Hadamard Matrix Analysis and Synthesis pp 109-115 | Cite as

# A Naturalness Preserving Transform

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## Abstract

The use of (284) has been suggested as the basis for a “naturalness preserving” transform. (J. Hershey and R. Yarlagadda, 1983; R. Yarlagadda and J. Hershey, 1985.) Consider that we have an image encoded as a square matrix, If α is close to unity, then we will still be able to recognize the original image We can gain some further insight into Π where C is an orthogonal matrix of order 2

*P*_{n}, that consists of*N*x*N*pixels where*N*= 2^{ n }. We form$$ {\Pi_n} = ({\pi_{{ij}}}) = {\Psi_n}(\alpha ){P_n}{\Psi_n}(\alpha ) $$

(287)

*P*_{ n }in Π_{ n }. By using (285d) we can recover*P*$$ {P_n} = {\Psi_n}(\frac{\alpha }{{2\alpha - 1}}){\Pi_n}{\Psi_n}(\frac{\alpha }{{2\alpha - 1}}),\alpha \ne \frac{1}{2} $$

(288)

_{ n }by exploring the Euclidean norm of (287). First we note that H^{’}$$ {{\rm H}^{'}}_n = {C^{{\rm T}}}(\begin{array}{*{20}{c}} {{I_2}n - 1} 0 \\ 0 { - {I_2}n - 1} \\ \end{array} )C $$

(289)

^{ n }and*I*_{2}^{ n-1}is an identity matrix of order 2^{ n-1}.## Keywords

Original Image Orthogonal Matrix Inverse Transforming Iterative Technique Spectral Norm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1997