Deep Functional Maps: Structured Prediction for Dense Shape Correspondence (1704.08686v2)

Abstract: We introduce a new framework for learning dense correspondence between deformable 3D shapes. Existing learning based approaches model shape correspondence as a labelling problem, where each point of a query shape receives a label identifying a point on some reference domain; the correspondence is then constructed a posteriori by composing the label predictions of two input shapes. We propose a paradigm shift and design a structured prediction model in the space of functional maps, linear operators that provide a compact representation of the correspondence. We model the learning process via a deep residual network which takes dense descriptor fields defined on two shapes as input, and outputs a soft map between the two given objects. The resulting correspondence is shown to be accurate on several challenging benchmarks comprising multiple categories, synthetic models, real scans with acquisition artifacts, topological noise, and partiality.

• The paper introduces a deep residual network that learns structured functional maps for dense shape correspondence.
• It embeds correspondence as structured prediction in functional space, eliminating reliance on fixed reference points or extensive post-processing.
• Experiments on FAUST and SHREC’16 benchmarks show error rates as low as 2.34 cm, demonstrating significant improvements over traditional methods.

Deep Functional Maps: A Structured Prediction Model for Dense Shape Correspondence

The paper "Deep Functional Maps: Structured Prediction for Dense Shape Correspondence" introduces an advanced framework for establishing dense correspondence between deformable 3D shapes. Distinct from existing approaches that treat shape correspondence as a labeling problem, this research redefines the task as a structured prediction problem within the domain of functional maps. The authors leverage a deep residual network to model the learning process, providing accurate predictions of dense descriptors that effectively encapsulate the structural similarities between target shapes.

Existing approaches to shape correspondence often focus on developing robust point descriptors or label spaces that link a query shape to a fixed reference model. However, these methods encounter challenges, such as reliance on computationally expensive post-processing or a fixed number of reference points, resulting in potentially inaccurate solutions due to flexibility constraints. This work addresses the pitfalls of these previous solutions by embedding the computation of functional maps—interpreted as linear operators between function spaces on shapes—directly into the learning architecture, thus offering more precise and efficient correspondence mappings.

The paper presents novel contributions that reflect both theoretical and practical enhancements over prior art. The idea of moving from point-to-point labeling to structured prediction in functional space is a significant shift. The approach employs a residual network architecture that iteratively refines input descriptors, ultimately generating a soft map—a probabilistic distribution representing correspondence over the target shape's surface. The performance underpins the ability of the framework to generalize across categories with diverse scanning conditions, providing robust solutions even amidst synthetic models and real-world topological noise.

Experimentally, the framework demonstrates superior results on standard benchmarks including the FAUST and SHREC'16 datasets, comprising synthetic and real-world scans of various categories. The architecture outperforms prevalent descriptor learning methods by a substantial margin. The authors report average error rates as low as around 2.34 cm on specific challenging datasets, showcasing their model's robustness against scanning artifacts and mesh inconsistencies.

These advances have significant implications. Practically, improved shape correspondence directly benefits applications like 3D animation, medical imaging, and object recognition in autonomous systems. Theoretically, the shift towards functional spaces opens new avenues for research into manifold learning, offering refined abstractions for representing complex structures. Future developments could further adapt these techniques to accommodate high-degree deformations and the corresponding non-linearities intrinsic to such scenarios.

In summary, the paper enriches the field of shape correspondence by transitioning from traditional descriptor-based approaches to structured prediction within functional map space. The utilization of deep learning architectures in this way enhances both the accuracy and efficiency of dense correspondence tasks, with vast implications for practical applications and theoretical inquiries within computer graphics and computer vision.