Summation invariant and its application to object recognition

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Summation invariant and its application to object
recognition

• Wei-Yang Lin, Nigel Boston, Yu Hen Hu
• March 21, 2005

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Agenda of today

• Introduction
• Transformation group
• Invariant of transformation group
• Differential invariant
• Integral invariant
• Summation invariant
• Novel shape descriptor
• Experiment on fish recognition

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Introduction

• A fundamental issue in computer vision is to deal
  with images of objects under geometrical
  transformations
• Our objective is to develop robust shape
  descriptor that is invariant to certain
  geometrical transformation

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Transformation Group

• Image of objects are subject to geometric
  distortion.
• Transformation groups shown here are important in
  computer vision

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Invariant of transformation group

• Traditional method
• Moment invariant, Fourier descriptors, Wavelet
  invariant, Differential invariant, Integral
  invariantetc.
• We propose a method based on summation operation
  and demonstrate its superior discriminating power.

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Differential Invariant

• Affine transformation acting on plane
• We define Jet space as
• Find invariant in higher dimensional space

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Differential Invariant

• By method of normalization, we can solve for
  moving frame.
• We can find invariant by substituting moving
  frame into

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Differential Invariant

• Pro you can easily find invariant for specific
  transformation
• Con Higher order derivatives are sensitive to
  noise and of little practical use.

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Integral Invariant

• Hann and Hickman 2002 prolong group action to
  space defined by potentials.
• New coordinates are defined by
• where (x0 ,y0) is initial point and (x1 ,y1)
  is end point

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Integral Invariant

• For affine transformation acting on plane
• Its integral invariant

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Integral Invariant

• Pro Compared with differential invariant, it
  will be less sensitive to noise.
• Con Given a shape, its difficult to find its
  analytical expression. Numerical approximation of
  integral is easier but it produces error.

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Summation Invariant

• Define potential as adding up finite number of
  points.

• Prolong group action to potential jet space J
  (x0, y0, xN-1, yN-1, a, b, Ixx, )

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Summation Invariant

• For affine transformation
• Its summation invariant

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Summation Invariant

• Its not simply discretization of integral
  invariant because
• Shape is defined differently.
• Jet space is defined differently.
• Like integration, summation operation will
  decrease noise level.

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Semi-local Summation Invariant

• Summation invariant is a map from to .
• Problem
• Space is too small to perform accurate
  recognition.
• Solution
• Define summation invariant locally to extract
  local feature and also expand the space of
  feature vector

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Semi-local Summation Affine Invariant
For affine transformation, we can define
semi-local summation invariant as Where  
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Semi-local Summation Affine Invariant
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Experiment Results

• Recognize fish under affine transformation.
• Use fish contour from SQUID database.
• Resample fish contour such that N 512
• Calculate semi-local summation invariant with M
  51.

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Generate Fish Database

• Choose first 100 fish contours from SQUID
  database.
• Apply affine transformation to fish contour.
• parameters are randomly generated
• initial point is also randomly chosen
• Quantization and remove redundant points
• Resample fish contour such that the length of
  contour is 512

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Fish Database

• 100 distinct types
• For each type, we apply affine transformations to
  generate 20 variations.

20 distinct types from database
Each type has 20 variations
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Fish Database

• Add Gaussian distributed noise to fish contour

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Error Rate of Fish Recognition

• denote the standard deviation of Gaussian
  distributed noise,
• which is added on fish contours.

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Conclusion

• New solution to problem of recognizing objects
  under transformation
• Affine invariant is derived
• Compared to traditional method
• Superior discriminating power
• Better noise immunity