements and the fitted curve. The shaded band is the fitted megapixel range.

Use in LeafMachine3

Stage mp_conversion_factor, the first pipeline stage (no dependencies, CPU only, no image decode). It evaluates the model on the working image's megapixels and stores specimen.cf_px_per_cm_predicted_by_mp. ruler_cf uses that value as the megapixel anchor: a conversion factor measured from a ruler is published only if it agrees with the anchor within anchor_tol (default 0.25) or rests on enough ruler length to outvote it.

import json, math
from huggingface_hub import hf_hub_download
m = json.load(open(hf_hub_download("phyloforfun/lm3_mp_conversion_factor__sqrt_fit", "json/model.json")))
cf_px_per_cm = m["k"] * math.sqrt(width * height / 1e6)

Citation

Weaver, W. N. (2026) LeafMachine3. https://leafmachine.org