Object Recognition

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Object Recognition

• Concettina Guerra

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Matching objects
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Model-based Object RecognitionShape Retrieval
Images from http//www.cs.princeton.edu/gfx/proj/
shape/
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Issues

• How to represent a shape?
• Variety of shape descriptors
• Shape histograms
• Spin images, shape contexts, shape
  distributions,..
• Spherical harmonics
• Reflective Symmetry Descriptor
• Skeletal Graphs
• Aspect Graphs
• Tradeoff simplicity vs accuracy

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Issues

• How to match shapes?
• Graph-based approaches
• Interpretation trees
• .
• How to efficiently retrieve a shape from a
  database of objects?
• Indexing techniques

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A simple object representation

• An object is given as a set of points (features)
• The object can be in 2D space or 3D space
• (for instance, the set of points is extracted
  from an intensity image or from a range image)

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Geometric Matching ofPoint Clouds
Two input sets
Two sets after superimposition
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Transformations

• Translation
• Translation and Rotation
• Rigid Motion (Euclidian Trans.)
• Translation, Rotation Scaling

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Inexact Matching Simple case two closely
related sets with the same number of points .
Assume transformation T is given
Question how to measure a matching error?
( slides by Prof. Haim Wolfson).
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Distance Functions

• Two point sets Aai i1n
• Bbj j1m
• Pairwise Correspondence
• (ak1,bt1) (ak2,bt2) (akN,btN)

(1) Bottleneck max aki bti (3) RMSD
(Root Mean Square Distance) Sqrt(
Saki bti2/N)
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Hausdorff DistanceAnother distance function

• Let A a1, a2, ..., am B b1, b 2, ..., bn
  be sets of either points or segments.
• Definition. (Hausdorff Distance)
• H(A, B) max (h(A, B), h(B, A))
• where the one-way Hausdorff distance is
• h(A, B) maxa minb r (a, b)
• where a (b) is a point of A (B) and r (a, b), is
  a metric.

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Correspondence is Unknown
Given two configurations of points in the
three dimensional space,
find those rotations and translations of one
of the point sets which produce large
superimpositions of corresponding 3-D
points.
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Largest Common Point Set (LCP) problem
Given egt0 and two point sets A and B find a
transformation T and equally sized subsets A (a
subset of A) and B (a subset of B) of maximal
cardinality such that dist(A,T(B)) e.
Bottleneck metric optimal solution in O(n32.5)
C. Ambuhl et al. 2000
RMSD metric open problem
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Exact solution in 2Dusing Hausdorff Distance

• This problem is generally solved as a problem of
  intersection of unions of disks in the
  transformation space.
• Time complexity O( m3 n3 log2nm) in R2

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Two instances of the problem

• Similarity of the two sets of atoms with known
  correspondences
• Aai , Bbi , i1,,n
• ai ?? bi
• Similarity of the two sets of atoms with unknown
  correspondences
• Aai , Bbj , i1,,n j1,,m
• ai(k) ?? bj(k) k1,,Kltn,m

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Superposition RMSD

• Given two sets of 3-D points with known
  correspondences
• Aai , Bbi , i1,,n
• find a 3-D rotation R and translation T that
  minimizes
• D2minR,T Si Rai T - bi 2
• RMSDD / sqrt(n)
• A closed form solution exists for this task.

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Orthogonal Procrustes problem

• The Solution is based on Singular Value
  Decomposition (SVD) of the correlation matrix A
  of the points
• Aij Sk ak ibk j
• where ak i is the ith component of the vector
  ak
• The solution involves eigenvalue analysis of a
  correlation matrix of the points.

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Finding Correspondences

• Geometric Hashing, Indexing
• (Wolfson et al., 1998)
• Two steps
• Obtain Hypotheses of Correspondences
• Verify Hypotheses

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Hashing function

• From an Object
• To invariant Features
• To t-ples of numbers
• To indeces
• Use the t indeces to access a t-dimensional hash
  table

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Indexing Methodsfor Fast retrieval of 3D
patterns

• Select a set of target objects
• Create and store a hash table indexed by
  invariant geometric properties of the selected
  objects
• Update the databases as new objects are found
• Use the table to identify the most similar
  object for a target object.

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Reference Frame

• A 3-D reference frame (r. f.) can be defined by
  three non collinear points
• Invariant
• the coordinates of any other point in the r.f.

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Geometric Hashing PreprocessingTable
Construction

• Pick a reference frame.
• Compute the coordinates of all the other points
  in this reference frame.
• Use the triplet of coordinate as an index to
  the hash (look-up) table and record in that entry
  the record
• (object, ref. Frame)
• Repeat above steps for each reference frame.

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A note

• The coordinates are invariant under rigid
  transformations (obvious) as well as under affine
  transformations
• In other words, the coordinates are independent
  of the view (i.e. if we compute them in the model
  plane or in some affine view we obtain the same
  value)

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Geometric Hashing - Recognition

• For the target object do
• Pick a reference frame satisfying pre-specified
• constraints.
• Compute the coordinates of all other points in
• the current reference frame.
• Use each coordinate to access the hash-table
• to retrieve all the records (prot., r.f., shape
• sign., pt.).

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Geometric Hashing - Recognition

• For records with matching shape sign. vote for
  the (object, r.f.).
• Compute the transformations of the high
  scoring hypotheses.
• Repeat the above steps for each r.f.

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Issues

• For the query object how many reference frames do
  we need to consider?
• What is a good selection for a reference frame?
• How does the distribution of geometric invariants
  affect the matching?

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Complexity of Geometric Hashing

• Preprocessing
• O(Nn4)
• Match Detection/Recognition
• O(Rns).
• N- number of objetcs.
• O(n)- no. of features in an object.
• s - size of a hash-table entry.
• R size of the hash table