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 \(g(T_2)\), from an input time domain signal \(s(t)\). In the case of a CPMG experiment, we have:
where \( K(t,T_{2,i}) = \exp \left( -\frac{t}{T_{2,i}}\right) \) 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 \(T_2\). The information about the positions of the \(T_2\) peaks and their areas reflects properties of the analyzed material, such as pore size distribution, water/oil content, meat/fat ratio, among others.
Download sample files to test ILT and guide your own data format.
More information can be found in this video about ILT, or in our papers:
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Developed by M. C. B. Cardinali, G. V. Von Atzingen, T. B. Moraes
