go with the flow a new manifold modeling and learning framework for image ensembles richard g....
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Go With The FlowA New Manifold Modeling and Learning Framework for Image Ensembles
Richard G. BaraniukRice University
Aswin Sankaranarayanan Chinmay Hegde Sriram Nagaraj
Go With The FlowA New Manifold Modeling and Learning Framework for Image Ensembles
Richard G. BaraniukRice University
Sensor Data Deluge
Sensor Data Deluge
Sensor Data Deluge
Concise Models
• Efficient processing / compression requires concise representation
• Sparsity of an individual image
pixels largewaveletcoefficients
(blue = 0)
Concise Models
• Efficient processing / compression requires concise representation
• Our interest in this talk: Collections of images
Concise Models
• Our interest in this talk:Collections of images parameterized by q 2 Q
– translations of an object q: x-offset and y-offset
– rotations of a 3D object: q pitch, roll, yaw
– wedgeletsq: orientation and offset
Concise Models
• Our interest in this talk:Collections of images parameterized by q 2 Q
– translations of an object q: x-offset and y-offset
– rotations of a 3D object: q pitch, roll, yaw
– wedgeletsq: orientation and offset
• Image articulation manifold
Image Articulation Manifold• In practice: N-pixel images:
• In theory:
• K-dimensional articulation space
• Thenis a K-dimensional manifoldin the ambient space
• Very concise model
articulation parameter space
Smooth IAMs• In practice: N-pixel images:
• In theory:
• If images are smooththen so is
• Local isometry: image distance parameter space distance
• Linear tangent spacesare close approximationlocally
articulation parameter space
Smooth IAMs• In practice: N-pixel images:
• In theory:
• If images are smooththen so is
• Local isometry: image distance parameter space distance
• Linear tangent spacesare close approximationlocally
articulation parameter space
Ex: Manifold Learning
LLEISOMAPHE …
• K=1rotation
Ex: Manifold Learning
• K=2rotation and scale
Theory/Practice Disconnect: Smoothness
• Practical image manifolds are not smooth!
• If images have sharp edges, then manifold is everywhere non-differentiable [Donoho, Grimes]
Local isometryLocal tangent approximations
Theory/Practice Disconnect: Smoothness
• Practical image manifolds are not smooth!
• If images have sharp edges, then manifold is everywhere non-differentiable [Donoho, Grimes]
Local isometryLocal tangent approximations
Failure of Local Isometry
• Ex: translation manifold
• Local isometry
all blue imagesare equidistantfrom the red image
0 20 40 60 80 1000
0.5
1
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3.5
Translation in [px]
Euc
lidea
n di
stan
ce
Failure of Tangent Plane Approx.
• Ex: cross-fading when synthesizing / interpolating images that should lie on manifold
Input Image Input Image
Geodesic Linear path
Image Articulation Manifold
• Linear tangent space at is K-dimensional
– provides a mechanism to transport along manifold
– problem: since manifold is non-differentiable, tangent approximation is poor
• Our goal: replace tangent spacewith new transport operator that respects the nonlinearity of the imaging process
Tangent space at 0I
Articulations
0
1 2
2I1
I0I IAM
Optical Flow Transport• Brightness constancy: Given two images I1 and I2,
we seek a displacement vector field f(x, y) = [u(x, y), v(x, y)] such that
Ex:
• Linearized brightness constancy (LBC)
)),(),,((),( 12 yxvyyxuxIyxI
),()(),()(),(),( 1112 yxvIyxuIyxIyxI YX
History of Optical Flow
• Dark ages (<1985): special cases solved– LBC an under-determined set of linear equations
• Horn and Schunk (1985): – Regularization term: smoothness prior on the flow
• Brox et al (2005)– shows that linearization of brightness constancy is
a bad assumption– develops optimization framework to handle BC directly
• Brox et al (2010), Black et al (2010), Liu et al (2010)– practical systems with reliable code
Optical Flow Manifold
0
1 2
IAM
OFM at 0I
2I1
I0I
Articulations
• Consider a reference imageand a K-dimensional articulation
• Collect optical flows fromto all images reachable by aK-dimensional articulation
• Provides a mechanism to transport along manifold
Optical Flow Manifold
0
1 2
IAM
OFM at 0I
2I1
I0I
Articulations
• Consider a reference imageand a K-dimensional articulation
• Collect optical flows fromto all images reachable by aK-dimensional articulation
• Provides a mechanism to transport along manifold
• Theorem: Collection of OFs is a smooth, K-dimensional manifold(even if IAM is not smooth)[N,S,H,B,2010]
Flow Metric• Replace intensity-based image distance
– root cause of IAM non-differentiability
… with distance along the OF field from I1 to I2
• Enables differential geometric tools– ex: vector fields, curvature, parallel transport, …
curve c
IAMFlow Field Vt
OFM is Smooth (Translation)
Euclidean distance between image intensities
Flow metric between images(globally linear)
OFM is Smooth (Rotation)
00 30 60 30 60
-80 -60 -40 -20 0 20 40 60 800
50
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Articulation in [ ]
Inte
nsity
-80 -60 -40 -20 0 20 40 60 80-20
-10
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Articulation in [ ]
Opt
ical
flow
vx i
n pi
xels
Articulation θ in [⁰]
Inte
nsi
ty I(θ
)O
p. flow
v(θ
)Euclidean distance between image intensities
Flow metric between images(nearly linear)
The Story So Far…
0
1 2
IAM
OFM at 0I
2I1
I0I
Articulations
flow metric
Euclidean metric
Tangent space at 0I
Articulations
0
1 2
2I1
I0I IAM
Input Image Input Image
Geodesic Linear pathIAM
OFM
OFM Synthesis
ISOMAP embedding error for OFM and IAM
2D rotations
Reference image
Manifold Learning
Manifold Learning
-6 -4 -2 0 2 4 6-5
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-2
-1
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1
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5Two-dimensional Isomap embedding (with neighborhood graph).
Embedding of OFM
2D rotations
Reference image
Data196 images of two bears moving linearlyand independently
IAM OFM
TaskFind low-dimensional embedding
OFM Manifold Learning
IAM OFM
Data196 images of a cup moving on a plane
Task 1Find low-dimensional embedding
Task 2Parameter estimation for new images(tracing an “R”)
OFM ML + Parameter Estimation
• Point on the manifold such that the sum of geodesic distances to every other point is minimized
• Important concept in nonlinear data modeling, compression, shape analysis [Srivastava et al]
Karcher Mean
10 images from an IAM
ground truth KM OFM KM linear KM
• Goal: build a generative model for an entire IAM/OFM based on a small number of base images
• El CheapoTM algorithm:– choose a reference image randomly– find all images that can be generated from this image by OF– compute Karcher (geodesic) mean of these images– compute OF from Karcher mean image to other images– repeat on the remaining images until no images remain
• Exact representation when no occlusions
Manifold Charting
Manifold Charting
IAM
• Goal: build a generative model for an entire IAM/OFM based on a small number of base images
• Ex: cube rotating about axis
• All cube images can be representing using 4 reference images + OFMs
• Many applications– selection of target
templates for classification– “next-view” selection for
adaptive sensing applications
Summary
• IAMs a useful concise model for many image processing problems involving image collections and multiple sensors/viewpoints
• But practical IAMs are non-differentiable– IAM-based algorithms have not lived up to their promise
• Optical flow manifolds (OFMs)– smooth even when IAM is not– OFM ~ nonlinear tangent space– support accurate image synthesis, learning, charting, …
• Barely discussed here: OF enables the safe extension of differential geometry concepts– Log/Exp maps, Karcher mean, parallel transport, …
Related Work
• Analytic transport operators– transport operator has group structure [Xiao and Rao 07]
[Culpepper and Olshausen 09] [Miller and Younes 01] [Tuzel et al 08]
– non-linear analytics [Dollar et al 06]
– spatio-temporal manifolds [Li and Chellappa 10]
– shape manifolds [Klassen et al 04]
• Analytic approach limited to a small class of standard image transformations (ex: affine transformations, Lie groups)
• In contrast, OFM approach works reliably with real-world image samples (point clouds) and broader class of transformations
Open Questions
• Our treatment is specific to image manifolds
• What are the natural transport operators for other data manifolds?
dsp.rice.edu
Open Questions
• Theorem: random measurements stably embed aK-dim manifoldwhp [B, Wakin, FOCM ’08]
• Q: Is there an analogousresult for OFMs?