Sampling and Reconstruction
with Adaptive Meshes
Outline
Problem
Background
-- Numerical Grid Generation
Dynamic Meshes
Adaptive Meshes
Algorithm
Results
Advantages & Disadvantages
Demetri Terzopoulos and M. Vasilescu
Presented by Yuan Chen
Problem
Background
Dynamic Mesh
Dynamic Mesh
Essential component of adaptive mesh
Constructed from physically-based nodes
and springs
(Numerical Grid Generation)
A tool for numerical solution of partial differential
equations on arbitrarily shaped regions.
The numerical solution is built on those grids.
Make best use of those finite grids to represent the
physical solution with sufficient accuracy.
The grid is influenced by both the geometric
configuration and by the physical solution being
done thereon.
Adaptively sample the data to lower
resolution
--- for compression or simplification
Conventional approaches: uniformly
sampling
Finite elements: spring and its attached
nodes
(Node)
Node: i, i = 1, ….N
-- mass: mi
-- position : Xi (t ) = [ xi (t ), yi (t ), zi(t )]
-- velocity: vi = dXi / dt
-- acceleration: ai = d 2 Xi / dt 2
n
-- force f i (t )
n
Be subject to a net nodal force: f i (t )
1
Dynamic Mesh
Spring: ij
-- connect node j to node i
-- natural length lij
-- stiffness cij
Force exerted on node i: i
sij =
Dynamic Mesh
(Spring)
For each node:
mi
j
cijeij
rij ,
rij
(Motion equations)
d 2 Xi
dt
2
+γi
dX i
+ g i = f i ; i = 1,..., N
dt
mi : mass X i : position i : damping coefficien t
g i : the total force exerted from all connected springs
f i : an external force applied at node i
rij = X j − X i , eij = rij − lij
Dynamic Mesh
(Numerical time-integration)
Dynamic Mesh
Quadrilateral elements are assembled into
bounded surfaces (actually rectanglar)
At each time step t:
1. f i
nt
t
t
= f i − i vi − g i
Bi fint
d 2Xi
m
+γ
t
t
2 .a i =
i
dt 2
mi
t
t
t+ t
3.v i
= v i + ta i
t+ t
t
t+ t
4. X i
= X i + tv i
dXi
i dt + gi = f i ; i = 1,..., N
Boundary condition: The boundary nodes can
only move along the boundary in (x,y) visual
domain
Adaptive Meshes
Adaptive Meshes (Adaptation functions)
Dynamic meshes with an adaptation function
Adaptation function,
-- stiffness of springs are changed according
to the adaptation function during the process.
G is a normalized spatial filter; H is some function of
partial derivative of the input data field.
(Utilized in this paper)
Nodal Observation,
-- stiffness of spring increases in regions
with rapid variation.
2
Adaptive Meshes (Feedback procedure)
Algorithm
Feedback procedure
t
c ij = ( 1 −
Algorithm
t
t
ij )c min + ij c max ,
1
t
t
t
(O i + O j )
ij =
2
More interesting area
Higher
larger
larger
mi
d 2Xi
dX
+ γ i i + gi = f i ; i = 1,..., N
dt
dt 2
Input: A 2D scalar-valued intensity or range
image d(k,l)
At each time step t :
1. Evaluate the adaptation function and nodal
observation for each node i
2. Adjust the stiffness for each spring ij
3. Compute the total force from springs for each
node i
4. Evaluate the current nodal forces, acceleration,
the new velocity, and the new position.
(Data forces)
External data force at time t:
f i t = [ 0,0, α ( d (Π X it ) − zit )]
The only trigger to shift nodes in z
direction.
For surface reconstruction, it deflects the
mesh to fit the input data.
Results
(Image Reconstruction)
Results
(Surface Reconstruction)
Image from D. Terzopoulos & M. Vasilescu
3
Advantages
Applicable to arbitrary dimensional data
sets
Nodes are optimally distributed to preserve
the interesting properties of the input data.
Interactive rates (no actual timing data)
Image from D. Terzopoulos & M. Vasilescu
Disadvantages
Not suitable for arbitrary mesh in 3D
-- Assume a planar surface as initial guess
No theoretical justification
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