A Probabilistic Model for Component-Based Shape Synthesis

Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen Koltun

Stanford University

SP2-2

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 Burglary

AlarmRadio

Pearl, 1988

Call

Goal

• A tool that automatically synthesizes a variety of new, distinct shapes from a given domain.

Sailing ships vary in:

- Size

- Type of hull, keel and mast

- The number and configuration of masts.

Various geometric, stylistic and functional relationships influence the selection and placement of individual components to ensure that the final shape forms a coherent whole.

Probabilistic Reasoning for Assembly-Based 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 tabletopsSleg=narrow column-like legs

For 1-legged tableNtop=1Nleg=1Stop=roughly circular tabletopsSleg=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]:

a fictitious dataset that comprises the training data O and approximate statistics for the values of the latent variables.

the parameters estimated for a given G

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 2

5/12 7/12

CPT of Stop (R=1)

1 2

1.0 0.0

CPT of Stop (R=2)

1 2

0.0 1.0

CPT of Sleg (R=1)

1 2

1.0 0.0

CPT of Sleg (R=2)

1 2

0.0 1.0

…

Shape Synthesis

• Step 1: Synthesizing a set of components

1-legged or 4-legged

Rect or circular?

column-like or split

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.