Deformable Model with Adaptive Mesh and Automated Topology Changes Jacques-Olivier Lachaud Benjamin Taton Laboratoire Bordelais de Recherche en Informatique (LaBRI) Deformable Model with Adaptive Mesh and Automated Topology Changes – p.1/21
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 Deformable Model with Adaptive Mesh and Automated Topology Changes – p.2/21
� � Motivations Segmentation/Reconstruction of large 3D images. steady technical improvements of acquisition devices, increase of image resolution and hence of image size. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.3/21
� � 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. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.3/21
� � Motivations Segmentation/Reconstruction of large 3D images. Deformable templates, superquadrics, not generic enough Fourier snakes. . . 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. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.3/21
� � � Motivations Segmentation/Reconstruction of large 3D images. Deformable templates, superquadrics, not generic enough Fourier snakes. . . Fully generic models (T-Snakes, Simplex computationally meshes, Level-sets. . . ) expensive Objective To build a deformable model that can recover objects with any topology, with costs more independent from the size of input data. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.3/21
� � � � � Model Description Explicit model Closed triangulated surface, Dynamics of a mass-spring system that undergoes image forces, regularizing internal forces, any other additional force. . . Deformable Model with Adaptive Mesh and Automated Topology Changes – p.4/21
� � � � � 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 of the model mesh Deformable Model with Adaptive Mesh and Automated Topology Changes – p.4/21
� � � � � Model Description Explicit model Closed triangulated surface, Dynamics of a mass-spring system that undergoes image forces, regularizing internal forces, any other additional force. . . Transformed into adaptive sampling Regular sampling by changing metrics of the model mesh Deformable Model with Adaptive Mesh and Automated Topology Changes – p.4/21
� � � � � Model Description Explicit model Closed triangulated surface, Dynamics of a mass-spring system that undergoes image forces, regularizing internal forces, any other additional force. . . Transformed into adaptive sampling Regular sampling by changing metrics of the model mesh Automated topology changes Deformable Model with Adaptive Mesh and Automated Topology Changes – p.4/21
✞ ✡ ✁ ✂ ☞ � ✠ ✡ ☛ ✂ ☞ ✁ ✁ � � � 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. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.5/21
☛ ✂ ✁ ✂ ✁ ☞ ✠ ✡ Regular Sampling Regular sampling using distance constraints ✝✟✞ ✄✆☎ Restoring constraints Edge too short: contraction Edge too long: split (+ special case. . . ) Deformable Model with Adaptive Mesh and Automated Topology Changes – p.5/21
✁ ✂ ✠ ✡ ☛ ✂ ☞ ✁ Resolution Adaptation Euclidean distance replaced by a Riemannian distance ✝✟✞ ✄✆✌ Deformable Model with Adaptive Mesh and Automated Topology Changes – p.6/21
☞ ✁ ✁ ✂ ✁ ✌ ✠ ✡ ☛ ✂ ✄ Resolution Adaptation Euclidean distance replaced by a Riemannian distance ✝✟✞ ✄✆✌ If underestimates distances edge lengths fall under the threshold edges contract and vertex density decreases. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.6/21
✌ ✄ ☞ ✁ ✁ ✌ ✄ ✁ ☞ ✂ ☛ ✡ ✠ ✂ ✁ 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 overestimates distances edge lengths exceed the threshold edges split and vertex density increases. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.6/21
✁ ✄ ✁ ✂ � � ✠ ✡ ☛ ✂ ☞ ✌ Resolution Adaptation Euclidean distance replaced by a Riemannian distance ✝✟✞ ✄✆✌ The new distance should overestimate distances in interesting parts of the image to increase accuracy, underestimate distances elsewhere to decrease accuracy. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.6/21
☛ ✍ ✍ ✍ ✑ ☎ ✝ ✍ ✔ ✓ ✒ Riemannian Metrics Euclidean length of an elementary displacement ✄✏✎ ✄✏✎ ✄✏✎ ✄✏✎ Deformable Model with Adaptive Mesh and Automated Topology Changes – p.7/21
✠✘ ✘ ✘ ✠✘ ✕ ✗ � ✕ � ✗ ✍ ✘ ✔ ✖ ✙ ☛ � � ✕ ☛ ✍ ✑ ✍ ✓ ✝ ✍ ✍ ☛ ✑ ✌ ✝ ✍ ✙ ☛ ✒ ✖ ☛ ✓ ✕ ✘ ✗ ✠✘ ✘ ✘ ✖ ✙ ✌ 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 ✝✟✖ of the displacement. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.7/21
✝ ✛ ✜ ✚ ✑ ✌ ✍ ✚ ☛ ✒ ✗ ✔ ✍ ✚ ✓ ☛ ✓ ✕ ✝ ✚ ✝ ✣ ✚ ✔ ☛ ✍ ✍ ✥ ✑ ✌ ✝ ✍ ✣ ☛ ✒ ✄ ✓ ✓ ✕ ☛ ✗ ✠✘ ✘ ✘ ✖ ✙ ☛ ☛ Riemannian Metrics Riemannian length of an elementary displacement ✄✏✎ ✝✟✖ ✄✏✎ ✄✏✎ ✄✏✎ Length of a path ✍✢✜ ✝✤✣ ✝✤✣ Sum of the lengths of the elementary dis- Length of a path placements it is composed of. Deformable Model with Adaptive Mesh and Automated Topology Changes – p.7/21
✣ ✓ ☛ ✒ ✚ ✜ ✍ ✔ ☛ ✣ ☛ ✞ ✝ ✚ ✝ ✕ ✓ ✄ ☛ ✠ ✒ ✝ ✌ ✑ ✪ ✩ ✚ ☛ ✞ ✡ ✠ ✞ ✝ ✝ ✡ ✬ ☛ ✚ ☛✫ ✒ ✗ ☛ ✕ ✓ ✒ ✍ ☛ ✘ ✡ ✍ ✝ ✌ ✑ ✮ ✍ ✠✘ ✘ ✚ ✑ ✛ ✗ ✒ ☛ ✚ ✝ ✌ ✚ ✖ ✝ ✭ ✍ ✔ ✓ ☛ ✙ ✚ Riemannian Metrics Riemannian length of an elementary displacement ✄✏✎ ✝✟✖ ✄✏✎ ✄✏✎ ✄✏✎ Length of a path ✍✢✜ ✝✤✣ ✝✤✣ Distance between two points and ✄✆✌ ✦★✧ Deformable Model with Adaptive Mesh and Automated Topology Changes – p.7/21
✍ ✴ ✹ ✠ ✗ ✡ ✝ ✝ ✸ ✵ ✲ ☛ ✍ ✶ ✵ ✴ ✲ ✠ ✹ ✺ ☛ ✗ Geometrical Interpretation of Metrics Euclidean unit ball Local Riemannian unit ball 1 ✯✱✰✳✷ ✯✱✰✳✲ 1 , local eigen decomposition of the metric. ✡✳✺ Deformable Model with Adaptive Mesh and Automated Topology Changes – p.8/21
✶ ✲ ✕ ✸ ✵ ✴ ✲ ✴ ✵ Geometrical Interpretation of Metrics Euclidean unit ball Local Riemannian unit ball 1 ✯✱✰✳✷ ✯✱✰✳✲ 1 Changing the Euclidean metric with a Riemannian metric Locally expanding/contracting the space Deformable Model with Adaptive Mesh and Automated Topology Changes – p.8/21
✗ ✭ ✡ ✍ ✹ ✗ ✕ ✸ ✵ ✴ ✲ ✶ ✵ ✴ ✲ 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 , Deformable Model with Adaptive Mesh and Automated Topology Changes – p.8/21
✍ ✵ ✗ ✡ ✗ ✍ ✡ ✕ ✺ ✸ ✴ ✭ ✲ ✭ ✶ ✵ ✴ ✲ ✹ ✺ ✹ 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 ,. . . Deformable Model with Adaptive Mesh and Automated Topology Changes – p.8/21
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