# Mathematical applications in sculpture and retail

What do a sculpture of a human body on a coast of the Netherlands and a virtual changing room for online clothes shopping have in common? Answer: the mathematical foundations of computer vision. Fundamental research by Professor Peter Giblin and collaborators has taken unexpected turns, leading to significant cultural, economic and societal impacts.

## The challenge

In the case of pure mathematics research, sometimes it seems that real-world applications are hiding in another dimension. However, Giblin’s work with singularity theory, which describes how objects depend on parameters, is a prime example that multidisciplinary collaboration within and outside academia can lead to impacts that exceed expectations.

Specifically, the application of singularity theory to computer vision looked at how to reconstruct a 3D model of an object, but based only on information from a video without knowing the exact track of the camera, or from a set of 2D photographs.

## Research action

Giblin and colleagues realised that the secret in forming a 3D model from a 2D image lies in surveying the outlines of the object, and its so-called ‘frontier points’.

A ball and outline (or ‘apparent contour’) would appear as a circle, and a frontier point is a fixed feature point on the surface that is visible from multiple directions. Naturally, more complex objects have more complex outlines, and as you walk round the object its outline changes.

When using suitable optimisation techniques, the geometry of the object can be recovered by measuring apparent contours and frontier points.

## Working in partnership

Giblin has worked in collaboration with mathematicians, computer scientists and engineers across Europe and the US. Specifically, the impact is underpinned by research undertaken from 1995-2000, with the initial ideas dating back to 1986. The work was carried together with collaborators such as Roberto Cipolla, Kalle Ǻström, Frank Pollick and research students.

The digital blueprint of the human body, in both cases, was achieved using case-specific software developed by Professor Cipolla at Cambridge, based on the fundamental research on singularity theory carried out with Giblin. The theory was described in the book Visual motion of curves and surfaces, by Cipolla and Giblin.

## Outputs and outcomes

Extensive numerical work through innovative software engineering put the theory into practice. For example, Metail is an online clothing shop that allows the customer to create an avatar of themselves based on just single 2D image to give an idea how the clothes fit their body. The company is collaborating with a commercial partner, supermarket giant Tesco, and has raised over £4M in investment, filed patents and won sizeable grants in 2012 and 2013 from the UK Government's Technology Strategy Board.

The work also fed into artist Anthony Gormley’s huge striking sculpture of a crouching man ‘Exposure’, which is made out of interlocking pylons and situated on the coast of the Netherlands. The sculpture has a great cultural value which is recognised internationally, winning the BCSA Structural Steel Design Award in 2011.

For the past 40 years, Giblin has also actively engaged with schools, delivering various mathematical outreach activities such as ‘Saturday Maths Club’, as well as serving on national mathematics committees. He was awarded an OBE for 'services to mathematical sciences' in the 2018 Queen’s Birthday Honours.

Pioneering research on singularity theory in mathematics to map objects in 3D from 2D images has led to far-reaching impacts in the academic and commercial worlds.

Professor Peter Giblin