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 Hausdorff Distance

  • Registration methods

    • Correlation and sequential methods

    • Fourier methods

    • Point mapping

    • Elastic model-based matching


The hausdorff distance
The Hausdorff Distance 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • Voronoi surface d(x) = {(x, d(x))|xR2}


Computing hausdorff distance1
Computing Hausdorff distance Hausdorff Distance

  • The forward distance for a transformation t = (tx, ty, sx, sy)


Comparing portion of shapes
Comparing Portion of Shapes Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • Restrictions where A[k,l], 0kma, 0lna


Rasterizing transformation space2
Rasterizing transformation space Hausdorff Distance

  • Forward distance hK(t(B),A)

  • Bidirectional distance


Reverse distances in cluttered images
Reverse distances in cluttered images Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • 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 Hausdorff Distance

  • Camera images

    • Edge detection


Examples1
Examples Hausdorff Distance

  • Engineering drawing

    • Find circles


Conclusion
Conclusion Hausdorff Distance

  • 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