Let X be an N(= 2 n ) × M (= 2 m ) dimensional data matrix and let Y be the N × M Hadamard transform matrix as $$ \begin{gathered} X = \left( \begin{gathered} {x_{{11}}}\quad {x_{{12}}}\quad \cdots \quad {x_{{12}}} \hfill \\ {x_{{21}}}\quad {x_{{22}}}\quad \cdots \quad {x_{{12}}} \hfill \\ \;\, \vdots \quad \quad \vdots \quad \;\;\; \vdots \quad \;\; \vdots \hfill \\ {x_{{N1}}}\quad {x_{{N2}}}\;\; \cdots \quad {x_{{NM}}} \hfill \\ \end{gathered} \right) \hfill \\ \quad = ({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_1}\quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_2}\quad \cdots \quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_M}) \hfill \\ \end{gathered} $$ (89) and $$\begin{gathered} Y = \left( \begin{gathered} {y_{{11}}}\quad {y_{{12}}}\quad \cdots \quad {y_{{12}}} \hfill \\ {y_{{21}}}\quad {y_{{22}}}\quad \cdots \quad {y_{{12}}} \hfill \\ \;\, \vdots \quad \quad \vdots \quad \;\;\; \vdots \quad \;\; \vdots \hfill \\ {y_{{N1}}}\quad {y_{{N2}}}\;\; \cdots \quad {y_{{NM}}} \hfill \\ \end{gathered} \right) \hfill \\ \quad = ({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_1}\quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_2}\quad \cdots \quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_M}) \hfill \\ \end{gathered} $$ (90) where {x i } and {y i } are N-dimensional column vectors. The 2-D Hadamard transform is given by $$ Y = {H_n}X\,{H_m} $$ (91) where H n and H m are 2 n × 2 n and 2 m × 2 m Hadamard matrices. Since n may not necessarily be equal to m, we will use superscripts (n) and (m) to denote the columns h i (n) and h i (m) in H n and H m respectively.