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

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
• Elastic model-based matching
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
• 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 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
• Voronoi surface d(x) = {(x, d(x))|xR2}
Computing Hausdorff distance
• The forward distance for a transformation t = (tx, ty, sx, sy)
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
• Claim 1.

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

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

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 Approach
• 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
• 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
• 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 space
• Restrictions where A[k,l], 0kma, 0lna
Rasterizing transformation space
• Forward distance hK(t(B),A)
• Bidirectional distance
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
• 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 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
• Camera images
• Edge detection
Examples
• Engineering drawing
• Find circles
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