A better way to match 3D volumes

Date:
May 24, 2023
Source:
Massachusetts Institute of Technology
Summary:
Researchers developed an algorithm that can align two 3D shapes by
mapping their volumes, which is more effective than other methods
that align shapes by only mapping their surfaces.


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FULL STORY ==========================================================================
In computer graphics and computer-aided design (CAD), 3D objects are
often represented by the contours of their outer surfaces. Computers
store these shapes as "thin shells," which model the contours of the
skin of an animated character but not the flesh underneath.

This modeling decision makes it efficient to store and manipulate 3D
shapes, but it can lead to unexpected artifacts. An animated character's
hand, for example, might crumple when bending its fingers -- a motion
that resembles how an empty rubber glove deforms rather than the motion
of a hand filled with bones, tendons, and muscle. These differences
are particularly problematic when developing mapping algorithms, which automatically find relationships between different shapes.

To address these shortcomings, researchers at MIT have developed
an approach that aligns 3D shapes by mapping volumes to volumes,
rather than surfaces to surfaces. Their technique represents shapes
as tetrahedral meshes that include the mass inside a 3D object. Their
algorithm determines how to move and stretch the corners of tetrahedra
in a source shape so it aligns with a target shape.

Because it incorporates volumetric information, the researchers' technique
is better able to model fine parts of an object, avoiding the twisting
and inversion typical of surface-based mapping.

"Switching from surfaces to volumes stretches the rubber glove over
the whole hand. Our method brings geometric mapping closer to physical reality," says Mazdak Abulnaga, an electrical engineering and computer
science (EECS) graduate student who is lead author of the paper on this
mapping technique.

The approach Abulnaga and his collaborators developed was able to align
shapes more effectively than baseline methods, leading to high-quality
shape maps with less distortion than competing alternatives. Their
algorithm was especially well-suited for challenging mapping problems
where the input shapes are geometrically distinct, such as mapping a
smooth rabbit to LEGO-style rabbit made of cubes.

The technique could be useful in a number of graphics applications. For instance, it could be used to transfer the motions of a previously
animated 3D character onto a new 3D model or scan. The same algorithm
can transfer textures, annotations, and physical properties from one
3D shape to another, with applications not just in visual computing but
also for computational manufacturing and engineering.

Joining Abulnaga on the paper are Oded Stein, a former MIT postdoc who
is now on the faculty at the University of Southern California; Polina
Golland, a Sunlin and Priscilla Chou Professor of EECS, a principal investigator in the MIT Computer Science and Artificial Intelligence
Laboratory (CSAIL), and the leader of the Medical Vision Group; and
Justin Solomon, an associate professor of EECS and the leader of the
CSAIL Geometric Data Processing Group. The research will be presented
at the ACM SIGGRAPH conference.

Shaping an algorithm Abulnaga began this project by extending
surface-based algorithms so they could map shapes volumetrically, but each attempt failed or produced implausible maps. The team quickly realized
that new mathematics and algorithms were needed to tackle volume mapping.

Most mapping algorithms work by trying to minimize an "energy," which quantifies how much a shape deforms when it is displaced, stretched,
squashed, and sheared into another shape. These energies are often
borrowed from physics, which uses similar equations to model the motion
of elastic materials like gelatin.

Even when Abulnaga improved the energy in his mapping algorithm to better
model volume physics, the method didn't produce useful matchings. His team realized one reason for this failure is that many physical energies --
and most mapping algorithms -- lack symmetry.

In the new work, a symmetric method doesn't care which order the shapes
come in as input; there is no distinction between a "source" and "target"
for the map.

For example, mapping a horse onto a giraffe should produce the same
matchings as mapping a giraffe onto a horse. But for many mapping
algorithms, choosing the wrong shape to be the source or target leads to
worse results. This effect is even more pronounced in the volumetric case.

Abulnaga documented how most mapping algorithms don't use symmetric
energies.

"If you choose the right energy for your algorithm, it can give you maps
that are more realizable," Abulnaga explains.

The typical energies used in shape alignment are only designed to map
in one direction. If a researcher tries to apply them bidirectionally to
create a symmetric map, the energies no longer behave as expected. These energies also behave differently when applied to surfaces and volumes.

Based on these findings, Abulnaga and his collaborators created a
mathematical framework that researchers can use to see how different
energies will behave and to determine which they should choose to create
a symmetric map between two objects. Using this framework, they built
a mapping algorithm that combines the energy functions for two objects
in a way that guarantees symmetry throughout.

A user feeds the algorithm two shapes that are represented as tetrahedral meshes. Then the algorithm computes two bidirectional maps, from one
shape to the other and back. These maps show where each corner of each tetrahedron should move to match the shapes.

"The energy is the cornerstone of this mapping process. The model tries to align the two shapes, and the energies prevent it from making unexpected alignments," he says.

Achieving accurate alignments When the researchers tested their
approach, it created maps that better aligned shape pairs and which
were higher quality and less distorted than other approaches that work
on volumes. They also showed that using volume information can yield
more accurate maps even when one is only concerned with the map of the
outer surface.

However, there were some cases where their method fell short. For
instance, the algorithm struggles when the shape alignment requires
a great deal of volume changes, such as mapping a shape with a filled
interior to one with a cavity inside.

In addition to addressing that limitation, the researchers want to
continue optimizing the algorithm to reduce the amount of time it
takes. The researchers are also working on extending this method to
medical applications, bringing in MRI signals in addition to shape. This
can help bridge the mapping approaches used in medical computer vision
and computer graphics.

This research is funded, in part, by the National Institutes of Health,
Wistron Corporation, the U.S. Army Research Office, the Air Force Office
of Scientific Research, the National Science Foundation, the CSAIL Systems
that Learn Program, the MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint
Research Center, Adobe Systems, the Swiss National Science Foundation,
the Natural Sciences and Engineering Research Council of Canada, and a Mathworks Fellowship.

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========================================================================== Story Source: Materials provided by
Massachusetts_Institute_of_Technology. Original written by Adam
Zewe. Note: Content may be edited for style and length.


========================================================================== Journal Reference:
1. S. Mazdak Abulnaga, Oded Stein, Polina Golland, Justin
Solomon. Symmetric
Volume Maps: Order-invariant Volumetric Mesh Correspondence with
Free Boundary. ACM Transactions on Graphics, 2023; 42 (3): 1 DOI:
10.1145/ 3572897 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2023/05/230524181836.htm

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