Transcript Two mathematic models to describe 3D shape(s)

A Probabilistic Model for
Component-Based Shape Synthesis
Evangelos Kalogerakis, Siddhartha Chaudhuri,
Daphne Koller, Vladlen Koltun
Stanford University
Bayesian networks
• Directed acyclic graph (DAG)
– Nodes – random variables
– Edges – direct influence (“causation”)
• Xi ? Xancestors | Xparents
• e.g., C ? {R,B,E} | A
• Simplifies chain rule by using
conditional independencies
Earthquake
Radio
Burglary
Alarm
Call
Pearl, 1988
SP2-2
Goal
• A tool that automatically synthesizes a variety
of new, distinct shapes from a given domain.
Various geometric, stylistic and functional
relationships influence the selection and
placement of individual components to ensure
that the final shape forms a coherent whole.
Sailing ships vary in:
- Size
- Type of hull, keel and mast
- The number and configuration of masts.
Probabilistic Reasoning for AssemblyBased 3D Modeling
Siddhartha Chaudhuri, Evangelos Kalogerakis,
Leonidas Guibas, and Vladlen Koltun
ACM Transactions on Graphics 30(4) (Proc. SIGGRAPH), 2011
The probabilistic model is flat!!!!
It describes the relationship among components, but it does NOT
tells us how these components form the whole structure.
Offline Learning
Online shape synthesis
The model structure
R - shape style
S = {Sl} - component style per category l
N = {Nl} - number of components from category l.
C = {Cl} - continuous geometric feature vector for components from category l. (curvature
histograms, shape diameter histograms, scale parameters, spin images, PCA-based descriptors, and lightfield
descriptors)
D = {Dl} - discrete geometric feature vector for components from category l. (encode adjacency
information.)
For 4-legged table:
Ntop=1 or 2;
Nleg=4;
Stop=rectangular tabletops
Sleg=narrow column-like legs
For 1-legged table
Ntop=1
Nleg=1
Stop=roughly circular tabletops
Sleg=split legs
Learning
• The input:
– A set of K compatibly segmented shapes.
– For each component, we compute its geometric
attributes.
– The training data is thus a set of feature vectors:
– O = {O1,O2, . . . ,Ok}, where Ok = {Nk,Dk,Ck}.
• The goal
– learn the structure of the model (domain sizes of
latent variables and lateral edges between observed
variables) and the parameters of all CPDs in the model.
The desired structure G is the one that has highest
probability given input data O [Koller and Friedman 2009].
By Bayes’ rule, this probability can be expressed as
Max P(G|O) -- > Max P (O | G)
Assume prior distributions over the parameters Θ of the model.
parameter priors
summing over all possible assignments to the latent
variables R and S:
the number of integrals is exponentially large !!!
To make the learning procedure computationally tractable, they use
an effective approximation of the marginal likelihood known as the
Cheeseman-Stutz score [Cheeseman and Stutz 1996]:
the parameters estimated for a given G
a fictitious dataset that comprises the training data O and
approximate statistics for the values of the latent variables.
The score defines a metric to measure how good a model is.
The goal is to search a G maximize the score!
What does the G mean?
The number of table styles (R)
Whether a category of components belongs to a specific
style? What is the number? (S)
Greedy Structure search
• Initially, set the domain size of 1 for R (a single shape
style).
• for each category l,
– Set the component style as 1, compute the score, then 2,
3, …, stop when the score decreases. The local maximal
value is the style number of l. Move the next category.
• After the search iterates over all variables in S, increase
the domain size of R and repeat the procedure.
• terminates when the score reaches a local maximum
that does not improve over 10 subsequent iterations;
Domain size of R =1
• All tables belong to the same style.
• For leg:
– Compute the score for case 1: all legs are of the same style;
– Compute the score for case 2: narrow column-like legs and
split legs.
– Compute the score for case 3: three styles of legs. Score
decreases so stop.
• For table-top:
–…
CPT of R
1
CPT of Stop (R=1)
2
5/12
1
7/12
CPT of Sleg (R=1)
1
1.0
2
1.0
1
0.0
0.0
CPT of Sleg (R=2)
1
0.0
2
CPT of Stop (R=2)
0.0
2
…
1.0
2
1.0
Shape Synthesis
• Step 1: Synthesizing a set of components
1-legged or 4-legged
column-like or split
Rect or circular?
Pruning: Branches that contain assignments that have extremely low probability density
Shape Synthesis
• Step 1: Synthesizing a set of components
• Step 2: Optimizing component placement
“slots” specify
where this
component can be
attached to other
components.
Shape Synthesis
• Step 1: Synthesizing a set of components
• Step 2: Optimizing component placement
penalizes
discrepancies of
position and relative
size between each
pair of adjacent slots
Shape Synthesis
• Step 1: Synthesizing a set of components
• Step 2: Optimizing component placement
Application: Shape database
amplification
• synthesize all instantiations of the model that
have non-negligible probability
– identify and reject instantiations that are very
similar to shapes in the input dataset or to
previous instantiations. (by measuring the feature
vectors of corresponding components)
Application: Constrained shape
synthesis
• Give partial assignments to constrained
random variables assume values only from the
range corresponding to the specified
constraints.
4-leg
split
Results
• Learning took about 0.5 hours for construction vehicles,
3 hours for creatures, 8 hours for chairs, 20 hours for
planes, and 70 hours for ships.
• For shape synthesis, enumerating all possible
instantiations of a learned model takes less than an
hour in all cases, and final assembly of each shape
takes a few seconds.
Can it generate models like below?
or
The probability should
be very low.
or
Inspiration- Variability vs plausibility
• To maintain plausibility: should be similar to the existing
ones;
• To increase variability: should be as different as possible
from the existing ones.
• This work is good for maintaining plausibility but the
variability seems low.
• How to pursue large variability while maintaining plausibility?
or
Topic? Generating shape variation by
variability transfer
• learn the varying model in the dataset rather
than the shape model.
• Use the varying model to synthesize new
shape in another dataset.
Topic? function-preserved shape
synthesize
• The function of a component is not taken into
account in the current model…
• By considering function, we can create
variations with high dissimilarity on geometric
looking while preserve the function.