Quasi-Newton Solver for Robust Non-Rigid Registration (2004.04322v1)

Abstract: Imperfect data (noise, outliers and partial overlap) and high degrees of freedom make non-rigid registration a classical challenging problem in computer vision. Existing methods typically adopt the $\ell_{p}$ type robust estimator to regularize the fitting and smoothness, and the proximal operator is used to solve the resulting non-smooth problem. However, the slow convergence of these algorithms limits its wide applications. In this paper, we propose a formulation for robust non-rigid registration based on a globally smooth robust estimator for data fitting and regularization, which can handle outliers and partial overlaps. We apply the majorization-minimization algorithm to the problem, which reduces each iteration to solving a simple least-squares problem with L-BFGS. Extensive experiments demonstrate the effectiveness of our method for non-rigid alignment between two shapes with outliers and partial overlap, with quantitative evaluation showing that it outperforms state-of-the-art methods in terms of registration accuracy and computational speed. The source code is available at https://github.com/Juyong/Fast_RNRR.

• The paper introduces a novel quasi-Newton solver that applies Welsch’s function to robustly handle outliers and noise in non-rigid registration.
• It employs a majorization-minimization strategy with the L-BFGS method to decompose the complex problem into efficient least-squares optimizations.
• Empirical results from diverse datasets confirm its superior convergence and reduced error compared to state-of-the-art non-rigid registration techniques.

Quasi-Newton Solver for Robust Non-Rigid Registration: An Expert Overview

The paper "Quasi-Newton Solver for Robust Non-Rigid Registration" by Yuxin Yao and colleagues introduces a novel approach for solving the non-trivial problem of non-rigid registration (NRR) in computer vision. Non-rigid registration, which involves aligning two distinct 3D shapes where one or both may undergo non-rigid deformations, presents significant challenges due to the intricacies introduced by noise, outliers, and partial overlaps in data. This paper proposes a robust solution featuring a quasi-Newton method within an optimization framework underpinned by Welsch's function, providing advancements both in computational performance and registration accuracy.

Methodological Insights

The central contribution of this paper lies in utilizing Welsch's function in the optimization formulation for non-rigid registration. Employing a robust estimator such as Welsch's function allows for the efficient handling of outliers and reduces susceptibility to noise, a common pitfall in earlier approaches reliant on $\ell_p$-norm formulations. Specifically, the proposed method integrates Welsch's function in both the alignment error and regularization terms, thereby promoting robustness against large errors that conventional $\ell_2$-norm based models might struggle with.

To solve the non-convex optimization problem effectively, the authors deploy a majorization-minimization (MM) strategy, reducing the task to a series of simpler least-squares problems. The use of L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) method further enhances computational efficiency. This new formulation exhibits superior convergence properties when compared to existing first-order solvers like ADMM, as substantiated through their experiments, demonstrating faster convergence and robustness against various types of data imperfections.

Empirical Validation

Through extensive experimentation, the authors provide substantive evidence of the algorithm's efficacy. Tested across diverse datasets, including the MPI Faust dataset and human motion datasets, this approach demonstrates marked improvements in both the computational speed and accuracy of the registration process. The results indicate a high resilience to noise and partial overlaps — scenarios where many traditional methods tend to falter. Quantitatively, the method consistently outperformed state-of-the-art techniques such as N-ICP (Non-rigid Iterative Closest Point) and RPTS (Robust Point Set Tracking) in terms of reduced root mean square error (RMSE) and faster runtime.

Implications and Future Research Directions

The implications of this paper are manifold. Practically, this method can provide significant improvements in applications requiring real-time 3D registration, such as in augmented reality, robotics, and medical imaging. Its robustness to noise and partial data invariance renders it particularly valuable in dynamic, real-world environments where data imperfections are prevalent.

Theoretically, the successful integration of Welsch's function in optimization for non-rigid registration opens intriguing avenues for further exploration. Future research could explore the extension of this approach to other domains where robust estimation is required, potentially adapting the method for different types of data beyond point clouds, such as volumetric data or across different sensory modalities.

Moreover, the quasi-Newton strategy offers fertile ground for exploring more sophisticated forms of regularization and alternative robust functions that might enhance performance or extend applicability. As datasets and hardware continue to evolve, another promising direction would include harnessing advancements in parallel processing to resolve the limitations associated with scaling this approach to higher dimensional data or larger datasets.

In conclusion, this paper makes a significant contribution to the field of non-rigid registration, merging robust statistical estimation with effective optimization strategies to address long-standing challenges in 3D shape alignment. Its demonstrated benefits in speed and accuracy mark a noteworthy advancement in the domain, providing a strong foundation for both practical implementations and further academic inquiry.