  A hybrid classical-quantum approach for evaluation of multi-dimensional
Walsh-Hadamard transforms and its applications to quantum image processing are
proposed. In this approach, multidimensional Walsh-Hadamard transforms are
obtained using quantum Hadamard gates (along with state-preparation, shifting,
scaling and measurement operations). The proposed approach for evaluation of
multidimensional Walsh-Hadamard transform has a considerably lower
computational complexity (involving $O(N^d)$ operations) in contrast to
classical Fast Walsh-Hadamard transform (involving $O(N^d~\log_2 N^d)$
operations), where $d$ and $N$ denote the number of dimensions and degrees of
freedom along each dimension. Unlike many other quantum image representation
and quantum image processing frameworks, our proposed approach makes efficient
use of qubits, where only $\log_2 N $ qubits are sufficient for sequential
processing of an image of $ N \times N $ pixels. Selected applications of the
proposed approach (for $ d=2 $) are demonstrated via computational examples
relevant to basic image filtering and periodic banding noise removal and the
results were found to be satisfactory.
