  We study the quasiparticle interference (QPI) patterns caused by scattering
off nonmagnetic, magnetic point impurities, and edge impurities, separately, in
a two dimensional helical liquid, which describes the surface states of a
topological insulator. The unique features associated with hexagonal warping
effects are identified in the QPI patterns of charge density with nonmagnetic
impurities and spin density with magnetic impurities. The symmetry properties
of the QPI patterns can be used to determine the symmetry of microscopic
models. The Friedel oscillation is calculated for edge impurities and the decay
of the oscillation is not universal, strongly depending on Fermi energy. Some
discrepancies between our theoretical results and current experimental
observations are discussed.
