Download Image Processing and Data Analysis : The Multiscale Approach by Jean-Luc Starck Fionn D. Murtagh Albert Bijaoui PDF

By Jean-Luc Starck Fionn D. Murtagh Albert Bijaoui

ISBN-10: 051156435X

ISBN-13: 9780511564352

ISBN-10: 0521590841

ISBN-13: 9780521590846

ISBN-10: 0521599148

ISBN-13: 9780521599146

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Extra info for Image Processing and Data Analysis : The Multiscale Approach

Example text

Initialize f 1 with the given image, / . Initialize the multiresolution coefficients at each scale j , wj, to 0. 2 Using the Pyramidal Multi-Median Transform, determine the set of transform coefficients, wj1. 3 w / < _ w / + w/<. 4 Reconstruct image fi+\ from wj (using the interpolation algorithm described in the previous section). 5 Determine the image component which is still not reconstructible from the wavelet coefficients: / l + i <— / — / l + i . 6 Set i <— i + 1, and return to step 2. The number of iterations is governed by when fi+\ in step 5 approaches a null image.

The weight functions Ph{v) and pg(v) permit a general solution to the restoration of c ;(v). 78) It is easy to see that these filters satisfy the exact reconstruction eqn. 27). In fact, eqns. 78) give the general solution to this equation. In this analysis, the Shannon sampling condition is always respected. 26) is not necessary. 79) In Fig. 30 the chosen scaling function derived from a 5-spline of degree 3, and its resulting wavelet function, are plotted in frequency space. Their conjugate functions are plotted in Fig.

In addition, there is additive Gaussian read-out noise. The Anscombe transformation (eqn. 8)) has been extended (Bijaoui, 1994) to take this combined noise into account: see Appendix 1. 1 Noise modeling a Gaussian variable, y, of mean g and standard-deviation o\ and a Poisson variable, n, of mean mo: we set /(x,y) = y + an where a is the gain. 9) -GCg With appropriate values of a, a and g, this reduces to Anscombe's transformation (eqn. 8)). 2 shows an update of Fig. 1 for the combined Poisson and Gaussian case.

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