Direct Fitting of Gaussian Mixture Models

Leonid Keselman, Martial Hebert

Robotics InstituteCarnegie Mellon University

May 29, 2019

https://github.com/leonidk/direct_gmm1

Representations of 3D data

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Point Cloud Point Cloud+ Normals

Triangular Mesh

Nearest Neighbor Plane Fit Screened Poisson Surface Reconstruction

Gaussian Mixture Models for 3D Shapes

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GMM fit to object surface

Benefits• Closed-form expression• Can represent contiguous surfaces• Easy to build from noisy data• Sparse

Gaussian Mixture Model (GMM)

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! " = $%&'

()* +("; .*, Σ*) $

*)* = 1)* ≥ 0

Σ* is symmetric, positive-semidefinite

Gaussian Mixtures as a shape representation

W. Tabib, C. O’Meadhra, N. MichaelIEEE R-AL (2018)

B. Eckart, K. Kim, A. Troccoli, A. Kelly, J. Kautz. CVPR (2016)

B. Eckart, K. Kim, J. Kautz. ECCV (2018)

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Efficient Representation Mesh Registration Frame Registration

Fitting a Gaussian Mixture Model

1. Obtain 3D Point Cloud2. Select Initial Parameters3. Iterate Expectation & Maximization

i. E-Step: Each point gets a likelihoodii. M-Step: Each mixture gets parameters

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The E-Step (Given GMM parameters)

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!"# =1&#'" ((*#; ,", Σ")

&# =01'1 ((*#; ,1, Σ1)

Normalization constant for point j

Affiliation between point j & mixture i

The M-Step (Given point-mixture weights)

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!" = $%&'

($)&'

*+)%log /) 0(2%; 4), Σ))

8!"8/)

= 0

mixturespoints

8!"84)

= 0 8!"8Σ)

= 0

lower-bound loss

To get new parameters: takes derivatives, set equal to zero, and solve

4) =1;)$

%<)%2%/) =

;)=

Σ) =1;)$

%<)%(2%−4))(2%−4))?

<)% = +)%

;) =$%<)%

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Geometric Objects in a Probability Distribution

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Known curve in a given 2D probability distribution

Geometric Objects in a Probability Distribution

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Consider sampling N points from this curve

ℓ curve ≅()*+

,-(/))

Geometric Objects in a Probability Distribution

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Take a geometric mean to account for sample number

ℓ curve ≅ ()*+

,-(/))

+,

Geometric Objects in a Probability Distribution

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The curve will be the value in the limit

ℓ curve ≅ ()*+

,-(/))

+,

ℓ curve = lim,→6 ()*+

,-(/))

+,

= lim,→6exp log ()*+

,-(/))

+,

= lim,→6exp1< =

)*+

,log(-(/)))

Geometric Objects in a Probability Distribution

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ℓ curve ≅ ()*+

,-(/))

+,

ℓ curve = lim,→6 ()*+

,-(/))

+,

= lim,→6exp log ()*+

,-(/))

+,

= lim,→6exp1< =

)*+

,log(-(/))) = exp > log(-(/)) ?/

Geometric Objects in a Probability Distribution

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!= exp & log(+(,)) .,

1. If +(,) = 0 on curve, then L= 02. Invariant to reparameterization

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!" Area of each triangle#" Centroid of each triangle$", &", '" Triangle vertices

The E-Step (Given GMM parameters)

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!"# =1&#'" ((*#; ,", Σ")

&# =01'1 ((*#; ,1, Σ1)

Normalization constant for point j

Affiliation between point j & mixture i

2# Area of each triangle,# Centroid of each triangle3#, 4#, &# Triangle vertices

The New E-Step (Given GMM parameters)

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!"# =1&#'"(# )(+#; +", Σ")

&# =01'1(1)(+#; +1, Σ1)

Normalization constant for object j

Affiliation between object j & mixture i

(# Area of each triangle+# Centroid of each triangle2#, 3#, &# Triangle vertices

Taylor Approximation(2 terms)

The M-Step (Given point-mixture weights)

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!" =1%"&

'("')'

*" =%"+

Σ" =1%"&

'("'()'−!")()'−!")0

("' = 1"'

%" =&'("'

The New M-Step (Given point-mixture weights)

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!" =1%"&

'("')'

*" =%"+

Σ" =1%"&

'("' ()'−!")()'−!")0 + Σ'

("' = 2'3"'

%" =&'("'

2' Area of each triangle!' Centroid of each triangle4', 6', 7' Triangle vertices

Σ' =112 4'4'0 + 6'6'0 + 7'7'0 − 3 !'!'0

What is Σ"?

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ResultsDid all that math actually help us fit better/faster GMMs?

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Using different inputs

classic algorithm• Vertices of the mesh• Triangle centroids

our method• Approximate (E only)• Exact (E + M steps)

Measure the likelihood of a high-density point cloud (higher is better)

Evaluate across a wide range of mixtures(6 to 300)

Full E+M method works in all cases

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Stable under even random initialization!

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ApplicationsAre these models actually more useful?

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Mesh Registration (P2D)

Method1. Apply a random rotation + translation to the point cloud2. Find transformation to maximize the likelihood of the points

• Perform P2D with GMMs fit toi. mesh verticesii. mesh triangles

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Eckart, Kim, Kautz. “HGMR: Hierarchical Gaussian Mixtures for Adaptive 3D Registration.” ECCV (2018)

Mesh-based GMMs are more accurate

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Across multiple models

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0

50

100

150

Armadillo Bunny Dragon Happy Lucy

Rotation Error(% of ICP)

points mesh

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100

200

Armadillo Bunny Dragon Happy Lucy

Translation Error(% of ICP)

points mesh

Frame Registration (D2D)

Method1. Use a sequence from an RGBD Sensor

• 2,500 frame TUM sequence from a Microsoft Kinect

2. Pairwise registration between t & t-1 frames• Optimize the D2D L2 distance• Build GMMs using square pixels as the geometric object

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W. Tabib, C. O’Meadhra, N. Michael.“On-Manifold GMM Registration” IEEE R-AL (2018)

Representing points using pixel squares

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D2D Registration Results

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Compared to standard GMM• 2.4% improvement in RMSE• 22% faster D2D convergence

Questions?

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! = exp & log(+(,)) .,

The End!

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Extra Slides

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How to fit a Gaussian Mixture Model?

1. Obtain any collection of objects2. Perform Expectation + Maximization

i. E-Step: Each point gets a likelihoodii. M-Step: Each mixture gets new parameters

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Extension to arbitrary primitives

39Vasconcelos, Lippman. "Learning mixture hierarchies.” Advances in Neural Information Processing Systems (1999)

Approximation

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area-weighted geometric mean using the primitive’s centroids

Product Integral Formulation

• Product integrals provide a resampling-invariant loss function• Given S samples, of M primitives, with N mixture components

• This can be evaluated in the limit of samples (with a geometric mean)

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For GMMs we will use the lower bound

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P2D Registration Results

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Model Rotation Error(% of ICP)

Translation Error(% of ICP)

points mesh points meshArmadillo 127 37 161 33Bunny 50 28 41 17Dragon 68 25 40 19Happy 101 27 85 27Lucy 95 23 122 35

Mesh Registration with P2D

Method1. Apply a random rotation + translation to the point cloud2. Point-to-Distribution (P2D) registration of point cloud to GMM

• Perform tests with GMMs fit toi. mesh verticesii. mesh triangles

• Optimize the GMM likelihood with rigid body transformation (q & t)• BFGS Optimization using numerical gradients, starting from identity

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Eckart, Kim, Kautz. “HGMR: Hierarchical Gaussian Mixtures for Adaptive 3D Registration.” ECCV (2018)

Acknowledgements

• Martial Hebert• Robotics Institute• Reviewers• Funding?

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Representing points using pixel squares

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