The Laplace Beltrami operator (¢S)

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Surface representations: Introducing the co-mmon frameworks

A two dimensional surface embedded on R3 can be represented in several manners. It can be deflned as the solution of a simple equation.
For instance, the set of points S1 = ‘ (x; y; z) 2 R3 j x2 + y2 + z2 = 1“ deflnes the unit sphere and S1 is an implicit representation of this surface.
The unit sphere can also be represented using a parametric function on the space. Take f(µ1; µ2) = (cos µ1 sin µ2; sin µ1; cos µ1 sin µ2) = (x; y; z); then the unit sphere can also be deflned by the set
S2 = ff(µ1; µ2) j ¡ … • µ1; µ2 < …g ; and in this case S2 is an intrinsic representation of the unit sphere.
Of course S1 and S2 represent the same set of points. Thus, we have two difierent possible representations of the same surface: one implicit and the other intrinsic.
Implicit representations of 3-d surfaces are of the form S1 = f X 2 R3 j `(X) = 0g:
In particular for closed surfaces, in the image processing fleld, the conven-tions require that the function `, that deflnes the implicit representation, should be negative inside and positive outside the (possibly multiply con-nected) region bounded by the surface. Let us call this region ›. For simplicity, we may also ask ` to be at least C2. Usually, we use for ` the signed distance function deflned
> ¡Dist(X; S) 2
`(X) := 8 if X 2 ›
+Dist(X; S) if X = ›
On the other hand, intrinsic representations are given under the general form S2 = ‘(x; y; z) = f(u; v)j(u; v) 2 D ‰ IR2“ f(u; v) = (fx(u; v); fy(u; v); fz(u; v))
These are the two most common representations of mathematical surfaces.
However, the surfaces that can be found in computer vision problems are rarely known through these ideal representations: most of the time we have to face a discrete representation (a form given by a flnite set of points) and the connections between these points. We also may have some additional information like the outing normal vectors at each points of the surface. This is the case for data obtained by laser scanners. Other ways of obtaining digital surfaces -digital approximations of real surfaces- take as input a set of photographs of the surface, which are taken with a calibrated camera (like the one proposed in [18]) or in more general conditions (as done in [73]). In this case, we get as an output a volumetric scalar image whose values approximate the distance to the surface, providing directly an implicit discrete representation on the surface. Surfaces can also be approximated as a result of segmentation from volumetric images (it is the case of most of cortical images) and in this case the quality of the output may depend on the method used in either representations. This list of surface sources is far from being exhaustive and we gave examples of only a few of the intricate ways some data, representing a particular surface of interest, can be given in practice.
Most of the time, digitalized surfaces are given in one of these two forms:
† Triangulated Surfaces, Surface Meshes or Explicit Surfaces. In this case, the digitalized surface is given as a set of indexed triplets representing the vertex and a second set of vertex triplets that rep-resent the triangles. Data on the surface is given as a set of values (scalars, vectors, tensors..) where each value corresponds to a vertex of the surface mesh. This is the most common representation for ob-jects in 3D modeling and medical imaging. The traditional way to deal with these data is using the parametric functions given by the intrinsic representation1. Only recently, authors discovered (see [41]) that a discrete exterior calculus can be used for such matter (we will touch this matter in section 3.1.2 of this thesis).
† Implicit Volumes or Level-Sets. In this case, the digitalized sur-face is given as a scalar volumetric image, where each pixel value in a narrow band is the signed distance function to the surface or some other scalar function that is negative inside the surface and positive outside. Since we are only interested in the zero level-set, we only need the information in a band surrounding the surface. Data is stored in another volumetric image (scalar or multivalued, depending on the na-ture of the data) and also in pixels corresponding to the narrow band around the surface.
The Implicit Volumes became commonly used as a result of the increasing interest in Level-Set methods. Efiorts are being made to flnd better ways of storing the surface images. These do not use an entire volume to store the information because this information is in fact only needed in a band around the surface: this is the case of the RLE (Run-Length-Encoded) sparse level-set structure ([43]), unstructured graphs, adaptive grids ([19]), etc. Another inconvenient is that flne surface structures are not always captured by level-sets, although it is possible to use adaptive and triangulated grids (see for example [7] and [74]).

Bibliographical discussion

In this section we present the main aspects of the classical literature con-cerning regularization on manifolds. We also present a list of useful works that are strongly related to this subject and that we used for this work.

Explicit Surfaces

We start with a review of the works on surface regularization. It is indeed often the surface regularization problem that is placed flrst and its solution is closely related to the regularization of functions over manifolds when us-ing PDE’s methods. Both methods need the estimation of surface-intrinsic operators as the Laplace-Beltrami or Divergence operators: the flrst esti-mation is needed over the geometry of the manifold, whereas the second is needed over the set of functions deflned on the manifold.
The surface regularization problem appears principally in two areas: error optimization and PDEs. For the flrst approach we can brie°y cite the works from: [38, 42, 44, 60, 78, 101, 51, 49, 102, 56]. As the title of this thesis suggests, we are more concerned by the second cat-egory. For a very clear explanation on PDE’s based regularization methods for °at multivalued images, we recommend [96], and some references that review the work on this area are [25, 3].
Following the path of the °at image regularization, the difierent works start from the use of a Laplacian operator over the surface. This PDE-based evolution technique was originally imported for image processing on °at images by [68], [74], and [100].
In [31], [21], and [30] this idea is extended to smooth noised surfaces, proposing to apply the surface-intrinsic Laplacian, namely the Laplace-Beltrami operator over the geometry of the manifold, rather than a function over the manifold which is our case of study (we introduce the Laplace-Beltrami op-erator in the next section, but for a more proper deflnition we refer to [34]). Numerical issues on the discretization of the Laplace-Beltrami operator are discussed in [95]. In [50] and [51] the authors propose discrete approxima-tions of the Laplace-Beltrami extended to arbitrary connectivity in a way to deal with multi-resolution problems, far from the simple case of trian-gulations that is studied in this thesis. In [29] an anisotropic technique is presented for denoising height flelds and bivariate data.
In [31], the authors used an implicit discretization of geometric difiusion to obtain a strongly stable numerical regularization scheme. Afterward, in [30], they propose flner computational methods of normals and curvatures for discrete data with the objective to enhance and smooth triangulated surfaces. In [21], anisotropic geometric difiusion is introduced to enhance features while the smoothing efiect occurs.
Note that we have not said anything about the anisotropic difiusion yet. It is su–cient for the moment to say that this regularization technique permits to smooth small irregularities of the function (in this case the geometric rep-resentation) while it preserves and even accentuates the strong irregularities that deflne its principal features (corners, edges, etc). This is a major con-tribution in surface regularization techniques since earlier methods resulted on images that shrunk as the evolution went on and some important edges were not preserved.
The problems of regularizing the geometry of a surface and the data over a surface are closely related, and the way the corresponding discretization problems are solved is very similar. In [6], the authors propose an original method for regularizing the surface and the data on the surface simultane-ously, taking advantage of the fact that they share the same stifiness matrix. This is useful when the noise in the surface and the data are related, a sit-uation that is often seen when the surface and functional data is extracted from the same original set, on the same process, or with the same equipment. It is also useful when the data on the surface is related with the geometry on the surface: for example, the color on the lips’ surface limits itself to the lips’ corners. In [22] they use the data information to regularize the surface anisotropically in order to follow discontinuities on the data.
Few works have dealt with the regularization of functions over surfaces, but we can cite [20], where the authors propose the two estimations of the Laplace-Beltrami operator presented in the next section. More recently, in [82] and [54] (a work that can be found in the second and last chapters of this thesis), we presented the extension of the Beltrami Flow technique to the case of triangulated surfaces.
It is also worth mentioning the recent work [106] where the convergence of discrete Laplace-Beltrami operators over surfaces is studied, and where it is concluded that the best discretization known for the Laplace-Beltrami operator is the one obtained in [30], which is precisely the cotangents weights formula (2.7) that we retrieved in the previous section. This formula can also be found in [41] with a new approach: we focus ourselves on this new approach in the last chapter of this work. Finally, in [92, 90] the author shows segmentation of images deflned on surfaces by an extension of the geodesic active contours.

Level-Set methods

The level-set methods were flrst introduced in 1979 by Dervieux and Tho-masset for applications on °uid mechanics ([26, 27]) , and later in 1987 by Sethian and Osher ([65]) for computer vision applications. Level-set methods provide a very smart way to deal with moving interfaces such as curves and surfaces, as the model allows changes in the topology without major problems. Because of the importance of this discovery, these methods began to be used in many other applications from the image processing fleld such as segmentation ([45, 23, 62, 70]), motion estimation and tracking ([73, 61, 63]), front propagation ([65, 24, 36, 40, 71]), and many others. The level set method also ofiers an easier alternative to solve problems on flxed surfaces which before needed the entire and relatively complicated discretization of all the triangles.
Just like what we did for the explicit methods, we start citing the works on surface smoothing and reconstruction: [18, 66, 11, 75] (anisotropic geometric difiusion), [107] and [108]. More recently, in [94], an anisotropic difiusion of surface normals is investigated for features preserving surface reconstruction. An evolution PDE (such as the one presented in the previous section) deflned this time on the whole volume governs the behavior of the level surface.
Concerning non-°at image regularization, the framework showed in [17], [8], [10], and [9] was one of the starting points of this thesis: the authors present a level-set method to perform isotropic (see [17] for more details) regularization on scalar images that was extended afterward to anisotropic regularization for scalar and directional images in [8].
In [33], the authors show the use of anisotropic difiusion for vector fleld visualization on surfaces.
In [83], [81] and [82] we have generalized the Beltrami Flow regularization method for °at images for the case on non-°at implicit surfaces, and we presented numerical methods and applications for the case of scalar images. Another interesting contribution of these papers lies in the clariflcation of the relationship between the implicit and explicit approaches.

The Laplace Beltrami operator (¢S)

A classical method to erase the noise of °at scalar images in order to obtain a nice smoother function is the one based on the following PDE: @u = ¢u; ut=0 = u0: (2.1)
Here, the scalar data u0 : R2 ! R stands for the original noisy image deflned on the plane and the solution u : R2 ! R of the PDE stands for the regularized image: this is the one we are looking for. This PDE is borrowed from physics, it is known as the heat equation. In image processing, this regularization method is called isotropic difiusion, because the process smooths the image with the same intensity in all spatial directions without considering the fact that it blurs features of the image as well as the noise. The anisotropic regularization, which we introduce in section 3.2, on the contrary, privilege the smoothing efiect in certain particular directions that are determined by some other PDE regularization model.
The isotropic method is equivalent to the convolution of the image function with a normalized gaussian kernel of variance ¾ = p2t . It is also equivalent to the minimization of the energy (variational approach) deflned as:1 ZR2 kruk2 dxdy;
This is due to the fact that Equation (2.1) actually deflnes the difierential gradient descent used to minimize the energy.
The regularization algorithm consists in transforming the original image using PDE (2.1). The resulting process progressively smooths the image. Small irregularities flrst disappear, the image then starts to appear blurred and flnally it converges to a constant image as t ! 1 (see flg. 2.3). In practice, a human observer is needed in order to decide when to stop the evolution.
We have chosen this classical regularization method to show how both dif-ferent frameworks could be applied. Of course, we flrst have to deflne this PDE on a manifold, which is the idealistic mathematical object for the rep-resentation of surfaces embedded in R3.

The Laplace-Beltrami on explicit surfaces

A discretized surface is formed of a set of points in R3 and facets between them. We will consider the case where these facets are triangles: the case of triangulated surfaces. The function on the surface is deflned as a value attached to each vertex. These data can be the result of the application of a scanner from an object, numerical simulations retrieved by 2D images or various other extraction methods. It is the most simple and commonly used representation for objects in 3D modeling and medical imaging. Its portability and precision to represent detailed 3D objects are some of its major qualities compared to the implicit methods.
We represent mathematically the surface given by this kind of data using a parametrization. A parametrization of a certain surface S is a function
X:D!S; X2C2(D);
X(y) = ‘x1(y); x2(y); x3(y) : y = (y1;y2) 2 D ‰ R2“
such that (X1(p); X2(p)) = (Dy1 X(p); Dy2 X(p)) is a base for the tangent plane TpS at the point p 2 S.
Original noisy image Result of (2.1) after 3 iterations
Result of (2.1) after 10 iterations Result of (2.1) after 50 iterations
For example, if the surface is of the form x3 = fl0 + fl1x1 + fl2x2 + fl3x21 + 2fl4x1x2 + fl5x22 + : : : the Riemannian metric tensor g and the tensor l become 1 + fl2 fl1fl2 g = 1 1 + fl22 ; fl1 fl2
In this context, the Laplace-Beltrami operator ¢X can be expressed as ([20]): u : S!R; rX u = DyXT g¡1Dy u = X ¡ g¡1 @u Xi; i;j=1 i;j @yj
¢X u = j j X @ g ¡ g¡1 ¢ @u :
XX u = 11 2
r ¢ r µ j j 1 ¶
i;j=1 @yi i;j @yj
Using the Laplace-Beltrami operator, we can generalize PDE (2.1) from °at images to parametrization in the following manner@u = ¢X u; ut=0 = u0; (2.2) where again u0 : S ! R stands for the original noisy image and u : S ! R stands for the regularized image. To be able to perform flnite space-time difierences methods and compute the evolution of the PDE we need to estimate or discretize the Laplace-Beltrami operator. We now brie°y explain two known solutions (quoted from [20]).

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Parametric method

This is a rather tricky method that makes use of a special intrinsic property of the Laplace-Beltrami operator. Let us explain the idea.
If (y1; y2) is a conformal coordinate system, the Laplace-Beltrami operator becomes locally the planar Laplacian at y, ¢X =‚ µ@ (y1)2 + @ (y1)2 ¶
For an arbitrary surface S and a flxed point p 2 S, we can always flnd a conformal coordinate system by applying the a–ne transformation:
y := Q (x ¡ p) 1 ¡ 0 0 1n2 n1 0 1
Q := 0 2
y3 := z (y1; y2) = fl0 + fl1x1 + fl2x2 + fl3x1 + 2fl4x1x2 + fl5x2
where the fli are found by a standard least square estimation procedure using the neighboring points. We explain brie°y this standard method below:
let p1; :::; pm be the m neighbors of the point p, and let yj = Q (pi ¡ p). Set
0 y12 y22 ¡y12 ¢2 y12y22 ¡y22 ¢2 1
y1 y1 y1 2 y1y1 y1 2
B 1 2 ¡ 1 ¢ 1 2 ¡ 2 ¢ C
fl is then obtained by solving the linear system ,^ ¡T¢¡1¡T¢ fl = X X X Y :
Now, if we have the parametrization W (y1; y2) := p + Q0(y1; y2; z(y1; y2)), then, applying the linear transformation V (v1; v2) = W (Av) where A = g1=2, we obtain a conformal coordinate system such that the Laplace-Beltrami operator becomes rX = °1 + °3 where °1; °3 are deflned as the solutions of z(v1; v2) = °1(v1)2 + °2v1v1 + °3(v2)2 + :::
This allows us to retrieve proper data and start the computations.

Finite element method

A linear barycentric interpolation is used to flnd an approached solution.
Let u~ : (t; x) ! u~(t; x) be the approached solution to the difiusion equation.
Take the function u~ : (t; x) over each triangle:
NT 4 NT
Xi u~jTi (t; x) = X
u~ ! u~(t; x) = u~i(t; x)
=1 i=1
On each triangle Ti (i 2 f1 : : : NT g) composed by the nodes fpi1 ; pi2 ; pi3 g we have the barycentric coordinates functions f‡i1 ; ‡i2 ; ‡i3 g :
We look for u~i(t; 🙂 of the form u~i(t; x) = ‡i1 (x)~u(t; pi1 ) + ‡i2 (x)~u(t; pi2 ) + ‡i3 (x)~u(t; pi3 ) that is, we express u~i(t; 🙂 in terms of a linear combination of the barycentric coordinates functions, that we will later use as tests functions.
Now, let us focus on the linear system on each triangle Ti.
The functions
XNT `(x) =,i=1 are used as test functions. ‡i1 (x)`pi1 + ‡i2 (x)`pi2 + ‡i3 (x)`pi3
We take the integral of the difiusion equation ZTi @u ZTi ` @ti dT = ¡ < ru; r` > dT (2.3) where ui is replaced by the approached solution u~i in (2.3). Since we have the expressions for f‡i1 ; ‡i2 ; ‡i3 g and the equations holds for all ˆ, the linear system is verifled for the vector [~ui] = (~ui(t; pi1 ); u~i(t; pi2 ); u~i(t; pi3 ))
The critical value of – above which the numerical resolution will be unstable for the difiusion equation depends on thermal conductivity, which can be seen as conceptually equivalent to node distance: the closer the pair of nodes is, the stronger their mutual in°uence, and this mimics the situation of high thermal conductivity and fast heat propagation. Irregular node spacing, thus, parallels a situation of varying thermal conductivity across the fleld.
The critical temporal iteration step is also directly proportional to the min-imal node distance, because instability tends to spread over the whole lat-tice, and therefore a single pair of nodes that are too close together will be enough to globally afiect the result. Lattices possessing nodes in these con-ditions require very small iteration steps, and this may lead to exceedingly long processing times. Hence, a double constraint operates at the level of node separation: small inter nodal distance imply a small temporal iteration step and long computation times; on the other hand, very large inter nodal distances will eventually afiect the reliability of the Laplacian estimations. There is thus, a trade-ofi between processing speed and estimation accu-racy. If computation time is not unreasonably large, a conservative value for – should be chosen.
Note that the flnite element method is local in the sense that it is possible to estimate the Laplace-Beltrami operator on a vertex, using the information lying locally on its flrst ring neighboring triangles. On the other hand, with the parametric method it is necessary to compute the Laplace-Beltrami estimator for all the vertex at the same time, which can be more costly if we only want to work on part of the surface. The parametric method does not provide local estimators.
For this work, we have implemented both approaches (appendix 10.5) to work on general triangulated surfaces (in [20] an implementation is done for closed triangulations where each vertex has a constant number of neighbors).

The Laplace-Beltrami on implicit surfaces

We will now introduce the implicit framework explaining the generalization of the heat equation for scalar images deflned on implicit surfaces, as done by Bertalmio et al in [10] and [9].
Preliminaries for the implicit function approach
For a given vector w 2 R3, we deflne Pw as the orthogonal projection matrix Pw = I ¡ kwk2 :
As a consequence, the components of the matrix are (Pw)ij = –ij ¡ kwwiwk2j ; with –ij standing as usual for the Kronecker delta.
Let S ‰ R3 a surface, and ” its normal at x 2 R3. P” is an operator that projects vectors onto the tangent plane of S at point x.
Now, for X a vector fleld in R3, we may deflne the difierential operator PX r as the projection on the X-tangent plane, i.e. (PX r)i = j=1 µ –ij ¡ Xi 2 é@xj where @xj is the gradient vector operator in R3.
Given a real valued function u on R3 and given a vector fleld Y on R3 we will indistinctly use the notations (PX r) u = PX (ru) = PX ru
Projectors are useful tools when applied for the vector fleld X = rˆ because if we flx x 2 R3, Prˆ projects vectors into the level-set surface of ˆ passing through x. Therefore, if x 2 S, Prˆ projects vectors onto S at x.
Particularly, Prˆru evaluated on S is the projection of the gradient of u onto S. As a matter of fact, we deflne through this procedure what will be called in the remaining of this thesis the surface gradient of u on S or the intrinsic gradient: rSu := Prˆru:
Let us now list some useful properties of this tool:
1. Pwv ¢ z = v ¢ Pwz = Pwv ¢ Pwz,
2. (PX r)i u = ru ¢ PX ei,
3. Prur ¢ (PruX) = r ¢ (PruX kruk) kr1uk .
We represent the surface S by the zero level-set of a higher dimension function ˆ : R3 ! R; positive outside S and negative inside S, so that S = ‘x 2 R3 : ˆ (x) = 0“.
In our implementation the function ˆ is the signed distance function to S.
The scalar or vector data u (x) ; x 2 R3 is then smoothly extended to a band surrounding S:
Data in Implicit form Usually, the input image data is deflned only in the surface, to extend the initial image u0 over R3, a method proposed by Chen et al (1999) is to look for a u such as:
ru ¢ r’ = 0;
solving the PDE:
dudt + sign(’)(ru ¢ r’) = 0
on a narrow band surrounding S.
Computation of the isotropic difiusion equation
If we want to smooth the data u (x) : S ! R, where S is a surface, a way to do it is to follow the variational approach which consists in minimizing the corresponding 2-harmonic energy (see [10] and [9]): 1 ZS krSuk2 dS; which is equivalent to 1 Z›2R3 kPrˆruk2 – (ˆ) krˆk dx: 2
This equivalence can be derived from the fact that rSu = Prˆ noticed in the previous section) combined with the fact that ru (as we – (ˆ) krˆk dx = dS: › S
The equation that minimizes this energy, i.e. the corresponding harmonic map is @u = ¢Su (2.10) @t = rS ¢ rSu (2.11) = 1 r ¢ (Prˆru) krˆk (2.12) krˆk rS and ¢S stand here as usual for the intrinsic gradient and intrinsic Lapla-cian (the Laplace-Beltrami operator). This can be obtained computing the gradient descent of (2.9) (See [8] for details on this particular question).
Considering Z›2R3 kPrˆruk2 – (ˆ) krˆk dx E (u) = 2 and „ a perturbation of u, we have that (for the complete computation see appendix 10.2): E (u + t„) = ¡ S krˆkr ¢ (Prˆru krˆk) „dS (2.13) and since the last term must be identically equal to zero for all „, we can conclude that at the zero level-set ˆ,
1 krˆkr ¢ (Prˆru) krˆk = 0.
We can now extend this to the whole domain ›, as long as we assume that krˆk =6 0 at least in a band surrounding the surface.
Note that this result could have been intuitively predicted. Indeed, if we replace the expressions in the PDE (2.1) with the intrinsic ones and we arrive at the same result as in the Euclidean space case.
A numerical algorithm can be implemented using widely known numerical schemes for 3D operators without special considerations of the fact that we are dealing with surfaces. Everything is made on the cartesian grid. This is one of the reasons that makes this approach so attractive to implement for more complicated PDEs.

Summary and conclusions

In this chapter we have introduced the implicit and explicit surface repre-sentations, and discussed the related literature. We explained the general-ization of the isotropic regularization of °at images to surfaces to images deflned for the two principal surface representations. We chose to explain in detail the explicit (flnite element and parametric approximation, [20]) and implicit ([8, 10, 9]) Laplace-Beltrami regularization, which in the addition to the implicit anisotropic regularization, were the state of the art of image surface regularization methods when this work began. In the next chapter, we will show a new, simpler estimation for the Laplace-Beltrami operator deflned on triangulated surfaces, and two novel regularization methods: the anisotropic regularization and the Beltrami °ow for images deflned on tri-angulated surfaces.

Regularization on explicit surfaces

In this chapter we develop our work done1 on scalar images regularization over triangulated surfaces. We begin with the isotropic regularization, pre-senting a new numerical approach for the estimation of the Laplace-Beltrami operator. In the second part of the flrst section, we introduce the Discrete Exterior Calculus approach ([41]), which results relate to our approach. In the second section we use the same framework to deal with anisotropic reg-ularization, and generalize the numerical method to estimate a whole class of operators. In the third section we introduce the Beltrami framework and generalize it for images deflned on triangulated surfaces, proposing an nu-merical implementation based on the previous sections. We also show results on synthetic images.
We begin with the isotropic difiusion and present a difierent approach from those shown in chapter 2. We then discuss the anisotropic case together with the Beltrami °ow regularization.

Isotropic difiusion

In this subsection, we present an original procedure to obtain a discrete Laplace-Beltrami operator. We later extend this procedure to more general difierential operators and show that it is equivalent to the classical estimation shown in section ss:fem. We then compare this procedure to recent results that use the Discrete Exterior Calculus ([41]) as a major tool.

Table of contents :

1 Introduction 
2 State of the art 
2.1 Surface representations: Introducing the common frameworks
2.2 Bibliographical discussion
2.2.1 Explicit Surfaces
2.2.2 Level-Set methods
2.3 The Laplace Beltrami operator (¢S)
2.3.1 The Laplace-Beltrami on explicit surfaces
2.3.2 The Laplace-Beltrami on implicit surfaces
2.4 Summary and conclusions
3 Regularization on explicit surfaces 
3.1 Isotropic di®usion
3.1.1 An area-averaged Laplace-Beltrami estimation
3.1.2 Discrete Laplace-Beltrami via Discrete Exterior Calculus (DEC)
3.2 Anisotropic di®usion
3.2.1 Anisotropic di®usion on °at images
3.2.2 Area-averaged estimation
3.3 Extension to a more general case
3.4 Beltrami °ow
3.4.1 Introducing the Beltrami framework
3.4.2 Beltrami °ow over non-°at surfaces
3.4.3 Example: Gray image on the sphere
3.4.4 Estimation of the di®erential operator
3.4.5 Implementation details
3.4.6 Example
3.5 Summary and conclusions
4 Regularization on implicit surfaces 
4.1 Anisotropic di®usion
4.2 Beltrami °ow
4.2.1 Scalarfeld defened on a curve: Geometric derivation
4.2.2 The L1 and L2 limits
4.2.3 Implementing the regularization of scalarfelds on surfaces
4.2.4 Examples
4.3 Summary and conclusions
5 Comparison of methods and relationships 
5.1 From explicit to implicit via the Beltrami °ow
5.1.1 Implicit formulation
5.1.2 Intrinsic formulation
5.1.3 The implicit-explicit correspondence
5.2 Comparison of the two methods
5.2.1 From explicit to implicit
5.2.2 From implicit to explicit
5.2.3 Comments on the geometry of the mesh
5.2.4 Further comments and comparison for a particular application
5.3 Summary and conclusions
6 Vector image regularization 
6.1 Unconstrained vector regularization
6.1.1 Vector regularization: Explicit approach
6.1.2 Bi-dimensional vectorfeld on a surface: Implicit approach
6.2 Constrained regularization: Color images
6.2.1 Beltrami °ow, explicit-intrinsic approach
6.2.2 Isotropic and anisotropic regularization of color images on implicit surfaces
6.3 Summary and conclusions
7 Application to cortical images 
7.1 Occipital retinotopic areas extraction
7.1.1 Biological visual system
7.1.2 Extraction method
7.2 Regularization methods
7.2.1 Implicit Beltrami °ow scalar regularization
7.2.2 Explicit regularization
7.2.3 Explicit vector regularization
7.3 Summary and conclusions
8 Conclusions 
9 Author publications 
10 Appendix 
10.1 Stereographic direction di®usion
10.2 Computation of the implicit isotropic smoothing PDE
10.3 Implicit isotropic smoothing for color images
10.4 Numerical schemes for implicit regularization
10.5 Algorithm for explicit anisotropic smoothing

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