Geometric Hashing: A General and Efficient Model-Based Recognition Scheme

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Geometric Hashing A General and Efficient
Model-Based Recognition Scheme

• Yehezkel Lamdan and Haim J. Wolfson
• ICCV 1988
• Presented by Budi Purnomo
• Nov 23rd 2004

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Motivation

• Object recognition (ultimate goal of most
  computer vision research).
• Inputs
• A database of objects.
• A scene or image to recognize.
• Problems
• Objects in the scene undergo some
  transformations.
• Objects may partially occlude each other.
• Computationally expensive to retrieve each object
  from database and compare it against the observed
  scene.

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Problem Statement

• Recognition under Similarity Transformation
• Is there a transformed (rotated, translated and
  scaled) subset of some model point-set which
  matches a subset of the scene point-set?

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Outline

• Key idea
• General Framework
• Recognition under Various Transformations
• Recognition of 3D Objects from 2D Images
• Recognition of Polyhedra Objects
• Comparisons
• Alignment
• Generalized Hough Transform

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Key Idea (1/8)
Recognizing a pentagon in an image
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Key Idea (2/8)
Blue 1
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Key Idea (3/8)
Red 1
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Key Idea (4/8)
Green 5
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Key Idea (5/8)
Purple 1
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Key Idea (6/8)
Brown 1
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Key Idea (7/8)
Blue 1 Red 1 Green 5 Purple 1 Brown 1
Object is a pentagon!
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Key Idea (8/8)
Blue 1 Red 2 Green 2 Purple 1 Brown 1
Object is NOT a pentagon!
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Brute Force Recognition

• Let m points on the model,
• n points on the scene.
• Recognize a single model O((m x n)2 x t)
• where t is the complexity to verify the
• model against the scene.
• If mn, and tn, then we have O(n5) to recognize
  a single model.

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General Framework (1/2)

• Two stages algorithm
• Preprocessing (for each model)
• For each feature points pair
• Define a local coordinate basis on this pair.
• Compute and quantize all other feature points in
  this coordinate basis.
• Record (model, basis) in a hash table.

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General Framework (2/2)

• Online recognition (given a scene, extract
  feature points)
• Pick arbitrary ordered pair
• Compute the other points using this pair as a
  basis.
• For all the transformed points, vote all records
  (model, basis) appear in the corresponding entry
  in the hash table, and histogram them.
• Matching candidates (model, basis) pairs with
  large number of votes.
• Recover the transformation that results in the
  best least-squares match between all
  corresponding feature points.
• Transform the features, and verify against the
  input image features (if fails, repeat to 1).

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Two Stages Algorithm (1/2)
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Two Stages Algorithm (2/2)
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Complexity

• Assume mn, and k is the number of point to
  define the basis.
• Preprocessing O(nk1) for a single model.
• Recognition O(nk1) against all objects in the
  database.

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Under Various Transformations (1/2)

• Translation in 2D and 3D.
• 1-point basis.
• O(n2).
• Similarity transformation in 2D.
• 2-point basis.
• O(n3).
• Similarity transformation in 3D.
• 3-point basis.
• O(n4).

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Under Various Transformations (2/2)

• Affine transformation
• 3-point basis.
• O(n4)
• Projective transformation
• 4-point basis.
• O(n5)

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Recognition of 3D Objects from 2D Images (1/5)

• Correspondence of planes
• Preprocessing consider planar sections of the 3D
  object which contain three of more interest
  points.
• Hash (model, plane, basis) triplet.
• Use either projective transformation or affine
  transformation.
• Once the planes correspondence have been
  established, the position of the entire 3D body
  is solved.

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Recognition of 3D Objects from 2D Images (2/5)

• Singular affine transformation
• A x b U where
• A 2x3 affine matrix
• x 3x1 3D vector
• b 2x1 2D translation vector
• U 2x1 image

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Recognition of 3D Objects from 2D Images (3/5)

• A set of four non-coplanar points in 3D defines
  a 3D affine basis
• One point as origin
• The vectors between origin and the other three
  points as the unit (oblique) coordinate system.
• Preprocess the model points in this four-basis
  point.

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Recognition of 3D Objects from 2D Images (4/5)

• Recognition
• Pick four points p0, p1, p2, and p3 --gt three
  vectors v1, v2, and v3 in the 2D image.
• Exists ? v1 ? v2 ? v3 0, where (?, ?, ?) ?
  0
• A point p in the image, with v be the vector from
  p0 to p.
• Vote for all t ? 0 (a line with parameter t)
• v (?t?) v1 (? t?) v2 (t?) v3, where (?,
  ?) is the coordinate of v in the v1, v2 basis.

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Recognition of 3D Objects from 2D Images (5/5)

• Establishing a viewing angle with similarity
  transformation.
• Tesselate a viewing sphere (uniform in spherical
  coordinates).
• Record (model, basis, angle) in the hash table.
• 2-point basis O(n3) (the same order as without
  viewing angle because the viewing angle
  introduces only a constant factor -- independent
  of the scene).

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Recognition of Polyhedral Objects

• Polygonal objects
• Choose an edge as the basis, record (model, basis
  edge) in the hash table.
• Preprocessing and recognition is O(n2).

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Comparisons (1/2)

• With alignment method.
• Use exhaustive enumeration of all possible pairs
  in the objects and the images.
• Geometric hashing can process all models
  simultaneously, while the alignment method
  processes models sequentially.
• The alignment method does not require any
  additional memory, while geometric hashing
  requires a large memory to store hash table.
• Geometric hashing more efficient if
• The scene contains enough features (6-10) for
  efficient recognition by voting.
• There are many models.

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Comparisons (2/2)

• With Generalized Hough Transform (GHT).
• GHT quantizes all possible (continuous)
  transformations between the model and the scene
  into a set of bins, while
• Geometric Hashing quantizes just the (discrete)
  transformation represented by the basis.

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Summary

• Ability to recognize objects that have undergo an
  arbitrary transformation.
• Can perform partial matching.
• Efficient and can be parallelized easily.
• Use transformation-invariant access key to the
  hash table.
• Two phases (preprocessing and recognition).
• Require a large memory to store hash table.

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References

• 1 Yehezkel Lamdan and Haim J. Wolfson,
  Geometric Hashing A General and Efficient
  Model-Based Recognition Scheme, ICCV, 1988.