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

btaABthtf

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))

],[],[],[],[ 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

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highx

lowxc ssssttttt

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

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

jnijmi

xay

jj

axy

yjys

xjxs

00

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

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

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 iybjixbjDbx

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