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...th the BV seminorm which is formally given by � |u|BV = |∇u|, also referred to as the total variation (TV) of u. Problem (2.1) is called the ROF model, introduced to the field of image restoration in =-=[28]-=-. In [24] the authors showed that an iterative procedure (which turned out to be equivalent to Bregman’s relaxation method, cf. [7], and proximal point algorithms, cf. [12]) could be used to improve t...

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...ing scales, in particular those smoothing small scale features faster than large scale ones, so-called scale space methods. Scale space methods are obtained for example by nonlinear diffusion filters =-=[26]-=- of the form ∂u ∂t = div(γ(|∇u|2 )∇u), (1.1) in Ω×R+ with u(x,0) = f(x), where f : Ω → R denotes the given image intensity (Ω being a bounded open subset in R 2 ) and u : Ω×R+ → R the flow of smoothed...

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... flow for the minimal t∗ such that �f − u(t∗)� L 2 = σ. This stopping criterion is well-known in the theory of iterative regularization of inverse problems as the so-called discrepancy principle (cf. =-=[5, 10]-=- for a detailed discussion). 4 Direct Solution for Regularized Total Variation in 1D In the following we discuss the numerical solution of (7) in 1D. We recall here that p(t) ∈ ∂J(u(t)) and u ∈ ∂J ∗ (...

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...andard condition like J(0) = 0 by the inverse function theorem.sA possibility to invert the equation � for u is the � use of the the dual functional 〈u, p〉 − J(u) . Then one can easily show that (cf. =-=[4]-=-), defined by J ∗ (p) := supu p = ∂uJ(u) is equivalent to u = ∂pJ ∗ (p) and we obtain an explicit relation for u(p) provided we can compute the dual functional J ∗ . Under the above conditions, we can...

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...is is the ROF model, introduced to the field of image restoration in [11]. In [8] the authors showed that an iterative procedure (which turned out to be equivalent to Bregman’s relaxation method, cf. =-=[1]-=-, and proximal point algorithms, cf. [3]) could be used to improve the quality of regularized solutions to inverse problems, based on regularization functionals as in (2). Given a convex functional J(...

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...he method is stopped at a suitable final time, we may expect that noise is smoothed while large-scale features are preserved to some extent. For some examples of linear and nonlinear scale-spaces see =-=[2, 15, 19, 26, 33, 34]-=- and the references therein. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping time...

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... that D(g,u) = 0 implies that g = R(u), R being a nondecreasing function. This means that g and u are the same up to a contrast change. For a discussion of this kind of morphological equivalence, see =-=[1]-=-. The proof can be outlined as follows: D(g,u) = 0 implies ∇g = |∇g| ∇u |∇u| . When taking the curl of this equation, the resulting linear partial differential equation for u has the general solution ...

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...n particular for low-frequency components of w −g and large t, the evolution is very slow, and this is the reason why the reconstruction obtained from (2.1) can lose a lot of the image variation (cf. =-=[21]-=-) if λ < ∞. 4. Inverse Scale Space for Signals In the following we discuss the numerical solution of (3.3) in spatial dimension one. We recall here that p(t) ∈ ∂J(u(t)) and u(t) ∈ ∂J ∗ � � (p(t)). We ...

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...scale) ones. This again explains to some point why large scale features are incorporated earlier than small scale ones. We also mention that by a standard inequality for dual norms (cf. [21, p.32] or =-=[3]-=-) we have for ”clean images” f ∈ BV (Ω) t1 = �f�L2 ≤ �f�G J(u) , �f�L2 which yields a similar interpretation of scales in terms of the ratio of total variation and the L2-norm, e.g., in the disk examp...

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... fidelity term for removal of impulsive noise or for structure-texture decomposition [11, 23, 35]. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm =-=[4, 21, 32]-=-, H −1 norm [25] or on Gabor functions [5]. A scale-space, as opposed to the variational setting, naturally introduces a continuous set of solutions. Whereas for denoising usually a single solution is...

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...he method is stopped at a suitable final time, we may expect that noise is smoothed while large-scale features are preserved to some extent. For some examples of linear and nonlinear scale-spaces see =-=[2, 15, 19, 26, 33, 34]-=- and the references therein. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping time...

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...g. Over the last decades, a variety of methods have been proposed ranging from filtering methods over variational approaches to techniques based on the solution of partial differential equations (cf. =-=[10]-=-). Since the noise in images is usually expected to be a small scale feature, particular attention has been paid to methods separating scales, in particular those smoothing small scale features faster...

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...val of impulsive noise or for structure-texture decomposition [11, 23, 35]. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm [4, 21, 32], H −1 norm =-=[25]-=- or on Gabor functions [5]. A scale-space, as opposed to the variational setting, naturally introduces a continuous set of solutions. Whereas for denoising usually a single solution is selected, for d...

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...he method is stopped at a suitable final time, we may expect that noise is smoothed while large-scale features are preserved to some extent. For some examples of linear and nonlinear scale-spaces see =-=[2, 15, 19, 26, 33, 34]-=- and the references therein. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping time...

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... fidelity term for removal of impulsive noise or for structure-texture decomposition [11, 23, 35]. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm =-=[4, 21, 32]-=-, H −1 norm [25] or on Gabor functions [5]. A scale-space, as opposed to the variational setting, naturally introduces a continuous set of solutions. Whereas for denoising usually a single solution is...

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...he method is stopped at a suitable final time, we may expect that noise is smoothed while large-scale features are preserved to some extent. For some examples of linear and nonlinear scale-spaces see =-=[2, 15, 19, 26, 33, 34]-=- and the references therein. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping time...

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...v), vt = α(f −u). (5.13)s14 NONLINEAR INVERSE SCALE SPACE METHODS • ROF model: J = � Ω |∇u|, H = 1 2 �f −u�2 2 . ut = div ut = div � ∇u |∇u| � ∇u |∇u| � +λ(f −u+v), vt = α(f −u). • T V −L1 model (cf. =-=[11]-=-): J = � |∇u|, H = �f −u�1. Ω � +λ(sign(f −u)+v), vt = αsign(f −u). (5.14) (5.15) [Note that H is not strictly convex or smooth here and sign is just the notation for an element in the subgradient of ...

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...f,u): • Linear model: J = 12‖∇u‖22, H= 12‖f−u‖22. ut = ∆u+λ(f−u+v), vt = α(f−u). (5.13) • ROF model: J =∫ Ω |∇u|, H= 12‖f−u‖22. ut = div ( ∇u |∇u| ) +λ(f−u+v), vt = α(f−u). (5.14) • TV −L1 model (cf. =-=[10]-=-): J =∫ Ω |∇u|, H=‖f−u‖1. ut = div ( ∇u |∇u| ) +λ(sign(f−u)+v), vt = αsign(f−u). (5.15) [Note thatH is not strictly convex or smooth here and sign is just the notation for an element in the subgradien...

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...or structure-texture decomposition [11, 23, 35]. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm [4, 21, 32], H −1 norm [25] or on Gabor functions =-=[5]-=-. A scale-space, as opposed to the variational setting, naturally introduces a continuous set of solutions. Whereas for denoising usually a single solution is selected, for decomposition or segmentati...

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...−u(t),p(t)〉 dt dt = −〈f −u(t),∂tp(t)〉 = −�u(t)−f� 2 L 2. We can also have the following convergence of p(t) to q ∈ ∂J(f) if we assume the stronger condition q ∈ L 2 (a so-called source condition, cf. =-=[9]-=-). From (3.3) we proceed formally to 1 d 2 dt �p(t)−q�2 L2 = 〈∂tp(t),p(t)−q〉 = 〈f −u,p(t)−q〉 = −D(f,u)−D(u,f). So for strictly convex smooth J and for J(f) < ∞ we have that the subgradient of u(t) mon...

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...on. A related procedure involving the ROF model using TikhonovMorozov rather than Bregman iteration which multiplies λ by two in each step yields a multiscale method suggested in [17] and analyzed in =-=[31]-=-. This procedure does not obviously extend to the nonlinear inverse scale-space method, discussed below, and does not appear to satisfy an estimate analogous to that of equation (2.4). 3. Inverse Scal...

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...ield of image restoration in [11]. In [8] the authors showed that an iterative procedure (which turned out to be equivalent to Bregman’s relaxation method, cf. [1], and proximal point algorithms, cf. =-=[3]-=-) could be used to improve the quality of regularized solutions to inverse problems, based on regularization functionals as in (2). Given a convex functional J(u), e.g., J(u) = |u|BV , the iterated re...

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...he total variation. In this paper we present a different version of constructing inverse scale space methods as the limit of an iterated refinement procedure previously introduced by the authors (cf. =-=[24]-=-) and demonstrate its applicability to image restoration. With the new approach we are able to easily implement nonlinear inverse scale space methods even for the total variation functional, and, in c...

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...preserved to some extent. For some examples of linear and nonlinear scale-spaces see [2, 15, 19, 26, 33, 34] and the references therein. Diffusion filters can be related to regularization theory (cf. =-=[30]-=-) with certain regularization functionals, but theoretical foundations of choosing optimal stopping times are still missing (see [16, 22] for two recent studies concerning the stopping time problem). ...

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...he method is stopped at a suitable final time, we may expect that noise is smoothed while large-scale features are preserved to some extent. For some examples of linear and nonlinear scale-spaces see =-=[2, 15, 19, 26, 33, 34]-=- and the references therein. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping time...

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... example, one may evolve a deconvolution scale-space (with H = 1 2 �f −Ku�2 L 2) or to have a flow based on the L 1 fidelity term for removal of impulsive noise or for structure-texture decomposition =-=[11, 23, 35]-=-. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm [4, 21, 32], H −1 norm [25] or on Gabor functions [5]. A scale-space, as opposed to the variation...

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..., ∂J(u) = {J ′ (u)} if J is Fréchet-differentiable. This procedure improves the quality of reconstruction for many problems with discontinuous solutions, e.g., deblurring and denoising of images (cf. =-=[18, 24]-=-) when the least-squares term �f −u� 2 L 2 is replaced by an appropriate fitting term for individual examples. Note that the regularization term used in the first step is a so-called generalized Bregm...

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... example, one may evolve a deconvolution scale-space (with H = 1 2 �f −Ku�2 L 2) or to have a flow based on the L 1 fidelity term for removal of impulsive noise or for structure-texture decomposition =-=[11, 23, 35]-=-. Other types of fidelity terms may be considered in the future, for instance ones based on the G-norm [4, 21, 32], H −1 norm [25] or on Gabor functions [5]. A scale-space, as opposed to the variation...

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...ndations of choosing optimal stopping times are still missing (see [16, 22] for two recent studies concerning the stopping time problem). Recently, inverse scale space methods have been introduced in =-=[29]-=-, which are based on a different paradigm. Instead of starting with the noisy image and gradually ∗ Preprint. To appear in Commun. Math. Sci., 4(1), March 2006 † Institut für Industriemathematik, Joha...

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... while small scale features (including the “noise”) are still missing. The inverse scale space method can also be related to regularization theory, in particular iterated Tikhonov regularization (cf. =-=[6, 12]-=-) with the same regularization functionals as for diffusion filters. The construction of inverse scale space methods in [12] worked well for quadratic regularization functionals, but did not yield con...

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...e flow for the minimal t∗ such that �f −u(t∗)� L 2 = σ. This stopping criterion is well-known in the theory of iterative regularization of inverse problems as the so-called discrepancy principle (cf. =-=[14, 27]-=- for a detailed discussion). This is a key justification for our denoising approach. We emphasize this result because the Bregman distance is stronger than L 2 for the regularizations we are consideri...

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...in. Diffusion filters can be related to regularization theory (cf. [30]) with certain regularization functionals, but theoretical foundations of choosing optimal stopping times are still missing (see =-=[16, 22]-=- for two recent studies concerning the stopping time problem). Recently, inverse scale space methods have been introduced in [29], which are based on a different paradigm. Instead of starting with the...

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...ze the behaviour of the flow for suitable f, and to show that the solutions converge to this steady state, which we will do in the linear case below. A general convergence proof by Lie and Nordbotten =-=[20]-=-, which can apply for general convex J and L2 squared fidelity term, is discussed in the next section. We examine the second order in time formulation (5.11). In the linear case the subgradient is uni...

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...(0) = 0. We assume � f = 0. Ω If the flow u(t) according to (7) exists and is well behaved (which can be shown under reasonable assumptions on the functional J, in particular for total variation, cf. =-=[2]-=-), it is an inverse scale space method in the sense of [6]. This means that the flow starts at u(0) = 0 and incorporates finer and finer scales (with the concept of scale depending on the functional J...