Volumetric Varifolds: Theory and Applications

• Volumetric varifolds are measure-theoretic representations that model both diffuse surfaces and full-dimensional mass distributions through volumetric discretization on meshes.
• Mesh-based discretization couples Lebesgue mass with best-fit tangent planes, quantifying error bounds and ensuring convergence through density controls and regularized mean curvature.
• Kernel metrics and RKHS embeddings enable diffeomorphic registration frameworks, facilitating advanced quantization and optimal control methods in computational anatomy.

Volumetric varifolds are varifold representations in which mass is distributed over volumetric cells or, in the full-dimensional case $k=n$, reduces essentially to a Radon measure on the ambient space. In the cited literature, the term appears in several closely related senses: as a diffuse discretization of a $d$-submanifold on a mesh, as a model for “diffused” surfaces with arbitrary weight measure, and as a full-dimensional registration object in RKHS- and LDDMM-based shape analysis; related mesh-based extensions also appear in image-varifolds on $\mathbb{R}^d\times\mathfrak{F}$ for spatial molecular data (Sagueni, 8 Sep 2025, Menne et al., 2016, Hsieh et al., 2019, Hsieh et al., 2021, Miller et al., 2022).

1. General definition and full-dimensional specialization

A $d$-varifold in $\mathbb{R}^n$ is a nonnegative Radon measure on $\mathbb{R}^n\times G_{d,n}$, where $G_{d,n}$ is the Grassmannian of $d$-planes. Its mass, or weight, measure is the projection onto $\mathbb{R}^n$: $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$ for $d$0. The support $d$1 projects to $d$2 (Sagueni, 8 Sep 2025).

This general definition includes rectifiable varifolds, for which one has

$d$3

with $d$4 countably $d$5-rectifiable, $d$6, and $d$7 the approximate tangent plane. It also includes arbitrary, non-rectifiable objects whose weight measure may be diffuse in the ambient domain (Buet, 2014, Menne et al., 2016).

In the full-dimensional case $d$8, the Grassmannian degenerates. One formulation states that $d$9 is a singleton, so a varifold on $\mathbb{R}^d\times\mathfrak{F}$0 is exactly a Radon measure on $\mathbb{R}^d\times\mathfrak{F}$1 alone, i.e. a usual volume or weight distribution $\mathbb{R}^d\times\mathfrak{F}$2. In the oriented $\mathbb{R}^d\times\mathfrak{F}$3-dimensional presentation, $\mathbb{R}^d\times\mathfrak{F}$4 records only orientation, since every full-dimensional plane is $\mathbb{R}^d\times\mathfrak{F}$5 itself (Hsieh et al., 2021, Hsieh et al., 2019).

A recurrent misconception is to identify varifolds only with sharp submanifolds. The diffused-surface literature makes the opposite point explicitly: one allows arbitrary varifolds so that $\mathbb{R}^d\times\mathfrak{F}$6 may be a diffuse measure in $\mathbb{R}^d\times\mathfrak{F}$7, which models “volumetric” or “diffused” surfaces (Menne et al., 2016). Conversely, the full-dimensional registration literature treats volumetric varifolds as genuine volume distributions rather than lower-dimensional interfaces (Hsieh et al., 2019, Hsieh et al., 2021).

2. Mesh-based volumetric discretization of submanifolds

For a bounded domain $\mathbb{R}^d\times\mathfrak{F}$8 with mesh $\mathbb{R}^d\times\mathfrak{F}$9 of maximal cell diameter $d$0, and a smooth $d$1-submanifold $d$2, the volumetric varifold discretization is

$d$3

where $d$4, $d$5, and

$d$6

In words, each cell carries a uniform density $d$7 of Lebesgue mass coupled with a best-fit tangent plane $d$8. The mass measure is therefore

$d$9

and its support is $\mathbb{R}^n$0 (Sagueni, 8 Sep 2025).

A closely related formulation in Buet’s rectifiability framework writes

$\mathbb{R}^n$1

with $\mathbb{R}^n$2 and

$\mathbb{R}^n$3

As the mesh size $\mathbb{R}^n$4, $\mathbb{R}^n$5 weak-$\mathbb{R}^n$6 in measures (Buet, 2014).

The approximation of the mass measure is quantitative. Proposition 1.8, cited from Buet et al., states that for any Lipschitz $\mathbb{R}^n$7 on $\mathbb{R}^n$8,

$\mathbb{R}^n$9

and a similar estimate holds for test functions $\mathbb{R}^n\times G_{d,n}$0 on $\mathbb{R}^n\times G_{d,n}$1 under an extra $\mathbb{R}^n\times G_{d,n}$2-regularity assumption on $\mathbb{R}^n\times G_{d,n}$3 (Sagueni, 8 Sep 2025).

These constructions are volumetric because a lower-dimensional object is represented through $\mathbb{R}^n\times G_{d,n}$4-dimensional Lebesgue mass inside cells. The rectifiability literature stresses that this feature forces scale restrictions: the Ahlfors-type density estimates must be imposed at radii $\mathbb{R}^n\times G_{d,n}$5 with $\mathbb{R}^n\times G_{d,n}$6 typically larger than the mesh size $\mathbb{R}^n\times G_{d,n}$7, so that balls see the correct $\mathbb{R}^n\times G_{d,n}$8-dimensional mass scaling rather than the $\mathbb{R}^n\times G_{d,n}$9-dimensional Lebesgue measure of a cell (Buet, 2014).

3. Regularized mean curvature and the Brakke approximate equality

A general varifold may not have bounded first variation, so the mean curvature is regularized by convolving both the first variation and the mass with kernels $G_{d,n}$0 supported in $G_{d,n}$1. For $G_{d,n}$2,

$G_{d,n}$3

The regularized first variation at $G_{d,n}$4 is

$G_{d,n}$5

and the regularized mass is

$G_{d,n}$6

Assuming $G_{d,n}$7, the $G_{d,n}$8-approximate mean curvature vector is

$G_{d,n}$9

Under mild assumptions on $d$0 and $d$1, this quantity enjoys stability and convergence to the classical mean curvature on $d$2-manifolds as $d$3 (Sagueni, 8 Sep 2025).

The principal consistency result concerns a $d$4 mean-curvature flow $d$5, $d$6, of a closed $d$7-manifold in a convex domain $d$8, with volumetric discretization $d$9 at each time. For $\mathbb{R}^n$0 and $\mathbb{R}^n$1, Theorem 2.1 gives constants $\mathbb{R}^n$2 (Ahlfors constant), $\mathbb{R}^n$3 (stability of $\mathbb{R}^n$4), $\mathbb{R}^n$5 (Lipschitz bound on tangent-map), and further mesh- and kernel-dependent constants such that, for sufficiently small $\mathbb{R}^n$6 and $\mathbb{R}^n$7,

$\mathbb{R}^n$8

Here $\mathbb{R}^n$9 is the bounded-Lipschitz distance between measures. In particular, if $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$0 and $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$1, the right-hand side can be made arbitrarily small (Sagueni, 8 Sep 2025).

The derivation splits into three error-producing substitutions. First, one replaces the classical mean curvature in the Brakke weak form by $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$2, producing an $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$3 error. Second, one replaces integrals over $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$4 by integrals over $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$5, using the volumetric approximation estimate to obtain an $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$6 error per unit time. Third, one replaces $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$7 by $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$8, and Lemma 2.7 together with Proposition 2.5 yields an $$\|V\|(\varphi):=\int_{\mathbb{R}^n\times G_{d,n}} \varphi(x)\,dV(x,S),$$9 error in the time integral (Sagueni, 8 Sep 2025).

The convergence statement is explicit. One may choose $d$00 slowly so that $d$01, for instance $d$02. Then the right-hand side tends to zero, the discrete mass curve $d$03 converges to the unique classical solution $d$04 of the exact Brakke equality, and compactness of varifolds yields convergence of $d$05 in the bounded-Lipschitz sense to the continuous Brakke flow $d$06 (Sagueni, 8 Sep 2025).

4. Diffused surfaces, density control, and rectifiability

The diffused-surface viewpoint studies arbitrary varifolds whose weight measure need not be concentrated on a rectifiable set. For an $d$07-varifold $d$08 in $d$09 with finite mass, the maximal-type density function is

$d$10

and the “diffused” region at scale $d$11 is

$d$12

The general isoperimetric inequality then states that for $d$13,

$d$14

with $d$15 depending only on $d$16. When $d$17 is supported in a ball $d$18, one recovers

$d$19

where $d$20 is the best isoperimetric constant (Menne et al., 2016).

This analytic framework is coupled to a notion of generalized weak differentiation. A $d$21-measurable function $d$22 belongs to $d$23 if there exists a $d$24-measurable map

$d$25

satisfying the stated integrability condition and the distributional identity involving $d$26, $d$27, and $d$28. In this setting, $d$29 is $d$30-a.e. well-defined and plays the role of the weak gradient. If $d$31 is Lipschitz on $d$32, then $d$33 and

$d$34

A calculus is developed, including closedness under limits, composition, products, and a coarea formula (Menne et al., 2016).

The same paper derives Sobolev-type inequalities on diffused varifolds. One form uses a local median $d$35 defined through $d$36 and yields

$d$37

while another, on the rectifiable part $d$38, gives

$d$39

A Poincaré-type inequality in a ball follows as a corollary (Menne et al., 2016).

Rectifiability of limits of volumetric approximations is addressed quantitatively through averaged height-excess. For a $d$40-varifold $d$41, point $d$42, plane $d$43, and scale $d$44,

$d$45

If a sequence $d$46 satisfies uniform density bounds

$d$47

for $d$48, together with

$d$49

and $d$50, then the limit $d$51 is $d$52-rectifiable (Buet, 2014).

A plausible implication is that volumetric varifolds occupy a dual position in geometric analysis: they permit diffuse or cellwise approximations at the discrete level, while quantitative density and flatness controls can still force rectifiable structure in the limit. That interpretation is consistent with both the diffused-surface and rectifiability frameworks (Menne et al., 2016, Buet, 2014).

5. Kernel metrics, quantization, and diffeomorphic dynamics

In the registration literature, volumetric varifolds are endowed with RKHS metrics. In the oriented full-dimensional $d$53-dimensional case, one considers a finite nonnegative Radon measure $d$54 on $d$55, with product kernel

$d$56

where $d$57 may be Gaussian and $d$58. If $d$59 is the RKHS with reproducing kernel $d$60, then

$d$61

For Dirac sums,

$d$62

Under mild regularity, this pseudo-metric is bounded by the bounded-Lipschitz distance and metrizes narrow convergence on sets of varifolds with uniformly bounded mass and support; if $d$63 is $d$64-universal, it is a true distance (Hsieh et al., 2019).

Quantization replaces a general volumetric varifold by

$d$65

There exist choices of $d$66 such that $d$67, and if $d$68 is compact then one can achieve

$d$69

For source quantizations $d$70, the approximate registration functionals

$d$71

$d$72-converge to the exact energy

$d$73

and minimizers of $d$74 accumulate to minimizers of $d$75 (Hsieh et al., 2019).

Diffeomorphic registration is formulated as optimal control. If $d$76 is an RKHS of vector fields embedded in $d$77, the flow $d$78 solves $d$79, and the inexact matching problem is

$d$80

If $d$81, $d$82, and $d$83 is compactly supported, a minimizer exists. Pontryagin’s Maximum Principle gives the Hamiltonian

$d$84

with

$d$85

and

$d$86

for an optimal solution (Hsieh et al., 2019).

The metamorphosis extension introduces a growth rate $d$87 through

$d$88

and energy

$d$89

The corresponding tangent metric is

$d$90

The paper proves that this defines a genuine right-invariant Riemannian metric on each orbit $d$91, and relaxed problems with an additional RKHS fidelity term admit minimizers under mild assumptions (Hsieh et al., 2021).

Taken together, these results show that volumetric varifolds support both approximation theory and large-deformation dynamics. The measure-theoretic representation, the kernel metric, the $d$92-convergence of discrete energies, and the Hamiltonian structure of registration all persist in the volumetric setting (Hsieh et al., 2019, Hsieh et al., 2021).

6. Image-varifolds on meshes and volumetric feature measures

A related but distinct extension is the image-varifold framework for spatial transcriptomics. A $d$93-dimensional image-varifold $d$94 is a finite Radon measure on $d$95, where $d$96 is a feature space such as genes, RNA-counts, or cell-types. It can be disintegrated into a spatial measure $d$97 and a family of transition probabilities $d$98 on $d$99: $\mathbb{R}^d\times\mathfrak{F}$00 Important examples include a continuum image $\mathbb{R}^d\times\mathfrak{F}$01, a point-cloud varifold $\mathbb{R}^d\times\mathfrak{F}$02, and a semi-discrete mesh-based form

$\mathbb{R}^d\times\mathfrak{F}$03

where $\mathbb{R}^d\times\mathfrak{F}$04 are small volumetric cells with center $\mathbb{R}^d\times\mathfrak{F}$05, volume $\mathbb{R}^d\times\mathfrak{F}$06, weights $\mathbb{R}^d\times\mathfrak{F}$07, and feature laws $\mathbb{R}^d\times\mathfrak{F}$08 (Miller et al., 2022).

Comparison is again performed by an RKHS embedding. If the reproducing kernel on $\mathbb{R}^d\times\mathfrak{F}$09 is the product of a spatial kernel $\mathbb{R}^d\times\mathfrak{F}$10 and a feature kernel $\mathbb{R}^d\times\mathfrak{F}$11, then

$\mathbb{R}^d\times\mathfrak{F}$12

and

$\mathbb{R}^d\times\mathfrak{F}$13

The chordal distance is $\mathbb{R}^d\times\mathfrak{F}$14. In semi-discrete form,

$\mathbb{R}^d\times\mathfrak{F}$15

This is a volumetric measure model because each cell contributes through its volume, center, density, and local feature histogram (Miller et al., 2022).

The diffeomorphic action is the “copy-and-paste” push-forward

$\mathbb{R}^d\times\mathfrak{F}$16

and the LDDMM energy is

$\mathbb{R}^d\times\mathfrak{F}$17

By Pontryagin’s maximum principle and the reproducing-kernel property, the optimal $\mathbb{R}^d\times\mathfrak{F}$18 admits the reduced form

$\mathbb{R}^d\times\mathfrak{F}$19

and an adjoint Hamiltonian system provides the gradient (Miller et al., 2022).

The practical pipeline is explicitly mesh-based: build a simplicial mesh of tetrahedra or triangles, prune empty cells, attach cell centers, cell volumes, weights from local particle densities, and feature laws from gene counts or cell-type frequencies; assemble Gram matrices for $\mathbb{R}^d\times\mathfrak{F}$20 and the feature inner products; compute the varifold norm by double sums; integrate the state equation for the node positions; compute the data attachment and its gradient via centers, volumes, and normals; solve the adjoint system backward; update the shooting variables by gradient descent or L-BFGS; and reconstruct the diffeomorphism once converged. The stated application is diffeomorphic registration of unstructured spatial transcriptomics or cell-type data at micron resolution (Miller et al., 2022).

This suggests that volumetric-varifold ideas extend beyond tangent-plane encoding. In one branch, a cell stores approximate geometric orientation $\mathbb{R}^d\times\mathfrak{F}$21; in another, the cell stores a feature law $\mathbb{R}^d\times\mathfrak{F}$22. The common structure is a geometric measure supported on volumetric elements and equipped with a kernel metric or weak geometric flow formalism (Sagueni, 8 Sep 2025, Miller et al., 2022).