Unconstrained Automatic Image Matching Using Multiresolutional CriticalPoint Filters Yoshihisa Shina

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Unconstrained Automatic Image Matching Using
Multiresolutional Critical-Point
FiltersYoshihisa Shinagawa, Tosiyasu L. Kunii

• Jo, Young-Gwan
• Computer Vision Lab.
• CSE. POSTECH

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Introduction

• Image Matching
• point, line, high-level feature matching
• generating other views, morphing, volume
  rendering, 3D reconstruction, tracking
• Hierarchical Matching
• multiresolution
• global matching

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Notation

• Level m the size of each image at the mth level
  is 2m?2m.
• a pixel at the location (i,j) of the
  image at mth level. s0,1,2,3 indicating the
  kind of the critical-point filter.
• intensity value of the pixel
• submapping
  between p(m,s) and q(m,s)
• is the parent of .
  is the child of .
• ?,? weighting parameters, ?, ? penalty for
  violating BC

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Critical-Point Filter

• Maximum point
• Minimum point
• Saddle points

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Overview of Image MatchingUsing
Multiresolutional Critical-Point Filters

• Making subimages
• critical-point filters
• Selecting candidates for mapping ( or matching )
• inherited quadrilateral
• checking BC
• if no pixel that satisfies BC exists
• expand quadrilateral
• abandon the third of BC
• abandon the first and second of BC
• Determining mapping
• computing candidate minimizing energy ( or cost )

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Bijectivity Conditions

• The edges of the quadrilateral f (m,s)(R) should
  not intersect one another.
• The orientation of the edges of should be the
  same as that of R ( clockwise or counterclockwise
  ).
• The length of one edge of f (m,s)(R) can be zero
  to allow mappings that are retractions i.e., f
  (m,s)(R) may be a triangle. It is not allowed,
  however, to be a point nor a line segment (
  figures of area 0 ).

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The Energy of the Mapping

• Difference in the intensity of matched points
• Difference in the location of matched points

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Cost Related to the Pixel Intensity

• The energy
• The total energy

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Cost Related to the Locationsof the Pixel for
Smooth Mapping

• The energy
• The total energy

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Total Energy of the Mapping

• The goal is to find a mapping that gives the
  minimum energy

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Determining the Mappingwith Multiresolution

• The value f (m,s)(i,j) (k,l) is determined
  using f (m-1,s) ( m1,2,,n ).
• should be inside the inherited
  quadrilateral.
• Inherited quadrilateral

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The New Energy E0

• The energy of submapping f(m,0)
• The energy of submapping f(m,s) (s1,2,3)

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Automatic Determinationof the Optimal Parameter
Values

• Dynamic determination of ?
• When we increase ?, a change in the mapping is
  not possible until becomes greater
  than one.
• If exceeds one, some pixels may move
  to a stabler state.
• When ? tries to go beyond the optimal value,
  however, the mapping becomes excessively
  distorted, and increases rapidly. In
  such a case, the total energy
  is dominated by . The system tries to
  reduce the total energy by decreasing
  regardless of . Thus, begins to
  increase.
• Dynamic determination of ?
• When ? is zero, is determined irrespective
  of the previous submapping and the present
  submapping can be elastically deformed and
  becomes too distorted.
• When ? is very large, is completely
  determined by the previous submapping i.e., the
  submappings are very stiff, and the pixels are
  mapped to the same locations.

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Implementation

• The values of f (m,s)(i,j) (m0,,n) that satisfy
  BC are chosen from the candidates (k,l) by
  awarding a penalty to the candidates that violate
  BC.
• In the implementation
• The penalty for violating the third condition ?2
• The penalty for violating the first and second
  condition ?100000
• If z-component of W is less than 0, the candidate
  is awarded a penalty by multiplying by ?.

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Problems

• Because of bijectivity condition, the pixels on
  the borders of the source image are mapped to the
  pixels on the borders of the destination image.
  This causes artifacts near the borders of the
  resulting images.
• When there are bifurcations of regions in the
  images, it is useful to abandon BC at bifurcation
  points.