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Among the approaches that have been proposed in the literature to implement the continuous RT in the discrete domain, the nite Radon transform (FRAT) that has been used in this work was originally proposed in [99]. It is both perfectly invertible and nonredundant, and it is de ned as summations of image pixels over a certain set of lines in a discrete 2-D space, de ned in a similar way as the continuous lines in the Euclidean space. Speci cally, given a real function f [i, j] de ned over a nite 2 grid ZP , where ZP = {0, 1, . . . , P 1}, its FRAT is 1 FRATf [k, l] = r[k, l] = P f [i, j],
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2 where Lk,l de nes the set of points that form a line on ZP :
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Lk,l = {(i, j) : j = ki + l (mod P), i ZP }, LP,l = {(l, j) : j ZP }, where k ZP+1 is the line direction and l is its intercept. (20.14)
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The FRAT can be inverted using a nite back-projection (FBP) operator, de ned as the sum of Radon coef cients of all the lines that go through a given point, that is, 1 FBPr [i, j] = f [i, j] = P r[k, l],
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2 where Oi,j denotes the set of indices of all the lines that go through a point (i, j) ZP , that is,
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Oi,j = {(k, l) : l = j ki (mod P), k ZP+1 } {(P, i)}.
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The proposed watermark embedding domains, which stem from the Radon transform domain, have been designed in order to allow an energy compaction for each Radon projection in few representative coef cients. Speci cally, we employ the following: r The ridgelet transform [99] obtained by applying a wavelet decomposition to the Radon projections. Watermark embedding in the ridgelet domain was already proposed in [100] by the authors and in [101]; r The Radon-DCT (R-DCT) transform obtained by applying the discrete cosine transform (DCT) to each Radon projection. 20.3.1.1 Ridgelet Domain
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Given an integrable bivariate function f (x) = f (x1 , x2 ), its continuous ridgelet transform (CRT) de ned in [99] can be evaluated by applying the wavelet transform in the Radon domain. Speci cally, the CRT can be obtained by applying a 1-D wavelet transform to Rf ( , t) as follows: CRTf (a, c, ) = a 1/2
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Rf ( , t) dt.
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From Eq. (20.17), it can be seen that an invertible nite ridgelet transform (FRIT) [99] can be derived from the application of a 1-D discrete wavelet transform on each FRAT projection sequence (r[k, 0], r[k, 1], . . . , r[k, P 1]), for each direction k ZP+1 : FRITf [k, q] = g[k, q], q ZP . (20.18)
Thanks to the wavelets properties, the FRIT is able to concentrate the energy of each Radon projection sequence in its rst coef cients. 20.3.1.2 Radon-DCT Domain
As an alternative to wavelet analysis, the DCT can be used to obtain energy compaction. A novel embedding domain is thus de ned, indicating with Radon-DCT (R-DCT) the transform derived from application of the DCT on each FRAT projection
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sequence (r[k, 0], r[k, 1], . . . , r[k, P 1]), k ZP+1 :
R-DCTf [k, q] = c[k, q] = [l]
r[k, l] cos
(2l + 1)q , 2P
(20.19)
where q ZP , [0] = 1/N and [l] = 2/N, l = 0. Coef cients R-DCT / [k, 0] = c[k, 0], k ZP+1 , represent the DC component of each projection k, and f are therefore connected with the mean value of each Radon projection.