  The robust Chinese remainder theorem (CRT) has been recently proposed for
robustly reconstructing a large nonnegative integer from erroneous remainders.
It has found many applications in signal processing, including phase unwrapping
and frequency estimation under sub-Nyquist sampling. Motivated by the
applications in multidimensional (MD) signal processing, in this paper we
propose the MD-CRT and robust MD-CRT for integer vectors. Specifically, by
rephrasing the abstract CRT for rings in number-theoretic terms, we first
derive the MD-CRT for integer vectors with respect to a general set of integer
matrix moduli, which provides an algorithm to uniquely reconstruct an integer
vector from its remainders, if it is in the fundamental parallelepiped of the
lattice generated by a least common right multiple of all the moduli. For some
special forms of moduli, we present explicit reconstruction formulae. Moreover,
we derive the robust MD-CRT for integer vectors when the remaining integer
matrices of all the moduli left divided by their greatest common left divisor
(gcld) are pairwise commutative and coprime. Two different reconstruction
algorithms are proposed, and accordingly, two different conditions on the
remainder error bound for the reconstruction robustness are obtained, which are
related to a quarter of the minimum distance of the lattice generated by the
gcld of all the moduli or the Smith normal form of the gcld.
