Managing the Level of Detail in 3D Shape Reconstruction and Representation Leila De Floriani, Paola Magillo Department of Computer and Information Sciences University of Genova, Italy Enrico Puppo Institute for Applied Mathematics National Research Council, Genova, Italy

Contribution Reconstruction and representation of the volume of 3D objects at multiple levels of detail Reconstruction based on a set to sculpturing updates of a tetrahedral mesh Rapresentation based on a multiresolution structure (called Multistage) Applications: - CAD/CAM - Virtual reality - Computer vision - Robotics

3D Object Reconstruction Known information: a set of points on the object boundary Volumetric representation: a mesh of tetrahedra filling the interior of the object

3D Object Reconstruction Approach: Build a Delaunay tetrahedralization of the points Remove tetrahedra to bring more (all) data points to the boundar The Delaunay tetrahedralization may not contain a sub-mesh with all points on its boundary Several heuristics for removing tetrahedra exist

3D Object Representation Given a sequence of sculpturing updates of a tetrahedral mesh some updates depend on other updates some pairs of updates are mutually independent updates C and B depend on A updates C anb B are mutually independent Dependency is a partial order relation.

3D Object Representation A Multi-Shape (MS) encodes a partially ordered set of updates through a DAG

3D Object Representation Every cut of the DAG provides a valid object representation Representations at variable resolution are found in cuts An "importance" is associated with each update The user can define resolutions variable in different ports of the object refine only top side refine only bottom-left corner refine uniformly at importance=1

Formalization A tetrahedral mesh T is a valid object representation iff T is connected the boundary of T is a 2-manifold T contains all data points inside or on its boundary Ideal situation (not always possible): all data points on the boundary

Formalization A sculpturing update of a mesh removes a connected set of tetrahedra having at least one face exposed on the mesh boundary Genus-preserving update: empties a cavity Genus-increasing update: creates some hole Genus-decreasing update: breaks some handle

Formalization An update U’ depends on an update U iff U blocks U’ (U exposes some face of the tetrahedron removed by U’) A blocks B B depends on A

The Multi-Shape (MS) each mesh update U: a node, labeled with the set of removed tetrahedra an additional root node: creation of the initial mesh T an additional drain node: deletion of the final sculptured mesh dependency of U’ from U: an arc (U,U’), labeled with the set of faces belonging to tetrahedra removed by U’ and exposed by U An initial tetrahedral mesh T + A partially ordered set of mesh updates, represented as a DAG

Variable Resolution Meshes from an MS Each node in an MS stores some resolution parameters describing its importance. For instance: type of update subtracted volume change in the area of the boundary surface of the object weighted importance of its descendant (= of the updates it blocks)

Variable Resolution Meshes from an MS Resolution threshold: - Depends on resolution parameters and location in space - Boolean predicate defined on MS nodes - True iff the update is relevant Goal: - Extract from the MS a mesh where all relevant updates have been performed - Equivalent to determine a subset of MS updates closed w.r.t. the partial order (a set of nodes bounded by a cut) containing all relevant updates. A and C are relevant Resulting mesh

Algorithm for Variable Resolution Mesh Extraction Top-down traversal of the DAG. Can build a volumetric representation of the object, a surface- based one, or both.

Algorithm for Variable Resolution Mesh Extraction Initialization: nodeset := root cut := out-arcs of root tetrahedra := initial mesh T0 triangles := triangles labeling out-arcs of root enqueue the root

Algorithm for Variable Resolution Mesh Extraction Generic step: - U := next node in the queue - add U to nodeset - delete in-arcs of U from cut, and add out-arcs of U - delete tetrahedra labelling U from tetrahedra - delete triangles labelling in-arcs of U from triangles, add triangles labelling out-arcs of U - enqueue* every child U’ of U which satisfies the threshold - (*) if some parent of U’ is not in nodeset, then recursively enqueue them before enqueueing U’

Algorithm for Variable Resolution Mesh Extraction The algorithm is interruptible The time complexity is linear in the size of the output mesh + that of number of tetrahedra removed to obtain it

Generation of an MS through Sculpturing Produce an initial tetrahedral mesh and a sequence of updates. Arrange them into a DAG according to their mutual dependencies. Input data: a set of points on the object surface Initial mesh: Delaunay tetrahedralization of the points Sequence of mesh updates: generated through a sculpturing algorithm based on an  -tool

Generation of an MS through Sculpturing Alpha tool: Virtual carving tool Sphere of radius =  Can erase a tetrahedron t iff t has a face on the mesh boundary whose circumcircle has a radius >= alpha

Generation of an MS through Sculpturing A sequence of updates is generated by starting from a big  and decreasing  as soon as no more tetrahedra can be erased with the current tool. …. Erased with  =10Erased with  =9Erased with  =8

Sculpturing Algorithm if removing t leaves the mesh valid, then remove t (genus-preserving update) if removing t disconnects the mesh or isolates a data point, then do not remove t (no update) Maintain all tetrahedra having some face on the current mesh boundary in a priority queue. Iteratively pick the next tetrahedron t (largest radius of circumcircle):

Sculpturing Algorithm if removing t causes a non-manifold condition, then try to remove some tetrahedron near t to restore the manifold condition: - if success, then remove t along with the other tetrahedra (genus- increasing or -decreasing update)

Sculpturing Algorithm - if failure since the  -tool is too big to continue carving, then reinsert t into the queue with a smaller  - if failure for disconnecting the mesh or isolating some point, then do not remove t (no mesh update)

Sculpturing Algorithm Adjacency relations between tetrahedra removed in successive updates determine the dependency relations mesh updates and their dependencies are recorded in the nodes and arcs of the MS, respectively

Conclusions The Multi-Shape: a model for the efficient encoding and retrieval of selectively sculptured tetrahedral meshes: - both volumetric and a surface-based representations provided - nesting property - variable-resolution mesh extraction according to user criteria - more powerful than existing hierarchical representations since based on a partial order An iterative sculpturing algorithm to build an MS: - for shapes of arbitrary genus - can sculpture with tools of different sizes in different parts of the object