Functional Data Morphometrics

• Functional Data Morphometrics is the application of functional data analysis to morphometric data, modeling shapes as continuous functions on geometric domains.
• It leverages tools such as square-root velocity functions, FPCA, and quasi-conformal mappings to achieve invariant alignment and effective shape registration.
• This approach overcomes limitations of landmark-based methods by providing smoother, interpretable representations and actionable insights in biological and imaging studies.

Searching arXiv for papers on Functional Data Morphometrics and related frameworks. Functional Data Morphometrics (FDM) is the application of functional data analysis to morphometric problems in which shape, form, or morphometric images are represented as functions on a common domain or on geometric domains carrying metric structure. Across the recent literature, the term encompasses several closely related but nonidentical formulations: planar outline and interior analysis via square-root velocity functions (SRVFs) and quasi-conformal mappings (Li et al., 7 May 2026); voxelwise morphometry treated as a spatial random function with smoothly varying distributional parameters (Palma et al., 2024); outline-based species classification using multivariate functional principal component analysis (MFPCA) (Pillay et al., 2023); and broader metric-geometric frameworks for functions on geometric domains (Anbouhi et al., 2022). In all of these formulations, the central move is the same: morphometric objects are not reduced to isolated landmarks or raw pixel arrays, but are modeled as continuous functions or function-valued geometric objects, enabling alignment, invariance handling, statistical analysis, and, in some settings, explicit morphing or normative deviation mapping.

1. Conceptual scope and definitions

FDM is defined in one strand of the literature as “the application of Functional Data Analysis (FDA) to morphometric data,” where each specimen’s outline is modeled as a smooth function on a common domain and analyzed by tools such as functional principal component analysis (Pillay et al., 2023). A more geometric formulation treats morphometric data as “functions defined on geometric domains, with the domains themselves carrying metric and possibly measure structure,” thereby embedding morphometrics into functional metric geometry and Gromov-type comparison frameworks (Anbouhi et al., 2022). In image-based work, FDM is also used to describe the representation of shape and texture as continuous random functions rather than high-dimensional pixel grids (Moindjié, 9 Jun 2026). In neuroimaging, tensor-based morphometry (TBM) images are treated as smooth three-dimensional functional objects with spatially varying distributional properties, yielding subject-specific normative maps and deviation scores (Palma et al., 2024).

These definitions differ in domain, codomain, and inferential objective, but they share several structural commitments. First, morphometric variation is modeled in a function space rather than in a finite coordinate vector space. Second, nuisance variability such as translation, rotation, scale, and parameterization is handled explicitly, either by quotient constructions, alignment procedures, or invariant descriptors. Third, statistical structure is imposed through functional covariance, geodesic metrics, or distributional models defined over the domain. This suggests that FDM is best understood not as a single algorithm but as a family of functional representations and comparison principles for morphometric data.

A persistent motivation across the literature is the limitation of classical landmark-based geometric morphometrics. Landmark methods depend on point selection and correspondence, whereas FDM methods often use the entire boundary, the full surface, the image interior, or the full voxel field (Pillay et al., 2023). Another recurring motivation is that discrete or pixelwise representations can be high-dimensional and computationally inefficient, while continuous functional representations can regularize the problem and yield interpretable low-dimensional summaries (Moindjié, 9 Jun 2026).

2. Functional representations of morphometric objects

A basic FDM representation for planar outlines models a regular closed curve $\beta:[0,1]\to\mathbb{R}^2$ by its square-root velocity function

$$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$

In the SRVF framework, translation invariance is immediate, the elastic metric becomes the standard $L^2$ metric, and the curve can be reconstructed up to translation by

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$

For closed curves, the closure condition is

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$

This representation is central to the FDA-QC formulation of planar morphometry (Li et al., 7 May 2026).

For outline-based multivariate morphometrics, a two-dimensional outline may be represented coordinatewise as

$$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$

with a basis expansion on a common one-dimensional domain, although in practice the functional structure may be learned from the data by FPCA rather than by fixing a basis a priori (Pillay et al., 2023). In image-based shape-texture analysis under a star-shaped assumption, the contour is modeled as a random closed curve $C:[0,1]\to\mathbb{R}^2$, while the texture is mapped to a common functional domain via a polar or disk parameterization (Moindjié, 9 Jun 2026).

For three-dimensional surface data, FDM represents shape as a function $f:S\to\mathbb{R}^3$ on a two-dimensional manifold $S$, usually interpreted in the Hilbert space $L^2(S)$ with inner product

$$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$0

This supports surface-integral formulations of registration, PCA, asymmetry, and group comparison (Katina et al., 2020).

In volumetric morphometry, each TBM image is a scalar field $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$1 over a common brain domain $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$2, where $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$3 is the determinant of the Jacobian of the deformation aligning a subject’s structural MRI to a minimal deformation template (Palma et al., 2024). In that setting, the functional object is not a boundary or surface but a smooth random field defined over voxels.

A more abstract formulation broadens the representation further: a morphometric datum may be a 1-Lipschitz map $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$4 between Polish metric spaces, where the domain $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$5 carries the geometry and possibly a probability measure, and the codomain $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$6 carries the functional values (Anbouhi et al., 2022). This places FDM within metric geometry rather than classical Euclidean shape space.

3. Registration, invariance, and alignment

One of the main technical distinctions among FDM approaches concerns how they remove nuisance variability while preserving morphologically meaningful variation.

In the SRVF framework for planar curves, reparameterization is modeled by the action of $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$7: $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$8 After removing translation and normalizing length, the elastic geodesic distance between unit-norm SRVFs $$q(t) = \frac{\dot{\beta}(t)}{\sqrt{\|\dot{\beta}(t)\|}}.$$9 and $L^2$0 is

$L^2$1

Equivalently, one may maximize the $L^2$2 inner product and define the distance by an arccosine formula on the unit sphere in SRVF space (Li et al., 7 May 2026). Algorithmically, the workflow removes translation, rescales to unit length, estimates optimal rotation by SVD, estimates optimal reparameterization by dynamic programming, and projects to the closed-curve subspace if needed. The induced map gives a monotone, orientation-preserving boundary correspondence (Li et al., 7 May 2026).

In outline-based craniodental morphometrics, alignment is less elaborate. Landmark sets are standardized by Generalized Procrustes Analysis (GPA), removing translation, rotation, and scaling before conversion to functional data; no explicit temporal warping is reported (Pillay et al., 2023). This means that reparameterization invariance is not fully built into that pipeline, although smooth functional representation and covariance smoothing still regularize variation.

In star-shaped image analysis, alignment is expressed through the decomposition

$L^2$3

where $L^2$4 is scale, $L^2$5 is rotation, $L^2$6 is reparameterization, and $L^2$7 is translation. The aligned contour is then mapped into a polar functional representation, which normalizes the interior domain for texture analysis (Moindjié, 9 Jun 2026).

For three-dimensional morphometrics based on SRVF and arc-length parameterization, full elastic alignment seeks

$L^2$8

with arc-length reparameterization used to stabilize sampling and curvature representation (Pillay et al., 31 Aug 2025). A softer alternative adds regularization terms to avoid excessive warping, balancing elastic matching and fidelity to the original landmark trajectory (Pillay et al., 31 Aug 2025).

Topological FDM provides a different form of invariance. Merge trees, constructed from the sublevel-set connectivity of a function, are invariant under homeomorphic reparameterizations: for any homeomorphism $L^2$9, the merge trees of $\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$0 and $\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$1 are isomorphic (Pegoraro et al., 2021). In that line of work, phase variability is quotiented out at the representation level rather than corrected by explicit alignment.

The broad metric-geometric framework goes further still by allowing comparison of functional data on distinct domains. Here invariance is formulated in terms of isometries, measure-preserving maps, and 1-Lipschitz morphisms rather than Euclidean transformations alone (Anbouhi et al., 2022).

4. Statistical structure and metrics

The statistical core of FDM varies substantially across applications, but nearly all approaches rely on metrics, eigenstructures, or distributional parameter fields defined in functional spaces.

For multivariate functional outlines, FPCA and MFPCA are central. After centering,

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$2

univariate FPCA is performed on each component, then the resulting scores are stacked into a block matrix

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$3

whose eigenanalysis yields multivariate eigenfunctions and scores (Pillay et al., 2023). The truncated multivariate Karhunen–Loève representation is

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$4

This provides low-dimensional features for classification or visualization (Pillay et al., 2023).

For 3D surfaces, functional PCA is expressed in terms of a covariance operator on vector-valued functions over the surface, with scores obtained by surface integration (Katina et al., 2020). In practice, the continuous integrals are approximated by area-weighted vertex sums on a common mesh. This surface-integral viewpoint also supports functional asymmetry measures and group-shape-space analyses (Katina et al., 2020).

In TBM-based normative mapping, the statistical model is distributional rather than purely geometric. At each spatial location $\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$5,

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$6

with smoothly varying skew-normal parameters over the brain domain (Palma et al., 2024). The moments are determined by

$\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$7

with corresponding formulas for mean, variance, and skewness. The fitted parameter fields are interpolated by Gaussian radial basis functions from a coarse spatial grid to the full domain. A probability integral transform then produces subject-specific normative $\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$8-maps: $\beta(t) = \beta(0) + \int_0^t q(s)\,\|q(s)\|\,ds.$9 These $\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$0-maps standardize local deviations relative to a covariate-adjusted normative population (Palma et al., 2024).

In planar FDA-QC morphometrics, two complementary quantities summarize variation. Boundary variation is measured by the elastic distance

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$1

while interior deformation is summarized by quasi-conformal distortion through the Beltrami coefficient $\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$2 (Li et al., 7 May 2026). The Teichmüller metric is

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$3

and a robust practical summary uses the mean magnitude of $\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$4: $\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$5 The two are then fused into a combined dissimilarity

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$6

This metric fusion is distinctive because it combines boundary and interior information in a single morphometric graph (Li et al., 7 May 2026).

In the metric-geometric formulation, the principal distances are functional analogues of Gromov–Hausdorff and Gromov–Wasserstein distances. For metric-measure fields, the functional Gromov–Wasserstein distance is defined by

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$7

coupling structural discrepancy in the domains with discrepancy in the functional values (Anbouhi et al., 2022). This is a foundational rather than application-specific metric, but it clarifies how FDM can be defined even when there is no common parameter domain.

5. Boundary–interior coupling, morphing, and local interpretation

A notable development in recent FDM is the explicit coupling of outline geometry and interior deformation. FDA-QC is the clearest example. After obtaining an optimal boundary correspondence by SRVF-based elastic registration, the correspondence is extended to the whole planar domain by a quasi-conformal map satisfying the Beltrami equation

$\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$8

The local anisotropy is encoded by $\int_0^1 q(s)\,\|q(s)\|\,ds = 0.$9, and the local eccentricity is summarized by

$$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$0

Given a boundary map and optional landmark constraints, the interior map is reconstructed by the Linear Beltrami Solver through the elliptic system

$$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$1

with Dirichlet boundary conditions and optional landmark constraints (Li et al., 7 May 2026).

Because incompatible constraints may lead to fold-overs with $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$2, FDA-QC uses an iterative truncation–smoothing–solve procedure. The current Beltrami coefficient is truncated at $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$3 where necessary, its magnitude is replaced by a domain-wide average in a Teichmüller-style step, and the system is re-solved until bijectivity is restored or a maximum of 100 iterations is reached (Li et al., 7 May 2026). This gives a practical route to interior correspondence with controlled anisotropic distortion.

The same framework yields morphing. Boundary morphing is defined by the SRVF geodesic

$$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$4

with reconstruction of the intermediate boundary by integration. Interior morphing is achieved by linearly interpolating the Beltrami coefficient,

$$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$5

and solving the corresponding elliptic system with the intermediate boundary condition (Li et al., 7 May 2026). The resulting family $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$6 couples boundary and interior evolution continuously.

For temporal morphogenesis, local descriptors can be extracted from the Jacobian $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$7 of the intermediate map. The local areal expansion $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$8 and the local orientation change $$x(t) = \sum_{k=1}^{K} c^{(x)}_k \phi_k(t), \qquad y(t) = \sum_{k=1}^{K} c^{(y)}_k \phi_k(t),$$9 summarize spatial growth and rotation (Li et al., 7 May 2026). This is an important shift from purely descriptive shape comparison to mechanistically interpretable local deformation analysis.

A conceptually related development appears in star-shaped image FDM, where the contour defines a common coordinate system for the interior texture. The mapping

$C:[0,1]\to\mathbb{R}^2$0

transfers the unit disk to the object interior, and the texture is analyzed on the common disk domain (Moindjié, 9 Jun 2026). Although this framework does not use quasi-conformal theory, it similarly binds boundary geometry and interior signal into a single functional representation.

6. Applications and empirical findings

The empirical literature shows that FDM has been applied to biological outlines, insect wings, neuroimaging, image classification, functional signals, and high-resolution surfaces.

In planar biological morphometry, FDA-QC was applied to leaves and insect wings. For leaf mapping, the method produced smooth correspondences, automatically aligning tips and lobes while concentrating interior deformation near high-curvature regions (Li et al., 7 May 2026). For Arabidopsis morphogenesis, geodesic-QC morphs produced temporally consistent deformations, and spatial growth analysis via $C:[0,1]\to\mathbb{R}^2$1 and $C:[0,1]\to\mathbb{R}^2$2 revealed heterogeneous areal expansion and outward rotations at later stages (Li et al., 7 May 2026).

The most detailed quantitative benchmark in that work concerns European honey bee wings. Using $C:[0,1]\to\mathbb{R}^2$3 specimens from AT, HR, MD, and SI, with $C:[0,1]\to\mathbb{R}^2$4 venation landmarks per wing, pairwise FDA-QC mappings with landmark constraints were fused into $C:[0,1]\to\mathbb{R}^2$5. Sweeping $C:[0,1]\to\mathbb{R}^2$6, the optimal

$C:[0,1]\to\mathbb{R}^2$7

maximized clustering agreement with region labels, yielding Adjusted Rand Index $C:[0,1]\to\mathbb{R}^2$8, Normalized Mutual Information $C:[0,1]\to\mathbb{R}^2$9, and purity $f:S\to\mathbb{R}^3$0 (Li et al., 7 May 2026). Boundary-only and interior-only baselines both gave ARI $f:S\to\mathbb{R}^3$1, NMI $f:S\to\mathbb{R}^3$2, and purity $f:S\to\mathbb{R}^3$3, indicating that coupled boundary–interior information captured morphological variation more effectively than either component alone (Li et al., 7 May 2026).

In craniodental morphometrics of shrews, the FDM pipeline converted landmark outlines from dorsal, jaw, and lateral views into functional data and used predicted MFPC scores for classification. The first two MFPCs explained 81.56% of variance for all views combined, compared with 62.94% for the first two classical PCs (Pillay et al., 2023). For the dorsal view, the first two MFPCs explained 86.4% of variance, and GLM on the predicted MFPC scores achieved 95.4% accuracy, the highest among the evaluated models (Pillay et al., 2023). The dorsal view consistently showed the clearest separation of the three species.

In three-dimensional cranial morphometrics of kangaroos, eight pipelines combining GM, arc-length parameterization, FDM, and SRVF alignment were compared. Elastic-SRV-FDM achieved the highest classification accuracy across classifiers, especially LDA, while functional pipelines generally captured variance with fewer components and lower reconstruction error than GM-based pipelines (Pillay et al., 31 Aug 2025). Exact accuracy tables were not reported, but the simulation and application both supported the utility of arc-length and SRVF-based functional representations.

In normative neuroimaging, the skewed functional-data model was fit to the ADNI TBM dataset comprising $f:S\to\mathbb{R}^3$4 adults, with a normative training sample of 183 cognitively normal subjects (Palma et al., 2024). The fitted spatial parameter functions showed that both mean and standard deviation were highest in the lateral ventricles, while skewness was positive across most of the brain mask (Palma et al., 2024). The tail-mean deviation index $f:S\to\mathbb{R}^3$5 increased with disease severity from CN to MCI to AD, and deviation extremity correlated with ADAS13 cognitive scores (Palma et al., 2024). This use of FDM is distinct from shape-space methods: it focuses on normative deviation from a healthy distribution rather than on inter-shape registration alone.

In high-resolution 3D facial surface analysis, functional surface methods supported sex and population comparisons, asymmetry quantification, and comparison to closest controls (Katina et al., 2020). For British females, 10 principal components explained 82% of variance. Combined subspace morphs showed sex differences such as more prominent nasal tip, chin, and brow ridge in males, and more prominent cheeks in females (Katina et al., 2020). The same framework was extended to orthognathic surgery assessment, where asymmetry scores and closest-control morphs provided clinically interpretable patient-specific evaluations (Katina et al., 2020).

In image classification with joint shape and texture, a star-shaped FDM representation was applied to 2,148 RGB pistachio images. The joint multivariate functional model achieved ACC 0.77 and F1 0.74 under MFPCA, outperforming pixel-based scalar-on-image baselines, which achieved ACC 0.63 and F1 0.42 (Moindjié, 9 Jun 2026). This indicates that functional morphometric representations can also be competitive in supervised learning settings.

Finally, topological FDM via merge trees was evaluated on the misaligned Aneurisk65 dataset. A mixed merge-tree metric combining curvature and radius achieved 84.6% leave-one-out accuracy, slightly exceeding both a persistence-diagram baseline at 83.1% and an aligned FPCA benchmark at 81.5% (Pegoraro et al., 2021). This suggests that when phase variability is severe, representation-level invariance may compete effectively with explicit registration.

7. Relation to geometric morphometrics, controversies, and open directions

FDM is often presented in contrast to landmark-based geometric morphometrics, but the relationship is more nuanced than simple replacement. Many FDM pipelines still begin with GPA or with landmark-based sampling, as in the shrew and kangaroo studies (Pillay et al., 2023, Pillay et al., 31 Aug 2025). In such cases, FDM augments rather than eliminates classical morphometric preprocessing. Its principal gain lies in treating ordered coordinates or boundaries as smooth functions, thereby enabling covariance smoothing, functional PCA, or elastic alignment.

Several recurrent claims should therefore be interpreted carefully.

A common misconception is that FDM is intrinsically landmark-free. This is true for some formulations, such as dense boundary SRVF registration or star-shaped contour extraction, but false for others, including multiview craniodental classification and some 3D landmark-trajectory pipelines (Pillay et al., 2023, Pillay et al., 31 Aug 2025). Another misconception is that all FDM methods are automatically invariant to parameterization. Full reparameterization invariance is a property of SRVF-type elastic methods and certain topological summaries, not of every FDA-based outline model (Li et al., 7 May 2026, Pegoraro et al., 2021).

There are also important methodological limitations. FDA-QC assumes simply connected planar domains with regular closed boundaries and sufficient mesh quality; it is sensitive to discretization quality and to incompatible constraints that can induce fold-overs (Li et al., 7 May 2026). The normative TBM model assumes skew-normal voxelwise distributions, a choice the authors themselves note may be too restrictive for heavier tails or other mean–variance structures (Palma et al., 2024). High-resolution surface FDM depends on accurate model correspondence and can be affected by noisy or reflective regions (Katina et al., 2020). Star-shaped shape-texture analysis requires the object interior to be star-shaped with respect to a chosen center, which excludes more complicated geometries (Moindjié, 9 Jun 2026). The broader metric-geometric theory, while general, does not yet provide the rate-of-convergence or sample-complexity results that would make empirical deployment routine (Anbouhi et al., 2022).

A deeper open question concerns the statistical geometry of combined dissimilarities. In FDA-QC, the fused metric $f:S\to\mathbb{R}^3$6 is geometrically motivated, but its “deeper statistical properties warrant further study” (Li et al., 7 May 2026). This suggests a broader issue for FDM: once multiple functional channels or geometric components are combined, the resulting metric may be effective empirically without yet being fully characterized statistically.

Several future directions are explicitly suggested in the literature. These include extensions from planar objects to tensor-field representations and to 3D surfaces and volumes (Li et al., 7 May 2026); richer distributional families such as skew-$f:S\to\mathbb{R}^3$7 or gamma models in normative morphometry (Palma et al., 2024); Gaussian copula modeling of spatial dependence in z-maps (Palma et al., 2024); non-star-shaped generalizations for joint shape-texture analysis (Moindjié, 9 Jun 2026); and broader use of Gromov-type distances for functional data on non-common domains (Anbouhi et al., 2022). A plausible implication is that FDM is moving toward an overview of elastic shape analysis, distributional functional modeling, and metric geometry, with different application areas emphasizing different aspects of that synthesis.

Taken together, the recent literature portrays Functional Data Morphometrics as a technically heterogeneous but conceptually coherent field. Its unifying principle is the treatment of morphometric structure as a functional object with explicit geometry, enabling analyses that couple registration, invariance, statistical inference, and, in several settings, biologically interpretable deformation or deviation measures (Li et al., 7 May 2026, Palma et al., 2024, Anbouhi et al., 2022).