  Explicit coding schemes are proposed to achieve the rate-distortion function
of the Heegard-Berger problem using polar codes. Specifically, a nested polar
code construction is employed to achieve the rate-distortion function for the
doubly-symmetric binary sources when the side information may be absent. The
nested structure contains two optimal polar codes for lossy source coding and
channel coding, respectively. Moreover, a similar nested polar lattice
construction is employed when the source and the side information are jointly
Gaussian. The proposed polar lattice is constructed by nesting a quantization
polar lattice and a capacity-achieving polar lattice for the additive white
Gaussian noise channel.
