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Published bySydney Watson Modified over 9 years ago
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Roee Litman, Alexander Bronstein, Michael Bronstein
Diffusion-geometric maximally stable component detection in deformable shapes Diffusion-geometric maximally stable component detection in deformable shapes Roee Litman, Alexander Bronstein, Michael Bronstein
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MSER In a nutshell… Shape Diffusion Geometry MSER The Feature Approach
for Images Deformable Shape Analysis MSER Maximally Stable Extremal Region Shape MSER Diffusion Geometry
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Feature Approach in Images
Feature based methods are the infrastructure laid in the base of many computer vision algorithms: Content-based image retrieval Video tracking Panorama alignment 3D reconstruction form stereo
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Problem formulation Find a semi-local feature detector
High repeatability Invariance to isometric deformation Robustness to noise, sampling, etc. Add discriminative descriptor
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The “what” Results
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Visual Example
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Visual Example
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(Taken from the TOSCA dataset)
More Results (Taken from the TOSCA dataset) - part human, part horse - partial matching - how to match? Horse regions + Human regions
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1st, 2nd, 4th, 10th, and 15th matches
Region Matching -now let's test it on real data Query 1st, 2nd, 4th, 10th, and 15th matches
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3D Human Scans Taken from the SCAPE dataset
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Scanned Region Matching
Query 1st, 2nd, 4th, 10th, and 15th matches
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Volume & surface isometry
Volume vs. Surface Original Volume & surface isometry Boundary isometry
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Volumetric Shapes Usually shapes are modeled as 2D boundary of a 3D shape. Volumetric shape model better captures "natural" behavior of non-rigid deformations. (Raviv et-al) Diffusion geometry terms can easily be applied to volumes 2D Meshes can be voxelized
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Volumetric Regions Taken from the SCAPE dataset
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(The “how”) Methodology
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Original MSER (Matas et-al)
Show some epipolar lines :)
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MSER Popular image blob detector Near-linear complexity:
High repeatability [Mikolajczyk et al. 05] Robust to affine transformation and illumination changes
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MSER – In a nutshell Threshold image at consecutive gray-levels
Search regions whose area stay nearly the same through a wide range of thresholds Efficient detection of maximally stable regions requires construction of a component tree
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MSER – In a nutshell SKIP?
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Algorithm overview
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Algorithm overview Represent as weighted graph Component tree
Stable component detection Represent as weighted graph Component tree Stable component detection
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Algorithm overview Represent as weighted graph Component tree
Stable component detection Represent as weighted graph
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Image as weighted graph
An undirected graph can be created from an image, where: Vertices are pixels Edges by adjacency rule, e.g. 4-neiborhood
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Weighting the graph In images Gray-scale as vertex-weight
Color as edge-weight [Forssen] In Shapes Curvature (not deformation invariant) Diffusion Geometry
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Weighting Option For every point on the shape:
Calculate the prob. of a random walk to return to the same point. Similar to Gaussian curvature Intrinsic, i.e. – deformation invariant
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Weight example Color-mapped Level-set animation
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Diffusion Geometry Analysis of diffusion (random walk) processes
Governed by the heat equation Solution is heat distribution at point at time
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Heat-Kernel Given Solve using: i.e. - find the “heat-kernel”
Initial condition Boundary condition, if these’s a boundary Solve using: i.e. - find the “heat-kernel”
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Probabilistic Interpretation
The probability density for a transition by random walk of length , from to
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Spectral Interpretation
How to calculate ? Heat kernel can be calculated directly from eigen-decomposition of the Laplacain By spectral decomposition theorem:
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Laplace-Beltrami Eigenfunctions
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Deformation Invariance
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Computational aspects
Shapes are discretized as triangular meshes Can be expressed as undirected graph Heat kernel & eigenfunctions are vectors Discrete Laplace-Beltrami operator Several weight schemes for is usually discrete area elements
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Computational aspects
In matrix notation Solve eigendecomposition problem
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Scale Space The time parameter of the heat kernel spans different scales of transition length is not invariant to shape’s scale Commute-time kernel - scale invariant Probability of a transition by random walk of any length
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Auto-diffusivity Special case - The chance of returning to after time
Related to Gaussian curvature by Now we can attach scalar value to shapes!
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Weight example Color-mapped Level-set animation
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Algorithm overview Represent as weighted graph Component tree
Stable component detection Represent as weighted graph Component tree Stable component detection
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The Component Tree Tree construction is a pre-process of stable region detection Contains level-set hierarchy, i.e. nesting relations. Constructed based on a weighted graph (vertex- or edge-weight) Tree’s nodes are level-sets (of the graph’s cross-sections)
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Tree Example
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Preliminaries - 1/3 We focus on the undirected graph with the vertex set and edge set , denoted Vertices and are adjacent if The ordered sequence is a path if every consecutive pair is adjacent and are linked A graph is connected if every vertex pair is linked
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Preliminaries - 2/3 A graph is called a sub-graph of
Given , we take only vertices belonging to an edge A graph is a (connected) component of if it is a maximal connected sub-graph
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Preliminaries - 3/3 Vertex-weighted graph - Edge-weighted graph -
A cross-section is a sub-graph of a weighted graph, with all weights Level-set is a (connected) component of a cross-section Altitude of a level-set is the maximal weight it contains (can be smaller than ) vertex weight are scalar only edge weight are more general
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“Graphic” Example A graph Edge-weighted 7 Cross-Section
Two 5 level-sets (with altitude 4) Every level-set has Size (area) Altitude (maximal weight) 4 7 8 9 8 1 4 1
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Tree Construction Iterate over vertices by order of weight
Create a new component from vertex If vertex is adjacent to existing component(s) add exiting component’s vertices to new one Store component’s area & weight
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Tree Construction 1 4 7 8 9 4 7 8 9 8 1 4 1
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Algorithm overview Represent as weighted graph Component tree
Stable component detection Represent as weighted graph Component tree Stable component detection
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Detection Process For every leaf component in the tree:
“Climb” the tree to its root, creating the sequence: Calculate component stability Local maxima of the sequence are “Maximally stable components”
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The Detils Performance
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Benchmarking The Method
Method was tested on SHREC 2010 data-set: 3 basic shapes (human, dog & horse) 9 transformations, applied in 5 different strengths 138 shapes in total Scale Original Deformation Holes Noise
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Results
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Quantitative Results Vertex-wise correspondences were given
Regions were projected onto another shape, and overlap ratio was measured Overlap ratio between a region and its projected counterpart is Repeatability is the percent of regions with overlap above a threshold
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Repeatability 65% at 0.75
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Conclusion Stable region detector for deformable shapes
Generic detection framework: Vertex- and edge-weighted graph representation Works on surface and/or volume data Partial matching & retrieval potential Tested quantitatively (on SHREC10)
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Thank You Any Questions?
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