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Strong Maximum Principle (Chapter 3)
Our study began with an investigation of the Strong Maximum Principle for viscosity solutions of second-order non-linear parabolic integro-differential equations [Cio]. It is worth mention-ing that Strong Maximum Principle for linear elliptic equations goes back to Hopf in the 20s and to Nirenberg in 1953.
We study separately the propagation of maxima in the horizontal component of the do-main and the local vertical propagation in simply connected sets of the domain. For horizon-tal propagation of maxima we give two types of results: one coming from the structure of the nonlocal operator and the other as the natural extension of the classical results of local propa-gation of maxima. We point out that we do not restrict ourselves to the periodic setting.
To be more precise, by the Strong Maximum Principle for equation (1.1) in an open set Ω ×(0, T ) we mean the following.
SMaxP: Any u ∈ U SC (Rd × [0, T ]) viscosity subsolution of (1.1) that attains a maximum at (x0, t0) ∈ Ω×(0, T ) is constant in Ω×[0, t0].
Figure 1.1: Strong Maximum Principle follows from the horizontal and vertical propagation of maxima.
• horizontal propagation of maxima: if the maximum is attained at some point (x0, t0) then the function becomes constant in the connected component C (P0) of the domain Ω×{t0} which contains the point (x0, t0) ,
• local vertical propagation: if the maximum is attained at some point (x0, t0) then at any time t < t0 one can find another point (x, t ) where the maximum is attained.
The propagation of maxima in Ω×(0, t0) then follows by standard covering arguments.
The horizontal propagation of maxima requires two different perspectives.
Translations of measure supports. Using the structure of the nonlocal operator, we show that Strong Maximum Principle holds whenever the whole domain (not necessarily con-nected) can be covered by translations of measure supports, starting from a maximum point. Under quite general assumptions on the nonlinearity F , we show that Theorem 1.1.3.
Non-degeneracy assumptions. However, we manage to show that the propagation of maxima holds, under suitable non-degeneracy and scaling assumptions on the nonlinearity F .
Viscosity Solutions Theory for Integro-Differential Equations 7
Theorem 1.1.4. If a viscosity subsolution u attains a maximum at P0 = (x0, t0), then u is constant (equal to the maximum value) in the horizontal component of the domain, passing through point P0.
In this case, the non-degeneracy argument applies with respect to the variable x1, which is shown to be a sufficient condition for the mixed equation above to satisfy the Strong Maximum Principle.
Local vertical propagation of maxima occurs under softer assumptions on the non-degeneracy and scaling conditions. We establish the following result, under some specific assumptions on F .
Theorem 1.1.5. Let u ∈U SC (Rd ×[0, T ]) be a viscosity subsolution of (1.1) that attains a maxi-mum at P0 = (x0, t0) ∈ QT . Then any rectangle R0(x0, t0) = {(x, t )||xi − x0i| ≤ ai , t0 − a0 ≤ t ≤ t0} contains a point P 6=P0 such that u(P ) = u(P0).
As an application, we use the Strong Maximum Principle to prove a Strong Comparison Result of viscosity sub and supersolution for integro-differential equations of the form (1.1) with the Dirichlet boundary condition u = ϕ on Ωc ×[0, T ] (1.13).
where ϕ is a continuous function. Namely, for certain nonlinearities we have the sequel result.
Theorem 1.1.6 (Strong Comparison Principle). Assume the Lévy measure µ satisfies assump-tion (N ) with β > 1. Let u ∈U SC (Rd ×[0, T ]) be a viscosity subsolution and v ∈ LSC (Rd ×[0, T ]) a viscosity supersolution of (1.1), with the Dirichlet boundary condition (1.13). If u −v attains a maximum at P0 = (x0, t0) ∈ Ω×(0, T ), then u − v is constant in C (P0).
Each of these results hold under suitable assumptions on the nonlinearity F , that we make precise in each chapter. We mention that they extend to parabolic integro-differential equa-tions the results obtained by Da Lio in [DL04] for fully nonlinear degenerate elliptic equations, as well as the maximum principle for nonlocal operators generated by nonnegative kernels obtained by Coville in [Cov08].
Regularity of Solutions (Chapter 4)
There are two approaches for proving the Hölder regularity of viscosity solutions of integro-differential equations:
• either by Harnack inequalities (see Silvestre [Sil06] Caffarelli and Silvestre [CS09])
• or by Ishii-Lions’s method [IL90] (see Barles, Chasseigne and Imbert [BCI11]).
Recently there have been many papers dealing with C 0,α estimates and regularity of solutions (not necessarily in the viscosity setting) for fully nonlinear PIDEs and the literature has been considerably enriched. We give an overview of the existent results in the ongoing chapter, but send the reader to Chapter 2 for further details on the references.
The two above methods do not cover the same class of equations and each of it has its own advantages. The powerful Harnack approach for strictly elliptic fully nonlinear equations leads in general to further regularity such as C 1,α, but requires some integrability condition of the measure at infinity. On the other hand, viscosity methods apply under weaker ellipticity assumptions and therefore deal with a large class of degenerate, fully nonlinear equations, in particular with super-linear gradient growth, allow measures which are only bounded at infin-ity, but do not seem to yield further regularity.
Viscosity Solutions Theory for Integro-Differential Equations 9
We refer in the sequel to the direct viscosity method, introduced by Ishii and Lions in [IL90]. According to it, in order to establish Lipschitz or Hölder regularity of the solution u, we shift the function u and show that the corresponding difference can be uniformly controlled by some radial function φ(|z|) = L|z|α, for α ∈ (0, 1].
Figure 1.4: Uniformly controlling the shift of u by φ(|x − y|) = L|x − y|α, for all α ∈ (0, 1] .
This simple method, closely related to classical viscosity solutions theory, was recently ex-plored by Barles, Chasseigne and Imbert in [BCI11] in order to prove C 0,α estimates of solutions of PIDEs. They were able to deal with a large class of integro-differential equations as well as a general class of nonlocal operators, satisfying proper assumptions. The authors prove that the solution is α-Hölder continuous for any α < min(β, 1), where β characterizes the singularity of the measure associated with the integral operator. However, in the case β ≥ 1 the ad-literam estimates do not yield Lipschitz regularity.
In addition, they assume the equation is elliptic in a generalized sense, i.e. at each point of the domain, the ellipticity comes either from the second order term, or from the nonlocal term. We realized when studying the Strong Maximum Principle that another type of ellipticity might arise: at each point, the nonlinearity is degenerate individually in the second-order term, and in the nonlocal term, but the combination of the local and the nonlocal diffusions ren-ders the nonlinearity uniformly elliptic. We recall that we termed this type of equations mixed integro-differential equations since for example the diffusion term might give the ellipticity in one direction, whereas the nonlocal term in the complementary direction. For this type of non-degeneracy, the assumptions in [BCI11] are not satisfied and we provide new Hölder and Lipschitz regularity results in this framework.
In this case local diffusions occur only in x1-direction and nonlocal diffusions in x2-direction.
Figure 1.5: Mixed integro-differential equation: classical diffusion occurs in x1 direction and fractional diffusion in the complementary direction x2.
The three nonlinearities must satisfy suitable strict ellipticity and growth conditions, that we omit here for the sake of simplicity, but can be illustrated on the following example:
−a1(x1)Δx1 u−a2(x2)Ix2 [x, u] −I[x, u]+b1(x1)|Dx1 u1|k1 +b2(x2)|Dx2 u|k2 + |Du|n +cu = f (x) where the nonlocal term Ix2 [x, u] is non-degenerate (satisfying (N)), of fractional exponent β ∈ (0, 2) and ai (xi ) > 0, for i = 1, 2.
When β > 1, we show that the solution is Lipschitz continuous for mixed equations with gradient terms bi (xi )|Dxi u|ki having a natural growth ki ≤ β if bi bounded. If in addition bi are τ-Hölder continuous, then the solution remains Lipschitz for gradient terms up to growth k i ≤ τ +β. When β ≤ 1, the solution is α-Hölder continuous for any α < β. An open problem is the Lipschitz continuity of solutions in the critical case β = 1.
Viscosity Solutions Theory for Integro-Differential Equations 11
In particular, for the advection fractional diffusion equation ut +(−Δu)β/2 +b(x) •Du = f we obtain that the solution is Lipschitz continuous in the subcritical case β > 1, with b bounded and is Hölder continuous in the supercritical case β ≤ 1, whenever b is C 1−β+τ, where τ > 0. Recently, Silvestre obtained similar results to ours by Harnack techniques [Sila]. In ad-dition to our results, he showed [Silb] that in the supercritical case the solution becomes C 1,τ, with τ as before.
With this type of nonlinearities in mind, we establish Lipschitz and Hölder regularity re-sults at several stages. For the sake of precision, we illustrate them on the previous general example.
Partial regularity. We first give regularity estimates of the solution with respect to each of its variables, in which case we use classical regularity arguments in one set of variables, and uniqueness type arguments in the other variables. Then the following holds.
The Lipschitz/Hölder constant depends on ||u||∞, on the dimension of the space d and only on the constants associated to the Lévy measures and on the functions a2 and b2.
If we want to extend the regularity result with respect to all its variables, the unique-ness requirements seem rather restrictive, as gradient terms are restrained to sublinear growth.
A priori estimates. Nevertheless, the regularity results can be extended to superlinear cases, by a gradient cut-off argument. Roughly speaking, one should look at the approximated equations with |Du| replaced by |Du| ∧ R, for R > 0 and remark that their solutions are Lipschitz continuous, with the Lipschitz norm independent of R, thus the solution ob-tained by this approximation is Lipschitz continuous.
Global regularity then follows by interchanging the roles of x1 and x2. Subsequently, we can deal with gradient growths of orders respectively k1 = 2, k2 ≤ β.
We provide the detailed proof of the regularity result for periodic viscosity solutions of el-liptic integro-differential equations. In a separate section, we give the extensions to the non-periodic setting, parabolic case as well as fully nonlinear Bellman – Isaacs equations, pointing out each time the main differences that occur in the proof.
Work in Progress and Future Plans
The arguments are classical up to the point of solving the stationary ergodic problem. To prove the uniqueness and existence of λ and v we use the classical argument introduced by Lions, Papanicolau and Varadhan in [LV] and consider the approximated equations F (x, D vε, D2vε, I[x, vε]) +εvε = 0, in RN . (1.15)
and pass to the limit as ε → 0. Therefore it is necessary to show some compactness for this family of functions and this requires in particular equicontinuity, if posible uni-form C 0,α estimates. The inconvenient is that the gradient bound depends in general on the L∞ norm of the solution. Sometimes, it is possible to establish gradient bounds independent of the L∞ norm of the solution (see for example the work of Barles and Souganidis [BS01]). In the nonlocal case, we show that the renormalized functions v˜ε(x) = vε(x) − vε(0) are uniformly bounded, thus they are equicontinuous. We argue by contradiction and combine Hölder estimates and the Strong Maximum Principle for the equations satisfied by the renormalized functions wε = . The proof is different though for the superlinear and sublinear case. So far, the sublinear case is completed.
Almost Periodic Framework
We would like to consider in near future the stationary ergodic problem in almost pe-riodic domains. There have been some recent developments [Sch] and [GS] in this di-rection but to the best of our knowledge, their study concerns only a particular family of nonlinear equations and nonlocal operators.The ergodic problem is related both to the homogenization and to the long time behavior of solutions. The case of fully nonlinear degenerate second-order PDEs in almost periodic environment has been studied by Li-ons and Souganidis in [LS05], where they assume the ellipticity has the same order as the space oscillation. We would like to investigate if similar techniques can be adapted to nonlocal equations.
Regularity Theory for Fully Mixed Jump Diffusion Equations
An interesting problem would be the extension of the regularity results to the more gen-eral case of Mixed Jump Diffusion Equations
− a(x)Δx1 u +b(x)(−Δ)αx2/2u = 0. (1.16)
We have been already working on the approach given by classical viscosity methods, where we have obtained partial regularity of solutions: the problematic case seems to be when the gradient forms a π/4 angle with the xi -axis. At the same time, we are interested in the approach given by Harnack inequalities and ABP estimates, when a(x) and b(x) are bounded above and below but not necessarily continuous. We expect that similar estimates to those given by Caffarelli and Silvestre in [CS09] can be obtained.
Image Restoration and Visualization by Curvature Motions 13
Image Restoration and Visualization by Curvature Motions
The role of curvatures in visual perception goes back to the 50s and one of the first important contributions is due to Attneave [Att54]. Attneave argued on neurological grounds that the human brain could not possible use all the information provided by states of simulation, and showed that the important information, that stimulated the retina, is located at regions where color changes abruptly (contours), and furthermore at angles and peaks of curvature.
This explains why, in one of the first serious attempts to cope with this numerical chal-lenge, Asada and Brady [AB86] introduced the concept of multiscale curvature. This paper led to increasingly sophisticated attempts to analyze planar shapes by their curvatures.
These equations are of great interest in image processing since they can be axiomatically ob-tained from the image multiscale theories, as the partial differential equations satisfying some invariance properties: causality, locality, isometry and contrast invariance, and in the case of (ACM) affine invariance as well. This characterization was given in ’93 by Alvarez, Guichard, Lions and Morel in [AGLM93].
The PDEs above describe the evolution of a hypersurface (curve in 2D) moving according to its mean curvature. Osher-Sethian defined and studied in [OS88] the level set method for the motion of fronts by (mean) curvature, for which Chen-Giga-Goto [CGG91] and Evans-Spruck [ES91] provided rigorous justifications. Their arguments are based on the notion of viscosity solution of Crandall and Lions [CL83], that allows one to give a suitable meaning to the (MCM) and (ACM) equations, in the class of uniformly continuous functions. The undertaking is an-alytically subtle principally because the mean curvature evolution equation is nonlinear, de-generate, and even undefined at points where Du = 0. Their approach, recast slightly, is this. Given the initial curve Σ0 as above, select some function u0 such that Σ0 = {x; u0(x) = 0}.
Solve then the Dirichlet problem for the parabolic (MCM) or (ACM) with initial condition u0, which gives the solution u(x, t ) and define the evolution of Σ0 as Σt = {x; u(x, t ) = 0}.
The evolution is shown to be geometric, in the sense that it is uniquely defined and indepen-dent of the choice of the initial function u0 ∈ BUC (Ω). However, for semi-continuous initial data, may be more than one solution of the initial value problem as pointed out by Soner in [Son93]. In fact, this is the case whenever the level set develops a nonempty interior or be-comes ‘fat’. This difficulty was explained by Barles, Soner and Souganidis in [BSS93].
Figure 1.6: Checkerboard image (upper-left corner) evolving by mean curvature motion. (a) For bounded uniformly continuous initial data, the level set is uniquely defined. (b) For semi-continuous functions, the fattening phenomena discards uniqueness of solutions.
was one of the first versions of curve analysis proposed by Mackworth and Mokhtarian in [MM86]. Rigorous proofs were given by Gage and Hamilton for convex Jordan curves [GH86] and later extended to embedded curves by Grayson [Gra87]. The Affine Shortening equation.
is a surprising variant of curve shortening introduced by Sapiro and Tannenbaum [ST93a]. A remarkably fast and geometric algorithm for affine shortening was given by Moisan in [Moi98]. It was used for shape identification algorithms in the works of Cao, Lisani, Musé, Sur, [CLM+08]. Following the works of Mackworth and Mokhtarian [MM86] on curve smooth-ing and shape extraction, Caselles et al. realized the potential of using directly the image level lines instead of its edges. They proposed to perform contrast invariant image analysis directly on the set of level lines, or topographic map [CCM96]. A fast algorithm computing the topo-graphic map was developed by Monasse and Guichard in [MG98].
Algorithms for Curvature Motions (Chapter 5)
All sound shape smoothing algorithms in the computer vision literature perform curve short-ening or affine curve shortening. But the numerical variety of the underlying numerical algo-rithms is worth noticing. In the first part, we present a review, analysis and comparison of the main classes of curvature algorithms. We discuss their history, implementation, advantages and drawbacks. There are three kinds of initial data for the algorithms: digital curves, digital sets, or digital images. We shall examine each in turn.
From Curve Shortening to Image Curvature Motion (Chapter 6)
We were first intrigued to find out if an explicit connection can be given between the geometric approach for Curve/Affine Shortening and the viscosity approach for the Mean/Affine Curvature Motion. This lead us to set forth in [CMM10] a new image processing numerical chain, that we termed Level Lines (Affine) Shortening. This image processing pipeline yields in practice sub-pixel evolutions of images by mean, respectively affine curvature, but it also gives an accurate curvature estimate, based on a direct computation on the level lines. It, hopefully, advances Attneave’s program and proves that computer vision can get close enough to human vision. This was one reason ahead to rigorously justify that the Level Lines Shortening, and its Affine variant, computes explicitly a viscosity solution for the Mean Curvature Motion, respectively Affine Curvature Motion.
Evans and Spruck checked in [ES91] the consistency of the level set approach with the clas-sical motion by mean curvature. More precisely, they showed that the mean curvature motion agrees with the classical motion, if and as long as the latter exists. The result applies for a smooth hypersurface, given as the connected boundary of a bounded open set. Their argu-ments are based on comparison techniques with lower barriers for the approximated mean curvature motion and strongly use the fact that the hypersurface is the zero level set of its (signed) distance function. However, the result does not describe the complete behavior of all the level lines of a Lipschitz function. Namely, if we are given a Lipschitz function u0 which evolves by mean curvature in the viscosity sense, are all of its level lines evolving indepen-dently by curve shortening? For dimension n ≥ 3 the result is not true, since hypersurfaces can develop singularities and possible change topology. Thanks to Grayson’s theorem, the 2D case has a very peculiar structure which we will take advantage of and show that evolving indepen-dently and simultaneously by curve shortening all the level lines of a function is equivalent to applying directly a mean curvature motion to the functions itself.
The Level Lines Shortening (LLS) builds on the previous mentioned contributions and connects explicitly the geometric approach for curve shortening evolutions and the viscos-ity framework for curvature motions. Both in the continuous and the discrete case, Level Lines (Affine) Shortening is defined as an operator which first extracts all the level lines of an image, then it independently and simultaneously smooths all of its level lines by curve shortening (CS) (respectively affine shortening (AS)) and eventually reconstructs, at each step, a new im-age from the evolved level lines. The chain is based on a topological structure, the inclusion tree of level lines as a full and non-redundant representation of an image [CM10a], and on a topological property, the monotonicity of curve shortening with respect to inclusion. There-fore, the hierarchy of the level lines is maintained while performing the smoothing. Thus, the chain realizes the commutative diagram:
We prove that the image reconstructed from the evolved level lines is a viscosity solution of the mean curvature motion (MCM) (resp. affine curvature motion (ACM)) provided that the level lines at almost all levels evolve by curve shortening (resp. affine shortening).
A similar result holds for LLAS and the affine curvature PDE with initial condition u0.
The initial image will be considered as an element of a particular space of functions V S (Ω) that we term space of very simple functions and is related to fattening phenomena for Lips-chitz continuous functions. This class corresponds to bilinearly interpolated images defined on a rectangle Ω whose topographic maps contain only Jordan curves. The set of very simple functions arises naturally in image processing, since level lines corresponding to noncritical levels are sufficient to grant an exact reconstruction of the digital image.
In this way, the described chain corresponds exactly to its numerical implementation [CMMM] and has the advantage of satisfying both numerically and analytically all the invari-ance properties required by the scale space in question. Then, by stability properties of viscos-ity solutions and density arguments, the result is immediately extended to Lipschitz functions.
Level Lines Shortening Algorithm (Chapter 7)
The next chapter describes the discrete Level Lines Shortening (LLS) Algorithm and its variant the Level Lines Affine Shortening (LLAS).
Digital images are given in discrete sampled forms on a rectangle Ω. The underlying sub-stratum is assumed to be continuous and interpolated as such on Ω. Among the possible inter-polations, the bilinear interpolation presents two advantages: it is the most local of continuous.
Figure 1.7: Illustration of the LL(A)S numerical chain. (a) Original image. (b) Bilinear level lines extraction. (d) Simultaneous and independent smoothing of level lines by affine shortening. (c) Image reconstructed from shortened level lines.
interpolations, and it preserves the order between the gray levels of the image. For this reason, we deal with level lines in bilinearly interpolated images.
Level Lines Shortening. The discrete algorithm is performed in three steps
1. it first extracts all the bilinear level lines of a digital image, with a number of levels sufficient to grant an exact reconstruction of the initial image
2. then it independently and simultaneously smooths all of its level lines by discrete curve shortening (CS) (resp. affine shortening (AS)).
3. the evolved image is eventually reconstructed from its evolved level lines.
One of the difficulties in image processing was until recently the extraction of the con-tours at subpixel accuracy. In ’98, Monasse and Guichard gave a fast algorithm [MG98].
Figure 1.8: The four pairs present various implementations of the mean curvature motion on a checkerboard image (left column) and zooms at an X-junction, with its level lines overprinted on the image (right). From top to bottom : (a). original image (the zoom is by bilinear interpolation), (b). Level lines shortening, (c). Finite difference scheme, (d) FDS stack filter. Only LLS does not create new extrema.
for computing the tree of shapes of an image and accordingly, the topographic map. Caselles and Monasse describe in their recent work [CM10a] a direct extraction algo-rithm for the tree of bilinear level lines. But surprisingly, no algorithm for the inverse reconstruction at subpixel accuracy was given. We complete their work with a fast algo-rithm of image reconstruction from an arbitrary family of Jordan curves given at subpixel resolution, provided it is embedded in a tree structure.
Accurate evolution. The order preserving property or inclusion principle is the main structural requirement of LL(A)S algorithm. It basically prevents the crossing of two different level curves and therefore permits the construction of a unique image having a prescribed set of level lines. Some level lines may present multiple crossings at saddle points, in which case the level lines shortening develops a non-empty interior. Performing classical Fi-nite Difference Schemes (FDSs) for mean curvature spurious diffusions occur around the image extrema and at T-junctions or X-junctions. At saddle points FDSs algorithms create new extrema, and therefore spurious level lines. Only LLS resolves this issue, by separately evolving the level lines and then reconstructing the image.
An Image Curvature Microscope. Furthermore, we show that Level Lines Shortening yield accurate curvature estimates, based on direct computation on the level lines. Most of curvature algorithms have a common drawback: they are based on finite difference schemes (FDS) for the second order differential operator curv(u). In this case, the cur-vature depends on the gray values of the neighbor pixels and consequently, high os-cillations along transverse level lines appear. Noise and aliasing effects render direct methods of curvature computation on a raw image practically impossible.
However, whenever we talk about curvatures in a digital image, we actually refer to the curvatures of the level lines associated to the image. Therefore, the curvature must be computed as a 1D differential operator, directly on level lines. After short time smooth-ing the pixelized level lines become accurate and we deal with curves, having sub-pixel control points, whose curvature can be faithfully computed. The striking difference be-tween an FDS result and an LLS result is displayed in Figure 7.11.
(a) had its curvatures computed in two different ways: by a FDS (b), and by LLS (c). In the first case the curvature presents oscillations, whereas the second result is coherent with our perception.
Range of Applications. In the last part we give a wide range of illustrative examples of image restoration and visualization.For example, in Fig.7.25 we display bacterial morphologies and the corresponding curvature map. In this case, the curvature is an intrinsic geomet-rical descriptor, useful for shape discrimination. An accurate curvature filter permits to make curvature statistics.
Table of contents :
1 Introduction
1.1 Viscosity Solutions Theory for Integro-Differential Equations
1.1.1 Lévy Processes and Integro-Differential Equations (Chapter 2)
1.1.2 StrongMaximum Principle (Chapter 3)
1.1.3 Regularity of Solutions (Chapter 4)
1.1.4 Work in Progress and Future Plans
1.2 Image Restoration and Visualization by CurvatureMotions
1.2.1 Algorithms for CurvatureMotions (Chapter 5)
1.2.2 From Curve Shortening to Image CurvatureMotion (Chapter 6)
1.2.3 Level Lines Shortening Algorithm (Chapter 7)
1.2.4 Work in Progress and Future Plans
1.3 Publications related to the thesis
2 Levy Processes. Integro-Differential Equations
2.1 Introduction
2.2 Lévy processes
2.2.1 The Structure of Lévy processes. Lévy-Khintchine Formula
2.2.2 Examples of Lévy Processes
2.2.3 The Lévy-Itô Decomposition
2.2.4 Integro-Differential Operators
2.3 Integro-Differential Equations
2.3.1 Existence and Comparison Results
2.3.2 Regularity Theory for Fully Nonlinear Integro-Differential Equations
2.4 Discussion
3 StrongMaximum Principle
3.1 Introduction
3.2 StrongMaximum Principle – General Nonlocal Operators
3.2.1 Horizontal Propagation – Translations ofMeasure Supports
3.2.2 Horizontal Propagation – Nondegeneracy Conditions
3.2.3 Local Vertical Propagation
3.2.4 StrongMaximum Principle
3.3 StrongMaximum Principle for Lévy-Itô operators
3.4 Examples
3.4.1 Horizontal Propagation – Translations ofMeasure Supports
3.4.2 StrongMaximum Principle – Nondegenerate Nonlocal
3.4.3 StrongMaximum Principle coming from Local Diffusion Terms
3.4.4 Strong Maximum Principle for Mixed Differential-Nonlocal terms
3.5 Strong Comparison Principl
3.6 Appendix
3.7 Conclusion
4 Lipschitz RegularityMixed PIDES
4.1 Introduction
4.2 Notations and Assumptions
4.2.1 Viscosity Solutions for Integro-Differential Equations
4.2.2 Ellipticity Growth Conditions
4.2.3 LévyMeasures for General Nonlocal Operators
4.2.4 LévyMeasures for Lévy-Itô Operators
4.3 Lipschitz Continuity of Viscosity Solutions
4.3.1 Partial Regularity Results
4.3.2 Global Regularity
4.4 Examples and Discussion on Assumptions
4.4.1 Classical Nonlinearities
4.4.2 A Toy-Model for the mixed case
4.4.3 Mixed Integro-Differential Equations with First-Order Terms
4.5 Some Extensions of the Regularity Results
4.5.1 Non-periodic Setting
4.5.2 Parabolic Integro-Differential Equations
4.5.3 Bellman-Isaacs Equations
4.5.4 Multiple Nonlinearities
4.6 Estimates for Integro – Differential Operators
4.6.1 General Nonlocal Operators
4.6.2 Lévy-Itô Operators
4.7 Appendix
4.8 Conclusion
5 Curves and Curvatures
5.1 Introduction
5.2 Curvature Scale Spaces
5.2.1 Curvatures
5.2.2 Curve Evolutions
5.2.3 Image Evolutions
5.2.4 Connection
5.3 Curvature algorithms
5.3.1 Algorithms on Curves
5.3.2 Algorithms on Sets
5.3.3 Algorithms on Images
5.4 Discussion
6 Equivalence LLS -MCM
6.1 Introduction
6.2 Modeling Level Lines Shortening
6.2.1 Crowns of Jordan Curves
6.2.2 The Class of Very Simple Functions
6.3 Level Lines Shortening Operator
6.3.1 LLS Semigroup Operator
6.3.2 Properties of Level Lines Shortening
6.4 Equivalence with the CurvatureMotions
6.4.1 Mean CurvatureMotion
6.4.2 Affine CurvatureMotion
6.5 Conclusion
7 LLS algorithm and Curvature Estimates
7.1 Introduction
7.2 The Level Lines Shortening Algorithm
7.2.1 Bilinear Interpolation
7.2.2 Direct Extraction Algorithm of Bilinear Level Lines
7.2.3 Independent Evolution of All Level Lines
7.2.4 Image Reconstruction from a Tree of Level Lines
7.3 Numerical Properties of the LLS Numerical Chain
7.3.1 Fixed Point Property
7.3.2 Local Comparison Principle and Regularity
7.3.3 JPEG Artifacts Reduction on Color Images
7.3.4 AccurateMean Curvature Evolution
7.4 An Image CurvatureMicroscope
7.4.1 Discrete curvature for a polygonal line
7.4.2 Curvature map
7.4.3 Comparison with FDSes
7.4.4 CurvatureMicroscope
7.5 Image Restoration and Vizualization
7.5.1 Founding Example: Attneave’s cat
7.5.2 Geometric Shapes
7.5.3 Graphics and Aliasing
7.5.4 Pre-attentively Undiscriminable Textons
7.5.5 BacteriaMorphologies
7.5.6 Topography
7.5.7 Textures
7.5.8 Paintings
7.5.9 Text Processing
7.5.10 Fingerprints Restoration and Discrimination
7.6 Conclusion
