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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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Introduction Registration methods
Correlation and sequential methods Fourier methods Point mapping Elastic model-based matching
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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),(
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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))
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With transformations Forward distance
fB(t) = h(t(B), A) a hypothesize
reverse distance fA(t) = h(A, t(B)) a test method
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Computing Hausdorff distance Voronoi surface d(x) = {(x, d(x))|xR2}
xaxdbxxdAaBb
min)(',min)(
))('max),(maxmax(),( bdadBAHBbAa
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Computing Hausdorff distance The forward distance for a
transformation t = (tx, ty, sx, sy)
),('max
),(minmax
)(minmax)),(()(
yyyxxxBb
yyyxxxAaBb
AaBbB
tbstbsd
tbstbsa
btaABthtf
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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)),(()(
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A Multi-Resolution Approach Claim 1.
0bxxmax, 0byymax for all b=(bx,by)B,t = (tx, ty, sx, sy),
)'',''()',( maxmax yyyyxxxx ttyssttxsstt
)',(' tt
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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))
],[],[],[],[ highy
lowy
highx
lowx
highy
lowy
highx
lowx ssssttttR
))(2/)(),(2/)((
))(,(
maxmaxlowy
highy
lowy
highy
lowx
highx
lowx
highx
cLK
ttssyttssx
BtAH
)2/)(,2/)(,2/)(,2/)(( highy
lowy
highx
lowx
highy
lowy
highx
lowxc ssssttttt
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A Multi-Resolution Approach1. Start with a rectilinear region(cell) which
contains all transformations2. For each cell,
If
Mark this cell as interesting3. Make a new list of cells that cover all
interesting cells.4. Repeat 2,3 until the cell size becomes small
enough
))(2/)(),(2/)((
))(,(
maxmaxlowy
highy
lowy
highy
lowx
highx
lowx
highx
cLK
ttssyttssx
BtAH
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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
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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)
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Rasterizing transformation space Restrictions where A[k,l], 0kma, 0lna
yay
xax
y
x
yy
xx
jnijmi
xay
jj
axy
yjys
xjxs
00
max
maxmax
maxmax
max
maxmaxmin
maxmaxmin
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Rasterizing transformation space Forward distance hK(t(B),A)
Bidirectional distance
]/,/['],,,[ maxmax yyyxxxBb
thyxyxB iybjixbjDKjjiiF
]),,,[],,,,[max(],,,[ yxyxByxyxAyxyx jjiiFjjiiFjjiiF
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Reverse distances in cluttered images
],['],[],,,[),(
lkDlkALjjiiF
yyyxxxjilijiki
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
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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
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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 iybjixbjDbx
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Examples Camera images
Edge detection
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Examples Engineering drawing
Find circles
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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