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

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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

- Registration methods
- Correlation and sequential methods
- Fourier methods
- Point mapping
- Elastic model-based matching

- Given two finite point sets A = {a1,…,ap} and B = {b1,…,bq}
H(A, B) = max(h(A, B), h(B, A))

where

- 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))

- Forward distance
- fB(t) = h(t(B), A)
- a hypothesize

- reverse distance
- fA(t) = h(A, t(B))
- a test method

- Voronoi surface d(x) = {(x, d(x))|xR2}

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

- Partial distance
- The partial bidirectional Hausdorff distance
- HLK(A,t(B))=max(hL(A,t(B)), hK(t(B),A))

- Claim 1.
0bxxmax, 0byymax for all b=(bx,by)B,

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

- 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))

- 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

- A = {a1,…,ap} and B = {b1,…,bq} where each point aA, bB 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

- Translations
- An accuracy of one pixel

- Scales
- b=(bx,by) B,0bxxmax, 0byymax
- 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)

- Restrictions where A[k,l], 0kma, 0lna

- Forward distance hK(t(B),A)
- Bidirectional 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

- 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

- 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 ’

- Camera images
- Edge detection

- Engineering drawing
- Find circles

- 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