Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.

# Industrial and Applied Mathematics Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Transformation theory has long been known to be a mechanism for the design of metamaterials. It gives rise to the required properties of the material in order to direct waves in the manner desired. This talk will focus on the mathematical theory underpinning the design of acoustic and elastodynamic metamaterials based on transformation theory and aspects of the experimental confirmation of these designs. In the acoustics context it is well-known that the governing equations are transformation invariant and therefore a whole range of microstructural options are available for design, although designing materials that can harness incoming acoustic energy in air is difficult due to the usual sharp impedance contrast between air and the metamaterial in question. In the elastodynamic context matters become even worse in the sense that the governing equations are not transformation invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists of rigid rods that is (i) closely impedance matched to air and (ii) slows down sound in air. This is shown to be useful in a number of configurations and in particular it can be employed to half the resonant frequency of the standard quarter-wavelength resonator (or alternatively it can half the size of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are not transformation invariant one can employ the theory of waves in pre-stressed hyperelastic materials in order to create natural elastodynamic metamaterials whose inhomogeneous anisotropic material properties are generated naturally by an appropriate pre-stress. In particular it is shown that a certain class of hyperelastic materials exhibit this so-called “invariance property” permitting the creation of e.g. hyperelastic cloaks [2,3] and invariant metamaterials. This has significant consequences for the design of e.g. phononic media: it is a well-known and frequently exploited fact that pre-stress and large deformation of hyperelastic materials modifies the linear elastic wave speed in the deformed medium. In the context of periodic materials this renders materials whose dynamic properties are “tunable” under pre-stress and in particular this permits tunable band gaps in periodic media [4]. However the invariant hyperelastic materials described above can be employed in order to design a class of phononic media whose band-gaps are invariant to deformation [5]. We also describe the concept of an elastodynamic ground cloak created via pre-stress [6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N. (2018) “Deepening subwavelength acoustic resonance via metamaterials with universal broadband elliptical microstructure”. Applied Physics Letters 112, 251902.

[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane elastic waves”. Proc Roy Soc A 468 (2138) 563-580.

[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468 (2146) 2881-2903 [4] Bertoldi, K. and Boyce, M.C. (2008) “Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures”. Phys Rev B 77, 052105.

[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with deformation-independent band gaps” Proc Roy Soc A 473, 20160865.

[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground cloaking” J Acoust Soc America 143 (5)