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

tbstbsd

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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))(2/)(),(2/)((

))(,(

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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