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# Orthonormal Matrix Transforms

Module by: Ivan Selesnick. E-mail the author

Consider the transformation of the data vector xx: y=Qx y Q x y0y1yN-1=( q00q01q0N1 q10q11q1N1 qN10qN11qN1N1 )x0x1xN-1 y0 y1 yN-1 q 0 0 q 0 1 q 0 N 1 q 1 0 q 1 1 q 1 N 1 q N 1 0 q N 1 1 q N 1 N 1 x0 x1 xN-1 yk=nqknxn y k n q k n x n If QQ is an orthogonal matrix (meaning that QTQ=I Q Q I ) then the data vector xx can be recovered from yy using the transpose of QQ: x=QTy x Q y x0x1xN-1=( q00q10qN10 q01q11qN11 q0N1q1N1qN1N1 )y0y1yN-1 x0 x1 xN-1 q 0 0 q 1 0 q N 1 0 q 0 1 q 1 1 q N 1 1 q 0 N 1 q 1 N 1 q N 1 N 1 y0 y1 yN-1 xn=kqknyk x n k q k n y k

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