Hausdorff distance-based multiresolution maps
applied to image similarity measure
E. Baudrier*a, G. Millonb, F. Nicolierb, R. Seulinc and S. Ruanb
aLMA – University of La Rochelle, Avenue Cre´peau, 17000 La Rochelle, France
bCReSTIC – URCA, IUT, 9, rue de Que´bec, 10026 Troyes Cedex, France
cLe2i – CNRS UMR 5158, University of Burgundy – IUT, 12, rue de la fonderie, 71200 Le Creusot,
France
Abstract: Image comparison is widely used in image processing. For binary images that are
not composed of a single shape, a local comparison can be interesting because the features are
usually poor (colour) or difficult to extract (texture, forms). Thus a new binary image
comparison method that uses a windowed Hausdorff distance is presented. It enables local
dissimilarities to be quantified in a simple way. The comparison results in a dissimilarity map.
These maps are then used to evaluate the image similarity. The evaluation uses a classification
step that is based on a comparison of the dissimilarity map histogram with reference histograms.
The comparison is carried out at different scales of a multiresolution analysis, allowing the most
discriminating scale in a user-defined notion of dissimilarity to be chosen automatically in the
learning step. As an application, a database of digitalized ancient illustrations is successfully
processed by the new method.
Keywords: image comparison, binary images, dissimilarity map, multiresolution, mathema-
tical morphology
1 INTRODUCTION
Image retrieval is an active domain. Retrieving
images by their content, as opposed to metadata,
has become an important activity. It is classically
composed of two stages: first, information mining,
which results in an image signature and, secondly, a
signature distance measure which is used to decide on
the image similarity. In this process, the signature
must capture conspicuous features in order to be as
discriminating as possible in a user-defined sense. In
general, the signature contains colour, shape or
texture information.1 For the binary images that are
not composed of a single shape, the colour attributes
are poor. Often object shapes cannot be clearly
identified. In the scope of binary images, it is
proposed to replace this information mining by a
straight image comparison based on a modified
Hausdorff distance (HD),2 producing a dissimilarity
map (DM). The second step is then replaced by a
classification process based on the DM. While
information mining requires a priori knowledge on
discriminating features before comparing the images,
this process first expresses dissimilarities from the
image comparison before taking a decision. This
process developed for binary images can be adaptable
to pattern recognition. In the present paper, the
different stages of the measuring process are pre-
sented: first, a morphological multiresolution analysis
(MRA) (section 2) and, secondly, the construction of
DM between the two images (section 3) at each scale.
Then, the choice of the most discriminating MRA
scale and a decision on the similarity of the images
based on the DM classification at this scale is
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The MS was accepted for publication on 30 March 2006.
* Corresponding author: E. Baudrier, LMA – University of La
Rochelle, Avenue Cre´peau, 17000 La Rochelle, France; e-mail:
etienne.baudrier@univ-lr.fr
1
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presented at the end of section 5. Finally, some results
are presented in section 6 to show the efficiency of
this method.
2 MULTIRESOLUTION
The dissimilarity-measure definition depends on the
database considered and on the degree of similarity
chosen by the user (coarse or fine) at the same time.
These parameters are taken into account thanks to a
MRA during the learning step. This step determines
first the best discriminating MRA scale for the
classification and then the class reference histograms
at this scale. This section gives a general presentation
of the MRA, and then the chosen non-linear MRA
filter is introduced.
2.1 MRA for multidimensional signals
The MRA is directly transposed from the 1D case
described by Meyer3 and Mallat.4 In this paragraph,
the properties and definitions are valid whatever the
space dimension of the given functions (2D, 3D and
so on). The processed functions are square-integrable
over RN, so the signal f belongs to L2(RN) (N52 for
2D signals). A MRA is a sequence of successive
approximations spaces Vj satisfying the following
properties:
Vj[Z, Vj5Vj{1 with |
j[Z
Vj~L2(RN) and
\
j[Z
Vj~ 0f g (1)
f (x)[Vjuf (J jx)[V0,Vj[Z (2)
f (x)[V0[f (x{k)[V0,Vk[ZN (3)
AQ[V0, so that fQ0,k,k[Zg is an orthonormal
basis of V0 (4)
where Qj,k(x)5|detJ|
2j/2Q(J2jx2k), ;jZ, ;kZN
where J is defined by
x0
y0


~J
x
y


(5)
J is an N6N matrix (262 in 2D) with integer
coefficients (also for |det J|?J21). The scaling factor is
|det J|.
One can show that a base Wj is given by
translations of |det J|–1 families of wavelets func-
tions. So for one given scale, there are |det J|–1 detail
signals dj,i, and one approximation signal aj.
The multidimensional Mallat’s recursive algorithm
can be generalized without major modifications:
aj~
_
h
aj1
dj,k~
_
gk
aj1
(6)
where gk, h¯ are multidimensional kernels filters and *
denotes the convolution operator. h¯ is a low-pass
filter, and gk are high-pass filters.
2.2 Non-linear MRA
An extension of the MRA to the non-linear case is
presented here. Indeed, in the present study, the MRA
filter should satisfy conditions to preserve the binary
image main features. Many classical MRA filters use
Gaussian functions. They have good space-scale
properties, but they smooth transitions and could
yield to new structures at low resolutions. This
drawback is shared by the other linear filters. In
contrast, non-linear filters can avoid this problem.
Among them, morphological filters are good candi-
dates.5 Here, three criteria have been determined for
the morphological MRA filter t choice: t should be
edge-preserving, t should be auto-dual (i.e. t preserves
the black-to-white pixels ratio), and t should preserve
the ‘main’ features. Obviously, the last criterion is
subjective and is satisfied a posteriori. This led to the
choice of the morphological so-called median filter,
which fulfils these conditions.6 The morphological
MRA is thus described by the following process.
1. Non-linear median filtering of the approximation
aj
2. Down-sampling by a factor 2, giving aj21
3. Repeat the process up to scale J.
Let I be the image to treat; the analysis filter results in
an approximation image A and three detail images
Dv, Dd, Dh corresponding to the vertical, diagonal
and horizontal details. The coefficients A(k,l), Dv(k,l),
Dd(k,l), Dh(k,l) are computed with the following
operations:
b~sort½I (2i1,2j1), I (2i1,2j1),
I (i1,j), I (i,2j1), I (2i,2j) (7a)
A(k,l)~b (3) (7b)
Dv (k,l)~ I(2i1,2j1)I(2i1,2j)j j (7c)
Dd (k,l)~ I(2i1,2j1)I(2i,2j)j j (7d)
Dh (k,l)~ I(2i1,2j1)I(2i,2j1)j j (7e)
Two illustrations of the non-linear median filter are
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presented Figs 1 and 2. Figure 1 represents the same
image at four successive scales, from the finest
(Fig. 1a) to the coarsest (Fig. 1d). In the coarsest
one, the image quality is degraded; nevertheless the
main features are still visible. Figure 2 stands for the
first approximation of the same image and the three
detailed images at scale 1 (the detail images will not
be used afterwards).
3 OVERVIEW OF HAUSDORFF DISTANCE
3.1 Dissimilarity measure over binary images: choice
of Hausdorff distance
Among dissimilarity measures over binary images,
the HD has often been used in the content-based
retrieval domain and is known to have successful
applications in object matching2,7 or in face
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a b
c d
1 Four resolutions of same picture illustrating effect of non-linear median filter ((a) is original
image with all detail; (d) keeps only the image main features): (a) binarized original image at
resolution 0 (5126512 pixels); (b) resolution 1 (2566256 pixels); (c) resolution 2
(1286128 pixels); (d): resolution 2 (64664 pixels)
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recognition.8,9 It can be computed quickly using
Voronoi diagrams.10 A brief review of its definition
and properties follows.
Originally meant as a measure between two point
collections A and B in a metric space E (whose underlying
distance is d), it can be viewed as a dissimilarity measure
between two binary images A and B, considering A and
B, respectively, the black pixels of A and B. For finite sets
of points, the HD can be defined as:11
Definition 1. The Hausdorff distance
Given two finite sets of points A5(a1,…,an) and
B5(b1, …, bm), and an underlying distance d, the HD
is given by
DH(A,B)~max½h(A,B),h(B,A) (8)
where
h(A,B)~maxa[A minb[B d a,bð Þ½ f g (9)
h(A,B) is the so-called ‘directed HD’ and, in the
present paper, the underlying distance d(a,b) will be
the L‘ norm d(a,b)5||b2a||‘. For images, the same
notation is used: DH(A,B)5DH(A,B). The interest of
this measure comes first from its metric properties (in
this application, on the space of finite set of points):
non-negativity, identity, symmetry and triangle
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a b
c d
2 Result of non-linear median AMR: approximation 1 and corresponding three detail images: (a)
resolution 1 (2566256 pixels); (b) vertical detail; (c) horizontal detail; (d) diagonal detail
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inequality. These properties correspond to this
intuition for shape resemblance. Indeed, a pattern is
identical to itself. And the order of comparison does
not generally matter; for the case where the order of
comparison is important, the property of the HD
according to which the directed distance is not
symmetrical can be exploited. Finally, the triangular
inequality prevents some unknown patterns being
similar to two dissimilar patterns at the same time.
Moreover, the HD is a match methodology with-
out point-to-point correspondence, so it is robust to
local non-rigid distortions. Another source of interest
is the following property.
Proposition 1 (Translation): Let v be a vector of R2,
Tv translation of vector v, and A a non-empty finite
set of points, then
d(A,TvA)~ vk k (10)
It implies that for a small translation, the HD is
small, which matches the expectation for a dissim-
ilarity measure.
3.2 Some modified versions of the HD
The classical HD has good properties but it measures
the most mismatched points between A and B and, as
a consequence, it is sensitive to noise.12 Indeed
considering two images containing the same pattern
and one point added to the first image far from the
pattern, the HD will measure the distance between
the pattern and the point.
Several modifications of the HD have been
proposed to improve it, such as the partial HD,11
the modified HD (MHD),13 the censored HD
(CHD),12 the ‘doubly’ modified HD (M2HD),8 the
least trimmed squared HD (LTS-HD)14 and the
weighted HD (WHD).15 The next definitions are
detailed by Zhao et al.16
The directed distance of the partial HD is defined
by Huttenlocher et al.:11
hK(A,B)~K tha[Ad(a,B) (11)
where K tha[A denotes the Kth ranked value of d(a,B).
Thus, the PHD depends on a parameter p~ KNA
standing for the proportion of values taken into
account. The partial HD method yields good results
for an impulse noise case.
The directed distance of the MHD is defined by
Dubuisson and Jain:13
hMHD(A,b)~
1
NA
X
a[A
d(a,B) (12)
where NA5card(A). Unlike the partial HD, the MHD
measure does not require any parameter, and the
method adapts only to a Gaussian noise case.
The directed distance of the WHD is defined by
Zhao et al.:16
hWHD(A,B)~
1
NA
X
a[A
v(a):d(a) (13)
where
P
a[A
v(a)~NA. An image can be divided into
different parts, and the contribution of the different
parts to the image matching is different, therefore the
HD should be different. The WHD has been used in
the Chinese character image matching15,17 and in face
recognition.18
The directed distance of the CHD is defined by
Paumard:12
hk,l(A,B)~Ptha[AQthb[Bd(a,b) (14)
where Pth denotes the Pth ranked value for aA of
Qthb[Bd(a,b), with Q
th
b[Bd(a,b) representing the Qth
ranked value for bB of the underlying distance set.
Since the CHD ranks the underlying distance, the
effect of the impulse noise to the image is reduced.
The directed distance of the M2HD is defined by
Taka`cs:8
hM~
1
NA
X
a[A
d(a,B) (15)
with
d(a,B)~max(I :minb[NaB d(a,b),(1{I)P)
where NaB is a neighbourhood of point a in set B, and
I indicates whether a point b exists in NaB.
The directed distance of the LTS-HD is defined by
Sim et al.:14
hLTS(A,B)~
1
K
XK
1
d(a,B)(i) (16)
where K denotes k?NA, as in the partial HD case, and
d(a, B)(i) represents the ith distance value in the sorted
sequence d(a,B)(1)¡d(a,B)(2)¡ . . .¡d(a,B)(NA). The
measure of the LTS-HD is minimized by the
remaining distance values, after large distance values
are eliminated. Even if the object is occluded or
degraded by noise, this matching scheme yields good
results.
These measures are global and cannot account for
local dissimilarities. Indeed, the principle of HD is to
be a ‘max min’ distance, and this means that the value
of the HD between two images is reached for at least
a couple of points. But this does not say whether the
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value is reached in several parts or only for one pair,
which corresponds to different degrees of dissim-
ilarity. These remarks motivate us to design a local
HD in the next section.
4 HAUSDORFF DM
The main inconvenience of these HD measures lies in
the fact that they give a global dissimilarity measure
over images. To retrieve different images standing for
the same scene, the image comparison must be based
on the measure of local differences. To do so, a new
dissimilarity measure based on the HD is introduced
and designated DH,W. It consists in making a local
measure through a sliding window W. At each sample
point, the HD is computed on the part of the images
viewed through W. Thus, from two binary images to
be compared, a DM M is computed. M depends on
the parameters (wx,wy), the size of the window W and
p5(ph, pv), the step between the sample points.
4.1 Windowed HD
In this section, the notion of local dissimilarity is first
discussed, then a naive definition of a HD measure in
a local window is presented. The use of the HD in a
local window implies some modifications to make it
consistent when window size varies. A necessary
modification is brought. It finally leads to a fully
consistent definition. In this section, A and B depict
two non-empty finite sets of points of R2, and W a
convex closed subset of R2.
4.1.1 Naive definition
This consists in modifying the definition of the global
measure (Definition 1) by introducing a subset W
standing for the window.
Definition 2: HD in a window (naive)
HDW(A,B)~max(hW(A,B),hW(B,A)) (17)
where
hW(A,B)~ max
a[A\W
min
b[B\W
d a,bð Þ½ 
 
(18)
But this definition is not available for the possible
case where one set (for example B) has no point in W,
and the other one has. So as to make the value
consistent when W grows (see Fig. 3a,b), the distance
to the frontier of W must be taken into account.
4.1.2 Modification of the naive definition
Indeed, if there are points of A outside W and close to
its frontier (Fig. 3a), if W grows and includes them,
the new HDW will be equal to the distance to these
points. The most restrictive case will be when there
are points of A all over the frontier of W. Then, to
make it consistent with a bigger W, the computation
of HDW must include the distance of the points of the
two sets A and B to the edges of W. Thus, if B has no
point in W, the directed distances have the following
definition:
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3 Examples of critical cases for local HD: (a) B>W5Ø, naive measure is not defined; (b) to
define HDW and to make it consistent where W increases, distance to frontier is proposed; (c,
d) definition must then be modified to be consistent when W slides in R2.
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Definition 2: First modification
HDW(A,B)~max(hW(A,B),hW(B,A)) (19)
where there are two cases:
If A>W?Ø and B>W?Ø
hW(A,B)~ max
a[A\W
min
b[B\W
d a,bð Þ
 
(20)
If A>W?Ø and B>W5Ø
hW(A,B)~ max
a[A\W
min
w[Fr(W)
d a,wð Þ
 
(21)
If A>W5Ø
hW(A,B)~0 (22)
The frontier Fr(W) comes from the topology defined
by the metric d. In the application (section 6), the
distance is the one associated to the norm L‘.
4.1.3 Local HD
Once this case is clarified, the definition in the global
case must be modified to bring a consistency when the
window slides over the images. Indeed, taking two
cases in the definition will bring sharp variations of
the distance value when the window slides and would
not reflect the intuition (see Fig. 3b). Then the
distance to the frontier of W has to be taken into
account in the definition.
Definition 2. Final version: the HD in a window
Let A and B be two finites sets of R2
HDW(A,B)~max(hW(A,B),hW(B,A)) (23)
where there are three cases:
If A>W?Ø and B>W?Ø
hW(A,B)~ max
a[A\W
min
b[B\W
d a,bð Þ, min
w[Fr(W)
d a,wð Þ
 
(24)
If A>W?Ø and B>W5Ø
hW(A,B)~ max
a[A\W
min
w[Fr(W)
d a,wð Þ
 
(25)
If A>W5Ø
hW(A,B)~0 (26)
4.2 Properties
HDW has the following properties:
HDW(A,B)~HDW(B,A) (Symmetry) (27)
HDW(A,B)~ 0½ uA~B (Identity) (28)
These properties come from the metric properties of
the HD. In contrast, the triangular inequality is no
more verified.
If (wx,wy)~(1,1) and p~1, HDW(A,B)~ B{Aj j (29)
Property 25 means that, with the smallest window
size and the smallest step, the DM is equal the pixel-
to-pixel difference between A and B.
If (wx,wy)~(2m,2n),
then HDW(A,B)~HD(A,B) (30)
Here, this means that all the measures have the same
value (equal to the global HD) in the DM, when the
window includes the images.
4.3 Examples of DM
Figure 4 gives some examples of DM between image
1 and images 2, 3 and 4. Images 1 and 2 are very
similar, and they result in a DM with low distance
values. Image 3 illustrates the same scene as image 1,
but the grass and the helms have been represented
differently. In their DM, the highest values are
confined to these parts. Image 3 is completely
different from image 1 and, in their DM, high values
are more numerous.
It is possible to take different sizes for the sliding
window. For a given resolution, the larger the sliding
window, the coarser the measured dissimilarities. An
illustration is given in Fig 5.
Thus, a variation in the window size enables dif-
ferent sizes of dissimilarities to be highlighted. As the
human comparison process is a coarse-to-fine pro-
cess, it is interesting to begin the comparison with a
large sliding window. Nevertheless, it is time con-
suming. Another way of using this property is to keep
the same sliding window at different resolutions.
5 CLASSIFICATION
5.1 DM classification
At a fixed scale, the comparison process results in a
DM. It can be classified in two ways: Csim, which
includes the DM comparing similar images, and
Cdissim for those comparing dissimilar images. The
classification is based on the comparison of the DM
histogram to class reference histograms. The refer-
ence histograms are the average histograms over each
class. The DM histogram is then compared with
the reference histogram, using a nearest-neighbour
method with a Euclidean distance. Thus, a decision is
taken at each scale, the next paragraph presents the
contribution of the MRA to the decision process.
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5.2 Multiresolution and decision
5.2.1 Multiresolution and image quality
As the MRA is used in the classification process, it
must be shown that the MRA filter does not distort
the images too much from a classification point of
view. For this purpose, DM have been computed at
different scales, with a sliding window proportional
to the scale (so as to highlight the same kind of
dissimilarities). The classification (presented in the
next section) must then be applied to the DM to
evaluate the classification rates at each scales. The
results are presented in Table 1, which contains the
results of 100 comparisons done at resolution 1 (with
the original images), resolution 4 (size divided by 8)
and resolution 6 (size divided by 32). It shows that the
MRA does not affect (to some extent) the classifica-
tion: the results are obviously worse at resolution 6,
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4 Some pictures and their distance maps
5 565 sliding window shows fine dissimilarities, unlike 35635 sliding window which overtones
coarse ones (grass and helms)
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because information is damaged; nevertheless, the
classification remains quite right. For the present
database, resolution 4 offers better results, a property
that will be exploited in the decision process.
5.2.2 Best scale choice for the classification
Let I and J be two images 2m62n, t(I) and t(J) their
m6n approximations produced with the MRA
operator t. The sliding window size is kept the same
for both of the resolutions to produce the DM. If the
sliding window size is constant across scales, thus a
high-resolution DM highlights fine dissimilarities and
a coarse-resolution one highlights coarse dissimila-
rities. Figure 6 illustrates this property. Indeed, it
shows the DM of the same images at three different
scales, with a constant sliding window size equal to
10610. The pieces of detail highlighted in Fig. 6a are
finer than those in Fig. 6c. These DM are comparable
to the ones computed at scale 1 with different sliding
window sizes (Fig. 5). The dissimilarities highlighted
are similar, but the computation at a coarse resolu-
tion with a fixed sliding window size is faster.
Thus, for the decision process, the classification
rates are computed at each scale over the learning set.
Then the scale with the best rate is selected for the
global process (this choice depends on the database
and on the leaning set). In Table 2, the classification
rates at six scales are presented for the same learning
set of the application database. The results lead to the
choice of resolution 4.
6 RESULTS
The proposed method is tested on the database of
digitalized ancient illustrations provided by the
Troyes’ library within the framework of a collabora-
tion with the CReSTIC.19 These images, originally
printed on books with dark ink, have a strong
contrast, which allows them to be binarized with
almost no loss. This database is composed of 68
images, some of them illustrating the same scene. The
objective is to retrieve illustrations of the same scene.
First, 2380 DM were built, 103 of which are classified
in Csim thanks to a manual comparison of the
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a b c
6 Distance maps of images 1 and 2. Resolution 2 shows dissimilarities even in the riders and their
mounts; only the coarse dissimilarities (grass and helms) remain in resolution 4: (a) distance map
at resolution 2 (1286128), 10610 sliding window; (b) distance map at resolution 3 (64664),
10610 sliding window; (c) distance map at resolution 4 (32632), 10610 sliding window
Table 2 Results of classification of learning set of 100 DM at resolution 1–6 with 10610 sliding window size
Successful retrieval Resolution 1 Resolution 2 Resolution 3 Resolution 4 Resolution 5 Resolution 6
Found in Csim 78% 78% 76% 82% 70% 62%
Found in Cdissim 92% 86% 94% 94% 90% 72%
Table 1 Results of classification of 100 DM at resolution 1, 4 and 6 with sliding window size proportional to image size
Successful retrieval Scale 1 Scale 4 Scale 6
Found in Csim 72% 84% 62%
Found in Cdissim 86% 95% 72%
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impressions (and the others in Cdissim). The objective
is to test the method’s efficiency in assessing local
dissimilarities. The experiments was carried out by
the following way: first a learning is made on a set of
50 DM in Csim and 50 DM in Cdissim.
Then, the test is done on a distinct set of 50 DM of
Csim and 50 of Cdissim. The choice of the sets in each
class is randomized. Secondly, three classification
methods are applied. Finally, the results obtained are
compared manually and automatically. The three
classification methods are the following:
– the method based on the DM
– the one based on the global HD instead of the DM
– the method based on the SDW which uses also a
DM, but with the simple difference as local
measure instead of the HD: SDW(A,B)5
sum|A>W2B>W|.
The results are summarized in Table 3. They show
the efficiency of the DM both concerning spatial
information (comparison with the global HD) and
concerning the ability of the local HD to catch the
local dissimilarities (comparison with the SDW).
7 CONCLUSION
In the present paper, a new image comparison
method was presented, which preserves the measur-
ing properties of the HD and suppresses one of its
inconveniences, the outlier sensibility. At the same
time, the DM quantifies the dissimilarities between
the compared images. As a consequence, the DM
produced has a different distribution, whether the
images are similar or dissimilar. This dissimilarity
measure has been embedded in an AMR process,
which results in a coarse-to-fine process. The good
results of this method illustrate the benefits of the
DM. Moreover, it enables the final user to find the
dissimilarity zones between the compared images at a
glance. The perspectives of the study are to improve
the distance map by making the size of the sliding
window automatically chosen and adaptive, and then
to use a decision method taking in the whole distance
map, so as to exploit the spatial information in it.
Finally, it is planned that the improved method be
applied to different databases so as to test its
efficiency and robustness.
REFERENCES
1 Smeulders, A. W. M., Worring, M., Santini, S., Gupta,
A. and Jain, R. Content based image retrieval at the end
of the early years. IEEE Trans. Pattern Anal. Mach.
Intell., 2000, 22, 1349–1380.
2 Huttenlocher, D. P. and Rucklidge, W. J. A multi-
resolution technique for comparing images using the
Hausdorff distance. IEEE Trans. Pattern Anal. Machine
Intell., 1993, 705–706.
3 Meyer, Y. Ondelettes et ope´rateurs, 1990 (Hermann,
Paris).
4 Mallat, S. A theory for multiresolution signal decom-
position: the wavelet representation. IEEE Trans.
Pattern Anal. Machine Intell., 1989, 11, 674–694.
5 Vachier, C. Morphological scale-space analysis and
feature extraction, Proc. Int. Conf. on Image Processing
(ICIP), October 2001. ;
6 Heijmans, H. J. A. M. and Goutsias, J. Multiresolution
signal decomposition schemes. Part 2: Linear and
morphological pyramids. IEEE Trans. Image Process.,
2000, 9, 1897–1913.
7 Kwon, O.-K., Sim, D.-G. and Park, R.-H. Robust
Hausdorff distance matching algorithms using pyrami-
dal structures. Pattern Recognition, 2001, 34, 2005–2013.
8 Taka`cs, B. Comparing faces using the modified
Hausdorff distance. Pattern Recognition, 1998, 31,
1873–1881.
9 Jesorsky, O., Kirchberg, K. J. and Frischholz, R. W.
Robust face detection using the Hausdorff distance. In
Audio- and Video-based Person Authentification –
AVBPA 01 (Eds Bigun and Smeraldi), Halmstadt,
Sweden, 2001, Lectures Notes in Computer Science,
vol. 2091, pp. 90–95 (Springer, Berlin).
10 Borgefors, G. Distance transformations in digital
images. Comp. Vision Graph. Image Process., 1986, 34,
344–371.
11 Huttenlocher, D. P., Klanderman, D. and Rucklidge,
W. J. Comparing images using the Hausdorff distance,
IEEE Trans. Pattern Anal. Machine Intell., 1993, 15,
850–863.
12 Paumard, J. Robust comparison of binary images.
Pattern Recognition Lett., 1997, 18, 1057–1063.
13 Dubuisson, M.-P. and Jain, A. K. A modified
Hausdorff distance for object matching, Proc. 12th
Int. Conf. on Pattern Recognition (ICPR), Jerusalem,
Israel, 1994, pp. 566–568. <
14 Sim, D.-G., Kwon, O.-K. and Park, R.-H. Object
matching algorithms using robust Hausdorff distance
The Imaging Science Journal imsmp062.3d 24/7/07 14:12:08
The Charlesworth Group, Wakefield +44(0)1924 369598 - Rev 7.51n/W (Jan 20 2003)
Table 3 Results for DM, global HD and for DM based
on absolute difference measure
Successful retrieval DM HD SDW
Found in Csim 84% 46% 66%
Found in Cdissim 95% 70% 78%
10 E. BAUDRIER, G. MILLON, F. NICOLIER, R. SEULIN AND S. RUAN
The Imaging Science Journal Vol 55 IMAG mp062 # RPS 2007

measures. IEEE Trans. Image Process., 1999, 8, 425–
429.
15 Lu, Y., Tan, C., Huang, W. and Fan, L. An approach
to word image matching based on weighted Hausdorff
distance, Proc. 6th Int. Conf. on Document Anal.
Recogn, September 2001, pp. 921–925.=
16 Zhao, C., Shi, W. and Deng, Y. A new Hausdorff
distance for image matching. Pattern Recognition Lett.,
2004.>
17 Lu, Y. and Tan, C. L. Word spotting Chinese
document images without layout analysis, Proc. 16th
Conf. on Pattern Recognition, August 2002, vol. 3,
pp. 57–60. ?
18 Guo, B., Lam, K., Siu, W. and Yang, S. Human face
recognition using a spatially weighted Hausdorff
distance, IEEE Int. Symp. on Circuits and Systems,
2001, pp. 145–148. @
19 Seulin, R., Morel, O., Millon, G. and Nicolier, F. Range
image binarization: application to wooden stamps analy-
sis, in Int. Conf. on Quality Control by Artificial Vision
(QCAV) – IEEE, Gatlinburg, TN, May, Proc. SPIE,
Vol. 5132, 2003, pp. 252–258 (SPIE, Bellingham, WA). A
The Imaging Science Journal imsmp062.3d 24/7/07 14:12:08
The Charlesworth Group, Wakefield +44(0)1924 369598 - Rev 7.51n/W (Jan 20 2003)
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