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Journal of Lie Theory 32 (2022), No. 3, 709--736
Copyright Heldermann Verlag 2022

Differential Invariant Signatures for Planar Lie Group Transformations with Application to Images

Richard Brown
School of Information and Mathematical Sciences, Massey University, Palmerston North, New Zealand

Stephen Marsland
School of Mathematics and Statistics, Victoria University, Wellington, New Zealand

Robert McLachlan
School of Information and Mathematical Sciences, Massey University, Palmerston North, New Zealand

The actions of various Lie groups underlie the change of appearance of objects in images as the viewpoint changes, e.g., through camera motion. Despite significant advances in object recognition using machine learning in recent years, the question of how to recognise an object in an image as its appearance varies through camera motion and similar effects remains open. We demonstrate how differential invariant signatures can be derived for each of the transformation groups, and how the underlying invariances reflect the group-subgroup structure of the relevant Lie groups.
There are a variety of methods that can be used to identify differential invariants, and we provide examples of three of them: tensor contraction, transvectants, and the method of moving frames. We use the resulting invariants to construct practical sets to form three-dimensional invariant signatures. These signatures are not necessarily complete: the image cannot always be reconstructed uniquely up to transformation, but they are plottable, and depend at worst on third derivatives, although more channels of information, such as colour images, can reduce the highest order of derivative needed in some cases. We demonstrate the invariant signatures for each transformation group based on a simple smooth image. A full consideration of how these signatures could be used in practice will require effective methods to numerically approximate derivatives for images.

Keywords: Object recognition, planar transformation, planar Lie groups, invariants, differential invariant signature.

MSC: 65D18, 53A55, 68U10.

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