---
title: 'Volumetric Varifolds: Theory and Applications'
url: https://www.emergentmind.com/topics/volumetric-varifolds
type: topic
---

# Volumetric Varifolds: Theory and Applications

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 [2509.06440], [1612.03823], [1903.11196], [2112.04644], [2208.08376].

## 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 \(\varphi\in C^0_c(\mathbb{R}^n)\). The support \( \operatorname{spt}V\subset \mathbb{R}^n\times G_{d,n}\) projects to \(\operatorname{spt}\|V\|\subset\mathbb{R}^n\) [2509.06440].

This general definition includes rectifiable varifolds, for which one has
\[
V=v(M,\theta)=\theta\,\mathcal{H}^d\!\llcorner M\otimes \delta_{T_xM},
\]
with \(M\) countably \(d\)-rectifiable, \(\theta(x)>0\), and \(T_xM\) the approximate tangent plane. It also includes arbitrary, non-rectifiable objects whose weight measure may be diffuse in the ambient domain [1409.4749], [1612.03823].

In the full-dimensional case \(k=n\), the Grassmannian degenerates. One formulation states that \(G(n,n)\) is a singleton, so a varifold on \(\mathbb{R}^n\times G(n,n)\) is exactly a Radon measure on \(\mathbb{R}^n\) alone, i.e. a usual volume or weight distribution \(\rho\,dx\). In the oriented \(3\)-dimensional presentation, \( \widetilde{G}_3^3=\{+1,-1\}\) records only orientation, since every full-dimensional plane is \(\mathbb{R}^3\) itself [2112.04644], [1903.11196].

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 \(\|V\|\) may be a diffuse measure in \(U\), which models “volumetric” or “diffused” surfaces [1612.03823]. Conversely, the full-dimensional registration literature treats volumetric varifolds as genuine volume distributions rather than lower-dimensional interfaces [1903.11196], [2112.04644].

## 2. Mesh-based volumetric discretization of submanifolds

For a bounded domain \(\Omega\subset\mathbb{R}^n\) with mesh \(\mathcal{K}_h\) of maximal cell diameter \(h\), and a smooth \(d\)-submanifold \(M\subset\Omega\), the volumetric varifold discretization is
\[
V_h=\sum_{K\in\mathcal{K}_h} \frac{m_K}{|K|}\,\mathcal{L}^n\!\llcorner K \otimes \delta_{P_K},
\]
where \(|K|=\operatorname{Lebesgue}^n(K)\), \(m_K=\mathcal{H}^d(M\cap K)\), and
\[
P_K\in \arg\min_{S\in G_{d,n}} \int_{M\cap K} |T_yM-S|\,d\mathcal{H}^d(y).
\]
In words, each cell carries a uniform density \(m_K/|K|\) of Lebesgue mass coupled with a best-fit tangent plane \(P_K\). The mass measure is therefore
\[
\|V_h\|=\sum_K (m_K/|K|)\,\mathcal{L}^n\!\llcorner K,
\]
and its support is \(\bigcup K\) [2509.06440].

A closely related formulation in Buet’s rectifiability framework writes
\[
V_{\mathcal K}=\sum_{K\in\mathcal K} (m_K/|K|)\,\mathcal{L}^n\!\llcorner K\otimes\delta_{P_K},
\]
with \(m_K\approx \mathcal{H}^d(M\cap K)\) and
\[
P_K\in \arg\min_P \int_{M\cap K} |T_xM-P|^2\,d\mathcal{H}^d(x).
\]
As the mesh size \(\delta\to 0\), \(V_{\mathcal K}\rightharpoonup V\) weak-\(*\) in measures [1409.4749].

The approximation of the mass measure is quantitative. Proposition 1.8, cited from Buet et al., states that for any Lipschitz \(\varphi\) on \(\Omega\),
\[
|\|M\|(\varphi)-\|V_h\|(\varphi)| \le h\,\operatorname{Lip}(\varphi)\,\|M\|(\operatorname{spt}\varphi),
\]
and a similar estimate holds for test functions \(\varphi(x,S)\) on \(\Omega\times G_{d,n}\) under an extra \(C^1\)-regularity assumption on \(M\) [2509.06440].

These constructions are volumetric because a lower-dimensional object is represented through \(n\)-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 \(r\ge \beta_i\) with \(\beta_i\) typically larger than the mesh size \(\delta_i\), so that balls see the correct \(d\)-dimensional mass scaling rather than the \(n\)-dimensional Lebesgue measure of a cell [1409.4749].

## 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 \(\rho,\xi\) supported in \([0,1]\). For \(\varepsilon\in(0,1]\),
\[
\rho_\varepsilon(r)=\varepsilon^{-n}\rho(r/\varepsilon),\qquad
\xi_\varepsilon(r)=\varepsilon^{-n}\xi(r/\varepsilon).
\]
The regularized first variation at \(y\in\mathbb{R}^n\) is
\[
\delta V*\rho_\varepsilon(y)
=\int_{\mathbb{R}^n\times G_{d,n}} S[\nabla\rho_\varepsilon(z-y)]\,dV(z,S),
\]
and the regularized mass is
\[
\|V\|*\xi_\varepsilon(y)=\int_{\mathbb{R}^n}\xi_\varepsilon(z-y)\,d\|V\|(z).
\]
Assuming \(C_\rho=C_\xi=1\), the \(\varepsilon\)-approximate mean curvature vector is
\[
H_{\rho,\xi,\varepsilon}^V(y)
:= - \frac{\delta V*\rho_\varepsilon(y)}{\|V\|*\xi_\varepsilon(y)}.
\]
Under mild assumptions on \(\rho\) and \(\xi\), this quantity enjoys stability and convergence to the classical mean curvature on \(C^2\)-manifolds as \(\varepsilon\to 0\) [2509.06440].

The principal consistency result concerns a \(C^3\) mean-curvature flow \(M(t)\), \(t\in[0,T]\), of a closed \(d\)-manifold in a convex domain \(\Omega\), with volumetric discretization \(V_h(t)\) at each time. For \(\varphi\in C^2_c(\mathbb{R}^n;\mathbb{R}^+)\) and \(0\le t_1\le t_2\le T\), Theorem 2.1 gives constants \(C_0\) (Ahlfors constant), \(C_1\) (stability of \(H_\varepsilon\to H\)), \(C_2\) (Lipschitz bound on tangent-map), and further mesh- and kernel-dependent constants such that, for sufficiently small \(h\) and \(\varepsilon\),
\[
\begin{aligned}
&\Bigg|\|V_h(t_2)\|(\varphi)-\|V_h(t_1)\|(\varphi) \\
&\qquad + \int_{t_1}^{t_2}\int_{\mathbb{R}^n}
\left[-\varphi\,|H_{\rho,\xi,\varepsilon}^{V_h(t)}|^2
+\nabla\varphi\cdot H_{\rho,\xi,\varepsilon}^{V_h(t)}\right]
\,d\|V_h(t)\|\,dt\Bigg| \\
&\le 2\,\operatorname{Lip}(\varphi)\,\max_{i=1,2}\Delta(M(t_i),V_h(t_i))
+ \|\varphi\|_\infty\,C_1[\|M(t_1)\|(\mathbb{R}^n)-\|M(t_2)\|(\mathbb{R}^n)] \\
&\qquad + \|\varphi\|_{C^2}\,C\,(t_2-t_1)\,(\varepsilon+h/\varepsilon^3).
\end{aligned}
\]
Here \(\Delta(\cdot,\cdot)\) is the bounded-Lipschitz distance between measures. In particular, if \(\varepsilon\to 0\) and \(h/\varepsilon^3\to 0\), the right-hand side can be made arbitrarily small [2509.06440].

The derivation splits into three error-producing substitutions. First, one replaces the classical mean curvature in the Brakke weak form by \(H_{\rho,\xi,\varepsilon}^{M(t)}\), producing an \(O(\|\varphi\|_{C^1}(t_2-t_1)\varepsilon)\) error. Second, one replaces integrals over \(M(t)\) by integrals over \(\|V_h(t)\|\), using the volumetric approximation estimate to obtain an \(O(h)\) error per unit time. Third, one replaces \(H_{\rho,\xi,\varepsilon}^{M(t)}\) by \(H_{\rho,\xi,\varepsilon}^{V_h(t)}\), and Lemma 2.7 together with Proposition 2.5 yields an \(O(h/\varepsilon^3)\) error in the time integral [2509.06440].

The convergence statement is explicit. One may choose \(\varepsilon=\varepsilon(h)\to 0\) slowly so that \(h/\varepsilon^3\to 0\), for instance \(\varepsilon=h^{1/4}\). Then the right-hand side tends to zero, the discrete mass curve \(t\mapsto \|V_h(t)\|(\varphi)\) converges to the unique classical solution \(t\mapsto \|M(t)\|(\varphi)\) of the exact Brakke equality, and compactness of varifolds yields convergence of \(V_h(t)\) in the bounded-Lipschitz sense to the continuous Brakke flow \(M(t)\) [2509.06440].

## 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 \(m\)-varifold \(V\) in \(\mathbb{R}^n\) with finite mass, the maximal-type density function is
\[
M(x)=\sup_{0<s<\infty}\frac{\|V\|(B(x,s))}{\alpha(m)\,s^m},
\]
and the “diffused” region at scale \(d>0\) is
\[
A(d)=\{x\in\mathbb{R}^n: M(x)\ge d\}.
\]
The general isoperimetric inequality then states that for \(1<m\le n\),
\[
\|V\|(\{x:M(x)\ge d\})^{1-\frac1m}
\le T_m\,d^{-1/m}\,\|V\|(\mathbb{R}^n),
\]
with \(T_m\) depending only on \(m\). When \(V\) is supported in a ball \(B(a,r)\), one recovers
\[
\alpha(m)^{-1/m}\,r^{-1}\,\|V\|(\mathbb{R}^n)\le y(m)\,|\delta V|(\mathbb{R}^n),
\]
where \(y(m)\) is the best isoperimetric constant [1612.03823].

This analytic framework is coupled to a notion of generalized weak differentiation. A \(\|V\|\)-measurable function \(f:U\to Y\) belongs to \(T(V,Y)\) if there exists a \(\|V\|\)-measurable map
\[
V\!Df:U\to \operatorname{Hom}(\mathbb{R}^n,Y)
\]
satisfying the stated integrability condition and the distributional identity involving \(\delta V\), \(D\varphi\), and \(Dy(f(x))\). In this setting, \(V\!Df\) is \(\|V\|\)-a.e. well-defined and plays the role of the weak gradient. If \(f\) is Lipschitz on \(U\), then \(f\in T(V,Y)\) and
\[
|V\!Df(x)|\le \lim_{r\to 0}\operatorname{Lip}(f|B(x,r))
\quad\text{for }\|V\|\text{-a.e. }x.
\]
A calculus is developed, including closedness under limits, composition, products, and a coarea formula [1612.03823].

The same paper derives Sobolev-type inequalities on diffused varifolds. One form uses a local median \(g(x)\) defined through \(B(x,r(x))\) and yields
\[
\left(\int_A |g|^q\,d\|V\|\right)^{1/q}
\le C(m,n)\,d^{-1/m}
\left(
\int f\,d\|V\|
+\int |V\!Df|\,d\|V\|
\right),
\]
while another, on the rectifiable part \(A_d=\{x:\Theta^m(\|V\|,x)\ge d\}\), gives
\[
\left(\int_{A_d}|f|^q\,d\|V\|\right)^{1/q}
\le y(m)\,d^{-1/m}
\left(
\int f\,d\|V\|
+\int |V\!Df|\,d\|V\|
\right),
\qquad q=\frac{m}{m-1}.
\]
A Poincaré-type inequality in a ball follows as a corollary [1612.03823].

Rectifiability of limits of volumetric approximations is addressed quantitatively through averaged height-excess. For a \(d\)-varifold \(V\), point \(x\), plane \(P\), and scale \(\alpha>0\),
\[
E_\alpha(x,P,V)
=
\int_{r=\alpha}^1
\frac{1}{r^d}
\int_{B_r(x)\cap\Omega}
\left(\frac{d(y-x,P)}{r}\right)^2\,d\|V\|(y)\,\frac{dr}{r}.
\]
If a sequence \(V_i\) satisfies uniform density bounds
\[
C_1r^d \le \|V_i\|(B_r(x)) \le C_2r^d
\]
for \(\beta_i<r<\operatorname{dist}(x,\Omega^c)\), together with
\[
\sup_i \int E_{\alpha_i}(x,P,V_i)\,dV_i(x,P)<\infty,
\]
and \(V_i\rightharpoonup V\), then the limit \(V\) is \(d\)-rectifiable [1409.4749].

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 [1612.03823], [1409.4749].

## 5. Kernel metrics, quantization, and diffeomorphic dynamics

In the registration literature, volumetric varifolds are endowed with RKHS metrics. In the oriented full-dimensional \(3\)-dimensional case, one considers a finite nonnegative Radon measure \(\mu\) on \(\Omega\times \widetilde{G}_3^3\), with product kernel
\[
K((x,T),(y,S))=k^{pos}(x,y)\,k^T(T,S),
\]
where \(k^{pos}\) may be Gaussian and \(k^T(T,S)=\gamma(\langle T,S\rangle)\). If \(W\) is the RKHS with reproducing kernel \(K\), then
\[
\|\mu-\nu\|_{W^*}^2
=
\iint K((x,T),(y,S))\,d(\mu-\nu)(x,T)\,d(\mu-\nu)(y,S).
\]
For Dirac sums,
\[
\langle \mu,\nu\rangle_{W^*}
=
\sum_{i,j} w_i w'_j\,\rho(\|x_i-y_j\|^2)\,\gamma(\langle T_i,S_j\rangle).
\]
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 \(K\) is \(C_0\)-universal, it is a true distance [1903.11196].

Quantization replaces a general volumetric varifold by
\[
\mu_n=\sum_{i=1}^n w_i\,\delta_{(x_i,T_i)}.
\]
There exist choices of \((w_i,x_i,T_i)\) such that \(\|\mu_n-\mu\|_{W^*}\to 0\), and if \(\operatorname{supp}\mu\) is compact then one can achieve
\[
\|\mu_n-\mu\|_{W^*}=O\bigl(n^{-1/(3+\dim \widetilde{G}_3^3)}\bigr)=O(n^{-1/4}).
\]
For source quantizations \(\mu_{0,n}\to \mu_0\), the approximate registration functionals
\[
E_n(v)=\tfrac12\int_0^1 \|v_t\|_V^2\,dt
+\lambda\|\varphi_1^v{}_\#\mu_{0,n}-\mu_1\|_{W^*}^2
\]
\(\Gamma\)-converge to the exact energy
\[
E(v)=\tfrac12\int_0^1 \|v_t\|_V^2\,dt
+\lambda\|\varphi_1^v{}_\#\mu_0-\mu_1\|_{W^*}^2,
\]
and minimizers of \(E_n\) accumulate to minimizers of \(E\) [1903.11196].

Diffeomorphic registration is formulated as optimal control. If \(V\) is an RKHS of vector fields embedded in \(C_0^1\), the flow \(\varphi_t^v\) solves \(\dot\varphi_t^v=v_t\circ\varphi_t^v\), and the inexact matching problem is
\[
\min_{v\in L^2([0,1],V)}
\left\{
\tfrac12\int_0^1 \|v_t\|_V^2\,dt
+\lambda\|\varphi_1^v{}_\#\mu_0-\mu_1\|_{W^*}^2
\right\}.
\]
If \(V\hookrightarrow C_0^2\), \(W\hookrightarrow C_0^1\), and \(\mu_0\) is compactly supported, a minimizer exists. Pontryagin’s Maximum Principle gives the Hamiltonian
\[
H(p,\varphi,v)=\langle p,v\circ\varphi\rangle-\tfrac12\|v\|_V^2,
\]
with
\[
\dot\varphi_t^v=\partial_pH,\qquad
\dot p_t=-\partial_\varphi H,\qquad
\partial_vH=0,
\]
and
\[
v_t=K_V*\bigl(\xi^*_{\varphi_t^v}p_t\bigr)
\]
for an optimal solution [1903.11196].

The metamorphosis extension introduces a growth rate \(r_t\) through
\[
\partial_t\mu_t+\operatorname{div}(v_t\mu_t)=r_t\mu_t,
\]
and energy
\[
E(v,r)
=\tfrac12\int_0^1 \|v_t\|_V^2\,dt
+\tfrac\gamma2\int_0^1\int_{\mathbb{R}^n} r_t(x)^2\,d|\mu_t|(x)\,dt.
\]
The corresponding tangent metric is
\[
G_\mu((v,r),(v,r))
=\|v\|_V^2+\gamma\|r\|_{L^2(|\mu|)}^2.
\]
The paper proves that this defines a genuine right-invariant Riemannian metric on each orbit \(\Theta(\mu_0)\), and relaxed problems with an additional RKHS fidelity term admit minimizers under mild assumptions [2112.04644].

Taken together, these results show that volumetric varifolds support both approximation theory and large-deformation dynamics. The measure-theoretic representation, the kernel metric, the \(\Gamma\)-convergence of discrete energies, and the Hamiltonian structure of registration all persist in the volumetric setting [1903.11196], [2112.04644].

## 6. Image-varifolds on meshes and volumetric feature measures

A related but distinct extension is the image-varifold framework for spatial transcriptomics. A \(d\)-dimensional image-varifold \(\mu\) is a finite Radon measure on \(\mathbb{R}^d\times\mathfrak{F}\), where \(\mathfrak{F}\) is a feature space such as genes, RNA-counts, or cell-types. It can be disintegrated into a spatial measure \(\mathfrak{m}\) and a family of transition probabilities \((\zeta_x)_{x\in\mathbb{R}^d}\) on \(\mathfrak{F}\):
\[
\mu(U\times A)=\int_U \zeta_x(A)\,d\mathfrak{m}(x).
\]
Important examples include a continuum image \(q:\mathbb{R}^d\to \mathfrak{F}\), a point-cloud varifold \(\sum_k \delta_{x_k}\otimes\delta_{f_k}\), and a semi-discrete mesh-based form
\[
\mu=\sum_{\gamma\in\Gamma} \alpha_\gamma\,|\gamma|\,\delta_{m_\gamma}\otimes \zeta_\gamma,
\]
where \(\gamma\) are small volumetric cells with center \(m_\gamma\), volume \(|\gamma|\), weights \(\alpha_\gamma\ge 0\), and feature laws \(\zeta_\gamma\) [2208.08376].

Comparison is again performed by an RKHS embedding. If the reproducing kernel on \(\mathbb{R}^d\times\mathfrak{F}\) is the product of a spatial kernel \(K_1(x,y)\) and a feature kernel \(K_2(f,g)\), then
\[
\langle \delta_x\otimes\delta_f,\delta_y\otimes\delta_g\rangle_{W^*}
=K_1(x,y)\,K_2(f,g),
\]
and
\[
\langle \mu,\nu\rangle_{W^*}
=
\int\!\!\int K_1(x,y)\,K_2(f,g)\,d\mu(x,f)\,d\nu(y,g).
\]
The chordal distance is \(\|\mu-\nu\|_{W^*}=\sqrt{\langle \mu-\nu,\mu-\nu\rangle_{W^*}}\). In semi-discrete form,
\[
\langle \mu,\nu\rangle_{W^*}
=
\sum_{\gamma,\gamma'} \alpha_\gamma|\gamma|\,\alpha'_{\gamma'}|\gamma'|
\,K_1(m_\gamma,m'_{\gamma'})
\,\langle \zeta_\gamma,\zeta'_{\gamma'}\rangle_{W_2^*}.
\]
This is a volumetric measure model because each cell contributes through its volume, center, density, and local feature histogram [2208.08376].

The diffeomorphic action is the “copy-and-paste” push-forward
\[
(\phi\cdot\mu)(F)=\mu(|D\phi|\,F\circ\phi),
\]
and the LDDMM energy is
\[
E(v)=\int_0^1 \|v(t)\|_V^2\,dt
+\frac{1}{\sigma^2}\|\mu_{\phi(1)\cdot \mathcal{T}^0}-\mu_1\|_{W^*}^2.
\]
By Pontryagin’s maximum principle and the reproducing-kernel property, the optimal \(v\) admits the reduced form
\[
v(t,\cdot)=\sum_{i\in I^0} K_V(\cdot,z_i(t))\,a_i(t),
\]
and an adjoint Hamiltonian system provides the gradient [2208.08376].

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 \(K_1\) 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 [2208.08376].

This suggests that volumetric-varifold ideas extend beyond tangent-plane encoding. In one branch, a cell stores approximate geometric orientation \(P_K\); in another, the cell stores a feature law \(\zeta_\gamma\). The common structure is a geometric measure supported on volumetric elements and equipped with a kernel metric or weak geometric flow formalism [2509.06440], [2208.08376].

Source: https://www.emergentmind.com/topics/volumetric-varifolds