# Inverse Laplace Transform (ILT)

This WebApp performs the Inverse Laplace Transform (ILT) of your data, with a focus on Nuclear Magnetic Resonance data, however, any signal composed of multiple exponential growth or decay components can be processed.

ILT is an ill-posed inverse problem that requires a regularization method to be solved and can admit multiple solutions, where the general objective is to obtain the distribution of relaxation times g(T), from an input time domain signal s(t). In the case of a CPMG experiment, we have:

where \( K(t,T_{2i}) = e^{-t/T_{2i}} \) is the kernel matrix for CPMG experiments.

The figure below shows an illustration representing the processing of the ILT process. On the left, the Carr-Purcell-Meiboom-Gill (CPMG) type signal is represented by the graph of echo amplitudes as a function of time. By processing this signal with the ILT, we obtain the graph on the right, which represents the distribution of relaxation times T2. The information about the positions of the T2 peaks and their areas reflects properties of the analyzed material, such as pore size distribution, water/oil content, meat/fat ratio, among others.

More information can be found in this video about ILT, or in our papers:

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Developed by T. B. Moraes, L. P. Mazzero, W. S. Mendes