Functional data registration is a critical challenge in modern statistics, essential for separating phase variability from amplitude variability. While derivative-based frameworks offer mathematically elegant solutions, their dependence on signal velocities renders them susceptible to additive noise. This study proposes and evaluates a family of robust, Sobolev-regularized objective functions for the pairwise alignment of functional data, operating entirely within the original function space to avoid the need for numerical differentiation of the data. We define our optimization over a second-order Sobolev space and utilize the Centered Log-Ratio (CLR) transform to represent the warping functions. By penalizing both the velocity and acceleration of the centered log-derivative, this geometric approach preempts degenerate "pinching" artifacts and ensures the resulting warps are strictly monotonic, valid diffeomorphisms. In practice, this allows for highly efficient, unconstrained optimization within a finite-dimensional space. We systematically investigate four distinct pairwise data mismatch formulations: a Standard L2 baseline, a Symmetric L2 formulation, an Isometry (L2-preserving) mapping, and a Jacobian-weighted L2 functional. We establish robust theoretical foundations for these methods, proving the existence of optimal warps and the asymptotic consistency of the finite-dimensional estimators. Our results demonstrate that this CLR-regularized framework offers a powerful, computationally scalable, and noise-robust alternative to traditional derivative-based registration.

翻译：函数数据配准是现代统计学中的关键挑战，对于分离相位变异与幅度变异至关重要。尽管基于导数的框架提供了数学上优雅的解决方案，但其对信号速度的依赖使其易受加性噪声影响。本研究提出并评估了一族鲁棒的Sobolev正则化目标函数，用于函数数据的成对对齐，该方法完全在原始函数空间内操作，从而避免对数据进行数值微分。我们将优化定义在二阶Sobolev空间上，并利用中心对数比（CLR）变换表示扭曲函数。通过惩罚中心对数导数的速度与加速度，该几何方法可防止退化的“收缩”伪影，确保生成的扭曲是严格单调的有效微分同胚。在实践中，这允许在有限维空间内进行高效的无约束优化。我们系统研究了四种不同的成对数据失配公式：标准L2基线、对称L2公式、等距（保L2）映射以及雅可比加权L2泛函。为这些方法建立了稳健的理论基础，证明了最优扭曲的存在性以及有限维估计量的渐近一致性。结果表明，该CLR正则化框架为传统的基于导数的配准方法提供了强大、计算可扩展且抗噪声的替代方案。