By Stéphane Jaffard, Patrice Abry, Stéphane Roux (auth.), Maïtine Bergounioux (eds.)

ISBN-10: 3642196039

ISBN-13: 9783642196034

The contributions showing during this quantity are a picture of the several themes that have been mentioned in the course of the convention. They normally problem, picture reconstruction, texture extraction and photo class and contain a number of diverse equipment and functions. for that reason it used to be very unlikely to separate the papers into commonly used teams that's why they're awarded in alphabetic order. besides the fact that they almost always hindrance : texture research (5 papers) with diverse ideas (variational research, wavelet and morphological part research, fractional Brownian fields), geometrical tools (2 papers ) for recovery and invariant characteristic detection, class (with multifractal analysis), neurosciences imaging and research of Multi-Valued Images.

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**Additional resources for Mathematical Image Processing: University of Orléans, France, March 29th - April 1st, 2010**

**Sample text**

By definition of EH1 ,H2 , they satisfy eλ ∼ 2−H1 j and fλ ∼ 2−H2 j ; by definition of d(H1 , H2 ), since we use cubes of the same width 2− j to cover EH1 ,H2 , we need about 2−d(H1 ,H2 ) j such cubes; therefore the corresponding contribution is of the order of magnitude of 2−d j 2d(H1 ,H2 ) j 2−(H1 p+H2 q) j = 2−(d−d(H1 ,H2 )+H1 p+H2 q) j . When j → +∞, the main contribution comes from the smallest exponent, so that η (p, q) = inf(d − d(H1 , H2 ) + H1 p + H2 q). H (50) In the next section, we will show that the scaling function η (p, q) is a concave function on R, which is in agreement with the fact that the right-hand side of (50) necessarily is a concave function (as an infimum of a family of linear functions) no matter whether d(H1 , H2 ) is concave or not.

Remark 5. When α is an integer, then the wavelet condition (29) characterizes a slightly different space, which implies a Tαq (x0 ) with a logarithmic loss on the modulus of continuity. This is reminiscent of the case of uniform H¨older spaces for s = 1, in which case the wavelet condition (15) characterizes the Zygmund class instead of the usual C1 space. 20 S. Jaffard et al. We now relate local square functions and local l q norms of wavelet coefficients. (The derivation that we propose slightly differs form the one of [31], since it is in the spirit of “wavelet leaders”, whereas the one performed in [31] relies on extensions of two-microlocal spaces which were proposed by Y.

First, recall that the local square functions at x0 are ⎛ S f ( j, x0 )(x) = ⎝ ⎞1/2 ∑ λ ⊂3λ j (x0 ) |cλ |2 1λ (x)⎠ . The following theorem is proved in [27]. Theorem 3. Let q ∈ (0, ∞), α > −d/q and assume that the wavelet basis used is r-smooth with r > sup(2α , 2α + 2d( 1q − 1)) ; if f ∈ Tαq (x0 ), then ∃C ≥ 0 such that ∀ j ≥ 0, S f ( j, x0 ) q ≤ C2− j(α +d/q) (29) (if q < 1, then . q denotes the Lq quasi-norm). q Conversely, if (29) holds and if α ∈ / N, then f ∈ Tα (x0 ). Remark 5. When α is an integer, then the wavelet condition (29) characterizes a slightly different space, which implies a Tαq (x0 ) with a logarithmic loss on the modulus of continuity.

### Mathematical Image Processing: University of Orléans, France, March 29th - April 1st, 2010 by Stéphane Jaffard, Patrice Abry, Stéphane Roux (auth.), Maïtine Bergounioux (eds.)

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