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It is a functional norm when p to the norm:
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1 When p + , the expression Lp (f ) tends
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L (f ) = sup |f (x)|
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Given a signal f de ned on [a, b] and the values x [a, b] and > 0, we apply this tool at any x to the local function difference de ned by f (x) f (x ) where Using the norm L , this gives supx [x ,x+ ] (|f (x) f (x )|) This x x quantity is equivalent to -oscillation, since 1 f, [x , x + ] 2
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x [x ,x+ ]
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It is therefore possible to replace the -variation of f by the integral over J of supx [x ,x+ ] (|f (x) f (x )|), without altering the theoretical result for However, it is also possible to use Lp norms Indeed, the oscillation (or the local norm L ) only takes into account the peaks of the function In practice, it can happen that these peaks are measured with a signi cant error, or even destroyed in the process of acquisition of data (pro les of rough surfaces, for example) It is preferable to use all the intermediate values and replace the -variation with the quantity: 1 2
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In this expression, large values of p allow us to emphasize the effect of local peaks, whereas if p = 1, all the values of function f have equal importance These integrals make it possible to rectify the corresponding logarithmic diagram and to calculate the slope with precision We can also replace the above integral on J by a norm Lq , with q > 1 If q is large, this will take into account the more irregular parts of the signal We can also change the integral in the window [x , x+ ] into a convolution product by a kernel of type K(x / ), so that the results are even smoother However, it should be noted that except for particular cases (Weierstrass functions, for example), we do not exactly calculate the dimension with these methods, but rather an index smaller than [TRI 99], which nevertheless remains relevant to the signal irregularity Let us develop an example of the index just referred to Let K be a kernel 1 t belonging to the Schwartz class, with integral equal to 1 Let Ka (t) = a K( a ) for a a > 0 For a function f de ned in a compact, let f be the convolution of f with Ka Since f a is regular, the length a of its graph is nite We de ne the regularization dimension dimR (f ) of the graph of f as: dimR (f ) = 1 + lim log( a ) log a (110)
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This dimension measures the speed at which the length of less and less regularized versions of the graph of f tend to in nity It is easily proved that if f is continuous, is always true An interesting aspect of the dimension of the inequality dimR regularization is that it is a well-adapted estimation tool Results obtained on usual signals (Weierstrass function, iterated function system and Brownian fractional motion) are generally satisfactory, even for small-sized samples (a few hundred points) Moreover, the simple analytical form of dimR allows us to easily obtain an estimator for data corrupted by an additional noise, which is particularly useful in signal processing (see [ROU 98] and the FracLab manual for more details)
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13 H lder exponents 131 H lder exponents related to a measure The dimensional analysis of a set is related to its local properties To go further into this study, it is convenient to use a measure supported by the set In many cases (self-similar sets, for example), E is de ned at the same time as If E is a curve, constructing a measure on E is called a parameterization Without a parameterization it is impossible to analyze the curve However, a given set can support very different measures Particularly interesting ones are the well-balanced measures, in a sense we will explain Given a measure of Rn , let us rst de ne the H lder exponent of over any measurable set F by (F ) = log (F ) log diam(F )
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By convention, 0/0 = 1 and 1/0 = + Given a set E, we use this notion in a local manner, ie on arbitrarily small intervals or cubes intersecting E A pointwise H lder exponent is then de ned using centered balls B (x) whose radius tends to 0: (x) = lim inf B (x)
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The symmetric exponent can also be useful: (x) = lim sup B (x)
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In addition, the geometric context sometimes induces a speci c analysis framework If a measure is de ned by its value on the dyadic cubes, it will be easier to use the following H lder exponents:
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