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Legendre Transform is a mathematical algorithm that maps a function defined in the linear space in another function defined in the dual space. The function in the dual space is generated by the original curve’s tangent and supporting line. Given the function y=f(x) in the space (x,y), its Legendre Transform is g=x·m-f(x), m=f´(x), that using x=(f´)⁻¹(m) can be mapped in the dual space (m,g(m)). Given the family of straight lines g= x·m-f(x), the function f=f(x) is the envelope curve of that family. The Legendre Transform is applicable only to convex functions (f´´(x)>0), but its extended version (Legendre-Fenchel) is applicable also to the non-convex one. The Fourier Transform, Laplace Transform and PCA, are well known in IR thermography. They are based on integral transformations that behave well against the noise of experimental data and are very useful in the reduction of dimensionality. On the contrary, the Legendre Transform is based on the derivative, and its tendency is that of amplify the noise of data and no easy way of reducing data dimensionality is devised. Nonetheless, there are at least a couple of reasons to experiment on the Legendre Transform applied to IR thermography data. The first is the growing interest of such a transformation in the field of image processing (e.g. Super-Resolution). The second relies on the alternative methods of solution of the heat conduction PDE, that is very often the base for thermographic analysis both in parameter estimation and in NDT. These alternative methods take advantage of the Lagrangian/Hamiltonian formalism that have been proposed for some computational advantage and, also for deeper insight of the physics underlining the processes. The Legendre Transform is well known in the frame of classical mechanics for being the transformation that brings the Lagrangian description to the Hamiltonian[S1.1] one, and back, thanks to its involutive nature. The aim of this paper is to investigate on the possible application/advantage (if any) of using the Legendre Transform in the field of IR thermography, with special attention on experiments dedicated to thermal parameter evaluation, and to Non-Destructive Testing for analysis/identification of buried defects in manufacts.
