Topology-Preserving Shape Reconstruction and Registration via Neural Diffeomorphic Flow (2203.08652v2)

Abstract: Deep Implicit Functions (DIFs) represent 3D geometry with continuous signed distance functions learned through deep neural nets. Recently DIFs-based methods have been proposed to handle shape reconstruction and dense point correspondences simultaneously, capturing semantic relationships across shapes of the same class by learning a DIFs-modeled shape template. These methods provide great flexibility and accuracy in reconstructing 3D shapes and inferring correspondences. However, the point correspondences built from these methods do not intrinsically preserve the topology of the shapes, unlike mesh-based template matching methods. This limits their applications on 3D geometries where underlying topological structures exist and matter, such as anatomical structures in medical images. In this paper, we propose a new model called Neural Diffeomorphic Flow (NDF) to learn deep implicit shape templates, representing shapes as conditional diffeomorphic deformations of templates, intrinsically preserving shape topologies. The diffeomorphic deformation is realized by an auto-decoder consisting of Neural Ordinary Differential Equation (NODE) blocks that progressively map shapes to implicit templates. We conduct extensive experiments on several medical image organ segmentation datasets to evaluate the effectiveness of NDF on reconstructing and aligning shapes. NDF achieves consistently state-of-the-art organ shape reconstruction and registration results in both accuracy and quality. The source code is publicly available at https://github.com/Siwensun/Neural_Diffeomorphic_Flow--NDF.

• The paper introduces Neural Diffeomorphic Flow (NDF) that models 3D shapes as conditional diffeomorphic deformations, ensuring intrinsic topology preservation.
• It employs Neural ODE blocks with a quasi time-varying velocity field to progressively and invertibly map shapes to a template, boosting registration precision.
• Extensive experiments on medical segmentation datasets demonstrate that NDF achieves state-of-the-art performance in accurate, topology-preserving shape modeling.

Topology-Preserving Shape Reconstruction and Registration via Neural Diffeomorphic Flow

This paper, authored by Shanlin Sun et al., introduces an innovative approach termed Neural Diffeomorphic Flow (NDF) aimed at addressing the challenges inherent in 3D shape reconstruction and registration. The paper builds upon the foundational concept of Deep Implicit Functions (DIFs), which efficiently represent 3D geometry via continuous signed distance functions learned through deep neural networks. Existing methods employing DIFs often excel in flexibility and accuracy for shape reconstruction and dense point correspondences but fall short in preserving the intrinsic topology of shapes, a critical requirement in domains with inherent structural constraints, such as medical imaging.

Key Contributions and Methodology

1. Neural Diffeomorphic Flow (NDF): NDF innovates by modeling shapes as conditional diffeomorphic deformations of a template, ensuring intrinsic topology preservation. This differentiates it from previous DIFs-based approaches such as DIT and DIF-Net, which do not maintain topology during deformations. The proposed framework employs an auto-decoder composed of Neural Ordinary Differential Equation (NODE) blocks, which facilitate mapping shape instances to implicit templates progressively and invertibly.
2. Quasi Time-varying Velocity Field: The paper introduces a quasi time-varying velocity field in the NODE framework to model diffeomorphic flows dynamically. This design enables the progressive and invertible modeling of shape deformations, crucially preserving the topology across various deformation stages. This step-based velocity modeling addresses and improves upon the limitations associated with both stationary and fully time-varying velocity fields in diffeomorphic transformation processes.
3. Extensive Empirical Validation: Experiments conducted on medical organ segmentation datasets demonstrate that NDF achieves state-of-the-art performance in both shape reconstruction and registration, reflecting both high accuracy and quality. This empirical evaluation underscores NDF's capability in effectively preserving topology while ensuring precise shape alignment, crucial for medical imaging applications where structural fidelity is non-negotiable.

Implications and Future Directions

The introduction of the NDF presents significant implications for both theoretical and practical applications in fields that rely heavily on 3D shape modeling. By ensuring topology preservation, NDF expands the potential applicability of DIFs to more complex medical imaging tasks, such as organ segmentation in CT and MRI, where accurate anatomical modeling is essential for diagnostic and therapeutic processes.

From a theoretical perspective, the development of NDF invites further exploration into incorporating advanced mathematical frameworks, like optimal transport, within NODE-based architectures to enhance the flexibility and efficiency of shape correspondence algorithms. Additionally, the authors note that the absence of extensive regularization, possible due to the intrinsic properties of neural ODEs, opens avenues for research into minimalistic yet effective regularization strategies that do not compromise geometrical fidelity.

Moreover, the integration of NDF with large-scale medical datasets could facilitate significant advancements in automated diagnosis systems, providing higher accuracy and consistency in organ shape modeling. The principles outlined could also be potentially extended beyond medical applications, including robotic navigation, where real-time, accurate 3D environment modeling is paramount.

In conclusion, the Neural Diffeomorphic Flow represents a notable advancement in the field of 3D geometry processing, presenting a substantial leap towards achieving reliable topology-preserving models in complex domains. As the technology matures, further refinement and application-specific adaptation could see NDF become a staple methodology in various interdisciplinary fields.