  The quasiparticle interference (QPI) technique is a powerful tool that allows
to uncover the structure and properties of electronic structure of a material
combined with scattering properties of defects at surfaces. Recently this
technique has been pivotal in proving the unique properties of the surface
state of topological insulators which manifests itself in the absence of
backscattering. In this work we derive a Green function based formalism for the
ab initio computation of Fourier-transformed QPI images. We show the efficiency
of our new implementation at the examples of QPI that forms around magnetic and
non-magnetic defects at the Bi$_2$Te$_3$ surface. This method allows a deepened
understanding of the scattering properties of topologically protected electrons
off defects and can be a useful tool in the study of quantum materials in the
future.
