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Richard G. Baraniuk Chinmay Hegde Sriram Nagaraj Go With The Flow A New Manifold Modeling and Learning Framework for Image Ensembles Aswin C. Sankaranarayanan Rice University

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Sensor Data Deluge

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Concise Models Efficient processing / compression requires concise representation Sparsity of an individual image pixels large wavelet coefficients (blue = 0)

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Concise Models Efficient processing / compression requires concise representation Our interest in this talk: Collections of images

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Concise Models Our interest in this talk: Collections of image parameterized by \in –translations of an object : x-offset and y-offset –rotations of a 3D object pitch, roll, yaw –wedgelets : orientation and offset

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Concise Models Our interest in this talk: Collections of image parameterized by \in –translations of an object : x-offset and y-offset –rotations of a 3D object pitch, roll, yaw –wedgelets : orientation and offset Image articulation manifold

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Image Articulation Manifold N-pixel images: K-dimensional articulation space Then is a K-dimensional manifold in the ambient space Very concise model articulation parameter space

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Smooth IAMs N-pixel images: Local isometry: image distance parameter space distance Linear tangent spaces are close approximation locally articulation parameter space

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Smooth IAMs N-pixel images: Local isometry: image distance parameter space distance Linear tangent spaces are close approximation locally articulation parameter space

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Ex: Manifold Learning LLE ISOMAP LE HE Diff. Geo … K=1 rotation

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Ex: Manifold Learning K=2 rotation and scale

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Theory/Practice Disconnect: Smoothness Practical image manifolds are not smooth! If images have sharp edges, then manifold is everywhere non-differentiable [Donoho and Grimes] Tangent approximations ? Isometry ?

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Theory/Practice Disconnect: Smoothness Practical image manifolds are not smooth! If images have sharp edges, then manifold is everywhere non-differentiable [Donoho and Grimes] Tangent approximations ? Isometry ?

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Failure of Tangent Plane Approx. Ex: cross-fading when synthesizing / interpolating images that should lie on manifold Input Image Geodesic Linear path

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Failure of Local Isometry Ex:translation manifold all blue images are equidistant from the red image Local isometry –satisfied only when sampling is dense

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Tools for manifold processing Geodesics, exponential maps, log-maps, Riemannian metrics, Karcher means, … Smooth diff manifold Algebraic manifoldsData manifolds LLE, kNN graphs Point cloud model

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The concept of Transport operators Beyond point cloud model for image manifolds Example

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Example: Translation 2D Translation manifold barring boundary related issues Set of all transport operators = Beyond a point cloud model –Action of the articulation is more accurate and meaningful

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Optical Flow Generalizing this idea: Pixel correspondances Idea:OF between two images is a natural and accurate transport operator (Figures from Ce Liu’s optical flow page) OF from I 1 to I 2 I 1 and I 2

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Optical Flow Transport IAM Articulations Consider a reference image and a K-dimensional articulation Collect optical flows from to all images reachable by a K-dimensional articulation

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Optical Flow Transport IAM OFM at Articulations Consider a reference image and a K-dimensional articulation Collect optical flows from to all images reachable by a K-dimensional articulation Theorem: Collection of OFs is a smooth, K-dimensional manifold (even if IAM is not smooth)

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OFM is Smooth (Rotation) Articulation θ in [ ⁰ ] Intensity I(θ) Op. flow v(θ) Pixel intensity at 3 points Flow (nearly linear)

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Main results Local model at each Each point on the OFM defines a transport operator –Each transport operator maps to one of its neighbors For a large class of articulations, OFMs are smooth and locally isometric –Traditional manifold processing techniques work on OFMs IAM OFM at Articulations

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Linking it all together IAM OFM at Articulations Nonlinear dim. reduction The non-differentiablity does not dissappear --- it is embedded in the mapping from OFM to the IAM. However, this is a known map

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The Story So Far… IAM OFM at Articulations Tangent space at Articulations IAM

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Input Image Geodesic Linear path IAM OFM

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OFM Synthesis

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ISOMAP embedding error for OFM and IAM 2D rotations Reference image Manifold Learning

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Embedding of OFM 2D rotations Reference image

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Data 196 images of two bears moving linearly and independently IAM OFM Task Find low-dimensional embedding OFM Manifold Learning

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IAM OFM Data 196 images of a cup moving on a plane Task 1 Find low-dimensional embedding Task 2 Parameter estimation for new images (tracing an “R”) OFM ML + Parameter Estimation

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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

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Goal: build a generative model for an entire IAM/OFM based on a small number of base images El Cheapo TM 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

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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

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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, …

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Open Questions Our treatment is specific to image manifolds under brightness constancy What are the natural transport operators for other data manifolds? dsp.rice.edu

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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

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Limitations Brightness constancy –Optical flow is no longer meaningful Occlusion –Undefined pixel flow in theory, arbitrary flow estimates in practice –Heuristics to deal with it Changing backgrounds etc. –Transport operator assumption too strict –Sparse correspondences ?

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Open Questions Theorem: random measurements stably embed a K-dim manifold whp [B, Wakin, FOCM ’08] Q: Is there an analogous result for OFMs?

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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 space with new transport operator that respects the nonlinearity of the imaging process Tangent space at Articulations IAM

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OFM Implementation details Reference Image

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Pairwise distances and embedding

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Flow Embedding

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Occlusion Detect occlusion using forward-backward flow reasoning Remove occluded pixel computations Heuristic --- formal occlusion handling is hard Occluded

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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 (BC) 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

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