![Page 1: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/1.jpg)
Machine Learning Techniques for Geometric Modeling
Evangelos Kalogerakis
![Page 2: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/2.jpg)
3D models for digital entertainment
Limit Theory
![Page 3: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/3.jpg)
3D models for printing
MakerBot Industries
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3D models for architecture
Architect: Thomas Eriksson Courtesy Industriromantik
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Geometric modeling is not easy!
Autodesk Maya 2015
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“Traditional” Geometric Modeling
Manipulating curvesManipulating polygons Manipulating 3D primitives
Digital SculptingManipulating control points, cages image from Mohamed Aly Rable
image from Blender
image from autodesk.com
image from autodesk.com
image from wikipedia (CSG)
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Think of a “shape space” traversed by “low‐level” operations
Images from flossmanuals.net
duplicate
addfaces
extrude
add more faces,extrude etc
extrude
“native” shape representationpolygons, points, voxels…
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“Traditional” Geometric Modeling
Impressive results at the hands of experienced users
Operations requires exact and accurate input
Creating compelling 3D models takes lots of time
Tools usually have steep learning curves
![Page 9: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/9.jpg)
An alternative approach to geometric modeling
• Users provide high‐level, possibly approximate input
• Computers learn to generate low‐level, accurate geometry
Machine learning!
![Page 10: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/10.jpg)
“Low‐level” Shape Space
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“Low‐level” Shape Space
Design Space
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“Low‐level” Shape Space
? Design Space
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“Low‐level” Shape Space
attributes (discrete, continuous)
modern
sturdy Design Space
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“Low‐level” Shape Space
modern
sturdy
attributes (discrete, continuous)
Design Space
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“Low‐level” Shape Space
lines, pixels
attributes (discrete, continuous) sketch (approximate, noisy) Design Space
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“Low‐level” Shape Space
attributes (discrete, continuous) sketch (approximate, noisy)
gestures natural language brain signals etc
? Design Space
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Machine learning for Geometric Modeling
• Learn mappings from design to “low‐level” space
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“Low‐level” Shape Space
?No unique solution necessarily!
Design Space
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Machine learning for Geometric Modeling
• Learn mappings from design to “low‐level” space
• Learn which shapes are probable (“plausible”) given input
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“Plausible” chairs
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“Plausible” chairs
(not a binary or even an objective choice!)
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Machine learning for Geometric Modeling
• Learn mappings from design to “low‐level” space
• Learn which shapes are probable (“plausible”) given input
• Learn design space (“high‐level” representation)
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“Low‐level” Shape Space
Design space mightnot be pre‐defined!Learn it from data!
? Design Space
?
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“Low‐level” Shape Space
? Design Space
?
Learning formulation:
y = f( x )
Input: training shapesOutput: design space x,
mapping f
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“Low‐level” Shape Space
Design Space
?
Learning formulation:Input: training shapes,
design labelsOutput: mapping f
(easier!)[0.1, 1.0]
[0.4, 0.8][0.9, 0.1]
y = f( x )modern
sturdy
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“Low‐level” Shape Space
Design Space
[0.1, 1.0][0.4, 0.8]
[0.9, 0.1]
Goal:Generalize from training data:
Given new design dataproduce new shapes
[0.9, 0.5]
y = f( x )modern
sturdy
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Fundamental challenges
• How do we represent the shape space?
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“Low‐level” Shape Space
?
?
“Low‐level” shape space representation
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“Low‐level” Shape Space
?
?
“Low‐level” shape space representation
Can we use the polygon meshes as‐is for our shape space?
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“Low‐level” Shape Space
?
?
“Low‐level” shape space representation
Can we use the polygon meshes as‐is for our shape space?No. Take the first vertex on each mesh. Where is it?Meshes have different number of vertices, faces etc
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“Low‐level” shape space representation –the “computer vision” approach
Learn from pixels & multiple views! Produce pixels! Include view information?
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“Low‐level” shape space representation –another “computer vision” approach
Learn from voxels! Produce voxels! Include orientation information?
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“Low‐level” shape space representation –correspondences
Find point correspondences between 3D surface points. Can do aligment.Can we always have dense correspondences?
Image from Vladimir G. Kim, Wilmot Li, Niloy J. Mitra, Siddhartha Chaudhuri, Stephen DiVerdi, and Thomas Funkhouser, “Learning Part-based Templates from Large Collections of 3D Shapes”, 2013
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“Low‐level” shape space representation –abstractions
Parameterize shapes with primitives (cuboids, cylinders etc)How can we produce surface detail?
Image from E. Yumer., L. Kara, Co-Constrained Handles for Deformation in Shape Collections, 2014
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Fundamental challenges
• How do we represent the shape space?
• What is the form of the mapping? How is it learned?
![Page 36: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/36.jpg)
Regression example (simplistic)
Leg
thic
knes
s y
sturdiness xTraining data point ( shape + design values )
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Regression example (simplistic)
Leg
thic
knes
s y
sturdiness xTraining data point ( shape + design values )
![Page 38: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/38.jpg)
Regression example (simplistic)
Leg
thic
knes
s y
sturdiness xTraining data point ( shape + design values )
![Page 39: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/39.jpg)
Regression example (simplistic)
Leg
thic
knes
s y
sturdiness xTraining data point ( shape + design values )
![Page 40: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/40.jpg)
Regression example (simplistic)
Leg
thic
knes
s y
sturdiness x
new datapoint
Training data point ( shape + design values )
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Regression example (simplistic)
Leg
thic
knes
s y
sturdiness x
2
'' [ 1]
y xx x x
w
Training data point ( shape + design values )
new datapoint
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Regression example (simplistic)
Leg
thic
knes
s y
sturdiness x
2
'' [ 1]
y xx x x
w
2
.
'( ) [ ]itra n i
ii
L wy xwnew datapoint
Training data point ( shape + design values )
![Page 43: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/43.jpg)
Regression example (simplistic)
Leg
thic
knes
s y
sturdiness x
2
'' [ 1]
y xx x x
w
2
.
'( ) [ ]itra n i
ii
L wy xwnew datapoint
Training data point ( shape + design values )
…linear least‐squares solution…
![Page 44: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/44.jpg)
OverfittingImportant to select a function that would avoid overfitting & generalize (produce reasonable outputs for inputs not encountered during training)
image from Andrew Ng’s ML class (?)
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Classification example (Logistic Regression)
Suppose you want to predict pixels or voxels (on or off).
Probabilistic classification function:
( | ) ( ) ( ):
1( )1 exp( )
Pwhere
x xf w
w
= 1
xw
x
x
y
xy
( ) w x
![Page 46: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/46.jpg)
Logistic regression: training
Need to estimate parameters w from training data.
Find parameters that maximize probability of training data
[ 1] [ 0]
1
max ( 1 | ) [1 ( 1 | )]N
ii iP P
i iy y
wy y xx
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Logistic regression: training
Need to estimate parameters w from training data.
Find parameters that maximize probability of training data
[ 1] [ 0]
1
max ( ) [1 ( )]i
N
ii
i iy
w
yw wx x
![Page 48: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/48.jpg)
Logistic regression: training
Need to estimate parameters w from training data.
Find parameters that maximize log probability of training data
[ 1] [ 0]
1
max log{ ( ) [1 ( )] }i i
N
i
i iy y
wwx xw
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Logistic regression: training
Need to estimate parameters w from training data.
Find parameters that maximize log probability of training data
1
max [ 1]log ( ) [ 0]log(1 ( ))i
N
ii
i iwy y xw wx
![Page 50: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/50.jpg)
Logistic regression: training
Need to estimate parameters w from training data.
Find parameters that minimize negative log probability of training data
1
min [ 1]log ( ) [ 0]log(1 ( ))i
N
ii
iw i x w xywy
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Logistic regression: training
Need to estimate parameters w from training data.
In other words, find parameters that minimize the negative log likelihood function
1
min [ 1]log ( ) [ 0]log(1 ( ))i
N
ii
iw i x w xywy L(w)
![Page 52: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/52.jpg)
Logistic regression: training
Need to estimate parameters w from training data.
In other words, find parameters that minimize the negative log likelihood function
1
min [ 1]log ( ) [ 0]log(1 ( ))i
N
ii
iw i x w xywy L(w),
( ) [ ( ) ]ii di
id
xL yw
xw w
(partial derivative for dth parameter)
![Page 53: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/53.jpg)
How can you minimize/maximize a function?
Gradient descent: Given a random initialization of parameters and a step rate η, update them according to:
See also quasi‐Newton and IRLS methods
( )new old L w w w
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Regularization
Overfitting: few training data and number of parameters is large!
Penalize large weights ‐ shrink weights:
Called ridge regression (or L2 regularization)
2min ( ) wd
dL ww
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Regularization
Overfitting: few training data and number of parameters is large!
Penalize non‐zero weights ‐ push as many as possible to 0:
Called Lasso (or L1 regularization)
min ( ) | |wdd
L ww
![Page 56: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/56.jpg)
Lasso vs Ridge Regression
1w
( )L w
lassow
ridgew
2w
2|| ||w
1|| ||w
Modified image from Robert Tibsirani, Regression shrinkage and selection via the lasso, 1996
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Case study: the space of human bodies
Training shapes: 125 male + 125 female scanned bodies
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003
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Matching algorithm
scantemplate
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003
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Matching algorithm
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003to access the video: http://grail.cs.washington.edu/projects/digital-human/pub/allen04exploring.html
![Page 60: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/60.jpg)
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003
Principal Component Analysis
![Page 61: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/61.jpg)
Dimensionality ReductionSummarization of data with many (d) variables by a smaller set of (k) derived (synthetic, composite) variables.
n
d
X n
k
Y
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-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8 10 12
Variable X1
Varia
ble
X2
PC 1
PC 2
Each principal axis is a linear combination of the original variables
Principal Component Analysis
![Page 63: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/63.jpg)
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
average male mean + PCA component #1
Principal Component Analysis
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003to access the video: http://grail.cs.washington.edu/projects/digital-human/pub/allen04exploring.html
![Page 64: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/64.jpg)
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
x0y0z0x1y1z1x2
average male mean + PCA component #3
Principal Component Analysis
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003to access the video: http://grail.cs.washington.edu/projects/digital-human/pub/allen04exploring.html
![Page 65: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/65.jpg)
Fitting to attributesCorrelate PCA space with known attributes:
-40
-20
0
20
40
60
1.5 1.7 1.9 2.1Height (m)
Prin
cipa
l com
pone
nt #
1
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003
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Fitting to attributes
Slides from Brett Allen, Brian Curless, Zoran Popović, Exploring the space of human body shapes, 2003to access the video: http://grail.cs.washington.edu/projects/digital-human/pub/allen04exploring.html
![Page 67: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/67.jpg)
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
Case study: content creation with semantic attributes
![Page 68: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/68.jpg)
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
`
ScaryStrong
Case study: content creation with semantic attributes
![Page 69: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/69.jpg)
Less aerodynamic More aerodynamic
More scaryLess scary
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
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to access the video: https://www.youtube.com/watch?v=U_XfYzy2c9w
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Rank‐SVM: Project shape space onto a subspace that best preserves pairwise orderings
Ranking
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
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Rank‐SVM: Project shape space onto a subspace that best preserves pairwise orderings
Ranking
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
![Page 73: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/73.jpg)
Rank‐SVM: Project shape space onto a subspace that best preserves pairwise orderings
Ranking
Learn attribute strength:
subject to crowdsourced constraints:
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
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“Old‐Fashioned”
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
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“Old‐Fashioned”
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
![Page 76: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/76.jpg)
“Old‐Fashioned”
Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with Semantic Attributes, 2013
![Page 77: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/77.jpg)
Case study: a probabilictic model for component‐based synthesis
Given some training segmented shapes:
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
… and more ….
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Describe shape space of parts with a probability distribution
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
base diameter
base avg. m
ean curvature
Case study: a probabilictic model for component‐based synthesis
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Learn relationships between different part parameters within each cluster e.g. diameter of table top is related to scale of base plus some uncertainty
table top diam
eter
base diameter
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
Case study: a probabilictic model for component‐based synthesis
![Page 80: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/80.jpg)
Learn relationships between part clusters e.g. circular table tops are associated with bases with split legs
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
Case study: a probabilictic model for component‐based synthesis
![Page 81: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/81.jpg)
Represent all these relationships within a structured probability distribution (probabilistic graphical model)
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
Case study: a probabilictic model for component‐based synthesis
![Page 82: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/82.jpg)
Shape Synthesis ‐ Airplanes
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
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Shape Synthesis ‐ Airplanes
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen KoltunA Probabilistic Model for Component-Based Synthesis, 2012
![Page 84: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/84.jpg)
Shape Synthesis ‐ Chairs
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen Koltun, A Probabilistic Model for Component-Based Synthesis, 2012
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Shape Synthesis ‐ Chairs
Slides from Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, Vladlen Koltun, A Probabilistic Model for Component-Based Synthesis, 2012
![Page 86: C:UsersVDesktopSGP grad schoolMachine Learning Techniques ... · Slides from Siddhartha Chaudhuri, Evangelos Kalogerakis, Stephen Giguere, Thomas Funkhouser, Content Creation with](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7d3ea3c114f27e687fd562/html5/thumbnails/86.jpg)
“Low‐level” Shape Space
? Design Space
From “swallow” to “deep” mappings:
Input: training shapesOutput: latent spaces x,
mappings f“mediating”representations
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From “swallow” to “deep” mappings (networks)
Images, shapes, natural language have compositional structure
Deep neural networks!
Note: Let’s discuss them in the case of 2D images for now! Also let’s map from images to high‐level representations. We’ll see how this can be reversed later.
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Motivation
+
pixel 1
pixel 2
‐+
+
‐‐
+ ‐+
+ Coffee MugNot Coffee Mug‐
+
pixel 1
pixel 2
‐+
+
‐‐
+ ‐+
Is this a Coffee Mug?
Learning Algorithm
modified slides originally by Adam Coates
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Motivation
+handle?
cylinder?
‐++‐‐
+
‐ +
+ Coffee MugNot Coffee Mug‐
cylinder?handle?
Is this a Coffee Mug?
Learning Algorithm +handle?
cylinder?
‐++‐‐
+
‐ +
modified slides originally by Adam Coates
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Fixed/engineered descriptors + trained classifier/regressor
"Traditional" recognition pipeline
car?"Hand‐engineered" Descriptor Extractor
e.g. SIFT, bags‐of‐words
Trainedclassifier/regressor
modified slides originally by Adam Coates
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Trained descriptors + trained classifier/regressor
"New" recognition pipeline
car?Trained
DescriptorExtractor
Trainedclassifier/regressor
modified slides originally by Adam Coates
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From “swallow” to “deep” mappings (networks)In logistic regression, output was a direct function of inputs. Conceptually, this can be thought of as a network:
x1 x2 x3 ... xd 1
y ( ) ( )y f x w x
modified slides originally by Adam Coates
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Basic idea
Introduce latent nodes that will play the role of learned representations.
(1)1 1( )h w x h2
(1)2 2( )h w x
(2)( )y w h
x1 x2 x3 ... xd 1
h1 1
y
modified slides originally by Adam Coates
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Neural network
Same as logistic regression but now our output function has multiple stages ("layers", "modules").
Intermediate representation Prediction
x (1)( ) W x h (2)( ) W h
( )
( )( )
( )
...
m
where
1
2
w
wW
w
y
modified slides originally by Adam Coates
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Biological Neurons
Axon
Terminal Branches of AxonDendrites
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Analogy with biological networks
Activation Function
Slide credit : Andrew L. Nelson
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Neural network
Stack up several layers:
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y
modified slides originally by Adam Coates
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Forward propagation
Process to compute output:
x1 x2 x3 ... xd 1
modified slides originally by Adam Coates
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Forward propagation
x1 x2 x3 ... xd 1
1h1 h2 h3 hm...
x h(1)( ) W x
Process to compute output:
modified slides originally by Adam Coates
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Forward propagation
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
x h 'h(1)( ) W x (2)( ) W h
Process to compute output:
modified slides originally by Adam Coates
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Forward propagation
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y
x h 'h y(1)( ) W x (2)( ) W h (3)( ') W h
Process to compute output:
modified slides originally by Adam Coates
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Multiple outputs
x h 'h y(1)( ) W x (2)( ) W h (3)( ') W h
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y1 y2… …
modified slides originally by Adam Coates
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How can you learn the parameters?
Use a loss function e.g., for classification:
For regression:
1
( ) [ 1]log ( ) [ 0]log(1 ( ))t ti ou
i itput t
L f f
i,t i,tx xy yw
,2( ) [ ( )]t
ii
outpit
ut t
L f xyw
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BackpropagationFor each training example i (omit index i for clarity):
(3) ( )t ty f x
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y1 y2 For each output:
(3)(3)
,
( )t n
t n
L hw
w
(3)1,1w
(3)2,1w
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x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y1 y2
Backpropagation
(2) (2) (3) (3),'( )n t n t
t
w nw h
'( ) ( )[1 ( )] Note:
For each training example i (omit index i for clarity):
(2)1,1w
(2)2,1w
(2)(2)
,
( )n m
n m
L hw
w
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(1) (1) (2) (2),'( )m m n m n
n
w w x
x1 x2 x3 ... xd 1
1
1
h1 h2 h3 hm...
h1' h2' h3' hn'
y1 y2
BackpropagationFor each training example i (omit index i for clarity):
(1)1,1w (1)
2,1w(1)
(1),
( )m d
m d
L xw
w
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Is this magic?
All these are derivatives derived analytically using the chain rule!
Gradient descent is expressed through backpropagation of messages δ following the structure of the model
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Training algorithm
For each training example [in a batch]
1. Forward propagation to compute outputs per layer2. Back propagate messages δ from top to bottom layer3. Multiply messages δ with inputs to compute derivatives per layer4. Accumulate the derivatives from that training example
Apply the gradient descent rule
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Yet, this does not work so easily...
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Yet, this does not work so easily...
• Non‐convex: Local minima; convergence criteria.
• Optimization becomes difficult with many layers.
• Hard to diagnose and debug malfunctions.
• Many things turn out to matter:• Choice of nonlinearities.• Initialization of parameters.• Optimizer parameters: step size, schedule.
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Non‐linearities
• Choice of functions inside network matters.• Sigmoid function yields highly non‐convex loss functions• Some other choices often used:
1
‐1
1
tanh(·) ReLu(·) = max{0, ·}
“Rectified Linear Unit” Increasingly popular.
1
abs(·)
[Nair & Hinton, 2010]
tanh'(·)= 1 - tanh(·)2 abs'(·)= sign(·) ReLu'(·)= [ · > 0 ]
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Initialization
• Usually small random values.• Try to choose so that typical input to a neuron avoids saturating
• Initialization schemes for weights used as input to a node:• tanh units: Uniform[‐r, r]; sigmoid: Uniform[‐4r, 4r].• See [Glorot et al., AISTATS 2010]
• Unsupervised pre‐training
1
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Step size
• Fixed step‐size• try many, choose the best...• pick size with least test error on a validation set after T iterations
• Dynamic step size• decrease after T iterations
• if simply the objective is not decreasing much, cut step by half
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Momentum
Modify stochastic/batch gradient descent:
“Smooth” estimate of gradient from several steps of gradient descent:• High‐curvature directions cancel out.• Low‐curvature directions “add up” and accelerate.
Before : ( ),Withmomentum : ( ),previous
L w wL w w
w
w
w w ww w w w
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Regularize!
• Adding L2 regularization term to the loss function:
• Adding L1 regularization term to the loss function:
2( ( ) || || )L ww w w
1( ( ) || || )L ww w w
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Yet, things will not still work well!
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Main problem• Extremely large number of connections.• More parameters to train.• Higher computational expense.
modified slides originally by Adam Coates
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Local connectivity
Reduce parameters with local connections!
modified slides originally by Adam Coates
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Neurons as convolution filtersThink of neurons as convolutional filters acted on small adjacent (possibly overlapping) windows
Window size is called “receptive field” size and spacing is called “step” or “stride”
modified slides originally by Adam Coates
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Extract repeated structure
, ( )p fh g f pw x
Apply the same filter (weights) throughout the image Dramatically reduces the number of parameters
modified slides originally by Adam Coates
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Response per pixel p, per filter f for a transfer function g:
Can have many filters!
modified slides originally by Adam Coates
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PoolingApart from hidden layers dedicated to convolution, we can have layers dedicated to extract locally invariant descriptors
[Scherer et al., ICANN 2010][Boureau et al., ICML 2010]
', max( )p f ph pxMax pooling:
', ( )p fp
h avg pxMean pooling:
',p f gaussianh w pxFixed filter (e.g., Gaussian):
Progressively reduce the resolution of the image, so that the next convolutional filters are applied on larger scales
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Convolutional Neural Networks
ImageNet system from Krizhevsky et al., NIPS 2012:Convolutional layersMax‐pooling layersRectified linear units (ReLu).Stochastic gradient descent, L2 regularization etc
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Application: Image‐Net
Top result in LSVRC 2012: ~85%, Top‐5 accuracy.
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Learned representations
From Matthew D. Zeiler and Rob Fergus, Visualizing and Understanding Convolutional Networks, 2014
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Multi‐view CNNs
Image from Hang Su, Subhransu Maji, Evangelos Kalogerakis, Erik Learned-Miller, Multi-view Convolutional Neural Networks for 3D Shape Recognition, 2015
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Multi‐view CNNsUse output of fully connected layer as a shape descriptor. Shape retrieval evaluation in ModelNet40:
Image from Hang Su, Subhransu Maji, Evangelos Kalogerakis, Erik Learned-Miller, Multi-view Convolutional Neural Networks for 3D Shape Recognition, 2015
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Sketch‐based 3D Shape Retrieval using Convolutional Neural Networks
Image from Fang Wang, Le Kang, Yi Li, Sketch-based 3D Shape Retrieval using Convolutional Neural Networks, 2015
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Sketch‐based 3D Shape Retrieval using Convolutional Neural Networks
Image from Fang Wang, Le Kang, Yi Li, Sketch-based 3D Shape Retrieval using Convolutional Neural Networks, 2015
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Sketch‐based 3D Shape Retrieval using Convolutional Neural Networks
Image from Fang Wang, Le Kang, Yi Li, Sketch-based 3D Shape Retrieval using Convolutional Neural Networks, 2015
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Learning to Generate ChairsInverting the CNN…
Image from Alexey Dosovitskiy, J. Springenberg, Thomas BroxLearning to Generate Chairs with Convolutional Neural Networks 2015 to access video: http://lmb.informatik.uni‐freiburg.de/Publications/2015/DB15/
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Learning to Generate ChairsInverting the CNN…
Image from Alexey Dosovitskiy, J. Springenberg, Thomas BroxLearning to Generate Chairs with Convolutional Neural Networks 2015
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Deep learning on volumetric representations
Image from Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang and J. Xiao3D ShapeNets: A Deep Representation for Volumetric Shapes, 2015
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Image from Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang and J. Xiao3D ShapeNets: A Deep Representation for Volumetric Shapes, 2015
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Summary
Welcome to the era where machines learn to generate 3D visual content!
Deep learning seems one of the most promising directions
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Summary
Welcome to the era where machines learn to generate 3D visual content!
Deep learning seems one of the most promising directions
Big challenges: • Generate plausible, detailed, novel 3D geometry from high‐level specifications, approximate directions
• What shape representation should deep networks operate on? • Integrate with approaches that optimize for function and human‐object interaction