the laplace-beltrami operator in a level set framework and its...

OutlineThe Laplace-Beltrami and Interface Motion

A Splitting Scheme and Its Stability and ConvergenceSegmentation with Corners

A Narrow Band Approximation of the Laplace-Beltrami Eigen Spectrum

The Laplace-Beltrami Operator in a Level SetFramework and Its Applications

Wenhua Gao

April 4, 2011

Wenhua Gao The Laplace-Beltrami Operator in a Level Set Framework and Its Applications




The Laplace-Beltrami and Interface MotionLaplace-Beltrami OperatorLevel Set RepresentationInterface Motion

A Splitting Scheme and Its Stability and ConvergenceA Splitting SchemeStability and ConvergenceNumerical Results

Segmentation with CornersExisting ModelsChan-Vese Segmentation Model with CornersNumerical Examples

A Narrow Band Approximation of the Laplace-Beltrami EigenSpectrum

IntroductionNarrow Band, Parametrization and ApproximationNumerical Results





Laplace-Beltrami OperatorLevel Set RepresentationInterface Motion

Laplace-Beltrami Operator

Laplace-Beltrami: surface version of the Laplace operator.

Γ: a closed surface embedded in ℝd+1 with metric g .

∀ p ∈ Γ, ∃ a neighborhood U isomorphic to V ∈ ℝd .

Laplace-Beltrami operator ΔΓ: C∞(Γ) −→ C∞(Γ)

ΔΓu =1√∣g ∣

d∑i=1

∂

∂xi(√∣g ∣

d∑j=1

g ij ∂u

∂xj) (1)

where x = (x1, ⋅ ⋅ ⋅ , xd) ∈ V , u ∈ C∞(Γ), (g ij) is the inverse matrixof (gij) and ∣g ∣ = det(gij).






Level set Representation

Surfaces or interfaces can be represented as the zero level set of afunction � ∈ C(ℝd+1).

Γ = {x ∈ ℝd+1 : �(x) = 0}.

Signed Distance Function (SDF):

�(x) =

⎧⎨⎩−miny∈Γ ∣∣x − y ∣∣ if x is inside Γ,

miny∈Γ ∣∣x − y ∣∣ if x is outside Γ.

Property of SDF: ∣∇Φ∣ = 1.






Geometric Features

Outer unit normal vector of the surface Γ:

n⃗ =∇�∣∇�∣

.

Mean curvature of the surface Γ:

h = divn⃗ = ∇ ⋅ ∇�∣∇�∣

.

For signed distance function:

n⃗ = ∇�,

h = ∇ ⋅ ∇� = Δ�.






Level Set Representation of Operators

Surface gradient operator:

∇Γu = ∇u − (∇u ⋅ n⃗)⃗n = ∇u − ∇u ⋅ ∇�∣∇�∣2

∇�. (2)

Surface divergence operator: divΓv⃗ = ∇Γ ⋅ v⃗ .

Laplace-Beltrami operator:

ΔΓu = ∇Γ ⋅ ∇Γu

= Δu −∇ ⋅(∇u ⋅ ∇�∣∇�∣2

∇�). (3)






Interface Motion Equations

For ∀ x ∈ Γ, let vn(x , t) = ∂x∂t ⋅ n⃗ be the normal speed.

Evolution equation under a level set framework:

�t + vn∣∇�∣ = 0.

Examples:

▶ vn = −h: motion by mean curvature.

▶ vn = ΔΓh: motion by surface Laplacian of mean curvature.

▶ vn = ΔΓG (h): general surface diffusion.






Interface Motion Equations

Volume conservation:

d

dt∣Ω(t)∣ =

∫Γ(t)

vdA = −∫

Γ(t)xt ⋅ n⃗dA

= −∫

Γ(t)divΓ(g(h)∇Γh)dA = −

∫Γ(t)

g(h)∇Γh ⋅ ∇Γ1dA = 0.

Area shrinkage:

d

dt∣Γ(t)∣ = −

∫Γ(t)

vhdA = −∫

Γ(t)hxt ⋅ n⃗dA

=

∫Γ(t)

hdivΓ(g(h)∇Γh)hdA = −∫

Γ(t)g(h)∇Γh ⋅ ∇ΓhdA ⩽ 0.





A Splitting SchemeStability and ConvergenceNumerical Results

A Splitting Scheme

General surface diffusion: vn = ΔΓG (h).

Evolution equation under a level set framework:

�t + ∣∇�∣ΔΓG (h) = 0,

where h = ∇ ⋅ ∇�∣∇�∣ .

Numerical difficulties: fourth order and nonlinear.

▶ Explicit schemes: very small time step, dt ∼ dx4.

▶ Implicit schemes: solving a complex system.






A Splitting Scheme

A semi-implicit scheme can be obtained by adding bilaplacianstabilization terms:

�t + �Δ2� = �Δ2�− ∣∇�∣ΔΓG (h).

Take the left hand bilaplacian implicit and take right hand explicit:

�n+1 − �n

dt+ �Δ2�n+1 = �Δ2�n − ∣∇�n∣ΔΓG (hn),

(1 + dt�Δ2)(�n+1 − �n) = −dt∣∇�n∣ΔΓG (hn).






Numerical Discretization

Discretization: use ∣∇�∣� =√�2x + �2

y + �2 to avoid division byzero.

n⃗� =∇�∣∇�∣�

, h� = div∇�∣∇�∣�

=∇�∣∇�∣�

− ∇�T∇2�∇�∣∇�∣3�

.

Modified equation and corresponding numerical scheme:

�t = −∣∇�∣�ΔΓG (hk� ), (4)

Φk+1 − Φk

dt+ �Δ2Φk+1 = �Δ2Φk − ∣∇Φk ∣�ΔΓG (hk

�

). (5)






Stability and Convergence

Theorem 1. Let � be the exact solution of (4) and �k = �(kdt)be the exact solution at time kdt for a time step dt > 0 andk ∈ ℕ. Let Φk be the kth iterate of (5). Assume that there exits aconstant L such that ∣G ′(s)∣ ⩽ L, ∣G ”(s)∣ ⩽ L, and the discretesolution exists up to time T , then we have the followingstatements:

(i) Under the assumption that ∥�tt∥−1, ∥∇Δ�t∥2, ∥∇�∥∞ and∥�t∥−1 are bounded, the numerical scheme (5) is consistent withthe modified continuous equation (4) and first order in time.






Stability and Convergence

(ii) Let further ek = �k − Φk be the discretization error. If

∥∂��k∥∞ ⩽ K , ∥∇Φk∥∞ ⩽ K ,

for a constant K > 0 and all ∣�∣ ⩽ 3, kdt ⩽ T , then the error ek

converges to zero with first order in time.

Remark:

▶ Spatial discretization may impose additional restrictions ontime step dt. Numerical tests show dt ∼ dx2.

▶ Reinitialization is required if derivatives of � are large.






Numerical Results

Motion by surface Laplacian of mean curvature: G (x) = x .

50 100 150 200

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Figure: Merging of a circle and an ellipse.






Numerical Results

Figure: Splitting of a 3D dumbbell.Wenhua Gao The Laplace-Beltrami Operator in a Level Set Framework and Its Applications




Existing ModelsChan-Vese Segmentation Model with CornersNumerical Examples

Active Contour Model - Snake

The classical snake active contour models use an edge-detector,depending on the gradient of the image u0, to stop the evolvingcurve on the boundary of the desired object.

The Snake model minimizes the following energy:

J1(C ) = �

∫ 1

0∣C ′(s)∣2ds + �

∫ 1

0∣C ′′(s)∣ds

−�∫ 1

0∣∇u0(C (s))∣ds.






Chan-Vese Segmentation Model

Level set representation: C = {(x , y) : �(x , y) = 0}. � > 0 insidethe contour and � < 0 outside the contour.

The Chan-Vese model minimizes the following energy:

F (�, c1, c2) = �

∫Ω�(�)∣∇�∣dxdy + �

∫Ω

H(�)dxdy

+�1

∫Ω

(u0 − c1)2H(�)dxdy + �2

∫Ω

(u0 − c2)2(1− H(�))dxdy .

H is the heaviside function: H(x) =

{1, x ⩾ 0

0, x < 0.






Chan-Vese Segmentation Model

� is the one dimensional Dirac measure: �(x) = ddx H(x).

The gradient descent equation is

c1 =

∫Ω u0H(�)dxdy∫

Ω H(�)dxdy, c2 =

∫Ω u0(1− H(�))dxdy∫

Ω(1− H(�))dxdy,

�t = �(�)[�∇ ⋅ ∇�

∣∇�∣− � − �1(u0 − c1)2 + �2(u0 − c2)2

]. (6)






Low Curvature Image Simplifier

Fourth order Low Curvature Image Simplifier (LCIS) PDE

ut + div(g(Δu)∇Δu) = 0. (7)

g ⩾ 0: a weight function satisfying g(0) = 1 and g(∞) = 0.Mass conservation:

d

dt

∫udx =

∫utdx = −

∫div(g(Δu)∇Δu)dx = 0.

H1 energy decay:

d

dt

∫∣∇u∣2dx =

∫2∇u∇utdx = −

∫2Δuutdx

=

∫2Δudiv(g(Δu)∇Δu)dx = −

∫2g(Δu)(∇Δu)2dx ⩽ 0.






Low Curvature Image Simplifier

LCIS denoising model:

ut + div(g(Δu)∇Δu) = �(f − u). (8)






A Geometric Variant of LCIS

A geometric variant of LCIS PDE:

xt − divΓ(g(h)∇ΓΔΓx) = 0. (9)

Note that ΔΓx = hn⃗. Equation (9) is very similar to the surfacediffusion equation:

xt − divΓ(g(h)∇Γh)⃗n = 0. (10)

Level set representation of equation (10):

�t = −v ∣∇�∣ = −∣∇�∣divΓ(g(h)∇Γh). (11)






Geometric LCIS Evolution

The evolution of a curve under equation (11).

Figure: Corner formation in early stages.






Chan-Vese Segmentation Model with Corners

The regularization term, or the curve length term can avoid thenoisy or undesired pieces, it rounds off the corners at the sametime.

We can add equation (11) in to the Chan-Vese model. The newequation becomes:

�t = �(�)[�∇ ⋅ ∇�

∣∇�∣− � − �1(u0 − c1)2 + �2(u0 − c2)2

]−�∣∇�∣divΓ(g(h)∇Γh). (12)






A Simple shape

(a) (b)(a) segmentation without corner term; (b) segmentation with

corner;






A Building


corner;






A Walmart


corner;






Laplace-Beltrami Eigen Structure

Eigen structure problem:

ΔΓu = −�u on Γ. (13)

▶ Reuter, Wolter and Peinecke: Triangular mesh based method.

Disadvantage: Good triangulation required.

▶ Brandman: embed the shape in 3D domain and approximatethe LB spectrum with the spectrum of a 3D elliptic operator.

Disadvantage: computational cost and accuracy.






Triangular Mesh Representation

Example: a sphere (left) and the triangular mesh (right).






Narrow Band and Parametrization

Narrow Band with width � > 0: Γ� = {x : ∣�(x)∣ < �}.

New parametrization of the narrow band:

▶ For any p ∈ Γ, we choose an isothemal coordinate (x1, x2)around p on the surface, i.e., ⟨∂xi , ∂xj ⟩ = �ij , where ∂xidenotes the tangent vector in xi direction. The normaldirection is n⃗ = ∂x1 × ∂xj .

▶ For any q ∈ Γ�, let p = ProjΓq, then q = p + �(q)⃗n. q can beparameterized as (x̃1, x̃2, x̃3), where x̃1 = x1, x̃2 = x2,x̃3 = �(q).

▶ This new coordinate is unique iff � < 1maxp �(p) .






Laplace-Beltrami and Laplace Operator

Under this parametrization, for any p ∈ Γ, the Laplace-Beltramioperator is

ΔΓu =( ∂2

∂2x1+

∂2

∂2x2

)u. (14)

The Laplace operator is

Δu =1√∣g ∣

3∑i ,j=1

∂

∂x̃i

(√∣g ∣ g ij ∂

∂x̃ju),

where g = (gij) is the Euclidean metric of the narrow bandexpressed in the parametrization (x̃1, x̃2, x̃3), gij = ⟨∂x̃i , ∂x̃j ⟩,(g ij) is the inverse matrix of g and ∣g ∣ = det(gij).







Under the parametrization, we have

∂x̃1 = ∂x1 + x̃3n⃗x1 , ∂x̃2 = ∂x2 + x̃3n⃗x2 , ∂x̃3 = n⃗.

Thus we have

⟨∂x̃1 , ∂x̃1⟩ = 1 + 2x̃3⟨∂x1 , n⃗x1⟩+ x̃23∥n⃗x1∥2,

⟨∂x̃1 , ∂x̃2⟩ = x̃3(⟨∂x1 , n⃗x2⟩+ ⟨∂x2 , n⃗x1⟩) + x̃23 ⟨⃗nx1 , n⃗x2⟩,

⟨∂x̃2 , ∂x̃2⟩ = 1 + 2x̃3⟨∂x2 , n⃗x2⟩+ x̃23∥n⃗x2∥2,

⟨∂x̃1 , ∂x̃3⟩ = ⟨∂x̃2 , ∂x̃3⟩ = 0,

⟨∂x̃3 , ∂x̃3⟩ = 1.






Laplace Operator Under New Parametrization

The second fundamental form is⎧⎨⎩n⃗x1 =− L

E∂x1 −

M

G∂x2 ,

n⃗x2 =− M

E∂x1 −

N

G∂x2 ,

where E = ⟨∂x1 , ∂x1⟩, G = ⟨∂x2 , ∂x2⟩, L = ⟨∂x1x1 , n⃗⟩,M = ⟨∂x1x2 , n⃗⟩, N = ⟨∂x2x2 , n⃗⟩.

Under the isothermal parameterization, we have E = 1,G = 1.







The Euclidean metric of the narrow band under the newparametrization is

g̃ = (g̃ij) =Id + x̃3

(A 00 0

)where the matrix A is

A =

(−2L + x̃3

(L2 + M2

)−2M + x̃3

(LM + MN

)−2M + x̃3

(LM + MN

)−2N + x̃3

(M2 + N2

) ).







Consequently we have√∣g̃ ∣ = det(g̃ij) = 1 +

1

2x̃3(A11 + A22) + O(x̃2

3 ),

g̃−1 =

(Id − x̃3A + O(x̃2

3 ) 00 1

).

We obtain

Δu =( ∂2

∂2x̃1+

∂2

∂2x̃2

)u +

∂2

∂2x̃3u − 2�

∂

∂x̃3u + O(x̃3). (15)







We define

Δ12u =(∂2

∂2x̃1+ ∂2

∂2x̃2

)u,

Δ̃u = Δ12u + ∂2

∂2x̃3u.

Consider the following Neumann boundary condition problem:⎧⎨⎩Δ̃u = Δ12u +

∂2

∂2x̃3u = −�u in Γ�,

∂u

∂x̃3= 0 for x̃3 = ±�.

(16)






Eigen Structure Decomposition

Theorem: The eigen structure for (16) has the followingdecomposition

um,n = uΓm(x̃1, x̃2)uI

n(x̃3),

�m,n = �Γm + �In,

(17)

where �Γm and uΓ

m(x̃1, x̃2) = uΓm(x1, x2) are the eigenvalues and

eigenfunctions for (13), while �In and uIn are the eigenvalues and

eigenfunctions for the one dimensional problem

d2

d2tu = − �u, t ∈ (−�, �)

d

dtu = 0, t = ±�.

(18)






Eigen Structure Decomposition

(18) has following eigenvalues and eigenfunctions

�I0 = 0, uI0 = const,

�In =(2n − 1)2�2

4�2, uI

n = sin((n − 1/2)�/�), n = 1, 2, ⋅ ⋅ ⋅

Since ΔΓ is elliptic on Γ, we can order the eigenvalues

0 = �Γ1 ⩽ �Γ

2 ⩽ ⋅ ⋅ ⋅ ⩽ �Γm ⩽ ⋅ ⋅ ⋅ .

For � small enough, the eigenvalues of Δ̃ can be ordered as

�Γ1 ⩽ �Γ

2 ⩽ ⋅ ⋅ ⋅ ⩽ �Γm ⩽ �I1 + �Γ

1 ⩽ �I1 + �Γ2 ⩽ ⋅ ⋅ ⋅ .

The corresponding eigenfunctions of Δ̃

uk,0 = uΓk ⋅ const, k = 1, ⋅ ⋅ ⋅ ,m.






Approximation

For � small enough,

Δuk,0 = Δ̃uk,0 + O(x̃3) = �k,0uk,0 + O(�). (19)

If � ∼ dx ,then O(�) = O(dx). Δ is a very good approximation toΔ̃ for the eigen structure problem.

Solve the following problem to approximate the Laplace-Beltramieigen structure: ⎧⎨⎩Δu = −�u in Γ�,

∂u

∂n⃗= 0 on ∂Γ�.

(20)






Weak Formulation

Definition: u is weak solution to

Δu = −�u in Γ�,∂u

∂n⃗= 0 on ∂Γ�

if and only if

(∇u,∇v) = −�(u, v) for any v ∈ H1(Γ�).

Implementation with finite element is easy.

Choose cubes instead of tetrahedra as basic elements and tri-linearbasis functions.






Weak Formulation

Figure: The cubic elements forming the narrow band of a sphere.






Numerical Results

Numerical experiments on anatomical shapes: caudate, putamenand hippocampus.

Comparison: narrow band method vs. mesh based method.

For comparison of eigenfunctions, we define the correlationcoefficient of two functions:

corr(u, v) =

∫Γ uvdΓ√∫

Γ u2dΓ√∫

Γ v 2dΓ. (21)

Attention: eigenvalues with multiplicity greater than 1 may occur!






Comparison: Anatomical Shapes

Figure: The shapes used in our experiments. Top: Caudate. Middle:Putamen. Bottom: Hippocampus.






Narrow Band vs. Mesh: Eigenvalues

Figure: The eigenvalues computed from the mesh and narrow bandrepresentation for the shapes.






Narrow Band vs. Mesh: Eigenfunctions

Figure: The 2nd,3rd,4th eigenfunctions of a caudate surface. Top row:results from the mesh representation. Bottom row: results from thenarrow band representation.







Figure: The 2nd,3rd,4th eigenfunctions of a putamen surface. Top row:results from the mesh representation. Bottom row: results from thenarrow band representation.







Figure: The 2nd,3rd,4th eigenfunctions of a hippocampal surface. Toprow: results from the mesh representation. Bottom row: results from thenarrow band representation.






Narrow Band vs. Mesh: Correlation Coefficients

Figure: The correlation coefficient between the eigenfunctions computedwith mesh based and narrow band approaches.






Narrow Band vs. Mesh: Adjusted Correlation Coefficients

Small correlation coefficients occur!

Adjustment: project narrow band eigenfunctions onto the span ofmesh based eigenfunctions.

Figure: The adjusted correlation coefficient between the eigenfunctioncomputed with the narrow band approach and the mesh based approach.






Thank you!


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