A new way of comparing shapes will enable scientists to analyse data in more flexible and powerful ways.

The approach has come from a collaboration between physicist Dr Vanessa Robins and mathematician Dr Katharine Turner, who have found a method for comparing shapes in a way that quantifies the differences between them.

“The big roadblock has been that previous technology could only compare shapes with the same numbers of pieces and holes,” said Dr Robins, from the Department of Fundamental & Theoretical Physics, in the Research School of Physics.

“So you could compare a jam donut with a baguette, but not with a cinnamon donut or a pretzel, because of the different number of holes.

“This innovation in theory means we can meaningfully compare shapes with different numbers of holes or pieces.”

The approach, published in the Journal of Applied and Computational Topology, is based on a theory called the extended persistent homology transform.

It will be useful not only for baked goods, but for any shape analysis, such as fingerprints and CT scans of bones. The theory could also be a significant boost in the study of the lung condition, Chronic Obstructive Pulmonary Disease (COPD), which may progress more rapidly for people with certain lung geometries, Dr Robins said.

“To analyse lungs you need to be able quantify what lung geometry means,” she said.

Dr Robins and colleagues in Melbourne have applied for a grant to fund further research into COPD.

The new theory sprang from an honours project in which Dr Turner and Dr Robins supervised student Bettina Hill on a project to analyse 2D leaf shapes. The researchers approached the problem by taking parallel cuts through the leaf, sweeping from one side to the other. They studied how each cut can result in multiple pieces (for example if there is a hole), and whether these pieces join up again in adjacent cuts. By repeating this process at a range of angles they get a set of numbers that characterise the shape.

However, the process struck a complication when leaves curled around on themselves to create an apparent hole in the shape – a second distinct edge. While thinking about how to keep track of the edges encountered during the sweeps, Dr Turner realised an approach already existed, in a theory known as extended persistence.

“It’s a largely overlooked tool, and quite hard to understand. Previously it had been used only for manifolds, in which it doesn’t give you much more information. But if you use it for something solid with boundaries, you really unlock its potential,” Dr Turner said.

Dr Robins  took Dr Turner’s adaptation of extended persistence and devised algorithms for computing it for 2D shapes.  Their student James Morgan turned these into a code package, which the group then tested on fonts, focusing on the letters A and g.

They were delighted when the code, unprompted, was able to identify serif and sans-serif fonts as related but different.

Similarly, separate groups of the letter g were identified, depending on whether their tail forms a closed loop or not – but the software still classed them as related, a result that would have been impossible with previous topological analysis.

As well as comparing two shapes the software can be used to analyse the symmetry of a single shape, by comparing the shape with reflections or rotations of itself.

“It’s not just a yes or no answer, which would be easy. We can quantify how far off symmetric the shape is, which is really nice,” Dr Turner said.

The true potential of symmetry measurement will come when the code is extended to three dimensions, but that will be a challenge, Dr Robins said.

“The theory doesn’t change, but the implementation is much harder!”