Transported Square-Root Vector Field (TSRVF)

• TSRVF is a mathematical framework that represents temporal shape sequences by mapping them from a nonlinear preshape manifold to a linear Euclidean space.
• It employs inverse-exponential maps, parallel transport via the Levi–Civita connection, and PCA for dimensionality reduction of shooting vectors.
• The framework enables the use of vector time series models, such as VAR, to analyze and classify dynamic morphological patterns in biological and shape evolution studies.

The transported square-root vector field (TSRVF) is a mathematical framework for representing, modeling, and comparing temporal sequences of shapes, particularly in contexts such as morphological evolution of biological entities. The TSRVF addresses the challenge of analyzing sequences of shapes—modeled as points on an infinite-dimensional preshape manifold—by transforming the nonlinear, manifold-valued trajectories into a linear, Euclidean vector time series suitable for statistical modeling and machine learning. The fundamental steps involve computing shooting vectors via inverse-exponential maps between shapes, performing parallel transport via the Levi–Civita connection, and dimensionally reducing the resulting field using principal component analysis. This representation enables conventional vector time series models, such as VAR, to characterize the dynamics of shape evolution, with demonstrated applications in cell migration analysis (Deng et al., 2021).

1. Mathematical Definition and Underlying Manifold Structure

Let $\beta: [0,1] \to \mathcal{C}$ denote a one-parameter family of shapes, where $\mathcal{C}$ is the preshape manifold of unit-length closed curves, equipped with the $L^{2}$-metric. For each $t$, the velocity $$\dot\beta(t) = \frac{D}{Dt}\,\beta(t)\in T_{\beta(t)}\mathcal{C}$$ captures the instantaneous deformation of $\beta$. However, as each $\dot\beta(t)$ lies in a different tangent space, direct comparisons are not feasible.

The TSRVF solves this problem by parallel-transporting each $\dot\beta(t)$ back to a fixed reference tangent space, typically $T_{\beta(0)}\mathcal{C}$. Formally, given a discrete shape sequence $\beta: \{0,1,\dots,T-1\}\rightarrow \mathcal{C}$, the TSRVF is defined as:

$$F_\beta(\tau) = (\dot\beta(\tau))_{\beta(\tau)\to\beta(0)} \in T_{\beta(0)}\mathcal{C}, \quad \tau = 1, \ldots, T-1,$$

where $(\dot\beta(\tau))_{\beta(\tau)\to\beta(0)}$ denotes the parallel transport of the velocity via the Levi–Civita connection on $\mathcal{C}$. The integrated TSRVF (I-TSRVF) is

$H_\beta(\tau) = \sum_{s=1}^{\tau} F_\beta(s),$

though for time-series modeling, $F_\beta$ is typically preferred (Deng et al., 2021).

2. Parallel Transport with the Levi–Civita Connection

The preshape manifold $\mathcal{C}$, an infinite-dimensional submanifold of the Hilbert space $L^{2}$, inherits a Riemannian metric from the ambient space. The Levi–Civita connection provides a unique, torsion-free, metric-preserving way to transport tangent vectors. For any $V(t)\in T_{\beta(t)}\mathcal{C}$, $V$ is parallel along $\beta$ if $\nabla_{\dot\beta(t)} V(t) = 0$ for all $t$.

For two points $p$, $q$ on the unit sphere in Hilbert space, the parallel transport of $v \in T_{p}S$ along the minimal geodesic from $p$ to $q$ is explicitly:

$$\mathrm{PT}_{p\to q}(v) = v - \frac{\langle v, q \rangle}{1+\langle p, q \rangle} (p+q).$$

This formula is recursively applied along the discrete curve $\beta(0)\to\beta(1)\to\ldots\to\beta(T-1)$ for efficient practical computation (Deng et al., 2021).

3. Explicit Construction of the TSRVF

Compute the shooting vector between consecutive shapes using the inverse exponential map:

$$\dot\beta(\tau) = \exp_{\beta(\tau)}^{-1}(\beta(\tau+1)) \in T_{\beta(\tau)}\mathcal{C}, \quad \tau = 0,\ldots,T-2.$$

Parallel transport each shooting vector to the reference tangent space:

$$F_\beta(\tau) = \mathrm{PT}_{\beta(\tau)\to\beta(0)}\left[ \exp_{\beta(\tau)}^{-1}(\beta(\tau+1)) \right], \quad \tau = 1, \ldots, T-1,$$

or, using abbreviated notation, $$q(\tau) = \mathcal{T}_{\beta(\tau)} [\dot\beta(\tau)]$$ with $$\mathcal{T}_{\beta(\tau)} = \mathrm{PT}_{\beta(\tau)\to\beta(0)}$$. The time series $\{q(\tau)\}$ then uniformly resides in $T_{\beta(0)}\mathcal{C}$.

4. Reduction to Linear Representation and Euclidean Time Series

After transport, all $q(\tau)$ occupy the same linear tangent space, collectively forming an element of the Hilbert space:

$$\mathcal{H} = L^2([1,T-1], T_{\beta(0)}\mathcal{C})$$

with inner product

$$\langle q_1, q_2 \rangle_{\mathcal{H}} = \sum_{\tau=1}^{T-1} \langle q_1(\tau), q_2(\tau) \rangle_{T_{\beta(0)}\mathcal{C}}.$$

Statistical modeling becomes feasible via linear operations. Dimensionality reduction is achieved by analyzing all training TSRVFs with principal component analysis (PCA) in $T_{\beta(0)}\mathcal{C}$. Retaining the first $d$ directions gives a projection:

$$x(\tau) = \Pi(q(\tau)), \quad x(\tau) \in \mathbb{R}^d,$$

yielding a $d$-dimensional Euclidean time series, termed the TSRVF–PCA representation.

5. Algorithmic Workflow for TSRVF Computation

The computational steps for the TSRVF representation are:

Step	Operation	Description
1	Shooting vector	Compute $v(\tau) = \exp_{\beta(\tau)}^{-1}(\beta(\tau+1))$
2	Parallel transport	$$F(\tau) = \mathrm{PT}_{\beta(\tau)\to\beta(0)}[v(\tau)] = v(\tau) - \frac{\langle v(\tau), \beta(0) \rangle}{1+\langle \beta(\tau), \beta(0)\rangle} (\beta(\tau)+\beta(0))$$
3	PCA	Form data matrix from all $F(\tau)$ and extract top $d$ directions by functional PCA
4	Projection	$x(\tau) = \Pi(F(\tau)) \in \mathbb{R}^d$

This process is applied across training sequences to learn the principal subspace and then used to project each sequence, completing the transformation from manifold-valued shape paths to finite-dimensional Euclidean time series (Deng et al., 2021).

6. Statistical Modeling with VAR and Applications

Having reduced the shape dynamics to a Euclidean time series $\{x(\tau)\}$, modeling shape evolution proceeds with vector autoregressive (VAR) models. For order $p$,

$$x(\tau) = c + \sum_{j=1}^p A_j x(\tau-j) + \epsilon(\tau), \quad \epsilon(\tau) \sim N(0,\Sigma).$$

The estimated parameters $\Theta = (c, \{A_j\}, \Sigma)$ succinctly encode the dynamic morphological patterns of each sequence. Euclidean distances between VAR-parameter vectors provide a metric for shape-dynamics-based classification, which can be utilized along with standard classifiers such as SVM, Random Forest, and CNN. Furthermore, shape sequence synthesis is achieved by generating new $\tilde x(\tau)$ from the VAR and inverting the TSRVF transformation (Deng et al., 2021).

7. Significance and Impact

The TSRVF framework provides a principled mathematical approach for extracting and modeling time-varying dynamics in shape analysis problems, especially where data naturally lies on highly nonlinear and infinite-dimensional manifolds. By facilitating the application of linear statistical methods after the transformation, TSRVF has proven effective in classifying cell migration videos using shape dynamics, representing a significant advance in the computational analysis of biological motility (Deng et al., 2021). The explicit connection between geometric analysis and time series models underpins its extensibility to various applications involving temporal shape evolution.