Objective

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Calculating 3D shapes correspondence is a central problem in the field of geometry processing that plays an important role in shapes reconstruction

object recognition and classification

and other tasks. Therefore

finding a meaningful and accurate correspondence among shapes has important research significance and application value. In recent years

deep-learning-based calculation of the correspondence among 3D shapes has attracted the attention of many scholars in the field of geometry processing. Using neural networks to learn the feature descriptors on the surfaces of 3D shapes can help us obtain accurate and comprehensive feature information and provides a solid foundation for building accurate correspondences among 3D shapes. Deep-learning-based methods for calculating the correspondences can be roughly divided into 1) methods based on the end-to-end depth functional maps network

and 2) methods based on other neural networks. The first method uses deep functional map networks to learn the feature descriptors of 3D shapes and then use these descriptors to analyze the spatial structure characteristics of shapes. Afterward

theory of functional maps is applied to solve the shape matching problem

find out the functional maps matrix among shapes

and determine the correspondence among shapes. Meanwhile

the second method uses neural networks to learn the feature descriptors of shapes

takes the calculation of correspondences as part of the learning process

uses the learned shape spatial geometric structure features for shape matching

and obtains the ideal correspondence among shapes. The feature descriptors and the network of learning descriptors of shapes play a crucial role in calculating 3D shapes correspondence calculation methods ignore the important influence of feature descriptors on the representation of 3D shapes

and the calculated shape descriptors contain relatively few information that cannot solve the problems related to shape symmetry and shape boundary descriptor distortion. Moreover

in the subsequent correspondence calculation process

these methods are unable to generate an accurate functional map of the symmetrical part of shapes

thereby leading to inaccurate correspondence calculations. The existing 3D shape correspondence calculation methods based on deep learning all adopt a supervision mechanism

which limits the universality of these methods to a large extent. To address these problems

this paper proposes a self-supervised deep residual functional maps network (SSDRFMN) to calculate 3D shapes correspondence.

Method

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The proposed method involves two steps. First

we calculate the feature descriptor of the 3D shape by combining the local coordinate system with a histogram

which is a signature of the histograms of orientations (SHOT) descriptor. We initially establish a local coordinate on the surface of shapes and then enhance the recognition ability of our descriptor by introducing the geometric information of the feature points. Afterward

we calculate the local histogram at a given point and use the calculated geometric information to form a histogram and a signature. Compared with traditional feature descriptors

hybrid feature descriptors can better represent the spatial structure and surface feature information of 3D shapes and provide high-quality inputs for network learning. Second

we use end-to-end SSDRFMN to calculate the correspondence among shapes. The SHOT descriptors of the source and target shapes are inputted into SSDRFMN

and the feature descriptor iteratively trains the neural network. The deep functional maps (DFM) layer is then used to calculate the function mapping matrix between two shapes. The corresponding relationship problem is transformed into solving the function mapping matrix problem

and the functional map relationship is converted into a point-to-point correspondence relationship through a fuzzy correspondence layer. The self-supervised loss function is then used to calculate the geodesic distance error between shapes and to evaluate the corresponding relationship. The loss function minimizes the geodesic distance error between shapes via network training and replaces the real correspondence between the manually labeled models with geometric constraints to achieve a self-supervised learning of the network.

Result

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Experimental results show that compared with the supervised deep functional maps (SDFM) and spectral upsampling (SU) algorithms

the proposed algorithm reduces the geodesic error of correspondences between the human model and the MPI-FAUST dataset by 1.45% and 1.67%

respectively. Meanwhile

in the TOSCA dataset

the proposed algorithm reduces the geodesic errors of the correspondences of dog

cat

and wolf models by 3.13% (2.81%)

0.98% (2.22%)

and 1.89% (1.11%) compared with SDFM (SU)

respectively. Therefore

apart from its applicability to different datasets and its high scalability

this algorithm effectively overcomes the shortcomings of extant depth functional maps methods that require the true correspondence among shapes to supervise.

Conclusion

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Experimental results show that the proposed method outperforms the existing methods for calculating the correspondence among 3D shapes. On the one hand

our proposed method trains the neural network through the SHOT descriptor of shapes

there by effectively solving the symmetry and boundary distortion of the model and producing a better representation of the surface features of 3D shapes. On the other hand

the proposed method uses a self-supervised deep neural network to learn the features of descriptors and to accurately calculate the correspondences among shapes. This method also shows excellent universality and accuracy

thereby highlighting its value in shapes matching

model recognition

segmentation

and retrieval.