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Ding et al. Art Int Surg 2024;4:109-38 https://dx.doi.org/10.20517/ais.2024.16 Page 113

However, the lack of explicit relations among cells limits their geometric interpretability and their use in
high-level processing tasks such as rigid/deformable surface representation and reconstruction.

Point-based representation
Instead of uniformly sampling the space, point-based representation samples key points represented in the
Cartesian coordinate system.

Point cloud
A point cloud is a discrete set of data points in the 3D space . It is the most basic point-based
[35]
representation that can accurately represent the absolute position in an infinite space for each point.
However, the sparsity of the points limits the accuracy of the representation at the object level. While it is
easy to acquire from sensors, the lack of explicit relationships between points makes it slow to render.
Similar to grid-based representation, the geometric interpretability of point clouds is limited due to the lack
of relationships between points.

Boundaries
Boundaries are an alternate geometric representation method that introduces explicit relationships between
[36]
points. An example in 3D space is the polygon mesh . It is a collection of vertices, edges, and faces that
define the shape of a polyhedral object. This type of representation is usually stored and processed as a
graph structure. Boundary-based representations offer the flexibility to represent complex shapes and are
widely used for 3D modeling. Compared to the point cloud, boundary-based representation offers faster
[36]
rendering speed and more control over appearance . A visual comparison is provided in Figure 3.
However, it shares the same limitation as point clouds due to the sparsity of the points. Additionally,
boundary-based representations have higher complexity and memory usage, as the potential connections
increase quadratically as a function of the number of points. For high-level processing, the relation among
points and explicit boundary representation provides basic geometric interpretability.

Latent space representation
Latent space representation is encoded from the original representation, such as point clouds, 2D images, or
3D volume, for dimension reduction or feature processing and aggregation. We divide the latent space
representation into two subcategories based on the encoding methods.

Ruled encoding
Ruled encoding employs handcrafted procedures to encode the data to the latent space. One example is the
[37]
principal component analysis (PCA) . PCA linearly encodes the data to new coordinates according to the
deviation of all samples. For geometric understanding, PCA can be used to extract the principal component
as a model template for a specific object, e.g., face [38,39] , and express the object as a linear combination of
these templates. The latent representation reduces feature dimensionality and also removes noise .
[40]
However, the encoding is usually hard to interpret in geometric aspects and the projection and dimensional
reduction usually come with information loss.

Neural encoding
Neural encoding extracts features via neural network architectures. The encoding is learned from specific
data distributions with specific geometric signals or geometry-related tasks. Neural networks’ ability to
[41]
extract higher-level features enhances the representation ability . This makes neural encoding-based latent
space representation work well when the downstream tasks are highly correlated to the pre-training proxy
tasks and the application domain is similar to the training domain. However, when these conditions are not