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Deformable Model with Adaptive Mesh and Automated Topology Changes

Jacques-Olivier Lachaud Benjamin Taton Laboratoire Bordelais de Recherche en Informatique (LaBRI)

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Outline

• 1. Motivations
• 2. Description of the deformable model

2.1 Resolution adaptation by changing metrics 2.2 Topology adaptation 2.3 Dynamics

• 3. Defining metrics with respect to images

3.1 Required properties 3.2 Building metrics from images

• 4. Results
• 5. Conclusion and perspectives

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Motivations

Segmentation/Reconstruction of large 3D images.

• steady technical improvements of acquisition devices,
• increase of image resolution and hence of image size.

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Motivations

Segmentation/Reconstruction of large 3D images. Deformable templates, superquadrics, Fourier snakes. . .

• reduced set of shape prameters

robust and efficient,

• lack of genericity: new problem

new model.

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Motivations

Segmentation/Reconstruction of large 3D images. Deformable templates, superquadrics, Fourier snakes. . . not generic enough Fully generic models (T-Snakes, Simplex meshes, Level-sets. . . )

• very wide range of shapes,
• number of shape parameters directly determined by image

resolution heavy computational costs.

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Motivations

Segmentation/Reconstruction of large 3D images. Deformable templates, superquadrics, Fourier snakes. . . not generic enough Fully generic models (T-Snakes, Simplex meshes, Level-sets. . . ) computationally expensive Objective

• To build a deformable model
• that can recover objects with any topology,
• with costs more independent from the size of input data.

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Model Description

Explicit model

• Closed triangulated surface,
• Dynamics of a mass-spring system that undergoes
• image forces,
• regularizing internal forces,
• any other additional force. . .

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Model Description

Explicit model

• Closed triangulated surface,
• Dynamics of a mass-spring system that undergoes
• image forces,
• regularizing internal forces,
• any other additional force. . .

Regular sampling

• f the model mesh

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Model Description

Explicit model

• Closed triangulated surface,
• Dynamics of a mass-spring system that undergoes
• image forces,
• regularizing internal forces,
• any other additional force. . .

Regular sampling

• f the model mesh

Transformed into adaptive sampling by changing metrics

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Model Description

Explicit model

• Closed triangulated surface,
• Dynamics of a mass-spring system that undergoes
• image forces,
• regularizing internal forces,
• any other additional force. . .

Regular sampling

• f the model mesh

Transformed into adaptive sampling by changing metrics Automated topology changes

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Regular Sampling

Regular sampling using distance constraints

Where

• ✞

,

are neighbour vertices,

• ✄✆☎

denotes the Euclidean distance,

• ✁

determines the global resolution of the model,

• ☞

is the ratio between the lengths of the longest and smallest edge on the mesh.

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Regular Sampling

Regular sampling using distance constraints

Restoring constraints Edge too short: contraction (+ special case. . . ) Edge too long: split

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Resolution Adaptation

Euclidean distance replaced by a Riemannian distance

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Resolution Adaptation

Euclidean distance replaced by a Riemannian distance

If

underestimates distances edge lengths fall under the

threshold edges contract and vertex density decreases.

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Resolution Adaptation

Euclidean distance replaced by a Riemannian distance

If

underestimates distances edge lengths fall under the

threshold edges contract and vertex density decreases. If

• verestimates distances

edge lengths exceed the

threshold edges split and vertex density increases.

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Resolution Adaptation

Euclidean distance replaced by a Riemannian distance

The new distance

should

• verestimate distances in interesting parts of the image to

increase accuracy,

• underestimate distances elsewhere to decrease accuracy.

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Riemannian Metrics

Euclidean length of an elementary displacement

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Riemannian Metrics

Riemannian length of an elementary displacement

Where

is a Riemannian metric, i.e.

• ✕

is a dot product,

• ✕

is continous. Which means that

depends on both

• the displacement

,

• the origin

• f the displacement.

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Riemannian Metrics

Riemannian length of an elementary displacement

Length of a path

Length of a path

Sum of the lengths of the elementary dis- placements it is composed of.

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Riemannian Metrics

Riemannian length of an elementary displacement

Length of a path

Distance between two points

and

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Geometrical Interpretation of Metrics

Euclidean unit ball 1 1 Local Riemannian unit ball

,

local eigen decomposition of the metric.

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Geometrical Interpretation of Metrics

Euclidean unit ball 1 1 Local Riemannian unit ball

Changing the Euclidean metric with a Riemannian metric

Locally expanding/contracting the space

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Geometrical Interpretation of Metrics

Euclidean unit ball Local Riemannian unit ball

Changing the Euclidean metric with a Riemannian metric

Locally expanding/contracting the space along

with the ratio

,

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Geometrical Interpretation of Metrics

Euclidean unit ball Local Riemannian unit ball

Changing the Euclidean metric with a Riemannian metric

Locally expanding/contracting the space along

with the ratio

, along

with the ratio

,. . .

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Topology Changes

Two stages :

• detect self collisions of the model,
• perform appropriate local reconfigurations of the mesh.

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Topology Changes

Two stages :

• detect self collisions of the model,
• perform appropriate local reconfigurations of the mesh.

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Topology Changes

Two stages :

• detect self collisions of the model,
• perform appropriate local reconfigurations of the mesh.

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Topology Changes

Two stages :

• detect self collisions of the model,
• perform appropriate local reconfigurations of the mesh.

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Topology Changes

Detection of self collisions

Vertex

crosses over the

face Vertex

enters one of the

,

• r

spheres.

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Topology Changes

Detection of self collisions

• Collision if

and

are not neighbours and

.

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Topology Changes

Detection of self collisions

• Collision if

and

are not neighbours and

. Local reconfigurations Collision between two parts of the mesh Special case of edge contraction

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Topology Changes

Detection of self collisions

• Collision if

and

are not neighbours and

. Local reconfigurations Collision between two parts of the mesh Special case of edge contraction If the metric is changed

• ❄

replaced with a new constant

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Dynamics

Motion equations with a Euclidean Metric

• ✪

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Dynamics

Motion equations with a Riemannian metric

• ✪

Addition of a corrective term that takes account of the metric

• ❚

(Christoffel’s symbols),

• ❲

are the coefficients of the

,

• ❲

are the coefficients of

,

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Dynamics

Motion equations with a Riemannian metric

• ✪

Addition of a corrective term that takes account of the metric

• ❚

(Christoffel’s symbols),

• ❲

are the coefficients of the

,

• ❲

are the coefficients of

, (corrective term neglected: second order in

+ no influence on the rest position)

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Summary

Euclidean Metric Riemannian Metric Distance estimation

Regular sampling

uniform resolution

adaptive resolution Collision detection

Local recon- figurations unchanged Motion equations

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Required Properties

Four kinds of situations no contour plane contour edge, tubular structure corner Choice of the metric

• Eigenvectors should correspond the normal and the principal

directions of the contour,

• Eigenvalues should correspond to the strength and the

principal curvatures of the contour.

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Required Properties

Structure in the image Expected resolution Eigen structure of the metric No contour

• low in all directions

Flat contour

• low in the direction of the

contour

• high in the orthogonal direc-

tion

Tubular structure

• low in the direction of the

structure

• high in both orthogonal direc-

tions

Corner

• high along all directions

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Structure Tensor (1/2)

Definition The matrix

that represents the mapping:

Which results in:

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Structure Tensor (1/2)

Definition The matrix

that represents the mapping:

Interpretation

• smoothes the input image,

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Structure Tensor (1/2)

Definition The matrix

that represents the mapping:

Interpretation

• smoothes the input image,
• characterizes direction and orientation of image gradient,

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Structure Tensor (1/2)

Definition The matrix

that represents the mapping:

Interpretation

• smoothes the input image,
• characterizes direction and orientation of image gradient,
• removes the orientation information,

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Structure Tensor (1/2)

Definition The matrix

that represents the mapping:

Interpretation

• smoothes the input image,
• characterizes direction and orientation of image gradient,
• removes the orientation information,
• integrates the direction information over a neighbourhood.

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Structure Tensor (2/2)

Properties of the eigenstructure of the structure tensor In the neighbourhood of a contour

• ✍

• rthogonal to image contours,

contour strength,

• ✍

,

principal directions of the contour,

and

qualitatively equivalent to principal curvatures.

no contour

flat contour

sharp edge

corner

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Results (1/3)

Computer generated image Object represented in the image Slices extracted from the image

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Results (1/3)

Eigen decomposition of the structure tensor Object represented in the image Isosurfaces of the second and third eigen values of the structure tensor of the image.

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Results (1/3)

Segmentation/Reconstruction results 2163 vertices,

,

3497 vertices,

,

,

8904 vertices

,

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Results (1/3)

One corner of the cube Hole in the cube

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Results (1/3)

Repartition of edge lengths

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6 7

fine model,

adaptive model,

coarse model

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Results (2/3)

Computer generated image Object represented in the image Slices extracted from the image

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Results (2/3)

Eigen decomposition of the structure tensor Object represented in the image Isosurfaces of the second and third eigen values of the structure tensor of the image.

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Results (2/3)

Segmentation/Reconstruction results 2107 vertices

,

4832 vertices

,

,

8542 vertices

,

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Results (2/3)

Repartition of edge lengths

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6 7

fine model,

adaptive model,

coarse model

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Results (3/3)

Biomedical image (Head CT-scan)

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Results (3/3)

Eigen decomposition of the structure tensor An isosurface of the 2nd eigenvalue. An isosurface of the 3rd eigenvalue. Object in the image.

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Results (3/3)

Segmentation/Reconstruction results 10.970 vertices

23.142 vertices

,

,

46.590 vertices,

,

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Results (3/3)

Left ear Orbit of the left eye viewed from behind left

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Results (3/3)

Repartition of the edge length

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

fine model,

adaptive model,

coarse model

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Conclusion and Perspectives

Conclusion

• Deformable model that achieves both
• adaptive topology,
• adaptive resolution

Perspectives

• Initialization with adaptive resolution,
• Different ways of building metrics. . .

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