A Fractional-Order Spatiotemporal Unified Energy Framework for Non-Repetitive LiDAR Point Cloud Registration

Abstract

Non-repetitive scanning LiDARs provide high coverage yet exhibit irregular sampling patterns, which destabilize local features and correspondences. To address this, we propose a novel spatiotemporal unified energy framework that integrates fractional calculus into rigid pose estimation. Spatially, we introduce a Riesz fractional regularization term to impose non-local smoothness constraints on the residual field, mitigating structural inconsistencies. Temporally, we design a Grünwald–Letnikov fractional dynamics solver that leverages long-memory effects of historical gradients to reduce the risk of being trapped in local minima. Comparative experiments on the Stanford 3D, MVTec ITODD, and HomebrewedDB (HB) datasets demonstrate that our method significantly outperforms state-of-the-art geometric and learning-based approaches. Specifically, it maintains a success rate exceeding 90% even under severe sampling perturbations where traditional methods fail. Ablation studies further validate that the introduction of non-local spatial constraints and historical gradient memory significantly reshapes the energy landscape, ensuring robust convergence. This work provides a rigorous theoretical foundation for applying fractional operators to point cloud processing.

1. Introduction

Point cloud registration is a fundamental problem in autonomous navigation, robotic vision, and industrial metrology. Recently, non-repetitive scanning LiDARs (e.g., the Livox family) have been deployed on mobile platforms. Their spiral or pseudo-random patterns can achieve near full field-of-view coverage after temporal accumulation, which helps detect small objects and capture fine details, but also leads to frame-wise inconsistent sampling, sparsity, and noise. Local structures are therefore hard to reuse across frames and stable correspondences become difficult to construct, so that robust registration and multi-frame fusion under non-repetitive sampling remain challenging [1].

Rigid registration is dominated by the Iterative Closest Point (ICP) algorithm and probabilistic variants such as Generalized-ICP (G-ICP), which iteratively refine a rigid transform from nearest-neighbor matches [2,3]. However, ICP-type methods solve a non-convex problem and are highly sensitive to the initial pose, overlap ratio, occlusions, and partial views, so that small local errors can easily accumulate into global drift. Non-rigid approaches such as Coherent Point Drift (CPD) estimate a smooth deformation field and perform well in medical imaging and deformable reconstruction, but may yield implausible distortions when data are severely missing [4]. Certifiable robust methods such as TEASER and TEASER++ tolerate extreme outlier rates, yet still rely on sufficiently good putative correspondences and incur non-negligible computational cost [5]. Overall, these methods remain difficult to tune for non-repetitive LiDAR scans with weak overlap and viewpoint-dependent missing data.

Fractional calculus offers an alternative modeling and optimization paradigm by introducing non-integer-order derivatives and integrals that naturally encode non-local couplings and long-memory effects. In control and multi-agent systems, fractional-order models have been used for singular system analysis, consensus, and event-triggered or distributed control [6,7,8,9]. In image processing and computer vision, fractional-order differentiators and enhancement models can sharpen structures, suppress noise, and adapt to varying local smoothness [10,11,12], while recent fractional prescribed-performance controllers for wheeled mobile robots and time-delay nonlinear systems show the benefit of embedding performance guarantees directly into robotics-related dynamics [13,14]. Despite these advances, standard registration frameworks remain bound by integer-order calculus. Such methods rely on local gradients that are ill-defined within the spatial voids of non-repetitive scanning, and typically employ optimization schemes with short-term (exponentially decaying) memory that struggle to navigate the rugged, non-convex energy landscapes. Consequently, fractional spatio-temporal formulations—with their inherent non-locality and heavy-tailed memory—provide a rigorous mathematical basis for overcoming these geometric and dynamic bottlenecks.

Motivated by these observations, this paper proposes a fractional-order spatio-temporal energy framework for rigid registration of non-repetitive LiDAR point clouds. On the spatial side, we introduce a Riesz fractional regularization term defined on the residual field. Unlike local integer-order operators, this term imposes non-local, power-law weighted smoothness to suppress local outliers and mismatches while preserving important geometric details. On the temporal side, we embed a Grünwald–Letnikov (GL) fractional derivative into the gradient-based optimization. This accumulates past gradients with binomial-type weights, yielding an explicit long-memory update that acts as a generalized momentum term superior to classical short-memory approaches. By coupling these two ingredients into a unified variational formulation, the proposed method jointly encodes non-local geometric constraints and historical gradient memory, aiming to improve robustness and convergence under irregular sampling, occlusions, and noise, without relying on large task-specific training datasets or handcrafted features. Figure 1 presents the qualitative registration results of a conventional baseline and our method under non-repetitive sampling perturbations.

2. Preliminaries

2.1. Overview of the Point Cloud Registration Problem

In the fields of 3D environmental perception and geometric reconstruction, the fundamental task of point cloud registration is to estimate the rigid body pose between a given source point cloud and a target point cloud, thereby aligning them as closely as possible within a unified coordinate system. Let the source point cloud and the target point cloud be denoted as ${\mathcal{P}}_s$ and ${\mathcal{P}}_m$, respectively:

$${\mathcal{P}}_s=\{{\mathbf{x}}_i\in {\mathbb{R}}^3\mid i=1,\dots ,N_s\},{\mathcal{P}}_m=\{{\mathbf{y}}_j\in {\mathbb{R}}^3\mid j=1,\dots ,N_m\},$$

where ${\mathbf{x}}_i$ and ${\mathbf{y}}_j$ represent the 3D spatial positions of the points. The variable to be solved in the registration process is the rigid transformation

$$T=(\mathbf{R},\mathbf{t})\in SE\left(3\right),\mathbf{R}\in SO\left(3\right),\mathbf{t}\in {\mathbb{R}}^3,$$

whose action on a point ${\mathbf{x}}_i$ is defined as

$$T\left({\mathbf{x}}_i\right)=\mathbf{R}{\mathbf{x}}_i+\mathbf{t}.$$

(1)

Upon determining a specific correspondence model (such as nearest-neighbor correspondence or feature-based soft correspondence), the registration problem can be formulated as an energy minimization problem with the pose T as the independent variable. A typical formulation is given by

$$E_{\mathrm{class}}(\mathbf{R},\mathbf{t})=\sum _{i=1}^{N_s}\rho \left({∥{\mathbf{y}}_{\pi \left(i\right)}-\left(\mathbf{R}{\mathbf{x}}_i+\mathbf{t}\right)∥}_2^2\right),$$

(2)

where $\pi \left(i\right)$ denotes the index of the target point selected for the i-th source point (e.g., the nearest neighbor), and $\rho (·)$ represents a potential robust loss function designed to suppress the influence of outliers and noise. Equation (2) encapsulates the fundamental principle underlying numerous registration methods, typified by the Iterative Closest Point (ICP) algorithm: it measures the degree of alignment between point clouds via the sum of squared weighted residuals and subsequently solves for the optimal pose using numerical techniques such as gradient-based methods or iterative closest point schemes.

2.2. Grünwald–Letnikov Fractional Derivative and Its Discretization

Unlike integer-order calculus, which involves only finite-order derivatives, fractional calculus relates the change at the current moment to the historical trajectory over a past period through non-integer-order differential operators, thereby introducing long-memory characteristics at the operator level.

Given a function $f\left(t\right)$ and an order $\alpha \in (0,1)$, a commonly used definition of the fractional derivative is the Riemann–Liouville (RL) type. With the left endpoint $t_0$ as the starting point, it is defined as

$$\left({}_{t_0}{}^{\mathrm{RL}}D_t^{\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\int }_{t_0}^t{\displaystyle \frac{f\left(\tau \right)}{{(t-\tau )}^{\alpha }}}\mathrm{d}\tau ,$$

(3)

where $\Gamma (·)$ is the Gamma function. As seen from Equation (3), the value of the fractional derivative at time t is jointly determined by the function values over the interval $[t_0,t]$ through a weighted integral. The integral kernel ${(t-\tau )}^{-\alpha }$ exhibits a power-law decay, reflecting a non-local memory of historical information.

In numerical computation and discrete-time modeling, the Grünwald–Letnikov (GL) fractional derivative is more frequently employed. Let the time step be $h>0$ and $t_n=t_0+nh$; the GL fractional derivative is defined as

$$\left({}_{t_0}{}^{\mathrm{GL}}D_t^{\alpha }f\right)\left(t_n\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)f\left(t_{n-k}\right),$$

(4)

where

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)={\displaystyle \frac{\alpha (\alpha -1)\cdots (\alpha -k+1)}{k!}}={\displaystyle \frac{\Gamma (\alpha +1)}{\Gamma (k+1)\Gamma (\alpha -k+1)}}$$

are the generalized binomial coefficients. Under appropriate regularity conditions, Equation (4) is equivalent to the Riemann–Liouville definition in function space and can thus be regarded as a discrete approximation form of the fractional derivative.

With a fixed time step h, truncating the limit in Equation (4) to a finite sum yields the commonly used discretized form of the GL fractional derivative:

$$\left({}_{t_0}{}^{\mathrm{GL}}D_t^{\alpha }f\right)\left(t_n\right)\approx {\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{L_{\mathrm{mem}}-1}{\alpha }_k^{\left(\alpha \right)}f\left(t_{n-k}\right),$$

(5)

where $L_{\mathrm{mem}}$ denotes the memory length, and ${\alpha }_k^{\left(\alpha \right)}$ represents the corresponding discrete weights. To facilitate numerical implementation, ${\alpha }_k^{\left(\alpha \right)}$ is typically generated via a binomial recurrence relation:

$${\alpha }_0^{\left(\alpha \right)}=1,{\alpha }_k^{\left(\alpha \right)}={\alpha }_{k-1}^{\left(\alpha \right)}·{\displaystyle \frac{\alpha -(k-1)}{k}},k=1,2,\dots ,L_{\mathrm{mem}}-1.$$

(6)

It can be observed that ${\left\{{\alpha }_k^{\left(\alpha \right)}\right\}}_{k\ge 0}$ is precisely the sequence of generalized binomial coefficients for order $\alpha$. This sequence possesses the following typical properties: First, the weights decay according to a power law as k increases, thereby assigning greater weight to recent history while gradually weakening the influence of the distant past. Second, when $0<\alpha <1$, the weights exhibit alternating signs, which helps numerically balance the effects across different time scales.

Viewing Equation (5) as a generalization of the integer-order forward difference operator, it is evident that when $\alpha \to 1$ and $L_{\mathrm{mem}}=1$, the GL fractional derivative degenerates into a first-order difference, further corresponding to the classical first-order derivative. Conversely, when $\alpha \to 0$, the fractional derivative gradually degenerates into the identity operator, implying no additional memory effect is introduced to the time series. Therefore, by adjusting the order $\alpha$ and the memory length $L_{\mathrm{mem}}$, a smooth transition between the “memoryless” integer-order case and the “long-memory” fractional-order case can be achieved, enabling the description of temporal correlations of varying intensities within a unified framework.

In subsequent sections, we will construct a fractional-order temporal dynamics update strategy based on the aforementioned GL fractional derivative. Specifically, the sequence of pose iterations will be treated as a state sequence evolving over time, and the weights from Equation (5) will be used to perform a weighted superposition of historical updates, achieving non-local temporal modulation of the optimization path.

2.3. Fractional Laplacian and Graph Representation

The previous subsection reviewed the basic forms of fractional calculus from the temporal dimension. This subsection introduces the fractional Laplacian operator and its representation on discrete graph structures from the spatial dimension, laying the foundation for constructing the fractional spatial regularization term for the point cloud residual field in the subsequent sections.

In the continuous case, given a scalar field u:$\Omega \to \mathbb{R}$ defined on $\Omega \subset {\mathbb{R}}^d$, the classical (integer-order) Laplacian operator is defined as

$$\Delta u\left(\mathbf{x}\right)=\nabla ·\nabla u\left(\mathbf{x}\right),\mathbf{x}\in \Omega ,$$

(7)

and the corresponding Dirichlet energy is given by

$${\mathcal{E}}_{\mathrm{loc}}\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{\Omega }{\parallel \nabla u\left(\mathbf{x}\right)\parallel }_2^2\mathrm{d}\mathbf{x}.$$

(8)

The Euler–Lagrange equation of (8) corresponds to the elliptic partial differential equation involving the Laplacian operator. Since Equation (8) involves only local gradient information, it corresponds to an integer-order, local smoothness constraint.

To introduce non-local spatial correlations, one can consider the Riesz-type fractional Laplacian operator ${(-\Delta )}^{s/2}$, where $s\in (0,1)$ is the fractional order. A common equivalent characterization is through the Gagliardo seminorm:

$${\left[u\right]}_{H^s(\Omega )}^2={\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{{\left(u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)\right)}^2}{{\parallel \mathbf{x}-\mathbf{y}\parallel }_2^{d+2s}}}\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y}.$$

(9)

This energy performs a weighted accumulation over all point pairs $(\mathbf{x},\mathbf{y})$, where the weight kernel ${\parallel \mathbf{x}-\mathbf{y}\parallel }_2^{-(d+2s)}$ exhibits a power-law decay, reflecting the non-local characterization of long-range interactions. Formally, ${\left[u\right]}_{H^s(\Omega )}^2$ can be regarded as the fractional Dirichlet energy, whose variational derivative is associated with the Riesz fractional Laplacian operator ${(-\Delta )}^{s/2}u$. As $s\to 1$, this seminorm converges in an appropriate sense to the local gradient energy (8), thereby achieving a unified formulation of smoothness constraints from local to non-local within the interval $s\in (0,1]$.

In the discrete case, a point cloud can be viewed as a set of sampling points from the continuous domain $\Omega$. Let ${\left\{{\mathbf{z}}_i\right\}}_{i=1}^N$ be N sampling points in the point cloud. We construct an undirected weighted graph

$$G=(V,E,W)$$

based on these points, where $V=\{1,\dots ,N\}$ is the set of vertices, $E\subseteq V\times V$ is the set of edges, and $W=\left(w_{ij}\right)\in {\mathbb{R}}^{N\times N}$ is a non-negative symmetric weight matrix. A typical approach establishes edges $(i,j)\in E$ via k-nearest neighbors or a distance threshold, and sets

$$w_{ij}=\phi \left(\parallel {\mathbf{z}}_i-{\mathbf{z}}_j{\parallel }_2\right),(i,j)\in E,$$

(10)

where $\phi (·)$ is a monotonically decreasing weight function, such as an inverse distance or Gaussian kernel, reflecting the spatial proximity between adjacent nodes on the graph.

On the graph structure, the discrete form of the integer-order (local) Dirichlet energy can be written as

$${\mathcal{E}}_{\mathrm{loc}}^{\mathrm{G}}\left(u\right)={\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{\left(u_i-u_j\right)}^2,$$

(11)

where $u_i$ denotes the value of function u at vertex i. Equation (11) is closely related to the graph Laplacian matrix

$$L=D-W,D=\mathrm{diag}\left(\sum _j{w}_{ij}\right),$$

(12)

and can be written in matrix form as ${\mathcal{E}}_{\mathrm{loc}}^{\mathrm{G}}\left(u\right)={\displaystyle \frac{1}{2}}{u}^{\top }Lu$, corresponding to integer-order Laplacian smoothing on the graph.

To characterize fractional smoothing on a graph structure, one can proceed from two complementary perspectives: one is via the spectral decomposition of the graph Laplacian, raising the eigenvalues $\left\{{\lambda }_{\ell }\right\}$ of L to ${\lambda }_{\ell }^s$ to define the spectral fractional graph Laplacian $L^s$; the other is to draw inspiration from the continuous Gagliardo seminorm and discretize Equation (9) into a non-local energy on the graph. Following the second approach, we consider defining

$${\mathcal{E}}_{\mathrm{frac}}^{\mathrm{G}}(u;s)={\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{\left|u_i-u_j\right|}^{2s},0<s\le 1,$$

(13)

where $u_i$ can be a scalar field, or a specific component or norm of a vector field at vertex i. Compared to Equation (11), the fractional energy (13) generalizes the quadratic form of differences to a power function of order $2s$. When $s=1$, ${\mathcal{E}}_{\mathrm{frac}}^{\mathrm{G}}(u;1)={\mathcal{E}}_{\mathrm{loc}}^{\mathrm{G}}\left(u\right)$, degenerating into the classical graph Dirichlet energy; when s takes a value close to 0, the penalty for large differences is relatively mitigated, demonstrating stronger non-local tolerance.

From a variational perspective, the first-order variation of the fractional graph energy (13) corresponds to a certain “Riesz-type” fractional Laplacian operator on the graph structure, the specific form of which depends on the selection of the weight matrix W and the exponent $2s$. By adjusting s, one can continuously regulate between strong smoothing (s close to 1) and weak smoothing (s close to 0), thereby controlling the constraint strength on the non-local structure of graph signals within a unified framework.

In subsequent sections, we will treat the residual field in point cloud registration as a vector field defined on the target point cloud graph and specialize $u_i$ in Equation (13) as the residual or its norm, constructing a fractional spatial regularization term suitable for non-repetitive LiDAR point clouds.

3. Unified Spatiotemporal Fractional Energy Framework

This section articulates the theoretical formulation of the Unified Spatiotemporal Fractional Energy Framework. We first construct a fractional spatial regularization term via graph-based Dirichlet energy, enforcing non-local constraints on the residual field to mitigate structural sparsity. Concurrently, a fractional-order temporal dynamic solving strategy is derived based on the Grünwald–Letnikov derivative, incorporating long-term memory to modulate the optimization trajectory and enhance convergence robustness. By synthesizing these fractional mechanisms with geometric priors into a unified energy functional, we establish a rigorous minimization paradigm on the $\mathrm{SE}\left(3\right)$ manifold. This framework theoretically generalizes integer-order methods and effectively addresses the ill-posedness inherent in non-repetitive LiDAR point cloud registration. Figure 2 illustrates the overall pipeline of our registration method.

3.1. Fractional Spatial Regularization Term

In this paper, we introduce a fractional spatial regularization term on the residual field of point cloud registration. Unlike classical Laplacian smoothing, which imposes quadratic penalties only on local gradients, we treat the residual as a vector field defined on the target point cloud graph based on the fractional Laplacian and graph representation discussed in Section 2.3. By characterizing its non-local spatial structure through fractional Dirichlet energy, we achieve robust global constraints suitable for non-repetitive LiDAR data within a unified energy framework.

3.1.1. Residual Vector Field and Target Point Cloud Graph

Continuing the notation from Section 2.3, let the target point cloud be

$${\mathcal{P}}_m=\{{\mathbf{y}}_i\in {\mathbb{R}}^3\mid i=1,\dots ,N_m\}.$$

We construct an undirected weighted graph $G=(V,E,W)$ on it, where $V=\{1,\dots ,N_m\}$ is the set of vertices corresponding to the target points ${\mathbf{y}}_i$; $E\subseteq V\times V$ is the set of edges established via k-nearest neighbors or a distance threshold; and $W=\left(w_{ij}\right)$ is a non-negative symmetric weight matrix reflecting the mutual proximity of ${\mathbf{y}}_i$ and ${\mathbf{y}}_j$ in geometric space.

Given the current pose $T=(\mathbf{R},\mathbf{t})\in SE\left(3\right)$, we consider defining a 3D residual vector field on the target point cloud:

$${\mathbf{r}}_i\left(T\right)\in {\mathbb{R}}^3,i=1,\dots ,N_m.$$

(14)

The specific form of this field is determined by the matching model. For instance, in the case of hard correspondence, ${\mathbf{r}}_i\left(T\right)$ can be defined as the difference between the target point and its corresponding transformed source point; in the case of soft correspondence, it can be defined as the linear combination of differences between the weighted mean of transformed source points and ${\mathbf{y}}_i$. Regardless of the specific correspondence mechanism employed, Equation (14) can be interpreted as “the local alignment error of the target point ${\mathbf{y}}_i$ under the current pose T”.

In classical registration frameworks, the optimization process typically minimizes the weighted sum of residual norms directly, such as

$$\sum _i\rho \left(\parallel {\mathbf{r}}_i{\left(T\right)\parallel }_2^2\right).$$

This form utilizes only the local residual information of each point and does not explicitly exploit the spatial correlation structure of residuals on the target point cloud. For sparse and non-uniform point clouds generated by non-repetitive LiDAR, such purely local residual modeling is often insufficient to suppress high-frequency disturbances caused by viewpoint differences, occlusions, and sampling biases.

3.1.2. From Continuous Fractional Dirichlet Energy to Graph Discrete Form

To introduce non-local smoothing constraints spatially, we extend the fractional Dirichlet energy on the continuous domain to the residual vector field on the graph structure. As described in Section 2.3, for a scalar field u:$\Omega \to \mathbb{R}$, the Gagliardo-type fractional seminorm

$${\left[u\right]}_{H^s(\Omega )}^2={\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{{\left(u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)\right)}^2}{{\parallel \mathbf{x}-\mathbf{y}\parallel }_2^{d+2s}}}\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y},0<s<1,$$

(15)

provides the fractional Dirichlet energy associated with the Riesz fractional Laplacian ${(-\Delta )}^{s/2}$. For a vector field $\mathbf{u}$:$\Omega \to {\mathbb{R}}^3$, one can naturally replace the scalar difference with the Euclidean norm of the vector difference, yielding

$${\left[\mathbf{u}\right]}_{H^s(\Omega )}^2={\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{{∥\mathbf{u}\left(\mathbf{x}\right)-\mathbf{u}\left(\mathbf{y}\right)∥}_2^2}{{\parallel \mathbf{x}-\mathbf{y}\parallel }_2^{d+2s}}}\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y},0<s<1.$$

(16)

Clearly, as $s\to 1$, Equation (16) converges in an appropriate sense to the local fractional energy based on ${\parallel \nabla \mathbf{u}\parallel }_2^2$; when s is smaller, the integral kernel assigns more significant weights to distant point pairs, thereby reinforcing non-local spatial interactions.

Discretizing Equation (16) onto the target point cloud graph G yields an approximation of the fractional Dirichlet energy on the graph structure. Letting ${\mathbf{u}}_i\approx \mathbf{u}\left({\mathbf{y}}_i\right)$ and approximating the continuous kernel and voxel volume factors with weights $w_{ij}$, we have

$${\mathcal{E}}_{\mathrm{frac}}^{\mathrm{G}}(\mathbf{u};s)\approx {\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{∥{\mathbf{u}}_i-{\mathbf{u}}_j∥}_2^{2s},0<s\le 1.$$

(17)

Compared to the graph fractional energy for the scalar case, Equation (17) acts directly on the Euclidean norm difference of the vector field. Specifically, when $s=1$,

$${\mathcal{E}}_{\mathrm{frac}}^{\mathrm{G}}(\mathbf{u};1)={\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{∥{\mathbf{u}}_i-{\mathbf{u}}_j∥}_2^2,$$

which is structurally consistent with the classical graph Dirichlet energy. When $0<s<1$, the power exponent $2s$ weakens the penalty for large residual differences, allowing the energy to tolerate local anomalies to some extent while maintaining global smoothness through non-local accumulation.

3.1.3. Fractional Spatial Regularization Term for the Residual Field

Based on the above derivation, we specialize the vector field in Equation (17) to the residual field from the registration process defined in Equation (14). Letting

$${\mathbf{u}}_i={\mathbf{r}}_i\left(T\right),i=1,\dots ,N_m,$$

the fractional spatial regularization energy for a given pose T is defined as

$$E_{\mathrm{frac}}(T;s)={\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{∥{\mathbf{r}}_i\left(T\right)-{\mathbf{r}}_j\left(T\right)∥}_2^{2s},0<s\le 1.$$

(18)

Physically, $E_{\mathrm{frac}}(T;s)$ measures the fractional “roughness” of the residual vector field along the target point cloud graph. If a pose T results in generally small differences in residuals between adjacent points, the corresponding energy is low. Conversely, if there are drastic changes or isolated mismatched pairs in local regions, their contribution to the energy will be amplified through the power function form. By adjusting the order s, a smooth transition between two extreme cases can be achieved:

• When s approaches 1, Equation (18) approximates a quadratic penalty on $\parallel {\mathbf{r}}_i\left(T\right)-{\mathbf{r}}_j{\left(T\right)\parallel }_2^2$, imposing strong smoothness constraints on the residual field. This is beneficial for obtaining highly smooth solutions in scenarios with relatively low noise and good coverage.
• When s approaches 0, the power exponent $2s$ decreases, making the energy less sensitive to large differences on individual edges. This emphasizes evaluating the global structure through the cumulative effect of multiple edges, resulting in stronger robustness against local anomalies and incomplete viewpoint overlaps.

It is worth emphasizing that Equation (18) is a key innovation in the registration framework proposed in this paper. Existing point cloud registration methods mostly rely on weighted sums of point-to-point or point-to-plane local residuals, at most superimposing integer-order local smoothing terms in the objective function. In contrast, this paper proceeds from the perspective of fractional Laplacians and graph signal processing, treating residuals as a vector field on a graph and introducing fractional Dirichlet energy. Through this construction, we incorporate a non-local spatial regularization mechanism with a tunable order into a unified energy model, enabling the algorithm to more fully utilize global structural information on sparse, non-uniform, and partially overlapping point clouds generated by non-repetitive LiDAR.

In the following subsections, we will combine $E_{\mathrm{frac}}(T;s)$ with the data term and normal consistency term to form a unified registration energy, and introduce fractional dynamics in the temporal dimension to establish a complete unified spatiotemporal fractional solving framework.

3.2. Fractional-Order Temporal Dynamics Solving Strategy

In the previous subsection, we constructed the residual field regularization term $E_{\mathrm{frac}}(T;s)$ from a spatial perspective based on fractional graph Dirichlet energy. Simultaneously, this paper introduces a temporal dynamics solving strategy driven by fractional calculus in the temporal (iteration) dimension. We treat the pose update as a state sequence evolving over “time” and utilize the Grünwald–Letnikov fractional derivative to reconstruct historical updates with weights, thereby achieving long-memory modulation of the optimization path within a unified framework. This strategy differs fundamentally from integer-order iterative methods that rely solely on the current gradient, making it particularly suitable for non-repetitive LiDAR registration scenarios characterized by highly non-convex energy landscapes and sensitivity to initial values.

3.2.1. From Integer-Order Gradient Flow to Fractional-Order Gradient Flow

Let $T\left(t\right)\in SE\left(3\right)$ denote the pose trajectory in continuous time $t\ge 0$, and represent its local coordinates using an appropriate parameterization (e.g., a vector $\mathit{\xi }\left(t\right)\in {\mathbb{R}}^6$ in the Lie algebra $\mathfrak{se}\left(3\right)$), such that

$$T\left(t\right)=exp\left(\widehat{\mathit{\xi }}\left(t\right)\right)T_0,$$

where $\widehat{\mathit{\xi }}$ is the matrix representation of $\mathit{\xi }$ in the Lie algebra, and $T_0$ is the initial pose. In classical integer-order optimization theory, the process of minimizing an energy function $E\left(T\right)$ can be viewed as a gradient flow in the parameter space, with its continuous form given by

$${\displaystyle \frac{\mathrm{d}\mathit{\xi }\left(t\right)}{\mathrm{d}t}}=-\mu {\nabla }_{\mathit{\xi }}E\left(T\left(t\right)\right),$$

(19)

where $\mu >0$ is the step size scale. Equation (19) indicates that the rate of change of the state is entirely determined by the energy gradient at the current moment, with historical information reflected only indirectly through the current state.

To explicitly introduce historical memory effects, we consider replacing the integer-order derivative with a fractional-order derivative. Let the order be $\alpha \in (0,1)$ and adopt the notation ${}_0{D}_t^{\alpha }$ for the Riemann–Liouville or Grünwald–Letnikov fractional derivative given in Section 2.2. The fractional gradient flow can then be written as

$$\left({}_0{D}_t^{\alpha }\mathit{\xi }\right)\left(t\right)=-\mu {\nabla }_{\mathit{\xi }}E\left(T\left(t\right)\right),0<\alpha <1.$$

(20)

Compared to the integer-order case, the fractional derivative ${}_0{D}_t^{\alpha }\mathit{\xi }$ in Equation (20) depends by definition on the entire trajectory $\mathit{\xi }\left(\tau \right)$ within the interval $[0,t]$ (see Equations (3) and (4)). Consequently, past update history continuously influences current state evolution through power-law decay; this constitutes the mathematical basis for introducing long memory in the temporal dimension using fractional calculus in this paper.

3.2.2. Discrete Time Update Based on GL Form

In numerical implementation, we are concerned with the discrete iteration sequence ${\left\{{\mathit{\xi }}_n\right\}}_{n\ge 0}$, corresponding to time grid points $t_n=nh$, where $h>0$ is a fixed time step. Utilizing the Grünwald–Letnikov discretization formula (Equation (5) and the weight recurrence relation Equation (6)) from Section 2.2, the fractional derivative can be approximated as

$$\left({}_0{}^{\mathrm{GL}}D_t^{\alpha }\mathit{\xi }\right)\left(t_n\right)\approx {\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{L_{\mathrm{mem}}-1}{\alpha }_k^{\left(\alpha \right)}{\mathit{\xi }}_{n-k},$$

(21)

where $L_{\mathrm{mem}}$ is the truncated memory length, and ${\alpha }_k^{\left(\alpha \right)}$ is generated by Equation (6). Substituting Equation (21) into the fractional gradient flow (20) and discretizing at $t=t_n$, we obtain

$$\sum _{k=0}^{L_{\mathrm{mem}}-1}{\alpha }_k^{\left(\alpha \right)}{\mathit{\xi }}_{n-k}=-h^{\alpha }\mu {\nabla }_{\mathit{\xi }}E\left(T_n\right),$$

(22)

where $T_n=exp\left({\widehat{\mathit{\xi }}}_n\right)T_0$. Equation (22) provides the coupling relationship between the current gradient and historical states: given a fixed $\alpha$ and $L_{\mathrm{mem}}$, it can be rewritten as an update rule for the current increment.

To facilitate integration with common “incremental” optimization frameworks, we express the state iteration as

$${\mathit{\xi }}_{n+1}={\mathit{\xi }}_n+\Delta {\mathit{\xi }}_n,$$

(23)

where $\Delta {\mathit{\xi }}_n\in {\mathbb{R}}^6$ is the state increment at step n, which can be further decomposed into a rotational part $\Delta {\omega }_n\in {\mathbb{R}}^3$ and a translational part $\Delta {\mathbf{t}}_n\in {\mathbb{R}}^3$. Substituting (23) into Equation (22) and performing algebraic rearrangement, we derive a fractional weighted form based on historical increments. To highlight the structure of fractional memory, we write it as

$$\Delta {\mathit{\xi }}_n^{\mathrm{eff}}=-\eta {\nabla }_{\mathit{\xi }}E\left(T_n\right)+\tau \sum _{k=1}^{L_{\mathrm{mem}}-1}{\beta }_k^{\left(\alpha \right)}\Delta {\mathit{\xi }}_{n-k},$$

(24)

where $\eta >0$ is the base step size, $\tau >0$ is the overall scaling factor for the memory term, and ${\left\{{\beta }_k^{\left(\alpha \right)}\right\}}_{k=1}^{L_{\mathrm{mem}}-1}$ is the normalized memory weight sequence linearly transformed from $\left\{{\alpha }_k^{\left(\alpha \right)}\right\}$. Here, $\Delta {\mathit{\xi }}_n^{\mathrm{eff}}$ in Equation (24) represents the “effective increment” actually used to update the state at step n, comprising two parts: the first term is the instantaneous update driven by the current gradient, and the second term is the fractional memory weighted superposition of the increments from the past $L_{\mathrm{mem}}-1$ steps.

Equation (24) represents one of the key innovative forms in the temporal dimension of this paper. It concretizes the non-local time convolution structure introduced by the GL fractional derivative into a power-law weighted accumulation of historical increments, such that the optimization path is no longer instantaneously determined solely by the current gradient but comprehensively considers the direction and magnitude of recent history updates. With different values of $\alpha$, $\tau$, and $L_{\mathrm{mem}}$, Equation (24) allows for a continuous transition between the following scenarios:

• When $\alpha$ is close to 0, $\tau$ is small, and $L_{\mathrm{mem}}$ is short, the contribution of ${\beta }_k^{\left(\alpha \right)}$ is negligible. $\Delta {\mathit{\xi }}_n^{\mathrm{eff}}$ is primarily controlled by the current gradient term, approximating classical Gradient Descent or Gauss–Newton updates.
• When $\alpha$ is close to 1, and $\tau$ and $L_{\mathrm{mem}}$ take larger values, the response of memory weights to historical increments is enhanced. The algorithm exhibits significant temporal long-memory characteristics, which helps smooth the optimization path and suppress high-frequency oscillations in scenarios with complex energy landscapes and numerous local minima.

3.2.3. Geometric Implementation of Pose Update with Fractional Time Memory

Based on the aforementioned fractional increment update strategy, we execute pose updates on $SE\left(3\right)$ in Lie algebra form. Let

$$\Delta {\mathit{\xi }}_n^{\mathrm{eff}}=\left[\begin{array}{c}\Delta {\mathit{\omega }}_n^{\mathrm{eff}}\\ \Delta {\mathbf{t}}_n^{\mathrm{eff}}\end{array}\right],$$

where $\Delta {\mathit{\omega }}_n^{\mathrm{eff}}$ corresponds to the rotational part and $\Delta {\mathbf{t}}_n^{\mathrm{eff}}$ to the translational part. The pose update from step n to step $n+1$ can be expressed as

$$T_{n+1}=exp\left(\widehat{\Delta {\mathit{\xi }}_n^{\mathrm{eff}}}\right)T_n=\left[\begin{array}{cc}exp\left(\widehat{\Delta {\mathit{\omega }}_n^{\mathrm{eff}}}\right)& \Delta {\mathbf{t}}_n^{\mathrm{eff}}\\ {\mathbf{0}}^{\top }& 1\end{array}\right]T_n.$$

(25)

Here, $\widehat{\Delta {\mathit{\omega }}_n^{\mathrm{eff}}}$ is the skew-symmetric matrix representation of $\Delta {\mathit{\omega }}_n^{\mathrm{eff}}$, and $exp(·)$ is the matrix exponential map on $SO\left(3\right)$. It is evident that fractional time memory acts indirectly on the updates of rotation and translation via $\Delta {\mathit{\xi }}_n^{\mathrm{eff}}$: the current pose depends not only on the instantaneous gradient direction but also on the combined influence of the update directions and scales from several historical steps.

From a geometric perspective, Equation (25) preserves the standard structure of updates via exponential mapping on $SE\left(3\right)$, while Equation (24) redefines the temporal evolution of the “step size” at the Lie algebra level. This design allows the fractional temporal dynamics proposed in this paper to be naturally embedded into existing registration frameworks without altering the pose representation and update paradigm, providing a mechanism for temporal modulation with long-memory characteristics for the subsequently constructed unified energy solver.

The distinction between this paper and existing point cloud registration methods lies in the introduction of the GL fractional derivative into pose iterative updates on $SE\left(3\right)$ and the construction of optimization dynamics through power-law weighted accumulation of historical increments. Existing works typically employ only integer-order momentum or adaptive step size strategies, whereas this paper proceeds from fractional calculus theory, explicitly parameterizing the intensity and range of temporal memory as the order $\alpha$ and memory length $L_{\mathrm{mem}}$. This establishes a unified theoretical connection between integer-order optimization and long-memory optimization. In the following subsections, we will integrate the spatial fractional regularization term (18) with the temporal fractional dynamics (24) and (25) into a unified energy framework, presenting the complete registration energy model and solution process.

3.3. Construction and Solution of the Unified Energy Model

In Section 3.1 and Section 3.2, we introduced fractional regularization and fractional temporal dynamics from spatial and temporal dimensions, respectively. This subsection presents the complete unified energy model and describes how to construct a registration solving framework that incorporates both fractional spatial constraints and fractional temporal memory. The unified energy integrates existing priors—such as data consistency, entropy regularization, normal consistency, and spherical normal distribution alignment—into a single objective function. Furthermore, it employs the fractional residual field energy as a novel spatial regularization term and organizes the iterative process via the fractional time update strategy. This constitutes the overall innovation of our method compared to existing point cloud registration works.

3.3.1. Unified Energy Model and Definition of Terms

Let the current pose be $T=(\mathbf{R},\mathbf{t})\in SE\left(3\right)$, and the source and target point clouds be

$${\mathcal{P}}_s={\left\{{\mathbf{x}}_i\right\}}_{i=1}^{N_s},{\mathcal{P}}_m={\left\{{\mathbf{y}}_j\right\}}_{j=1}^{N_m},$$

respectively. Let $G=(V,E,W)$ be the graph constructed from the target point cloud, and let ${\left\{{\mathbf{r}}_i\left(T\right)\right\}}_{i=1}^{N_m}$ be the residual vector field as defined in Equation (14). We consider a sparse set of source-target correspondence edges on G:

$${\mathcal{C}}_{\mathrm{loc}}\left(T\right)\subseteq \{(i,j)\mid i\in \{1,\dots ,N_s\},j\in \{1,\dots ,N_m\}\},$$

which is determined jointly by local nearest neighbor search and a scale gating mechanism. For each edge $(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)$, let the scale-normalized geometric cost be $c_{ij}\left(T\right)\ge 0$. This typically corresponds to the truncated, normalized residual under non-repetitive LiDAR settings.

On the local edge set ${\mathcal{C}}_{\mathrm{loc}}\left(T\right)$, we obtain sparse coupling weights ${\left\{{\pi }_{ij}\left(T\right)\right\}}_{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}$ via an entropy-regularized local optimal transport model, satisfying

$${\pi }_{ij}\left(T\right)\ge 0,\sum _{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}{\pi }_{ij}\left(T\right)=1,$$

which are solved numerically using Sinkhorn iterations. Based on this, we define the following four categories of local or global energy terms:

• Data Term $E_{\mathrm{data}}\left(T\right)$:

  $$E_{\mathrm{data}}\left(T\right)={\displaystyle \frac{1}{N_s}}\sum _{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}{\pi }_{ij}\left(T\right)c_{ij}\left(T\right).$$

  (26)
  This term measures the scale-normalized residual between the transformed source points and target points, serving as the dominant term reflecting geometric consistency in the unified energy.
• Entropy Regularization Term $E_{\mathrm{ent}}\left(T\right)$:

  $$E_{\mathrm{ent}}\left(T\right)=\sum _{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}{\pi }_{ij}\left(T\right)log\left({\pi }_{ij}\left(T\right)+\epsilon \right),\epsilon >0.$$

  (27)
  This term imposes entropy regularization on the local transport plan, encouraging correspondences to maintain a certain degree of spread within the local neighborhood. This prevents degeneracy into extremely few edges, thereby enhancing stability under sparse sampling conditions.
• Normal Consistency Term $E_{\mathrm{norm}}\left(T\right)$: Let ${\mathbf{n}}_i^s$ and ${\mathbf{n}}_j^m$ be the unit normals of the source and target points, respectively. We calculate their alignment on each edge as

  $${\delta }_{ij}\left(T\right)=\left|{\left(\mathbf{R}{\mathbf{n}}_i^s\right)}^{\top }{\mathbf{n}}_j^m\right|,(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right),$$

  and define

  $$E_{\mathrm{norm}}\left(T\right)=1-{\displaystyle \frac{{\sum }_{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}{\pi }_{ij}\left(T\right){\delta }_{ij}\left(T\right)}{{\sum }_{(i,j)\in {\mathcal{C}}_{\mathrm{loc}}\left(T\right)}{\pi }_{ij}\left(T\right)+\epsilon }}.$$

  (28)
  When the normals of most weighted correspondence edges are nearly aligned, $E_{\mathrm{norm}}\left(T\right)$ takes a small value; conversely, when there are numerous mismatched edges with inconsistent normals, this term increases, thereby suppressing registration results with inconsistent geometric structures.
• Spherical Normal Distribution Alignment (SNDA) Term $E_{\mathrm{snda}}\left(T\right)$: We construct direction histograms $h_s$ and $h_m$ on the unit sphere for the normals of the source and target point clouds, respectively. Under pose T, $h_s$ is mapped to $h_s^{\left(\mathbf{R}\right)}$ via rotation $\mathbf{R}$. The normalized correlation coefficient between them is defined as

  $$\kappa \left(T\right)=\langle h_m,h_s^{\left(\mathbf{R}\right)}\rangle ,0\le \kappa \left(T\right)\le 1,$$

  where the inner product is integrated discretely over the unit sphere. The SNDA energy is then defined as

  $$E_{\mathrm{snda}}\left(T\right)=1-\kappa \left(T\right).$$

  (29)
  This term measures the consistency of normal distributions at a global scale and is capable of counteracting misleading local matches, making it particularly suitable for scenarios with uneven local coverage caused by non-repetitive scanning.

Additionally, for the residual field $\left\{{\mathbf{r}}_i\left(T\right)\right\}$, Section 3.1 provided the fractional graph Dirichlet energy (18):

$$E_{\mathrm{frac}}(T;s)={\displaystyle \frac{1}{2}}\sum _{(i,j)\in E}{w}_{ij}{∥{\mathbf{r}}_i\left(T\right)-{\mathbf{r}}_j\left(T\right)∥}_2^{2s},0<s\le 1,$$

which imposes non-local smoothness constraints on the residual field over the target point cloud graph via a fractional Laplacian-type kernel. This is one of the core innovative terms in the unified energy proposed herein.

In summary, we linearly combine the five sub-terms above to construct the unified spatiotemporal fractional energy model:

$$E_{\mathrm{uni}}(T;s)={\lambda }_{\mathrm{data}}{E}_{\mathrm{data}}\left(T\right)+{\lambda }_{\mathrm{ent}}{E}_{\mathrm{ent}}\left(T\right)+{\lambda }_{\mathrm{norm}}{E}_{\mathrm{norm}}\left(T\right)+{\lambda }_{\mathrm{snda}}{E}_{\mathrm{snda}}\left(T\right)+{\lambda }_{\mathrm{frac}}{E}_{\mathrm{frac}}(T;s),$$

(30)

where the weights ${\lambda }_{\mathrm{data}},{\lambda }_{\mathrm{ent}},{\lambda }_{\mathrm{norm}},{\lambda }_{\mathrm{snda}},{\lambda }_{\mathrm{frac}}\ge 0$ are normalized such that their sum equals 1. In this unified model, $E_{\mathrm{data}},E_{\mathrm{ent}},E_{\mathrm{norm}},$ and $E_{\mathrm{snda}}$ comprehensively characterize local geometric consistency and global normal distribution priors, while $E_{\mathrm{frac}}(T;s)$ introduces non-local spatial constraints on the residual field in the form of fractional graph regularization. Specifically, when ${\lambda }_{\mathrm{frac}}\to 0$ and $s\to 1$, the model degenerates into a unified energy containing only integer-order local/global geometric priors. Conversely, when the weight ${\lambda }_{\mathrm{frac}}$ is significant, fractional spatial regularization plays a dominant role in non-repetitive point cloud scenarios.

3.3.2. Solving Framework Based on Fractional Temporal Dynamics

Under the unified energy (30), the registration task can be formulated as

$$\underset{T\in SE\left(3\right)}{min}{E}_{\mathrm{uni}}(T;s).$$

(31)

Executing integer-order gradient descent or Gauss–Newton updates directly on $SE\left(3\right)$ is often limited by the non-convexity of the energy landscape and local minima. Therefore, we adopt the fractional temporal dynamics strategy proposed in Section 3.2, parameterizing the pose as a Lie algebra vector $\mathit{\xi }\in {\mathbb{R}}^6$ and organizing the solution process via the fractional increment update Equation (24) and the pose update Equation (25).

Specifically, let the pose at the n-th iteration be $T_n$, with the corresponding Lie algebra coordinate ${\mathit{\xi }}_n$. In each iteration step:

• Local Linearization and Base Step Construction: In the vicinity of pose $T_n$, we perform a first-order or second-order approximation of the local residual terms (including $E_{\mathrm{data}}$ and $E_{\mathrm{norm}}$) in the unified energy $E_{\mathrm{uni}}(T;s)$. By constructing normal equations, we obtain a base increment $\Delta {\mathit{\xi }}_n$. This increment can be viewed as a one-step approximation of the integer-order gradient flow (19) in discrete time.
• Introduction of Fractional Time Memory: We input $\Delta {\mathit{\xi }}_n$ and its historical steps ${\{\Delta {\mathit{\xi }}_{n-k}\}}_{k=1}^{L_{\mathrm{mem}}-1}$ into the fractional increment update formula (24) to obtain the effective increment with long memory:

  $$\Delta {\mathit{\xi }}_n^{\mathrm{eff}}=-\eta {\nabla }_{\mathit{\xi }}{E}_{\mathrm{uni}}(T_n;s)+\tau \sum _{k=1}^{L_{\mathrm{mem}}-1}{\beta }_k^{\left(\alpha \right)}\Delta {\mathit{\xi }}_{n-k},$$

  where $\alpha \in (0,1)$ is the fractional order, $\left\{{\beta }_k^{\left(\alpha \right)}\right\}$ is the memory kernel generated by GL weights, and $\eta$ and $\tau$ control the relative influence of the instantaneous gradient and historical memory, respectively. This step embodies our fractional innovation in the temporal dimension: unlike traditional momentum methods, the memory kernel $\left\{{\beta }_k^{\left(\alpha \right)}\right\}$ is derived from the rigorous mathematical definition of fractional derivatives, causing the influence intensity and time scale of historical steps to decay according to a power law rather than simple exponential decay.
• Pose Update and Energy-Driven Line Search: We execute the update on $SE\left(3\right)$ using the effective increment $\Delta {\mathit{\xi }}_n^{\mathrm{eff}}$:

  $$T_{n+1}=exp\left(\widehat{\Delta {\mathit{\xi }}_n^{\mathrm{eff}}}\right)T_n,$$

  and perform a 1D line search along this direction. We evaluate $E_{\mathrm{uni}}\left(exp\left(\widehat{s_{\ell }\Delta {\mathit{\xi }}_n^{\mathrm{eff}}}\right)T_n;s\right)$ on a set of scaling factors ${\left\{s_{\ell }\right\}}_{\ell }$ to select a step size that satisfies sufficient decrease conditions. To enhance robustness in complex energy landscapes, we employ a non-monotonic line search with a finite window. This allows the energy to undergo finite non-monotonic changes within an iteration window and terminates the iteration at the current scale early upon detecting a “plateau” (via a threshold on the max/min energy difference within the window).

3.3.3. Complementarity of Spatial and Temporal Components

The robustness of the proposed framework stems from the fundamental complementarity between its spatial and temporal constituents. To appreciate this synergy, it is instructive to delineate the limitations of each component when functioning in isolation.

The Spatial Fractional Regularization acts as a static structural constraint. By enforcing non-local smoothness via the Riesz potential, it effectively “repairs” the jagged energy landscape caused by sparse LiDAR sampling. However, spatial regularization alone is local in nature; while it ensures that the local geometry is consistent, it cannot prevent the optimizer from converging to a wrong pose if the initial alignment is poor (i.e., it creates a smooth basin, but potentially the wrong one).

Conversely, the Temporal Fractional Dynamics function as a kinetic driver. The heavy-tailed memory provides the necessary inertia to escape shallow local minima, addressing the non-convexity of the problem. Yet, without spatial regularization, the temporal dynamics would operate on a degenerate, noise-filled landscape. In such cases, the “long memory” might drive the solution towards spurious geometric inconsistencies rather than the true alignment.

Therefore, the two components are fundamentally complementary: the spatial term ensures the geometric validity of the convergence basin, while the temporal term provides the navigational capability to reach that basin globally.

The above steps constitute a unified solving framework driven jointly by fractional spatial regularization and fractional temporal memory. Spatially, $E_{\mathrm{frac}}(T;s)$ in the unified energy (30) imposes non-local constraints on the target point cloud graph via fractional graph Dirichlet energy. Temporally, the update process accumulates historical increments via power-law weighting through fractional temporal dynamics, smoothing the optimization trajectory and suppressing high-frequency oscillations while maintaining local geometric consistency. In the extreme case where ${\lambda }_{\mathrm{frac}}\to 0$, $\alpha \to 0$, and the memory length $L_{\mathrm{mem}}$ takes a minimal value, this framework degenerates into traditional integer-order local optimization methods. However, in the general setting, the fractional spatiotemporal mechanism provides stronger convergence stability and global robustness for non-repetitive LiDAR point cloud registration.

3.3.4. Verification of Consistency Between Unified Energy and Pose Variation

To verify the consistency between the constructed unified energy and the actual pose error, we experimentally observe the variations in energy under different pose changes.

First, an asymmetric 3D model point cloud containing 1000 points is randomly generated and utilized as both the source and target point clouds. For each given translation magnitude $\left|\mathbf{t}\right|\in \{0.0,0.1,0.2,0.3\}\mathrm{m}$, a corresponding ground truth rigid transformation $T_t^{\star }=(R^{\star }=\mathbf{I},{\mathbf{t}}^{\star })$ is constructed, yielding the source point cloud $S_t=T_t^{\star }\left(M\right)$.

Subsequently, while keeping the translation ${\mathbf{t}}^{\star }$ fixed, we scan different rotation errors $\mathrm{RRE}=\left|\delta \theta \right|\in [0°,30°]$ solely within the neighborhood of the true pose. We then calculate the unified energy using an implementation identical to that of the registration algorithm (incorporating Optimal Transport (OT), normal consistency, SNDA, and the fractional regularization term). As shown in Figure 3, the overall energy increases with the magnitude of pose variation. This phenomenon demonstrates that the proposed unified energy exhibits excellent consistency with rotation and translation errors.

In conclusion, this paper integrates graph fractional residual regularization (18) and GL-type fractional temporal dynamics (24), (25) into a single registration energy and solving framework. Theoretically, this framework treats integer-order registration methods as special cases of the fractional model; practically, it offers a new viable path for processing sparse, locally inconsistent point clouds generated by non-repetitive LiDAR.

4. Experiments and Analysis

This section aims to comprehensively validate the effectiveness and robustness of the proposed fractional spatiotemporal unified energy registration algorithm. First, a data generation pipeline simulating non-repetitive scanning characteristics is constructed to establish quantitative evaluation benchmarks. Subsequently, through comparative experiments on datasets such as Stanford 3D and MVTec ITODD, we demonstrate the significant accuracy advantages of our method over traditional geometric and advanced learning-based methods in handling point position jitter and structural missingness. Finally, ablation studies are conducted to quantitatively analyze the necessity of temporal memory, spatial regularization, and the three-stage solving strategy.

4.1. Experimental Data Preparation

The core characteristic of non-repetitive scanning LiDAR (e.g., Livox) lies in its scanning pattern (such as rosette or spiral) that does not repeat over short durations, resulting in significant local density drift and point position jitter between consecutive frames. Traditional registration datasets (e.g., 3DMatch) are typically constructed based on structured light or repetitive scanning LiDARs and lack such characteristics. To quantitatively evaluate the algorithm’s performance under non-repetitive sampling in a controlled environment, we constructed a simulation data generation pipeline. This pipeline aims to reproduce two key features of non-repetitive sampling: (1) partial overlap caused by occlusion and viewpoint limitations; and (2) local point position perturbations caused by sampling inconsistency.

4.1.1. Viewpoint Simulation and Partial Point Cloud Generation

The data generation starts from a complete, high-density model point cloud (e.g., a model from the Stanford 3D or ITODD datasets). To simulate the locality of real scanning, we perform the following steps:

• Model Preprocessing: First, the model point cloud $\mathcal{M}$ is centered (setting its centroid to the origin) and normalized (scaling it within a unit bounding box) to eliminate scale and translation ambiguities.
• Virtual Camera Scanning: We define a virtual camera pose, including its position ${\mathbf{p}}_{cam}$, observation target ${\mathbf{p}}_{tgt}$ (usually the origin), and up vector ${\mathbf{u}}_{cam}$.
• Visibility and Occlusion Judgment: By simulating a pinhole camera model, we calculate the visibility of each point in $\mathcal{M}$. A point ${\mathbf{y}}_j$ is considered visible if and only if it satisfies the following conditions simultaneously:

  −

  Frustum Constraint: Point ${\mathbf{y}}_j$ lies within the preset depth range $[d_{min},d_{max}]$ of the camera, i.e.,

  $$d_{min}\le {\parallel {\mathbf{y}}_j-{\mathbf{p}}_{cam}\parallel }_2\le d_{max}.$$

  −

  Forward Constraint: Point ${\mathbf{y}}_j$ must be located in front of the camera, i.e.,

  $$({\mathbf{y}}_j-{\mathbf{p}}_{cam})·({\mathbf{p}}_{tgt}-{\mathbf{p}}_{cam})>0.$$

  −

  Back-face Culling: To simulate occlusion by opaque surfaces, we utilize the normal vector ${\mathbf{m}}_j$ at point ${\mathbf{y}}_j$. The point is deemed visible only if the angle between the normal and the viewing vector $({\mathbf{p}}_{cam}-{\mathbf{y}}_j)$ is less than a certain threshold ${\theta }_{thresh}$ (e.g., 90°).

• Partial Point Cloud Basis: All points passing the above filtering constitute the “visible model point cloud” ${\mathcal{M}}_{vis}\subset \mathcal{M}$. This forms the basis for subsequent perturbation.

4.1.2. Ground Truth Pose and Source Point Cloud Generation

To formulate the registration problem, we define a known rigid transformation (i.e., ground truth) $T_{gt}=({\mathbf{R}}_{gt},{\mathbf{t}}_{gt})\in SE\left(3\right)$. This transformation is composed of preset Euler angles $(\varphi ,\theta ,\psi )$ and a translation vector $(t_x,t_y,t_z)$. By applying this transformation to ${\mathcal{M}}_{vis}$, we generate the “original source point cloud” ${\mathcal{S}}_{orig}$:

$${\mathcal{S}}_{orig}=\left\{{\mathbf{x}}_i\mid {\mathbf{x}}_i={\mathbf{R}}_{gt}{\mathbf{y}}_j+{\mathbf{t}}_{gt},{\mathbf{y}}_j\in {\mathcal{M}}_{vis}\right\}.$$

At this stage, we possess the model point cloud $\mathcal{M}$, the source point cloud ${\mathcal{S}}_{orig}$, the perfect correspondence between them (indices of points in ${\mathcal{M}}_{vis}$), and the ground truth transformation $T_{gt}$.

4.1.3. Adaptive Gaussian Perturbation (Simulating Non-Repetitive Sampling)

To simulate the central feature of non-repetitive sampling—local point position jitter—we apply an adaptive Gaussian perturbation to ${\mathcal{S}}_{orig}$ (as shown in Figure 1b), generating the final “perturbed source point cloud” ${\mathcal{S}}_{dist}$. The design of this perturbation process aims to correlate the intensity of the perturbation with the local density of the point cloud.

• Local Density Estimation: Point position jitter (in terms of absolute distance) caused by non-repetitive sampling is typically greater in sparse regions than in dense regions. We employ the K-Nearest Neighbor (K-NN) distance to quantify the local sparsity of each point ${\mathbf{x}}_i$ in ${\mathcal{S}}_{orig}$. We calculate the Euclidean distance $d_K\left({\mathbf{x}}_i\right)$ from ${\mathbf{x}}_i$ to its K-th nearest neighbor. The choice of K (e.g., $K=28$) simulates the scale of the perturbation influence. A larger $d_K\left({\mathbf{x}}_i\right)$ indicates that the region where ${\mathbf{x}}_i$ is located is sparser.
• Adaptive Perturbation Radius: We treat $d_K\left({\mathbf{x}}_i\right)$ as the adaptive perturbation radius $r_{dist}\left({\mathbf{x}}_i\right)$ for point ${\mathbf{x}}_i$. This implies that points in sparse regions are allowed to jitter within a larger range.
• Gaussian Perturbation Generation: We randomly select a certain proportion (e.g., 100%) of points ${\mathbf{x}}_i$ from ${\mathcal{S}}_{orig}$ for perturbation. For each selected point ${\mathbf{x}}_i$, we generate a 3D Gaussian perturbation vector ${\delta }_i\sim \mathcal{N}(\mathbf{0},{\sigma }_i^2{\mathbf{I}}_3)$.

  −

  Adaptive Standard Deviation: The standard deviation ${\sigma }_i$ of the perturbation is proportional to the perturbation radius $r_{dist}\left({\mathbf{x}}_i\right)$ of that point, i.e.,

  $${\sigma }_i={\displaystyle \frac{r_{dist}\left({\mathbf{x}}_i\right)}{3}}.$$

  The choice of $\sigma =r/3$ is based on the $3\sigma$ rule of the Gaussian distribution, ensuring that approximately $99.7\%$ of the random perturbation vectors ${\delta }_i$ naturally fall within the radius $r_{dist}\left({\mathbf{x}}_i\right)$.

  −

  Perturbation Truncation: To strictly ensure that the perturbation remains within the radius, we truncate the generated ${\mathit{\delta }}_i$:

  $${\mathit{\delta }}_i^{\mathrm{final}}=\left\{\begin{array}{cc}{\mathit{\delta }}_i,\hfill & \mathrm{if}\parallel {\mathit{\delta }}_i{\parallel }_2\le r_{dist}\left({\mathbf{x}}_i\right),\hfill \\ {\mathit{\delta }}_i·{\displaystyle \frac{r_{dist}\left({\mathbf{x}}_i\right)}{\parallel {\mathit{\delta }}_i{\parallel }_2}},\hfill & \mathrm{if}\parallel {\mathit{\delta }}_i{\parallel }_2>r_{dist}\left({\mathbf{x}}_i\right).\hfill \end{array}\right.$$

  −

  Generating Final Source Point Cloud:

  $${\mathcal{S}}_{dist}=\left\{{\mathbf{x}}_i^{\prime }\mid {\mathbf{x}}_i^{\prime }={\mathbf{x}}_i+{\mathit{\delta }}_i^{\mathrm{final}},{\mathbf{x}}_i\in {\mathcal{S}}_{orig}\right\}.$$

Through the above pipeline, we generated the registration pairs $(\mathcal{M},{\mathcal{S}}_{dist})$ for the experiments. ${\mathcal{S}}_{dist}$ simulates a source point cloud observed from a specific viewpoint (causing partial overlap) that has undergone non-repetitive scanning (causing point position jitter). The selection of the K value (e.g., from $K=8$ to $K=24$) allows us to control the severity of the simulated perturbation systematically, thereby evaluating the algorithm’s performance under different perturbation levels.The datasets utilized in this paper include the Stanford 3D dataset [15], the MVTec ITODD dataset [16], and the HomebrewedDB (HB) dataset [17]. As illustrated in Figure 4, we present examples of non-repetitively sampled point clouds generated using selected models from these datasets.

4.2. Registration Evaluation Criteria

To quantitatively evaluate the registration accuracy and robustness of the proposed algorithm, we adopt two standard metrics widely used in the field of 3D registration: Rotation Error (RRE) and Relative Translation Error (RTE). Given a registration pair, the ground truth rigid transformation is known as $T_{gt}=({\mathbf{R}}_{gt},{\mathbf{t}}_{gt})\in SE\left(3\right)$, where ${\mathbf{R}}_{gt}\in SO\left(3\right)$ is the ground truth rotation matrix and ${\mathbf{t}}_{gt}\in {\mathbb{R}}^3$ is the ground truth translation vector. For the estimated transformation $\widehat{T}=(\widehat{\mathbf{R}},\widehat{\mathbf{t}})$ output by the algorithm, we define RRE and RTE as follows:

• Rotation Error (RRE): RRE measures the angular difference between the estimated rotation $\widehat{\mathbf{R}}$ and the ground truth rotation ${\mathbf{R}}_{gt}$. It is defined as the rotation angle of the relative rotation matrix ${\mathbf{R}}_{\mathrm{err}}={\mathbf{R}}_{gt}^{\top }\widehat{\mathbf{R}}$. This angle $\theta$ can be calculated via the trace of ${\mathbf{R}}_{\mathrm{err}}$. The RRE converts this radian value (or uses it directly in calculation) into degrees:

  $$\mathrm{RRE}={\displaystyle \frac{180}{\pi }}\theta ={\displaystyle \frac{180}{\pi }}arccos\left({\displaystyle \frac{tr\left({\mathbf{R}}_{gt}^{\top }\widehat{\mathbf{R}}\right)-1}{2}}\right),$$

  (32)

  where $tr(·)$ denotes the trace of a matrix (the sum of diagonal elements), ${\mathbf{R}}_{gt}^{\top }$ denotes the transpose (inverse) of ${\mathbf{R}}_{gt}$, and arccos is the arccosine function.
• Relative Translation Error (RTE): RTE measures the Euclidean distance between the estimated translation $\widehat{\mathbf{t}}$ and the ground truth translation ${\mathbf{t}}_{gt}$:

  $$\mathrm{RTE}={∥\widehat{\mathbf{t}}-{\mathbf{t}}_{gt}∥}_2,$$

  (33)

  where ${\parallel ·\parallel }_2$ denotes the $L_2$ norm (Euclidean distance) of a vector.
• Success Rate: In non-convex registration problems, comparing average RRE/RTE alone can be misleading, as a few extreme failure cases (where RRE approaches 180°) can severely skew the mean. Therefore, we introduce the Success Rate as a more robust macroscopic evaluation metric. A registration experiment is considered successful if and only if both its RRE and RTE are below preset thresholds ${\tau }_R$ and ${\tau }_t$. In the experiments of this paper, we adopt a set of strict thresholds:

  $$\mathrm{Success}\iff (\mathrm{RRE}\le 3.0°)\wedge (\mathrm{RTE}\le 0.03\mathrm{m}).$$

  (34)
  The Success Rate is defined as the percentage of successful trials out of all test cases. This metric is a key criterion for measuring the robustness of the algorithm under challenging conditions (such as the non-repetitive sampling perturbations simulated in this experiment).

4.3. Bayesian Parameter Optimization

In the fractional spatiotemporal registration framework proposed above, several critical hyperparameters significantly influence performance. These include: the spatial fractional order s, which controls the sensitivity of the Riesz fractional regularization to outlier residuals; the temporal fractional order ${\alpha }_{\mathrm{time}}$ and the memory scale factor ${\kappa }_{\mathrm{GL}}$, which jointly regulate the inertia and convergence trajectory of the fractional temporal dynamics; and the weight vector for the unified energy terms

$$\mathit{\lambda }={\left[{\lambda }_{\mathrm{data}},{\lambda }_{\mathrm{ent}},{\lambda }_{\mathrm{norm}},{\lambda }_{\mathrm{snda}},{\lambda }_{\mathrm{frac}}\right]}^{\top },$$

where the five components correspond to the data term, entropy regularization term, local normal consistency term, spherical normal distribution term, and fractional spatial regularization term, respectively.

Since the mapping from the aforementioned hyperparameters to the final registration error is a highly nonlinear, strongly non-convex black-box function lacking analytical gradients, relying on manual tuning or regular grid search makes it difficult to find a reasonable parameter configuration within a limited budget. Therefore, we model the hyperparameter identification problem as a global optimization problem on a bounded domain and employ Bayesian Optimization (BO) for the solution [18,19].

We denote the set of hyperparameters to be optimized as

$$\Theta =\left\{s,{\alpha }_{\mathrm{time}},{\kappa }_{\mathrm{GL}},\mathit{\lambda }\right\},$$

where the energy weights must satisfy the simplex constraint

$${\lambda }_i>0,\sum _i{\lambda }_i=1,$$

to ensure the physical clarity of the unified energy. For the fractional orders s and ${\alpha }_{\mathrm{time}}$, we restrict their search intervals to sub-intervals of $(0,1)$ (specifically, $s,{\alpha }_{\mathrm{time}}\in [0.30,0.95]$ in our implementation) to verify the performance advantages of non-integer derivatives compared to integer-order cases. The memory scale factor ${\kappa }_{\mathrm{GL}}$ is confined to a finite positive interval (e.g., $[0.50,2.50]$) to control the decay rate of the GL memory kernel.

The optimization objective is defined as the sum of the average Relative Rotation Error (RRE) and Relative Translation Error (RTE) on a validation dataset ${\mathcal{D}}_{\mathrm{val}}$:

$${\Theta }^{*}=\underset{\Theta \in \Omega }{argmin}\mathcal{J}(\Theta ),\mathcal{J}(\Theta )={\displaystyle \frac{1}{|{\mathcal{D}}_{\mathrm{val}}|}}\sum _{k\in {\mathcal{D}}_{\mathrm{val}}}\left({\mathrm{RRE}}_k(\Theta )+{\mathrm{RTE}}_k(\Theta )\right),$$

(35)

where $({\mathrm{RRE}}_k(\Theta ),{\mathrm{RTE}}_k(\Theta ))$ represent the rotation and relative translation errors obtained by running the complete registration process on the k-th validation sample under the current hyperparameters $\Theta$. If a registration fails to output a valid pose or encounters numerical anomalies, the error for that sample is assigned a sufficiently large constant to impose a penalty in the objective function. Since the magnitudes of RRE (in degrees) and RTE (in meters) are comparable at the error scale of interest in this paper, their direct sum effectively reflects registration quality in both rotation and translation dimensions. It is important to note that subsequent comparative and ablation experiments utilize the optimal hyperparameter configuration obtained by minimizing $\mathcal{J}(\Theta )$ in this section, without re-tuning for each experimental scenario.

To efficiently search for the optimum given the high evaluation cost of the objective function, we employ a Gaussian Process (GP) as the surrogate model. Assuming the scalar objective function $\mathcal{J}(\Theta )$ follows a Gaussian Process prior

$$\mathcal{J}(\Theta )\sim \mathcal{GP}\left(m(\Theta ),k(\Theta ,{\Theta }^{\prime })\right),$$

(36)

where $m(·)$ is the mean function and $k(·,·)$ is the covariance kernel function. Given the observed parameter-error pairs ${\left\{({\Theta }_i,y_i)\right\}}_{i=1}^n$, the GP posterior provides the predicted mean and variance of the objective value for any candidate point $\Theta$, thereby characterizing the model’s comprehensive judgment of that point’s quality and uncertainty.

Based on the GP posterior, we adopt Expected Improvement (EI) as the acquisition function to select the sampling point ${\Theta }_{n+1}$ for the next iteration. Let the current best observed objective value be

$${\mathcal{J}}_{min}=\underset{1\le i\le n}{min}{y}_i,$$

then EI is defined as

$$\mathrm{EI}(\Theta )=\mathbb{E}\left[max\left(0,{\mathcal{J}}_{min}-\mathcal{J}(\Theta )\right)\right].$$

(37)

A larger EI value indicates a more favorable balance between “potential to achieve a smaller objective value” and “high current model uncertainty.” By repeatedly maximizing $\mathrm{EI}(\Theta )$ in the search space and evaluating the new point, Bayesian optimization achieves an adaptive trade-off between exploiting known good regions and exploring unknown areas.

Under the above framework, we statistically identify a set of more suitable fractional orders $s^{*}$, ${\alpha }_{\mathrm{time}}^{*}$, unified energy weights, and memory scale factor for the current data distribution. In the parameter tuning process, targeting the minimization of RRE and RTE, we set the default initial iterations to 20 and Bayesian optimization iterations to 60. We selected models from the Stanford 3D dataset and generated registration datasets according to the method described in Section 4.1. For all models in the dataset, we averaged the results over 30 iterations as the basis for evaluation. As shown in Figure 5, the data fluctuations stabilized after 40 iterations. Furthermore, we observed that for the overall parameters, the spatial fractional order parameter (s) performed well in the range of $0.4–0.6$, and the temporal fractional order (${\alpha }_{\mathrm{time}}$) in the range of $0.4–0.7$. In subsequent experiments, we will select the parameter combination that achieved the lowest RRE and RTE during optimization as the default hyperparameters, keeping them unchanged to ensure the objectivity of the conclusions in comparative and ablation studies.

4.4. Comparative Experiments

This section aims to systematically compare the proposed fractional spatiotemporal unified energy framework with representative traditional methods and learning-based methods to verify the effectiveness and superiority of our algorithm.

4.4.1. Comparison Methods and Characteristics

To cover different paradigms ranging from traditional geometric approaches to learning-based front-ends, we selected the following five representative methods for comparison:

• RANSAC+FPFH+ICP: A classic three-stage pipeline comprising feature extraction (FPFH), global estimation (RANSAC), and point-to-plane ICP for superior local refinement. It serves as a traditional baseline but relies heavily on feature stability.
• FGR+ICP: An advanced non-learning framework that optimizes global pose via robust cost functions (e.g., GNC) followed by ICP. It offers higher efficiency and stability than RANSAC under moderate outlier ratios [20].
• RANSAC+FPFH+TEASER++: A state-of-the-art geometric solver employing Truncated Least Squares (TLS) with provable robustness bounds. It excels in scenarios with high outlier ratios and severe occlusions [21,22].
• PREDATOR: An advanced Transformer-based network that simultaneously predicts overlap and inliers. We utilize its pre-trained model to evaluate cross-dataset generalization and robustness under non-repetitive sampling [23].
• GeoTransformer: Encodes multi-scale geometry using GNNs and Transformers. It serves both as a strong learning-based baseline and as the front-end generator for our method, enabling a direct evaluation of different back-end solving strategies [24].

The methods mentioned above cover a range from classic geometric baselines and newer generations of fast/provably robust solvers (RANSAC+FPFH+ICP, FGR, TEASER++) to representative learning-based registration models (PREDATOR, GeoTransformer). This selection comprehensively reflects the performance upper bounds of current non-learning and learning paradigms in the scenarios focused on in this work. The fractional spatiotemporal unified energy registrar proposed in this paper introduces spatial fractional regularization and temporal fractional memory within a unified energy framework and employs a three-stage decoupled solving strategy. We focus on examining its robustness and accuracy advantages under conditions of non-repetitive sampling and structural missingness.

4.4.2. Experimental Settings

Comparative evaluations were conducted on the Stanford 3D, ITODD, and HB datasets. To ensure sufficient geometric complexity, simple primitives such as cubes and cylinders were excluded. The curated subset comprises 8 models from Stanford 3D, 11 typical industrial parts from ITODD, and 18 deformable or occluded models from HB, all exhibiting rich structural details.To simulate non-repetitive sampling, we followed the established protocol where the target point cloud remains fixed while the source undergoes viewpoint cropping, nearest-neighbor perturbation, and density unevenness processing. These perturbations are governed by the neighborhood size $K\in \{8,12,16,20,24\}$, where a larger K introduces severe local sparsity and structural missingness to mimic complex non-repetitive LiDAR characteristics.

For a fair comparison, all methods utilized identical input data regarding coordinate systems and scales. The learning-based baselines, PREDATOR and GeoTransformer, employed official pre-trained weights for inference without fine-tuning. Geometric hyperparameters were adaptively set based on the object bounding box diagonal D: voxel size $\delta =0.01D$, normal estimation radius $2\delta$, FPFH calculation radius $5\delta$, and ICP distance threshold $1.5\delta$. All stochastic methods underwent 30 independent trials per setting. Our framework adopted fixed energy weights and fractional orders derived from Bayesian optimization without dataset-specific tuning, with the solver energy threshold linearly decaying from $1.2$ to $0.35$. All experiments were executed on an Intel Core i5-12400KF CPU (Intel Corporation, Santa Clara, CA, USA) with an NVIDIA RTX 4060 GPU (NVIDIA Corporation, Santa Clara, CA, USA).The implementation was based on Python 3.8.20, PyTorch 2.3.0, CUDA 12.1, and FAISS 1.7.1 (GPU enabled) for k-NN acceleration.

4.4.3. Results and Analysis

The quantitative results in Figure 6 and

Table 1. Success rates (RRE < 3°, RTE < 0.03 m) of different registration methods on Stanford 3D, ITODD, and HB datasets under varying non-repetitive sampling levels (K). The highest success rate for each dataset at the same K level is highlighted in bold.

Dataset	K	Success Rate (RRE < 3°, RTE < 0.03 m) of the Corresponding Method %
		Ours	GeoTrans Former	RANSAC+ FPFH+ ICP	PREDATOR	FGR+ ICP	RANSAC+ FPFH+ TEASER++
Stanford 3D	8	100.00	68.33	86.11	52.78	76.67	11.11
	12	100.00	66.11	76.67	49.44	71.67	13.89
	16	100.00	67.78	50.56	54.44	71.67	11.67
	20	100.00	54.44	51.67	49.44	66.11	2.22
	24	100.00	55.00	45.00	42.22	60.56	5.00
ITODD	8	100.00	91.52	79.09	20.00	62.73	0.30
	12	100.00	83.33	63.33	17.27	50.91	0.00
	16	100.00	70.00	53.64	13.94	45.76	0.00
	20	99.70	53.33	46.67	12.73	37.27	1.82
	24	90.61	31.52	33.94	14.24	33.94	0.00
HB	8	100.00	88.15	92.78	17.22	83.33	0.00
	12	100.00	53.52	85.00	15.19	76.85	0.00
	16	99.44	53.33	77.59	14.81	73.70	0.19
	20	96.85	44.63	55.93	13.70	68.33	13.70
	24	96.11	30.56	44.44	12.78	66.11	0.37

demonstrate the superiority of our method.

As shown in Figure 6, while traditional geometric pipelines (e.g., FGR+ICP) deteriorate markedly as sparsity increases ($K\ge 16$), our method consistently achieves the lowest median errors with the tightest error distributions. Table 1 further corroborates this robustness: under severe conditions ($K=24$), the success rates of PREDATOR and GeoTransformer drop to 14.24% and 31.52%, respectively, whereas our method maintains a success rate above 90% (e.g., 90.61% on ITODD). Notably, although all hyperparameters and weights are obtained via Bayesian optimization on the Stanford 3D dataset, our method still performs well on datasets with markedly different point densities and geometric scales, including the industrial-part data in ITODD and the everyday-object data in HB. Together, these results indicate strong generalization and confirm that the proposed fractional spatiotemporal framework effectively mitigates the adverse effects of irregular sampling.

Empirical observations also reveal a degradation in performance under conditions of higher neighborhood sizes (K). An in-depth inspection of the experimental data indicates that registration failures are primarily attributed to three specific models: obj_000001 from the HB dataset, and obj_000015 and obj_000018 from the ITODD dataset, as shown in Figure 7. In the respective experimental groups, failures on these instances resulted in a marked escalation in the RRE and RTE metrics for our method, occurring predominantly at larger neighborhood settings ($K=20,K=24$). Notably, these models exhibit pronounced geometric characteristics: geometric symmetry (HB obj_000001, ITODD obj_000015, and obj_000018).

For symmetric objects, the inherent symmetry renders the gradient of the Spatially-Normalized Density-Aware (SNDA) energy effectively zero along the axis of symmetry, thereby inducing unconstrained rotational drift. Similarly, for repetitive structures, the failure arises from the non-local Riesz regularization when coupled with a large neighborhood K. While a large K enhances robustness against sparsity, it simultaneously functions as a low-pass filter that over-smooths the high-frequency boundaries separating repeating units. When combined with the “heavy-tailed” inertia of the Grünwald–Letnikov dynamics, the consequent reduction in energy barriers allows the solver to inadvertently overshoot the optimal alignment and settle into a phase-shifted local minimum. These cases underscore an intrinsic trade-off: the very long-memory and non-local mechanisms designed to bridge sparse voids can, in the presence of perfect symmetry or periodicity, precipitate unconstrained sliding or periodic skipping.

4.5. Ablation Studies

To quantitatively analyze the role of each key module in the fractional spatiotemporal unified energy framework, this section conducts ablation experiments on several representative models that exhibit distinct characteristics compared to other front-ends. We selected bunny from the Stanford 3D dataset, as well as obj_000001 and obj_000013 from the ITODD dataset as research objects. These three types of objects correspond to free-form surfaces with rich details, mechanical parts with planar and hole structures, and industrial components with complex contours, respectively, allowing for a comprehensive reflection of the impact of each module on registration performance under different geometric features. Moreover, we further introduce an order-oriented ablation study. Specifically, while keeping all other hyperparameters fixed, we perform a controlled sweep over the spatial fractional order and the temporal fractional order. This design enables a direct and interpretable characterization of how the registration error varies with fractional orders in space and time, thereby complementing the module-wise ablations with explicit evidence on the error–order relationship.

4.5.1. Experimental Purpose and Settings for Module-Wise Ablations

The purpose of the module-wise ablations experiments is to verify the necessity of each module in coping with non-repetitive sampling, local structural missingness, and initial pose deviations by turning off or replacing specific modules one by one while keeping the overall structure of the fractional spatiotemporal unified energy framework unchanged, and observing the changes in RRE, RTE distributions, and success rates. Specifically, in this section, we examine the following five ablation configurations:

• no_time_frac: Disables the temporal fractional memory, relying solely on geometric increments (based on the current gradient and energy descent) to update the pose in the main processing stage;
• no_riesz_frac: Removes the fractional spatial regularization term from the unified energy, ceasing to couple point cloud residuals across neighborhoods via the Riesz-type non-local operator;
• no_ot: Replaces the soft correspondence term in the unified energy with 1-NN weighted SVD/ICP edges, i.e., relying only on nearest-neighbor rigid registration without explicitly solving for sparse optimal transport weights;
• no_bracket: Disables the bracketing search mechanism in the three-stage solver (including the bootstrap stage and the main processing/polishing stages), using only fixed step sizes or local line searches for updates;
• no_snda_boot: Skips the global rotation search based on the Spherical Normal Distribution Alignment (SNDA) prior in the bootstrap stage, entering the subsequent optimization directly from the given initial pose.

The five configurations mentioned above were evaluated on non-repetitively sampled point clouds identical to those used in the full model, with the data construction process consistent with the comparative experiments. For each object, each ablation configuration, and each non-repetitive level, independent trials were repeated 20 times. We statistically analyzed the distributions of rotation error (RRE) and relative translation error (RTE).

In addition, we provide the maximum values of RRE and RTE for each configuration to characterize the error long-tail and overall stability. All hyperparameters (including the weights of the sub-terms in the unified energy and the fractional orders) were fixed to the optimal configuration obtained via Bayesian optimization in Section 4.3. Only the corresponding modules were “turned on/off” or replaced to ensure the fairness of the ablation comparison.

4.5.2. Module-Wise Ablation Results and Analysis

The quantitative results of the module-wise ablations studies are presented in the boxplots of Figure 8 (covering three representative models) and the worst-case statistics in

Table 2. Maximum Rotation Error (RRE) across all test samples under different non-repetitive levels (K). Bold indicates the lowest maximum error.

K	Maximum Rotation Error
	Ours	no_riesz_frac	no_time_frac	no_bracket	no_ot	no_snda_boot
8	0.17	0.94	0.69	0.94	0.81	0.64
12	0.38	1.51	1.87	1.53	6.30	1.53
16	0.37	1.78	35.90	0.78	41.96	2.83
20	0.92	0.96	38.10	2.15	129.76	113.25
24	0.96	1.44	177.73	178.02	167.78	178.05

and 

Table 3. Maximum Relative Translation Error (RTE) across all test samples under different non-repetitive levels (K). Bold indicates the lowest maximum error.

K	Maximum Relative Translation Error
	Ours	no_riesz_frac	no_time_frac	no_bracket	no_ot	no_snda_boot
8	0.002	0.014	0.011	0.014	0.014	0.010
12	0.004	0.021	0.026	0.023	0.075	0.023
16	0.005	0.019	0.790	0.016	0.672	0.032
20	0.012	0.019	0.571	0.036	0.936	0.925
24	0.012	0.020	1.405	1.688	1.568	1.712

. These data comprehensively verify the necessity of each module in the proposed framework.

First, regarding the fractional temporal dynamics (no_time_frac), its absence leads to a catastrophic degradation in stability under high non-repetitive levels. As shown in Table 2 and Table 3, when $K=24$, the maximum RRE of no_time_frac surges to $177.73°$, compared to only $0.96°$ for the full model. The boxplots further illustrate that without historical memory, the error distribution becomes extremely dispersed with numerous outliers, confirming that the fractional temporal memory effectively suppresses oscillation and prevents divergence in complex energy landscapes.

Second, the fractional spatial regularization (no_riesz_frac) plays a crucial role in maintaining geometric consistency. While its impact on the success rate is less severe than removing the temporal module, Figure 8 shows that removing this term results in a systematic “lifting” of the error medians and a widening of the interquartile ranges, especially on the ITODD industrial parts. Table 2 indicates that the maximum RRE increases from $0.96°$ to $1.44°$ at $K=24$, proving that the non-local spatial coupling effectively inhibits local drift caused by sampling jitter.

Third, the Optimal Transport module (no_ot) is vital for handling sparsity. The substitution of soft correspondences with hard nearest neighbors causes the algorithm to fail rapidly as K increases. In Table 2, no_ot exhibits a high maximum translation error of $1.568$ m at $K=24$, and the boxplots show extremely long tails, indicating that explicit modeling of many-to-many relationships is essential for non-repetitive point clouds.

Finally, the bracketing search (no_bracket) and SNDA bootstrap (no_snda_boot) ensure convergence quality. Without the bracketing strategy, the solver often gets stuck in local minima, leading to a maximum RRE of $178.02°$ at high perturbation levels. Similarly, skipping the SNDA global initialization results in failure to correct large initial deviations, yielding errors comparable to random guessing ($178.05°$). In summary, the full model (Ours) integrates these mechanisms to achieve the lowest worst-case errors ($\mathrm{Max}\mathrm{RRE}<1°$) and the most compact error distributions across all scenarios.

4.5.3. Experimental Purpose and Settings for Order-Oriented Ablations

The goal of the order-oriented ablation experiments is to explicitly characterize the relationship between registration error and the fractional orders in our spatiotemporal framework. Unlike the module-wise ablations that disable specific components, this study performs a controlled scan over the fractional orders while keeping all other hyperparameters unchanged, thereby isolating the effect of fractionalization from other factors.

We consider two fractional orders: the spatial order s and the temporal order ${\alpha }_{\mathrm{time}}$. For the temporal-order scan, we fix the spatial order to $s=0.5$ and vary ${\alpha }_{\mathrm{time}}\in [0.5,1.0]$ with a step size of $0.1$. For the spatial-order scan, we fix the temporal order to ${\alpha }_{\mathrm{time}}=0.5$ and vary $s\in [0.5,1.0]$ with the same step size. In both scans, the integer-order limit corresponds to the endpoint value $1.0$.

To obtain a single error measure that can clearly reflect the relationship between fractional orders and registration accuracy, we employ a unified evaluation metric in the order-oriented ablations to integrate the two standard error terms, RRE and RTE. Since these two quantities have different physical units, we unify them by normalizing with the same success criteria used in our evaluation. Specifically, following the success thresholds $\mathrm{RRE}\le 3°$ and $\mathrm{RTE}\le 0.03\mathrm{m}$, we define the dimensionless composite error

$$J={\displaystyle \frac{\mathrm{RRE}}{3}}+{\displaystyle \frac{\mathrm{RTE}}{0.03}},$$

(38)

where RRE is measured in degrees and RTE in meters. In this way, both terms are expressed on a comparable scale with respect to the success standard, and a smaller J indicates a better registration result.

To ensure consistency with the module-wise ablations, the order-oriented ablations are conducted on the same three representative objects: bunny from the Stanford 3D dataset, and obj_000001 and obj_000013 from the ITODD dataset. For each object and each scanned order setting, we perform the ablation only under the most challenging non-repetitive level ($K=24$). All other experimental settings follow exactly the protocol used in the module-wise ablations, and the evaluation is repeated with the same number of trials. The final result for each configuration is reported as the mean value of the composite metric J over repeated runs.

4.5.4. Order-Oriented Ablation Results and Analysis

The order-oriented scans reveal a clear and repeatable pattern: the composite error J is minimized at moderate fractional orders, whereas the integer-order limit is consistently suboptimal. In other words, the error–order relationship is not monotonic; pushing the order to $1.0$ does not improve registration under the hardest setting, and instead increases J across all tested objects.

As shown in

Table 4. Parameter sensitivity analysis on different datasets by varying the temporal fractional order ${\alpha }_{\mathrm{time}}$ with a fixed spatial order $s=0.5$.

Fixed Spatial Order $\mathit{s}=0.5$
Parameter Selection	Bunny	Obj_000001	Obj_000013
${\alpha }_{\mathrm{time}}=0.5$	0.5589	0.2437	0.2519
${\alpha }_{\mathrm{time}}=0.6$	0.5458	0.2422	0.2387
${\alpha }_{\mathrm{time}}=0.7$	0.5461	0.2413	0.2503
${\alpha }_{\mathrm{time}}=0.8$	0.5498	0.2479	0.2540
${\alpha }_{\mathrm{time}}=0.9$	0.5871	0.2483	0.2552
${\alpha }_{\mathrm{time}}=1.0$	0.5931	0.2499	0.2609

and 

Table 5. Parameter sensitivity analysis on different datasets by varying the spatial order s with a fixed temporal fractional order ${\alpha }_{\mathrm{time}}=0.5$.

Fixed Temporal Order ${\mathit{\alpha }}_{\mathbf{time}}=0.5$
Parameter Selection	Bunny	Obj_000001	Obj_000013
$s=0.5$	0.5589	0.2437	0.2519
$s=0.6$	0.5743	0.2367	0.2436
$s=0.7$	0.5260	0.2439	0.2430
$s=0.8$	0.4906	0.2448	0.2465
$s=0.9$	0.5084	0.2456	0.2484
$s=1.0$	0.5973	0.2528	0.2521

, the order-oriented ablation scans over the fractional orders exhibit clear performance trends for both ${\alpha }_{\mathrm{time}}$ and s. For the temporal order (with $s=0.5$ fixed), the best performance concentrates around ${\alpha }_{\mathrm{time}}\in [0.6,0.7]$: the minimum J is achieved at ${\alpha }_{\mathrm{time}}=0.6$ for bunny ($0.5458$) and obj_000013 ($0.2387$), and at ${\alpha }_{\mathrm{time}}=0.7$ for obj_000001 ($0.2413$). Moving to the integer-order limit ${\alpha }_{\mathrm{time}}=1.0$ consistently degrades the error for all three objects. A similar conclusion holds for the spatial order (with ${\alpha }_{\mathrm{time}}=0.5$ fixed): the lowest J occurs at $s=0.8$ for bunny ($0.4906$), at $s=0.6$ for obj_000001 ($0.2367$), and at $s=0.7$ for obj_000013 ($0.2430$), while $s=1.0$ yields the largest error in every case.

Overall, these results provide direct evidence that fractional orders are not merely interchangeable with their integer-order counterparts. Instead, there exists a practical operating interval (roughly $0.6$–$0.8$ in this study) where the fractional spatiotemporal mechanism is most effective, offering a stable accuracy advantage across different object geometries.

4.6. Real-World Scenario Testing

To validate the registration capability of the proposed method on genuine non-repetitive LiDAR data, we conducted experiments using a Livox Mid-360 LiDAR sensor (Livox Technology Company Limited, Shenzhen, China). The experimental procedure was as follows: First, in a stationary scene, continuous frames of point clouds were acquired and accumulated to form a dense target point cloud. Subsequently, the position of the target objects was altered, and a single frame was captured by the LiDAR to serve as the sparse source point cloud for registration.

As illustrated in Figure 9, the test subject consisted of two televisions positioned side-by-side with an offset. This setup introduces significant challenges due to the presence of repetitive geometric structures combined with the irregular sparsity inherent to non-repetitive scanning. Figure 9a displays the experimental setup. Figure 9b visualizes the relative position between the source (red) and target (blue) point clouds prior to registration.

Upon obtaining the data, we performed registration using both the baseline and our method. Figure 9c presents the result using GeoTransformer. It is evident that GeoTransformer fails to align the point clouds correctly; this failure is attributed to the ambiguity caused by the coupling of non-repetitive sampling patterns and repetitive geometric features, leading to feature mismatching. In contrast, Figure 9d displays the result obtained by our method. The source and target point clouds overlap almost perfectly, achieving highly accurate alignment. This outcome demonstrates the superior performance of our method in real-world scenarios, effectively overcoming the limitations of existing learning-based approaches in such complex environments.

5. Discussion

The experimental results presented in Section 4 demonstrate that the proposed fractional spatiotemporal unified energy framework significantly outperforms state-of-the-art geometric and learning-based methods in registering non-repetitive LiDAR point clouds. This performance advantage is attributed to two core mechanisms rooted in fractional calculus. Spatially, unlike integer-order gradients that are sensitive to high-frequency jitter in irregular sampling, the Riesz fractional regularization imposes non-local constraints. By coupling residuals across a wider neighborhood via power-law kernels, it effectively mitigates the structural discontinuities inherent in non-repetitive patterns. Temporally, the Grünwald–Letnikov dynamics introduce a “heavy-tailed” memory of historical increments. As verified in the ablation studies, this long-memory effect is crucial for traversing the highly non-convex energy landscape, preventing the solver from being trapped in local minima caused by partial overlaps.

However, compared with standard local methods, the introduction of non-local operators inevitably increases computational complexity.

Table 6. Results of accuracy and time under different perturbation levels with varying sampling point counts. Bold indicates the best performance in each metric category.

Points	K	FGR + ICP				GeoTransformer				Ours
		RRE/°	RTE/m	Time/s		RRE/°	RTE/m	Time/s		RRE/°	RTE/m	Time/s
6000	8	62.576	0.5249	2.1802		1.6087	0.0201	0.9203		0.2102	0.0019	8.2590
	12	48.991	0.3011	2.2303		3.3671	0.0408	0.9144		0.2558	0.0029	7.5648
	16	50.068	0.4015	2.3084		3.5291	0.0459	0.9817		0.2952	0.0032	9.6769
	20	91.159	0.7589	2.2940		3.8393	0.0541	0.9973		0.4997	0.0050	9.4807
	24	66.821	0.4199	2.2051		5.4106	0.0776	1.0118		0.6561	0.0086	10.351
8000	8	13.125	0.1166	2.1044		0.8258	0.0080	0.8663		0.1264	0.0013	8.2137
	12	33.905	0.2156	2.0067		1.1721	0.0106	0.9291		0.0872	0.0005	9.3213
	16	58.524	0.6787	2.3554		1.6468	0.0156	0.8666		0.2137	0.0008	7.6903
	20	82.654	0.6842	2.2675		2.5765	0.0286	0.8688		0.2535	0.0012	8.1750
	24	92.740	0.1166	2.1044		3.7679	0.0406	0.8650		0.3773	0.0035	8.8328
10,000	8	4.4965	0.0672	2.0245		0.9190	0.0140	1.0462		0.0368	0.0005	11.456
	12	53.947	0.4399	2.3068		1.7617	0.0240	0.8860		0.1100	0.0015	8.0418
	16	37.923	0.3259	2.0984		1.5712	0.0189	0.8604		0.1343	0.0020	8.7841
	20	68.486	0.5459	2.2597		1.3239	0.0151	0.8446		0.1802	0.0008	9.3973
	24	68.195	0.5274	2.3169		3.0085	0.0374	0.8918		2.2480	0.0138	11.374

reports the runtime and accuracy statistics on the Stanford Bunny model under varying non-repetitive levels and sampling densities. For each configuration, experiments were repeated 30 times to report the mean results. In terms of accuracy, our method is markedly superior to both the conventional ICP-style pipeline (FGR+ICP) and the representative learning-based method (GeoTransformer). It maintains mean RRE values around 1° (max $2.248°$) and mean RTE close to $0.001\mathrm{m}$ (max $0.0138\mathrm{m}$). Nevertheless, the associated overhead is evident. Compared to the faster GeoTransformer, our method typically requires approximately 10 s per registration. The primary costs arise from: (i) residual-field evaluation of the spatial fractional non-local term over graph edges; (ii) energy-driven candidate screening in the polishing stage; and (iii) history accumulation induced by the temporal Grünwald–Letnikov update.

5.1. Analysis of Solver Convergence Properties

Given the computational overhead highlighted above, a fundamental question arises: Is this complexity justified, and why not employ classical Nesterov momentum or second-order optimization methods to accelerate convergence? The answer lies in the distinct mathematical properties required to traverse the pathological energy landscapes of non-repetitive LiDAR.

(1)

Heavy-tailed vs. Exponential Memory (Comparison with Nesterov). Classical momentum methods, including Nesterov, rely on an exponentially decaying history ($w_t\propto e^{-\lambda t}$). This implies a “short-term memory” mechanism. While efficient for smooth convex basins, this momentum dissipates rapidly when the solver encounters extended “flat” plateaus or continuous jittery regions characteristic of sparse LiDAR scans. In contrast, the Grünwald-Letnikov fractional definition introduces a memory kernel that follows a power-law decay ($w_t\propto t^{-(1+\alpha )}$). This “heavy-tailed” long-term memory allows the solver to accumulate and retain directional inertia over a significantly longer horizon. It provides the sustained “kinetic energy” necessary to escape deep, wide basins of attraction where classical exponentially-decaying momentum would stagnate.

(2)

Infeasibility of Second-Order Methods. Second-order methods (e.g., Gauss-Newton) require the inversion of the Hessian matrix to normalize the gradient. As discussed in Section 3.2, the sampling pattern of non-repetitive LiDAR is spatially sparse and anisotropic, frequently rendering the Hessian matrix ill-conditioned or singular (degenerate geometry) due to the lack of constraints in directions perpendicular to the scan lines. Consequently, second-order updates become numerically unstable and computationally prohibitive ($O\left(N^3\right)$), whereas our fractional first-order approach maintains numerical stability with linear complexity while effectively mimicking curvature-aware benefits through the non-local Riesz operator.

5.2. Future Horizons: Optimization and Generalization

While the current implementation prioritizes solvability over speed, the comparative results indicate that our improvement is not merely obtained by trading time for accuracy; rather, when other methods exhibit pronounced failure, our framework remains capable of delivering high-accuracy registration. To bridge the gap towards real-time deployment, future work will focus on developing fast numerical approximations, such as frequency-domain acceleration or recursive-filter approximations, alongside deeper GPU-centric implementations.

Beyond optimization for speed, we envision that the proposed framework holds significant promise for generalization to broader perception tasks. Although explicitly validated on non-repetitive LiDAR, the underlying mathematical principles—specifically the handling of data sparsity via non-local operators and optimization instability via fractional dynamics—are applicable to other domains:

• Generalization to Sparse Modalities: The core efficacy of the fractional Riesz regularization lies in imposing structural constraints across disconnected regions. This property suggests immediate applicability to other sparse sensing modalities, such as 4D Imaging Radar or sparse visual feature matching in textureless environments, where non-local potentials can effectively “densify” geometric constraints better than integer-order Laplacians.
• Continuous-Time Estimation: The temporal component, currently employed for iterative updates, shares a theoretical isomorphism with continuous-time trajectory estimation (e.g., in CT-SLAM). The Grünwald–Letnikov memory kernel can be reinterpreted as a trajectory smoothing prior that enforces long-range temporal coherence, potentially reducing drift in large-scale mapping tasks.
• Beyond Rigid Transformation: Finally, the fractional Laplacian’s capability to maintain global topology while tolerating local irregularities offers a mathematically elegant avenue for non-rigid registration. In deformable scenarios, the global coupling introduced by the fractional order naturally enforces elastic constraints, preventing topological tearing during local warping.

6. Conclusions

This paper proposes a novel spatiotemporal unified energy framework for the registration of non-repetitive LiDAR point clouds, successfully bridging fractional calculus theory with 3D rigid pose estimation. By integrating Riesz fractional spatial regularization and Grünwald–Letnikov temporal dynamics, the proposed method effectively overcomes the convergence instability and geometric inconsistency caused by irregular sampling and partial overlaps. Extensive comparative experiments on the Stanford 3D, ITODD, and HB datasets demonstrate that our approach achieves state-of-the-art accuracy and robustness, maintaining a success rate of over 90% even under severe sampling perturbations where traditional methods fail. The ablation studies further confirm that the introduction of non-local spatial constraints and historical gradient memory significantly reshapes the energy landscape, avoiding local minima. This work provides a rigorous theoretical foundation for applying fractional operators to point cloud processing, offering a robust solution for high-precision perception in complex dynamic environments. Future research will explore the extension of this fractional framework to continuous-time trajectory estimation and its integration into real-time pose estimation.