This work proposes a method to measure the photon polarisation in $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decays and prepares the necessary elements for this measurement using the data sets collected by the LHCb experiment at CERN in $2011$, $2012$, $2016$ and $2017$.

In the Standard Model of particle physics, the photon emitted in $b \rightarrow s \gamma$ transitions is predicted to be mostly left-handed. Additional couplings that enhance the right-handed photon component could be introduced by some new physics models. A direct and precise experimental test of the photon handedness is still missing. Thanks to the interference between various hadronic resonances in the $K^+ \pi^- \pi^+$ system, $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decays give access to a photon polarisation parameter $\lambda_{\gamma}$, which is predicted to be $+1$ in the Standard Model up to small corrections of the order of $m_s^2/m_b^2$. The value of this parameter has never been measured in these decays and could help constrain new physics effects.

This thesis presents a method to measure the photon polarisation parameter using a complex amplitude analysis in the five-dimensional phase-space of $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decays. The performed studies demonstrate that, in the ideal case of a background-free sample without distortions due to experimental effects, and ignoring the differences between non-factorisable hadronic parameters between the resonances in the $K^+ \pi^- \pi^+$ system, this amplitude analysis allows the measurement of the photon polarisation with a statistical uncertainty of around $0.018$ with a sample of 14,000 $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decays, corresponding to the signal statistics for LHCb in $2011$ and $2012$.

The second part of this work consists in the selection and mass fit of $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decay candidates using a total of $6.6,fb^{-1}$ of data collected by LHCb at centre-of-mass energies of $7$, $8$ and $13$ TeV, in order to prepare for an amplitude analysis of these decays. After the selection and mass fit, a total of $47,449 \pm 321$ signal candidates are found, and a background-subtraction method is applied in order to obtain signal-like data sets that will be used in the amplitude fit.

A roadmap for the amplitude analysis of $B^+ \rightarrow K^+ \pi^- \pi^+ \gamma$ decays using LHCb data is also presented, and the main sources of systematic uncertainties that are expected to affect the measurement of $\lambda_{\gamma}$ are described, as well as suggested methods to evaluate them. Finally, the sensitivity of the measurement in the presence of background is evaluated and the resulting uncertainty on $\lambda_{\gamma}$ is of the order of $0.028$, indicating good prospects for the future amplitude analysis.