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Page 114 Ding et al. Art Int Surg 2024;4:109-38 https://dx.doi.org/10.20517/ais.2024.16

Figure 3. Illustration of grid-based and point-based geometric representations: voxel, point cloud, and polygon mesh. We take a pelvic
model as an example.

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satisfied, the poor generalizability of neural networks makes this type of representation less informative
for downstream tasks. This limitation is also difficult to overcome due to the lack of geometric
interpretability.

Functional representation
Functional representation captures geometric information using geometric constraints typically expressed
as functions of the coordinates, specifying continuous sets of points/regions. We divide the functional
representation into two subcategories based on the formation of the function.

Ruled functions
Ruled functions can be expressed in parametric ways or implicit ways. Parametric functions explicitly
represent the point coordinates as a function of variables. Implicit functions take the point coordinates as
[43]
input . A signed distance function (SDF) f of spatial points x maps x to its orthogonal distance to the
boundary of the represented object, where the sign is determined by whether or not x is in the interior of
the object. The surface of the object is implicitly represented by the set {x | f(x) = 0}. Level sets represent
geometric understanding by a function of coordinates. The value of the function is expressed as a level. The
set of points that generate the same output value is a level set. A level set can be used to represent curves or
surfaces. For primitive shapes like ellipses or rectangles, the ruled functional representation provides perfect
accuracy, efficient storage, predictable manipulation, and quantized interpretability for some geometric
properties. Thus, this representation form is commonly used in human-designed objects and simulations.
However, for natural objects, although effort has been made to represent curves and surfaces using
[44]
polynomials, as shown in the development of the Bezier/Berstein and B-spline theory , accurate
parametric representations are still challenging to design.

Neural fields
[45]
Neural fields use the neural network’s universal approximation property to approximate traditional
functions to represent geometric information. For example, neural versions of level-set and SDF are
[47]
[46]
proposed. Neural fields map the spatial points (and orientation) to specific attributes like colors (radiance
field) and signed orthogonal distances to the surface (SDF) . Unlike latent feature representation, where
[48]
[49]
the intermediate feature map extracted by the neural networks represents the geometric information, the
Neural fields encode the geometric information into the networks’ parameters. Compared to the traditional
[45]
parametric representation, the universal approximation ability of neural networks enables the
representation of complex geometric shapes and discontinuities learned from observations. The
interpretability of the geometric information from the neural function depends on the function it
approximates. For example, NeRF lacks geometric interpretability as it models occupancy, whereas
[48]
Neuralangelo offers better geometric interpretability as it models SDF. However, since the network is a
[49]
black box, geometric manipulation is not as direct as in ruled functional representations.