A Reciprocal-Orthogonal Parametric Transform and Its Fast Algorithm

In this letter, a reciprocal-orthogonal parametric transform and an efficient algorithm for its simple construction and fast computation are proposed. The algorithm is developed by introducing a recursive approach to decompose the transform matrix into a product of sparse matrices using the Kronecker product. It is shown that the structure of the resulting algorithm is very similar to that of the well-known Walsh-Hadamard transform, except for the multipliers introduced by the independent parameters. The transform has a large number of independent parameters that can be chosen arbitrarily from the complex plane. Thus, many interesting special cases can easily be obtained from the proposed transform. Moreover, we carry out a number of experiments to show that its independent parameters can successfully be used as an additional secret key for image encryption.