a multi resolution technique for comparing images using the hausdorff distance
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A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance. Daniel P. Huttenlocher and William J. Rucklidge Nov 7 2000 Presented by Chang Ha Lee. Introduction. Registration methods Correlation and sequential methods Fourier methods Point mapping

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a multi resolution technique for comparing images using the hausdorff distance

A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance

Daniel P. Huttenlocher and William J. Rucklidge

Nov 7 2000

Presented by Chang Ha Lee

introduction
Introduction
  • Registration methods
    • Correlation and sequential methods
    • Fourier methods
    • Point mapping
    • Elastic model-based matching
the hausdorff distance
The Hausdorff Distance
  • Given two finite point sets A = {a1,…,ap} and B = {b1,…,bq}

H(A, B) = max(h(A, B), h(B, A))

where

with transformations
With transformations
  • A: a set of image points
  • B: a set of model points
  • transformations: translations, scales
    • t = (tx, ty, sx, sy)
    • w = (wx, wy) => t(w) = (sxwx+tx, sywy+ty)
    • f(t) = H(A, t(B))
with transformations1
With transformations
  • Forward distance
    • fB(t) = h(t(B), A)
    • a hypothesize
  • reverse distance
    • fA(t) = h(A, t(B))
    • a test method
computing hausdorff distance
Computing Hausdorff distance
  • Voronoi surface d(x) = {(x, d(x))|xR2}
computing hausdorff distance1
Computing Hausdorff distance
  • The forward distance for a transformation t = (tx, ty, sx, sy)
comparing portion of shapes
Comparing Portion of Shapes
  • Partial distance
  • The partial bidirectional Hausdorff distance
    • HLK(A,t(B))=max(hL(A,t(B)), hK(t(B),A))
a multi resolution approach
A Multi-Resolution Approach
  • Claim 1.

0bxxmax, 0byymax for all b=(bx,by)B,

t = (tx, ty, sx, sy),

a multi resolution approach1
A Multi-Resolution Approach
  • If HLK(A, t(B)) = >, then any transformation t’ with (t,t’) < - cannot have HLK(A, t’(B))  
  • A rectilinear region(cell)
  • The center of this cell
  • If

then no transformation t’R have HLK(A, t’(B))  

a multi resolution approach2
A Multi-Resolution Approach
  • Start with a rectilinear region(cell) which contains all transformations
  • For each cell,

If

Mark this cell as interesting

  • Make a new list of cells that cover all interesting cells.
  • Repeat 2,3 until the cell size becomes small enough
the hausdorff distance for grid points
The Hausdorff distance for grid points
  • A = {a1,…,ap} and B = {b1,…,bq} where each point aA, bB has integer coordinates
  • The set A is represented by a binary array A[k,l] where the k,l-th entry is nonzero when the point (k,l) A
  • The distance transform of the image set A
    • D’[x,y] is zero when A[k,l] is nonzero,
    • Other locations of D’[x,y] specify the distance to the nearest nonzero point
rasterizing transformation space
Rasterizing transformation space
  • Translations
    • An accuracy of one pixel
  • Scales
    • b=(bx,by) B,0bxxmax, 0byymax
    • x-scale: an accuracy of 1/xmax
    • y-scale: an accuracy of 1/ymax
    • Lower limits: sxmin ,symin
    • Ratio limits: sx/sy > amax or sy/sx > amax
  • Transformations can be represented by
    • (ix, iy, jx, jy) represents the transformation (ix, iy, jx /xmax, jy /ymax)
rasterizing transformation space1
Rasterizing transformation space
  • Restrictions where A[k,l], 0kma, 0lna
rasterizing transformation space2
Rasterizing transformation space
  • Forward distance hK(t(B),A)
  • Bidirectional distance
reverse distances in cluttered images
Reverse distances in cluttered images
  • When the model is considerably smaller than the image
    • Compute a partial reverse distance only for image points near the current position of the model
increasing the efficiency of the computation
Increasing the efficiency of the computation
  • Early rejection
    • K = f1q where q = |B|
    • If the number of probe values greater than the threshold exceeds q-K, then stop for the current cell
  • Early Acceptance
    • Accept “false positive” and no “false negative”
    • A fraction s, 0<s<1 of the points of B
    • When s=.2
      • 10% of cells labeled as interesting actually are shown to be uninteresting
increasing the efficiency of the computation1
Increasing the efficiency of the computation
  • Skipping forward
    • Relies on the order of cell scanning
    • Assume that cells are scanned in the x-scale order.
    • D’+x[x,y]: the distance in the increasing x direction to the nearest location where D’[x,y]’
    • Computing D’+x[x,y]: Scan right-to-left along each row of D’[x,y]
    • FB[ix,iy,jx,jy],…,FB[ix,iy,jx+-1,jy] must be greater than ’
examples
Examples
  • Camera images
    • Edge detection
examples1
Examples
  • Engineering drawing
    • Find circles
conclusion
Conclusion
  • A multi-resolution method for searching possible transformations of a model with respect to an image
  • Problem domain
    • Two dimensional images which are taken of an object in the 3-dimensional world
    • Engineering drawings
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