## Two Dimensional Extension

For image processing applications, the ID structures discussed previously are simply extended to two dimensions. We first adopt the method proposed by Mallat and Zhong/[20], shown in Fig.. 11, where filter I (m) = 1 + | h (m)| 2, and h (m), k (m), and S (m) are the same filters as constructed for the 1D case. The left side of Fig. 10 corresponds to analysis (decomposition) while the right side is synthesis (reconstruction). The bar above some of the analytic forms of the synthesis filters refers to the complex conjugate.

If we simply modify the two oriented wavelet coefficients independently, orientation distortions are introduced. One way to avoid this severe artifact is first to apply denoising to the magnitude of the gradient coefficients, and then carry out nonlinear enhancement on the sum of the Laplacian coefficients, as shown in Fig. 11. For the two oriented gradient coefficients Sx and Sy, the magnitude M and phase P are computed as arctan ( — |,

respectively. The denoising operation is then applied to M, obtaining M'. The denoised coefficients are then simply restored as sX = M'* cos(P) and sy' = M' * sin(P), respectively. For the enhancement operation, notice that the sum of two Laplacian components is i50tr0pic. Therefore, we may compute the sum of the two Laplacian components L = lx + l and F = lx/ly. A nonlinear enhancement operator is then applied to only L, producing L. Thus, the restored components are l'x = L' * F and l'y = L' * (1 - F).

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