Artists who bring heroes and villains to life in animated movies and video games could have more control over their animations because of a brand new technique introduced by MIT researchers.

Their method generates mathematical functions referred to as barycentric coordinates that outline how 2D and 3D shapes can bend, stretch, and move through space. For example, an artist using his tool might select features that adjust the movements of a 3D cat’s tail to match his vision for the animated cat’s “look.”

This GIF shows how researchers used their technique to maneuver a cat’s tail more easily.

Image: Courtesy of the researchers

Many other techniques for this problem are inflexible and only provide a single option for the centroid coordinate functions for a given animated character. Each feature may or will not be the most effective for a specific animation. The artist would have to start out over with a brand new approach each time they wanted to attain a rather different look.

“As researchers, we are able to sometimes get stuck within the loop of solving artistic problems without consulting the artists. What matters to artists is flexibility and the “look” of their final product. “You don’t care concerning the partial differential equations that your algorithm solves behind the scenes,” says Ana Dodik, lead writer of a paper on the technique.

Beyond its artistic applications, this system could possibly be utilized in areas akin to medical imaging, architecture, virtual reality and even computer vision to assist robots work out how objects move in the actual world.

Dodik, an electrical engineering and computer science (EECS) graduate student, wrote the paper with Oded Stein, assistant professor on the University of Southern California’s Viterbi School of Engineering; Vincent Sitzmann, assistant professor of EECS and leader of the Scene Representation Group within the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL); and senior writer Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Computing Group. The research was recently presented at SIGGRAPH Asia.

A general approach

When an artist animates a 2D or 3D character, a typical technique is to surround the character’s complex shape with a less complicated set of points connected by line segments or triangles, called a cage. The animator drags these points to maneuver and deform the character within the cage. The primary technical problem is determining how the character moves when the cage is modified. This movement is set by the design of a particular center of gravity coordinate function.

Traditional approaches use complicated equations to search out cage-based movements which can be extremely smooth and avoid kinks that would occur in a shape when it is incredibly stretched or bent. However, there are a lot of ideas about how you can translate the artistic idea of ​​“smoothness” into mathematics, and every of them results in a special set of barycentric coordinate functions.

The MIT researchers were in search of a general approach that might allow artists to have a say in designing or choosing smoothing energies for every shape. The artist can then preview the deformation and choose the smoothing energy that most closely fits their taste.

Although the flexible design of centroid coordinates is a contemporary idea, the fundamental mathematical construction of centroid coordinates dates back centuries. Introduced by the German mathematician August Möbius in 1827, barycentric coordinates determine how each corner of a shape exerts influence on the inside of the form.

In a triangle, the form Möbius utilized in his calculations, centroid coordinates are easy to design – but when the cage isn’t a triangle, the calculations grow to be messy. Creating centroid coordinates for an advanced cage is especially difficult because with complex shapes, each centroid coordinate must meet quite a lot of constraints while being as smooth as possible.

In a departure from previous work, the team used a special sort of neural network to model the unknown centroid coordinate functions. A neural network, loosely based on the human brain, processes an input using many layers of interconnected nodes.

While neural networks are sometimes utilized in AI applications that mimic human considering, on this project neural networks are used for mathematical reasons. The researchers’ network architecture knows how you can output centroid coordinate functions that exactly fulfill all boundary conditions. You construct the constraints directly into the network so that they’re at all times valid when generating solutions. This construction helps artists design interesting centroid coordinates without having to fret about mathematical elements of the issue.

“The hard part was constructing within the restrictions. Standard tools didn’t get us there, so we actually needed to think outside the box,” says Dodik.

Virtual triangles

The researchers relied on the triangular barycenter coordinates introduced by Möbius almost 200 years ago. These triangle coordinates are easy to calculate and meet all of the crucial constraints, but modern cages are rather more complex than triangles.

To close this gap, the researchers’ method covers a shape with overlapping virtual triangles connecting triples of points on the skin of the cage.

“Each virtual triangle defines a legitimate centroid coordinate function. We just need a solution to mix them,” she says.

This is where the neural network comes into play. It predicts how the centroid coordinates of the virtual triangles can be combined to acquire a more complicated but smooth function.

Using their method, an artist could check out a function, take a look at the ultimate animation, after which adjust the coordinates to create different movements until they arrive at an animation that appears the best way they need.

“From a practical perspective, I believe the largest impact is that neural networks provide you with a whole lot of flexibility that you simply did not have before,” says Dodik.

The researchers demonstrated how their method could produce more natural-looking animations than other approaches, akin to a cat’s tail gently curving because it moves moderately than rigidly folding near the corners of the cage.

In the longer term, they wish to check out different strategies to hurry up the neural network. They also wish to integrate this method into an interactive interface that might allow an artist to simply iterate on animations in real time.

This research was funded partly by the US Army Research Office, the US Air Force Office of Scientific Research, the US National Science Foundation, the CSAIL Systems that Learn Program, the MIT-IBM Watson AI Lab, and the Toyota-CSAIL Joint Research Center, Adobe Systems, a Google Research Award, the Singapore Defense Science and Technology Agency and the Amazon Science Hub.

This article was originally published at