arxiv: 2605.05778 · v2 · submitted 2026-05-07 · 🧬 q-bio.QM · cs.CG· cs.NA· math.NA

Recognition: 2 theorem links

· Lean Theorem

Planar morphometry via functional shape data analysis and quasi-conformal mappings

Hangyu Li , Gary P. T. Choi

Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:16 UTC · model grok-4.3

classification 🧬 q-bio.QM cs.CGcs.NAmath.NA

keywords planar morphometryfunctional data analysisquasi-conformal mappingsshape registrationbiological shape analysisleaf morphologyinsect wing

0 comments

The pith

A method combining elastic boundary registration with quasi-conformal interior extension captures coupled morphological variations in planar biological shapes more effectively than boundary-only or interior-only approaches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces FDA-QC, which registers closed planar curves using square-root velocity functions and elastic matching, then extends those correspondences across the full domain with quasi-conformal maps that can incorporate optional landmarks. This produces a single framework that supports both shape morphing and quantitative variation analysis while respecting both boundary and interior geometry. On leaf and insect-wing examples the combined description reveals morphological differences that separate boundary or interior methods miss, because it keeps the coupling between outer silhouette and inner features intact. A sympathetic reader would care because many life-science questions about growth and form depend on measuring how boundary changes relate to interior changes rather than treating them separately.

Core claim

FDA-QC registers planar curves by their square-root velocity functions under elastic matching, extends the resulting boundary correspondence to the interior via a quasi-conformal map, and thereby yields a unified representation that quantifies shape variation while allowing controlled morphing; experiments on leaf and insect-wing datasets show this joint boundary-interior description outperforms purely boundary-based or interior-based alternatives.

What carries the argument

FDA-QC pipeline: elastic registration of square-root velocity functions on the boundary followed by quasi-conformal extension to the interior domain.

Load-bearing premise

That extending boundary correspondences via quasi-conformal mappings accurately preserves and reveals biologically meaningful interior variations without introducing artifacts or requiring extensive landmark constraints.

What would settle it

Apply the method to a set of planar shapes whose true interior deformation is known from independent imaging or simulation; if the recovered interior variation deviates systematically from the known deformation while boundary error remains small, the central claim fails.

Figures

Figures reproduced from arXiv: 2605.05778 by Gary P. T. Choi, Hangyu Li.

**Figure 1.** Figure 1: The proposed FDA-QC method integrating functional shape data analysis (FDA) and quasi-conformal (QC) mapping techniques for planar morphometry. (a) A Urtica dioica leaf (left) and a Euonymus japonicus leaf (right) adapted from [22]. (b) Boundary alignment achieved using the FDA technique, where homologous points along the two closed curves are matched through an optimal reparameterization. (c) Planar corre… view at source ↗

**Figure 2.** Figure 2: An illustration of the basic concepts in functional shape data analysis and quasi-conformal theory. (a) Two curves in the Euclidean space can be represented as the square-root velocity functions (SRVF) in the function space (also known as the pre-shape space), in which the optimal registration between them can be effectively obtained. (b) A quasi-conformal map f between two planar structures can be visuali… view at source ↗

**Figure 3.** Figure 3: Mapping a pair of Acer palmatum Japanese maple leaf shapes using the proposed FDA-QC method. (a) FDA-based elastic boundary registration between a source leaf shape (red) and target leaf shape (blue). The black straight lines show the correspondence of several sample points established by the SRVF elastic registration. (b) A triangle mesh of the source leaf shape and the FDA-QC mapping result of it onto t… view at source ↗

**Figure 4.** Figure 4: Shape morphing of Arabidopsis leaves by the proposed FDA-QC framework. Here, τ = 0 corresponds to the source leaf shape, τ = 0.2, 0.4, 0.6, 0.8, 1 show the morphing results of it towards a target leaf shape. The boundary at each τ is obtained from the SRVF geodesic in the function space, while the interior deformation is generated by reconstructing a QC map from the interpolated Beltrami coefficient µτ = τ… view at source ↗

**Figure 5.** Figure 5: Local area and orientation change during the temporal development of an Arabidopsis plant captured by our FDA-QC morphing framework. Each column corresponds to one temporal transition between different stages in view at source ↗

**Figure 6.** Figure 6: Boundary and landmark extraction for honey bee (Apis mellifera) wing shape analysis. (a) Given a honey bee wing specimen, we extract the outer boundary curve (highlighted in green) and 16 stable venation branch/intersection points (highlighted in red). (b) Four examples of honey bee wing shapes discretized as triangle meshes. For each wing, the wing boundary is highlighted in black, and the 16 landmarks ar… view at source ↗

**Figure 7.** Figure 7: FDA-QC workflow for honey bee wing (Apis mellifera) shape analysis. Given two wing shapes to be compared, we first extract the boundary and interior landmarks for each of them as described in view at source ↗

**Figure 8.** Figure 8: MDS embedding of the 48 honey bee (Apis mellifera) wing specimens using D(β ∗) with β ∗ = 0.80. Marker shapes indicate geographic labels (AT, MD, SI, HR), and marker colors represent the clustering result obtained by the FDA-QC framework. The embedding shows compact AT and MD groups and an overlapping SI–HR sector, consistent with the European honey bee ecology. obtained in our original formulation. This a… view at source ↗

**Figure 9.** Figure 9: MDS plots of the honey bee wing clustering results achieved using the spectral clustering method. (a) The result with k = 3 clusters (optimal β ∗ = 0.85). (b) The result with k = 4 clusters (optimal β ∗ = 0.9). Marker shapes indicate geographic labels (AT, MD, SI, HR), and marker colors represent the clustering result obtained by spectral clustering. boundary deformation, described through the FDA-based el… view at source ↗

read the original abstract

The study of shapes is one of the most fundamental problems in life sciences. Although numerous methods have been developed for the morphometry of planar biological shapes over the past several decades, most of them focus solely on either the outer silhouettes or the interior features of the shapes without capturing the coupling between them. Moreover, many existing shape mapping techniques are limited to establishing correspondence between planar structures without further allowing for the quantitative analysis or modelling of shape changes. In this work, we introduce FDA-QC, a novel planar morphometry method that combines functional shape data analysis (FDA) techniques and quasi-conformal (QC) mappings, taking both the boundary and interior of the planar shapes into consideration. Specifically, closed planar curves are represented by their square-root velocity functions and registered by elastic matching in the function space. The induced boundary correspondence is then extended to the entire planar domains by a quasi-conformal map, optionally with landmark constraints. Moreover, the proposed FDA-QC method can naturally lead to a unified framework for shape morphing and shape variation quantification. We apply the FDA-QC method to various leaf and insect wing datasets, and the experimental results show that the proposed combined approach captures morphological variation more effectively than purely boundary-based or interior-based descriptions. Altogether, our work paves a new way for understanding the growth and form of planar biological shapes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

FDA-QC pairs SRVF boundary registration with QC interior extension for a unified morphing and variation framework on planar shapes, but the superiority claim over separate methods lacks quantitative backing.

read the letter

The core contribution here is a pipeline that registers closed planar curves using square-root velocity functions and elastic matching, then extends the correspondence across the domain with a quasi-conformal map, optionally guided by sparse landmarks. This produces both a shape morph and a way to quantify variation in one go, applied to leaf and insect wing outlines. The combination itself is the main novelty; it is not a routine add-on to existing FDA or QC work, and it gives a clean way to move from boundary alignment to a full planar map without separate interior pipelines. The datasets are appropriate for the claim, and the optional landmark handling is a practical touch for biological cases where a few interior points are known. The math stays grounded in established techniques, which keeps the method reproducible if the code is released. The soft spot is the evaluation. The abstract states that the combined approach captures morphological variation more effectively than boundary-only or interior-only methods, yet it supplies no numbers, no baseline comparisons, no error bars, and no description of how effectiveness was scored. That makes the central advantage hard to judge from the given information. The stress-test concern also lands: the interior is filled by the QC extension rather than pulled from actual image data such as venation or texture, so any claimed interior advantage is really an assumption that the chosen map (minimal dilatation or zero Beltrami) aligns with biologically relevant features. If it does not, the result is smooth interpolation rather than a true hybrid gain. This work is aimed at researchers in geometric morphometrics and computational biology who already use registration tools and want a single framework for both morphing and variation analysis. A reader focused on planar shape methods could extract useful implementation ideas even if the validation needs strengthening. I would send it to peer review; the technical setup is coherent and the application area fits, but the experiments will require clearer metrics and direct comparisons to support the claims.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces FDA-QC, a planar morphometry framework that represents closed curves via square-root velocity functions (SRVF), registers them by elastic matching in function space, and extends the resulting boundary correspondence to the full planar domain using quasi-conformal mappings (optionally with sparse landmarks). It claims this hybrid method captures both boundary and interior morphological variation more effectively than purely boundary-based or interior-based approaches, enables unified shape morphing and variation quantification, and demonstrates superiority on leaf and insect wing datasets.

Significance. If the experimental superiority claim is substantiated with quantitative metrics, the method would supply a diffeomorphic, landmark-optional pipeline that couples boundary registration with interior extension, offering a practical advance for biological morphometry studies that require consistent interior correspondences without direct interior feature extraction.

major comments (2)

[Experimental results] Experimental results section: the claim that FDA-QC 'captures morphological variation more effectively' than boundary-only or interior-only methods is stated without any quantitative metrics (e.g., Procrustes distances, variance explained, registration error, or comparison to baselines such as pure SRVF or landmark-based TPS), error bars, or statistical tests; the superiority therefore rests on qualitative visual inspection alone.
[Method (QC extension)] Quasi-conformal extension subsection: the construction registers only the boundary via SRVF + elastic matching and extends via QC (minimal dilatation or zero Beltrami) without using any interior intensity, texture, or venation data; the assertion that the method 'takes both the boundary and interior into consideration' therefore depends on the unverified premise that the chosen QC map aligns with biologically meaningful interior features rather than merely producing a smooth diffeomorphism.

minor comments (1)

[Abstract] Abstract and dataset description: the number of samples, species, and exact leaf/wing datasets are not specified, making reproducibility and scope assessment difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments, which help us improve the clarity and rigor of the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: Experimental results section: the claim that FDA-QC 'captures morphological variation more effectively' than boundary-only or interior-only methods is stated without any quantitative metrics (e.g., Procrustes distances, variance explained, registration error, or comparison to baselines such as pure SRVF or landmark-based TPS), error bars, or statistical tests; the superiority therefore rests on qualitative visual inspection alone.

Authors: We agree that quantitative support is needed to strengthen the superiority claims. In the revised manuscript we will add direct comparisons using Procrustes distances between registered shapes, the percentage of variance explained by the first few principal components of the FDA-QC representations, and registration error metrics against baselines (pure SRVF boundary registration and landmark-based TPS). We will also report standard errors and perform paired statistical tests (e.g., Wilcoxon signed-rank) to assess significance. These additions will replace reliance on visual inspection alone. revision: yes
Referee: Quasi-conformal extension subsection: the construction registers only the boundary via SRVF + elastic matching and extends via QC (minimal dilatation or zero Beltrami) without using any interior intensity, texture, or venation data; the assertion that the method 'takes both the boundary and interior into consideration' therefore depends on the unverified premise that the chosen QC map aligns with biologically meaningful interior features rather than merely producing a smooth diffeomorphism.

Authors: The FDA-QC pipeline considers the interior by using the boundary correspondence to induce a diffeomorphic extension over the entire planar domain via quasi-conformal mapping. This guarantees a consistent, orientation-preserving mapping of interior points that can be used for subsequent morphing and variation analysis, even when interior features are not directly observed. The minimal-dilatation QC map is chosen precisely because it provides a canonical, smooth interpolation that respects the boundary registration. We will revise the relevant subsection to clarify this hybrid construction, explicitly state the modeling assumption that the boundary-driven extension is biologically plausible for the leaf and wing datasets, and add a short discussion of limitations when interior landmarks or textures are available. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines FDA-QC as boundary registration via SRVF elastic matching followed by QC extension to the domain (optionally with landmarks). The central claim that the hybrid captures morphological variation more effectively rests on experimental results from leaf and wing datasets rather than any equation or definition that reduces the output to the inputs by construction. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. The derivation remains self-contained against external benchmarks and established FDA/QC techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard properties of square-root velocity functions for elastic curve registration and of quasi-conformal mappings for domain extension; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Quasi-conformal mappings can extend a boundary correspondence to the interior domain while preserving orientation and allowing optional landmark constraints.
Invoked when the induced boundary correspondence is extended to the entire planar domain.

pith-pipeline@v0.9.0 · 5549 in / 1205 out tokens · 39531 ms · 2026-05-14T22:16:02.623579+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

closed planar curves are represented by their square-root velocity functions and registered by elastic matching... extended to the entire planar domains by a quasi-conformal map
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

shape dissimilarity... dQC = mean(|μ|)... maximal dilatation K(f)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

[1]

D. W. Thompson,On growth and form. Cambridge University Press, 1917

work page 1917
[2]

Shape manifolds, procrustean metrics, and complex projective spaces,

D. G. Kendall, “Shape manifolds, procrustean metrics, and complex projective spaces,”Bulletin of the London Mathematical Society, vol. 16, no. 2, pp. 81–121, 1984

work page 1984
[3]

Size and shape spaces for landmark data in two dimensions,

F. L. Bookstein, “Size and shape spaces for landmark data in two dimensions,” Statistical Science, vol. 1, no. 2, pp. 181–222, 1986

work page 1986
[4]

Zelditch, D

M. Zelditch, D. Swiderski, and H. D. Sheets,Geometric morphometrics for biologists: a primer. Academic Press, 2012

work page 2012
[5]

F. L. Bookstein,The measurement of biological shape and shape change, vol. 24. Springer Science & Business Media, 2013

work page 2013
[6]

D. X. Gu and E. Saucan,Classical and discrete differential geometry: theory, applications and algorithms. CRC Press, 2023

work page 2023
[7]

Leaf growth is conformal,

K. Alim, S. Armon, B. I. Shraiman, and A. Boudaoud, “Leaf growth is conformal,” Physical Biology, vol. 13, no. 5, p. 05LT01, 2016. 27

work page 2016
[8]

Conformal growth of Arabidopsis leaves,

G. Mitchison, “Conformal growth of Arabidopsis leaves,”Journal of Theoretical Biology, vol. 408, pp. 155–166, 2016

work page 2016
[9]

Minimizing the elastic energy of growing leaves by conformal mapping,

A. Dai and M. B. Amar, “Minimizing the elastic energy of growing leaves by conformal mapping,”Physical Review Letters, vol. 129, no. 21, p. 218101, 2022

work page 2022
[10]

Global constraints within the developmental program of the Drosophila wing,

V. Alba, J. E. Carthew, R. W. Carthew, and M. Mani, “Global constraints within the developmental program of the Drosophila wing,”Elife, vol. 10, p. e66750, 2021

work page 2021
[11]

Planar morphometry, shear and optimal quasi- conformal mappings,

G. W. Jones and L. Mahadevan, “Planar morphometry, shear and optimal quasi- conformal mappings,”Proceedings of the Royal Society A, vol. 469, no. 2153, p. 20120653, 2013

work page 2013
[12]

Planar morphometrics using Teichm¨ uller maps,

G. P. T. Choi and L. Mahadevan, “Planar morphometrics using Teichm¨ uller maps,”Proceedings of the Royal Society A, vol. 474, no. 2217, p. 20170905, 2018

work page 2018
[13]

Data-driven quasi-conformal morphodynamic flows,

S. Mosleh, G. P. T. Choi, and L. Mahadevan, “Data-driven quasi-conformal morphodynamic flows,”Proceedings of the Royal Society A, vol. 481, no. 2314, p. 20240527, 2025

work page 2025
[14]

Landmark matching via large deformation diffeomor- phisms,

S. C. Joshi and M. I. Miller, “Landmark matching via large deformation diffeomor- phisms,”IEEE Transactions on Image Processing, vol. 9, no. 8, pp. 1357–1370, 2000

work page 2000
[15]

Computing large deformation metric mappings via geodesic flows of diffeomorphisms,

M. F. Beg, M. I. Miller, A. Trouv´ e, and L. Younes, “Computing large deformation metric mappings via geodesic flows of diffeomorphisms,”International Journal of Computer Vision, vol. 61, no. 2, pp. 139–157, 2005

work page 2005
[16]

Computable elastic distances between shapes,

L. Younes, “Computable elastic distances between shapes,”SIAM Journal on Applied Mathematics, vol. 58, no. 2, pp. 565–586, 1998

work page 1998
[17]

On shape of plane elastic curves,

W. Mio, A. Srivastava, and S. Joshi, “On shape of plane elastic curves,”Inter- national Journal of Computer Vision, vol. 73, no. 3, pp. 307–324, 2007

work page 2007
[18]

2D-shape analysis using conformal mapping,

E. Sharon and D. Mumford, “2D-shape analysis using conformal mapping,” International Journal of Computer Vision, vol. 70, no. 1, pp. 55–75, 2006

work page 2006
[19]

Shape analysis of elastic curves in Euclidean spaces,

A. Srivastava, E. Klassen, S. H. Joshi, and I. H. Jermyn, “Shape analysis of elastic curves in Euclidean spaces,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 7, pp. 1415–1428, 2010. 28

work page 2010
[20]

Srivastava and E

A. Srivastava and E. P. Klassen,Functional and shape data analysis, vol. 1. Springer, 2016

work page 2016
[21]

Elastic shape analysis of planar objects using tensor field representations,

R. Zhang and A. Srivastava, “Elastic shape analysis of planar objects using tensor field representations,”Journal of Mathematical Imaging and Vision, vol. 63, no. 9, pp. 1204–1221, 2021

work page 2021
[22]

Leaf - UCI Machine Learning Repository

P. Silva and A. Maral, “Leaf - UCI Machine Learning Repository.” https: //archive.ics.uci.edu/dataset/288/leaf (accessed on October 18, 2025), 2014

work page 2025
[23]

Lehto,Quasiconformal mappings in the plane, vol

O. Lehto,Quasiconformal mappings in the plane, vol. 126. Springer-Verlag Berlin Heidelberg, 1973

work page 1973
[24]

L. V. Ahlfors,Lectures on quasiconformal mappings, vol. 38. American Mathe- matical Society, Providence, RI, 2006

work page 2006
[25]

Landmark-and intensity-based registration with large deformations via quasi-conformal maps,

K. C. Lam and L. M. Lui, “Landmark-and intensity-based registration with large deformations via quasi-conformal maps,”SIAM Journal on Imaging Sciences, vol. 7, no. 4, pp. 2364–2392, 2014

work page 2014
[26]

Tooth morphometry using quasi-conformal theory,

G. P. T. Choi, H. L. Chan, R. Yong, S. Ranjitkar, A. Brook, G. Townsend, K. Chen, and L. M. Lui, “Tooth morphometry using quasi-conformal theory,” Pattern Recognition, vol. 99, p. 107064, 2020

work page 2020
[27]

Teichm¨ uller mapping (T-map) and its applications to landmark matching registration,

L. M. Lui, K. C. Lam, S.-T. Yau, and X. Gu, “Teichm¨ uller mapping (T-map) and its applications to landmark matching registration,”SIAM Journal on Imaging Sciences, vol. 7, no. 1, pp. 391–426, 2014

work page 2014
[28]

Convergence of an iterative algorithm for Te- ichm¨ uller maps via harmonic energy optimization,

L. M. Lui, X. Gu, and S.-T. Yau, “Convergence of an iterative algorithm for Te- ichm¨ uller maps via harmonic energy optimization,”Mathematics of Computation, vol. 84, no. 296, pp. 2823–2842, 2015

work page 2015
[29]

F. P. Gardiner and N. Lakic,Quasiconformal Teichm¨ uller theory. No. 76, American Mathematical Society, 2000

work page 2000
[30]

Discovering communities through friendship,

G. Morrison and L. Mahadevan, “Discovering communities through friendship,” PLoS One, vol. 7, no. 7, p. e38704, 2012

work page 2012
[31]

Asymmetric network connectivity using weighted harmonic averages,

G. Morrison and L. Mahadevan, “Asymmetric network connectivity using weighted harmonic averages,”Europhysics Letters, vol. 93, no. 4, p. 40002, 2011. 29

work page 2011
[32]

A simple mesh generator in MATLAB,

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,”SIAM Review, vol. 46, no. 2, pp. 329–345, 2004

work page 2004
[33]

Computer vision-based recognition and distinction of Arabidopsis thaliana eco- types using supervised deep learning models,

R. Sari´ c, E.ˇCustovi´ c, A. Akagi´ c, M. Trt´ ılek, M. G. Lewsey, and J. Whelan, “Computer vision-based recognition and distinction of Arabidopsis thaliana eco- types using supervised deep learning models,”The Plant Phenome Journal, vol. 8, no. 1, p. e70041, 2025

work page 2025
[34]

A framework for few-shot language model evaluation

A. Oleksa, E. C˘ auia, A. Siceanu, Z. Puˇ skadija, M. Kovaˇ ci´ c, M. A. Pinto, and A. Tofilski, “Collection of wing images for conservation of honey bees (Apis mellifera) biodiversity in Europe, Zenodo.” https://doi.org/10.5281/zenodo. 7244070(accessed on October 18, 2025), 2022

work page doi:10.5281/zenodo 2025
[35]

Honey bee (Apis mellifera) wing images: a tool for identification and conservation,

A. Oleksa, E. C˘ auia, A. Siceanu, Z. Puˇ skadija, M. Kovaˇ ci´ c, M. A. Pinto, P. J. Rodrigues, F. Hatjina, L. Charistos, M. Bouga, J. Preˇ sern,˙I. Kandemir, S. Raˇ si´ c, S. Kusza, and A. Tofilski, “Honey bee (Apis mellifera) wing images: a tool for identification and conservation,”GigaScience, vol. 12, p. giad019, 2023

work page 2023
[36]

Molecular characterisation of indigenous Apis mellifera carnica in Slovenia,

S. Suˇ snik, P. Kozmus, J. Poklukar, and V. Meglic, “Molecular characterisation of indigenous Apis mellifera carnica in Slovenia,”Apidologie, vol. 35, no. 6, pp. 623–636, 2004

work page 2004
[37]

Population genetic structure of coastal Croatian honeybees (Apis mellifera carnica),

I. Mu˜ noz, R. Dall’Olio, M. Lodesani, and P. De la R´ ua, “Population genetic structure of coastal Croatian honeybees (Apis mellifera carnica),”Apidologie, vol. 40, no. 6, pp. 617–626, 2009

work page 2009
[38]

Morphological diversity of Carniolan honey bee (Apis mellifera carnica) in Croatia and Slovenia,

Z. Puˇ skadija, M. Kovaˇ ci´ c, N. Raguˇ z, B. Luki´ c, J. Preˇ sern, and A. Tofilski, “Morphological diversity of Carniolan honey bee (Apis mellifera carnica) in Croatia and Slovenia,”Journal of Apicultural Research, vol. 60, no. 2, pp. 326– 336, 2021

work page 2021
[39]

A review of methods for discrimination of honey bee populations as applied to European beekeeping,

M. Bouga, C. Alaux, M. Bienkowska, R. B¨ uchler, N. L. Carreck, E. Cauia, R. Chlebo, B. Dahle, R. Dall’Olio, P. De la R´ ua, A. Gregorc, E. Ivanova, A. Kence, M. Kence, N. Kezi´ c, H. Kiprijanovska, P. Kozmus, P. Kryger, Y. Le Conte, M. Lodesani, A. M. Murilhas, A. Siceanu, G. Soland-Reckeweg, A. Uzunov, and J. Wilde, “A review of methods for discriminati...

work page 2011
[40]

Comparison of apiculture and winter mortality of honey bee colonies (Apis mellifera) in Austria and Czechia,

R. Brodschneider, J. Brus, and J. Danihl´ ık, “Comparison of apiculture and winter mortality of honey bee colonies (Apis mellifera) in Austria and Czechia,” Agriculture, Ecosystems & Environment, vol. 274, pp. 24–32, 2019. 30

work page 2019
[41]

Ruttner,Biogeography and Taxonomy of Honeybees

F. Ruttner,Biogeography and Taxonomy of Honeybees. Springer, 1988

work page 1988
[42]

Influence of honey bee (Apis mellifera) breeding on wing venation in Serbia and neighbouring countries,

H. Kaur, N. Nedi´ c, and A. Tofilski, “Influence of honey bee (Apis mellifera) breeding on wing venation in Serbia and neighbouring countries,”PeerJ, vol. 12, p. e17247, 2024

work page 2024
[43]

Bakhchou, A

S. Bakhchou, A. Aglagane, A. Tofilski, F. Mokrini, O. Er-Rguibi, E. H. El Mouden, J. Machlowska, S. Fellahi, and E. H. Mohssine, “Morphological diversity of Moroc- can honey bees (Apis mellifera L. 1758): Insights from a geometric morphometric study of wing venation in honey bees from different climatic regions,”Diversity, vol. 17, no. 8, p. 527, 2025. 31

work page 2025