Manifold Learning in the Wild
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Manifold Learning in the Wild A New Manifold Modeling and Learning Framework for Image Ensembles Aswin C. Sankaranarayanan Rice University. Richard G. Baraniuk Chinmay Hegde. Sensor Data Deluge. Internet Scale Databases. Tremendous size of corpus of available data

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Manifold Learning in the Wild

A New Manifold Modeling and Learning Framework for Image EnsemblesAswin C. Sankaranarayanan

Rice University

Richard G. BaraniukChinmayHegde



Internet scale databases
Internet Scale Databases

  • Tremendous sizeof corpus of available data

    • Google Image Search of “Notre Dame Cathedral” yields 3m results 3Tbof data


Concise models
Concise Models

  • Efficient processing / compression requires concise representation

  • Our interest in this talk: Collections of images


Concise models1
Concise Models

  • Our interest in this talk:

    Collections of image

    parameterized by

q\inQ

  • translations of an object

    • q: x-offset and y-offset

  • wedgelets

    • q: orientation and offset

  • rotations of a 3D object

    • q: pitch, roll, yaw


Concise models2
Concise Models

  • Our interest in this talk:

    Collections of image

    parameterized by

q\inQ

  • translations of an object

    • q: x-offset and y-offset

  • wedgelets

    • q: orientation and offset

  • rotations of a 3D object

    • q: pitch, roll, yaw

  • Image articulation manifold


Image articulation manifold
Image Articulation Manifold

  • N-pixel images:

  • K-dimensional articulation space

  • Thenis a K-dimensional manifoldin the ambient space

  • Very concise model

    • Can be learnt using

      Non-linear dim. reduction

articulation parameter space


Ex manifold learning
Ex: Manifold Learning

LLE

ISOMAP

LE

HE

Diff. Geo…

  • K=1rotation


Ex manifold learning1
Ex: Manifold Learning

  • K=2rotation and scale


Smooth iams
Smooth IAMs

  • N-pixel images:

  • Local isometryimage distance parameter space distance

  • Linear tangent spacesare close approximationlocally

  • Low dimensional

    articulation space

articulation parameter space


Smooth iams1
Smooth IAMs

  • N-pixel images:

  • Local isometryimage distance parameter space distance

  • Linear tangent spacesare close approximationlocally

  • Low dimensional

    articulation space

articulation parameter space


Smooth iams2
Smooth IAMs

  • N-pixel images:

  • Local isometryimage distance parameter space distance

  • Linear tangent spacesare close approximationlocally

  • Lowdimensional

    articulation space

articulation parameter space


Theory practice disconnect isometry
Theory/Practice Disconnect Isometry

  • Ex: translation manifold

    all blue images are equidistant from the red image

  • Local isometry

    • satisfied only when sampling is dense


Theory practice disconnect nuisance articulations
Theory/Practice DisconnectNuisance articulations

  • Unsupervised data, invariably, has additional undesired articulations

    • Illumination

    • Background clutter, occlusions, …

  • Image ensemble is no longer low-dimensional


Image representations
Image representations

  • Conventional representation for an image

    • A vector of pixels

    • Inadequate!

pixel image


Image representations1
Image representations

  • Replace vector of pixels with an abstract bagof features

    • Ex: SIFT (Scale Invariant Feature Transform) selects keypoint locations in an image and computes keypoint descriptorsfor each keypoint

    • Very popular in many many vision problems


Image representations2
Image representations

  • Replace vector of pixels with an abstract bagof features

    • Ex: SIFT (Scale Invariant Feature Transform) selects keypoint locations in an image and computes keypoint descriptorsfor each keypoint

    • Keypoint descriptors are local; it is very easy to make them robust to nuisance imaging parameters


Loss of geometrical info
Loss of Geometrical Info

  • Bag of features representations hide potentially useful image geometry

Image space

Keypoint space

  • Goal: make salient image geometrical info more explicit for exploitation


Key idea
Key idea

  • Keypoint space can be endowed with a rich low-dimensional structure in many situations


Key idea1
Key idea

  • Keypoint space can be endowed with a rich low-dimensional structure in many situations

  • Mechanism: define kernels ,between keypoint locations, keypoint descriptors


Keypoint kernel
Keypoint Kernel

  • Keypoint space can be endowed with a rich low-dimensional structure in many situations

  • Mechanism: define kernels ,between keypoint locations, keypoint descriptors

  • Joint keypoint kernel between two images

    is given by


Many possible kernels
Many Possible Kernels

  • Euclidean kernel

  • Gaussian kernel

  • Polynomial kernel

  • Pyramid match kernel [Grauman et al. ’07]

  • Many others


Keypoint kernel1
Keypoint Kernel

  • Joint keypoint kernel between two images

    is given by

  • Using Euclidean/Gaussian (E/G) combination yields


From kernel to metric
From Kernel to Metric

Lemma: The E/G keypoint kernel is a Mercer kernel

  • enables algorithms such as SVM

    Lemma: The E/G keypoint kernel induces a metricon the space of images

  • alternative to conventional L2 distance between images

  • keypoint metric robust to nuisance imaging parameters, occlusion, clutter, etc.


Keypoint geometry
Keypoint Geometry

Theorem:Under the metric induced by the kernel

certain ensembles of articulating images formsmooth, isometric manifolds

  • Keypointrepresentation compact, efficient, and …

  • Robust to illumination variations, non-stationary backgrounds, clutter, occlusions


Keypoint geometry1
Keypoint Geometry

Theorem: Under the metric induced by the kernel

certain ensembles of articulating images formsmooth, isometric manifolds

  • In contrast: conventional approach to image fusion via image articulation manifolds (IAMs) fraught with non-differentiability (due to sharp image edges)

    • not smooth

    • not isometric



Application manifold learning1
Application: Manifold Learning

  • 2D Translation

  • IAM KAM


Manifold learning in the wild
Manifold Learning in the Wild

  • Rice University’s Duncan Hall Lobby

    • 158 images

    • 360° panorama using handheld camera

    • Varying brightness, clutter


Manifold learning in the wild1
Manifold Learning in the Wild

  • Duncan Hall Lobby

  • Ground truth using state of the art structure-from-motion software

Ground truth

IAM

KAM


Manifold learning in the wild2
Manifold Learning in the Wild

  • Rice University’s Brochstein Pavilion

    • 400 outdoor images of a building

    • occlusions, movement in foreground, varying background


Manifold learning in the wild3
Manifold Learning in the Wild

  • Brochstein Pavilion

    • 400 outdoor images of a building

    • occlusions, movement in foreground, background

IAM

KAM


Internet scale imagery
Internet scale imagery

  • Notre-dame cathedral

    • 738 images

    • Collected from Flickr

    • Large variations in illumination (night/day/saturations), clutter (people, decorations),camera parameters (focal length, fov, …)

    • Non-uniform sampling of the space


Organization
Organization

  • k-nearest neighbors


Organization1
Organization

  • “geodesics’

“zoom-out”

“Walk-closer”

3D rotation


Summary
Summary

  • Challenges for manifold learning in the wild are both theoretical and practical

  • Need for novel image representations

    • Sparse features

      • Robustness to outliers, nuisance articulations, etc.

      • Learning in the wild: unsupervised imagery

  • Promise lies in fast methods that exploit only neighborhood properties

    • No complex optimization required


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