a multi-resolution technique for comparing images using the hausdorff distance daniel p....
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A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance
Daniel P. Huttenlocher and William J. RucklidgeNov 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
baBAhBbAa
minmax),(
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))|xR2}
xaxdbxxdAaBb
min)(',min)(
))('max),(maxmax(),( bdadBAHBbAa
Computing Hausdorff distance
The forward distance for a transformation t = (tx, ty, sx, sy)
),('max
),(minmax
)(minmax)),(()(
yyyxxxBb
yyyxxxAaBb
AaBbB
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tbstbsa
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Comparing Portion of Shapes
Partial distance
The partial bidirectional Hausdorff distance HLK(A,t(B))=max(hL(A,t(B)), hK(t(B),A))
baKABthAaBtb
thK
min)),(()(
A Multi-Resolution Approach
Claim 1.0bxxmax, 0byymax for all b=(bx,by)B,
t = (tx, ty, sx, sy),
)'',''()',( maxmax yyyyxxxx ttyssttxsstt
)',(' tt
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))
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))(,(
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A Multi-Resolution Approach
1. Start with a rectilinear region(cell) which contains all transformations
2. For each cell,If
Mark this cell as interesting
3. Make a new list of cells that cover all interesting cells.
4. Repeat 2,3 until the cell size becomes small enough
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))(,(
maxmaxlowy
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The Hausdorff distance for grid points
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
Rasterizing transformation space
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)
Rasterizing transformation space
Restrictions where A[k,l], 0kma, 0lna
yay
xax
y
x
yy
xx
jni
jmi
x
ay
j
j
ax
y
yjys
xjxs
0
0
max
maxmax
maxmax
max
maxmaxmin
maxmaxmin
Rasterizing transformation space
Forward distance hK(t(B),A)
Bidirectional distance
]/,/['],,,[ maxmax yyyxxxBb
thyxyxB iybjixbjDKjjiiF
]),,,[],,,,[max(],,,[ yxyxByxyxAyxyx jjiiFjjiiFjjiiF
Reverse distances in cluttered images
],['],[],,,[),(
lkDlkALjjiiF
yyy
xxxjilijiki
lk
thyxyxA
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 ’
)1]/,/[',0max( maxmaxmax yyyxxxxx
x iybjixbjDb
x
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
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