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### Transcript of Normalized Cuts and Image Segmentation J. Shi and J. Shape Contexts 9 Shape Matching and Object...

• Shape Representation

Soma Biswas

Department of Electrical Engineering,

Indian Institute of Science, Bangalore.

• Shape-Based Recognition

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Analysis of anatomical structures Figure from Grimson & Golland

Pose

Recognition, detection Fig from Opelt et al.

• Applications

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• Geometric Transformations

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• Related Problems

 Shape representation and decomposition

 Finding a set of correspondences between shapes

 Transforming one shape into another

 Measuring the similarity between shapes

 Shape localization and model alignment

 Finding a shape similar to a model in a cluttered image

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• Comparing Images Using the Hausdorff Distance

6 D. Huttenlocher, G. Klanderman, and W. Rucklidge, 1993

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• Shape Contexts

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Shape Matching and Object Recognition Using Shape Contexts, S. Belongie, J. Malik, and J.

Puzicha, 2002

 Approach for measuring similarity between shapes and apply it for object recognition

 Solve for correspondences between points on the two shapes

● Using shape contexts – describe coarse distribution of the rest of the shape

w.r.t. a given point on the shape

 Use the correspondences to estimate an aligning transform

● Using regularized thin-plate splines

 Compute the distance between the two shapes

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- Not required to be landmarks/curvature

extrema, etc

- More samples -> better approximation of

underlying shape

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 SC: extremely rich descriptors  Finding correspondence between 2 shapes = for each

sample pt on one shape, find sample pt on the other

shape with most similar SC

 Maximizing similarities and enforcing uniqueness ->

bipartite graph matching problem / optimal assignment

 Can add local appearance similarity at the 2 points (gray scale images)

 Choice is application dependent

 For robust handling of outliers, add

dummy nodes to each pt set

 When there is no real match, a pt will be

matched to the dummy

• Invariance and Robustness

 Matching approach should be

 1) invariant under scaling and translation

 2) robust under small geometrical distortions,

occlusion & presence of outliers

 Invariant to translation -> since all measurements are

taken w.r.t. pts on the object

 Scale invariance: Normalize all radial distances by

mean distance between the pt pairs in the shape

 From expts: insensitive to small perturbations of parts

of the shape, small non-linear transformations,

occlusions and outliers

 Can provide complete rotation invariance: use relative

frame – tangent vector at each point as the x-axis

(not suitable for say 6 and 9)

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-a,b- sampled points

- correspondence found using

bipartite matching

• Thin-Plate Spline Model

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• Minimizing Bend Energy

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• Matching Process

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• Object Recognition

 Prototype based recognition: categories represented by ideal examples, rather than

logical rules

 Eg. Prototype for bird category: sparrow

 Soft category membership – as one moves further away from the ideal example, the

association with that prototype falls off

 3 distances:

 Shape Context Distance between shapes P and Q: sum of SC matching costs over

best matching points

 Local image appearance difference: texture and color

 Amount of transformation necessary to align the shapes: Bending energy in TPS

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• Results – Digit Recognition

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- 300 sample points

-Computational Needs:

-For 100 sample points - ~200ms

• Inner Distance

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Using the Inner-Distance for Classification of Articulated Shapes, H. Ling and

D. Jacobs, 2005

• Model of Articulated Objects

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• Inner Distance

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• Computing the Inner Distance

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• Example

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• Experiments

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• Hierarchical Matching of Deformable Shapes

25 P. Felzenszwalb and J. Schwartz, 2007

Use:

- Compare pair of objects

- Detect objects in cluttered images

• The Shape Tree

 A be an open curve (a1, . . . , an).

 ai be a midpoint on A.

 L(ai|a1, an) -> location of ai relative to a1 and an

 First & last sample points define a canonical scale

and orientation, so L invariant to similarity transf.

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o Left child of a node: describes the subcurve from the start to the midpoint

o Right child describes the subcurve from the midpoint to the end.

o Bottom nodes capture local geometric properties such as the angle formed at a point,

o Root nodes capture more global information encoded by the relative locations of points

that are far from each other.

o Representation invariant to similarity transformations : Since contains only the

locations of points relative to two other points.

o Given the tree representation for A, along with the location of its start and end points a1 and an, the curve can be recursively reconstructed – translated, rotated & scaled version of A

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Bookstein coordinate of B

w.r.t. A and C

• Deformation Model

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• Elastic Matching

 A and B be 2 open curves

 Build shape tree for A -> look for mapping from points in A to points in B such that

the shape tree of A is deformed as little as possible

 Total deformation = sum over deformations applied to each node in the A shape-tree

 Hierarchical nature of the shape-tree ensures that both local and global geometric

properties are preserved by a good matching.

 Allow larger deformations near the bottom of a shape-tree as these do not change the

global appearance of an object.

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• Experiments

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• Results

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