Gray Level Local Dissimilarity Map and
Global Dissimilarity Index for Quality of
Medical Images
Frederic Morain-Nicolier  Jerome Landre  Su ruan 
 CRESTIC - URCA, 9 rue de Quebec, 10026 Troyes Cedex
Abstract: In order to evaluate performance quality of coding techniques, it is needed to have
a good global index and a local index allowing the localisation of the distortions. In this study,
a local dissimilarity map is presented for gray-level images. Its application to the comparison
of a compressed image and its reference allows an excellent visual detection of the distortions.
A global dissimilarity index is computed from the local dissimilarity map. These new measures
are compared to the structural similarity index (SSIM). The results of the global measure are as
good as the SSIM. The results of the local measure are quite superior to the SSIM computed in
a local window. We claim these good results come from the consistency of the proposed index.
It is more consistent to compute a global measure from a local one, than a local measure from
a global one.
Keywords: image compression, medical systems, performance evaluation
1. INTRODUCTION
Actual use of digital medical images is of growing interest.
The volume of acquired data for a single hospital in only
one year can be counted in tera-bytes. Moreover current
regulations impose to store some images for several years.
Compression seems to be a good answer to these digital
archiving problems. For the present, images and volumes
coding is always done without any loss of information. DI-
COM has become a standard and relies on a lossless JPEG
coding. However, only low rates are obtained (typically 2
to 8) with this compression scheme. These rates are too
low to face the amount of acquired data. The use of lossy
coding seems unavoidable.
The main problem with images and volumes when lossy
compressed is the potential deletion of important details,
or the introduction of new ones. This may lead to errors
in diagnosis with eventuallly important care and legal
consequences. Only coding with controlled loss can be thus
applied. An essential point is thus to assess the quality
of the used lossy coding. A global evaluation allows an
overview of the image quality, leading to methods com-
parison. A local estimation of the image quality is a key
information, leading to a localisation and an identifcation
of the method errors.
The performances of the compression techniques are evalu-
ated using objective and subjective metrics. Typical objec-
tives measures are mean squared error (MSE) or peak sig-
nal to noise ratio (PSNR). These metrics are very general
and don't really allow to assess the reconstructed image
quality. They are global measure and thus, distortions
can't be localized. A overall quality is computed. These
metrics are mainly used to compare methods or coding
rates. PSNR and MSE are simple and easy to compute.
These measures do not provide information regarding the
type of loss that causes the deterioration. Subjective mea-
sures are produced from psychovisual tests or quiz with no-
tations by experts. Mean opinion score (MOS) or receiver
operating characteristics (ROC) are typical subjectives
measures, see Cosman et al. (2000). These metrics can't
be easily computed.
Wang et al. (2004) have recently proposed to evaluate
image quality by measuring a structural similarity index.
The Structural Similarity Index (SSIM) is based on the as-
sumption that the human visual system is highly adapted
to extract structures of objects in a scene. The SSIM is
thus based on the degradation of structural information .
In order to provide local information, they also proposed
to compute the index from a local region instead of using
pixels. This is a classical scheme to derive a local measure
from a global one. They suggested to estimate spatial
information from a local 8  8 sliding window. The main
criticism is the choice of the window size. It is a parameter
of the computation.
In previous works, we have introduced a local dissimilarity
measure in order to compute a local dissimilarity map
(LDM), see Baudrier et al. (2008) and Nicolier et al.
(2007). This map is computed using sliding window with
an auto-adaptative size. Here is the general idea: if the
pixels located in the sliding window belong to coarse
features, the window is growed to be big enough to grasp
the feature's distances. The main problem with the LDM
is that only binary images can be compared.
In a more general way, we propose in this paper to reverse
the global-to-local scheme. We propose to derive a global
measure from a local one. To achieve this, an extension
of the LDM to compare gray level images is presented
in section 2. This variant relies on a gray level distance
transform, which can be computed by several methods.
A short review of these methods is given. In section 3,
Proceedings of the 7th IFAC Symposium on Modelling
and Control in Biomedical Systems, Aalborg, Denmark,
August 12 - 14, 2009
ThET2.4
"The material submitted for presentation at an IFAC
meeting (Congress, Symposium, Conference, Workshop)
must be original, not published or being considered
281

an estimation of the local quality of lossy coded image
is proposed and compared to the local SSIM. A global
estimation from this local one is given and illustrated in
section 4. The proposed quality metric is compared to clas-
sical ones (SSIM and PSNR). The presented experiments
show the good behavior of the metric. In the conclusion,
the remaining steps to provide more complete proofs of
the proposed metric usefulness are assessed.
2. GRAY LEVEL LOCAL DISSIMILARITY MAP
In this section, the denition of the local dissimilarity map
for binary images is recalled, see Baudrier et al. (2008). It
is then generalized for gray level images.
2.1 Local Dissimilarity Map for Binary Images
Distance Measure: the Choice of the Hausdor Distance
Among distance measures over binary images, the Haus-
dor distance (HD) has often been used in the content-
based retrieval domain and is known to have successful ap-
plications in object matching by example in Huttenlocher
et al. (1993), or in face recognition in Takacs (1998). Let's
have a brief review of the denition and of some properties
related to the HD. Originally meant as a measure between
two point collections, for nite sets of points, the HD is
dened by Huttenlocher et al. (1993) as:
Denition 1. (Hausdor distance) : given two non-empty
nite sets of points A = (a1; : : : ; an) and B = (b1; : : : ; bn)
of R2, and an underlying distance d, the HD is given by :
DH(A;B) = max(h(A;B); h(B;A)); (1)
whereh(A;B) = max
a2A
(min
b2B
d(a; b)): (2)
h(A;B) is the so-called directed Hausdor distance.
The interest of this measure comes rstly from its metric
properties: non-negativity, identity, symmetry and triangle
inequality. Moreover, the HD is a match methodology
without point-to-point correspondence, so it is robust to
local non-rigid distortions. For a small translation, the
Hausdor distance is small, which matches the expectation
for a distance measure.
Local Dissimilarity Map The notion of local distance is
rst discussed, then denition of a HD measure in a local
window is presented, ending with the local dissimilarity
map denition. In all this section, A and B design two
non-empty nite sets of points of R2, and W a convex
closed subset of R2.
Producing locally a distance implies to compare the two
images locally. It can be done thanks to a sliding window.
The parts of both images viewed through this window
are compared based on a distance measure. The sliding-
window size plays an important role: it should t the local
distance so that the distance can give a local measure.
Nevertheless, here is the general idea: if the pixels located
in the sliding window belong to coarse features, the win-
dow should be big enough to grasp the feature's distances.
Similarly, for ne features, a window "bigger" than the
features will include unwanted information on distances.
These requirements are important to obtain robust local
measure. In particular it is important to be robust to
registration error. Therefore, it is necessary to adapt the
size of the window to obtain precise measures.
The local HD denition: the restriction of the HD to a
window implies to modify its denition. It is indeed not
available in the case that one of the sets is empty, which
can happen in a window. Then the distance to the window
border must be introduced, see Baudrier et al. (2008).
With the new denition of the windowed HD, it is possible
to dene an algorithm which makes the window t the
local distance for each pixel (see following algorithm).
It consists of a sliding three-dimensional window whose
radius is locally adapted to nd the local optimal radius.
From this denition, a local distance map can be computed
by the folowing algorithm:
 For each pixel p, do
(1) n := 1
(2) while HDp;n(A;B)  n and n  HD(A;B), do
n := n+ 1
(3) HDMapp(A;B) = HDp;n1(A;B)
where HDp;n(A;B) is the HD computed between local
windows of radius n from pixel p of images A and B.
Denition 2. The local dissimilarity map between two
images is an image where each pixel value is the local HD
between the two images.
It is possible to express the LDM with the following
formula in order to allow fast computations, for details
see Baudrier et al. (2008):
Denition 3. (local HD (fast version)): for x a pixel of the
images,
HDloc(x) = jB(x)A(x)jmax(dtA(x);dtB(x)); (3)
where dtX is the distance transform of image X.
The formula is faster to compute than the algorithm based
on the windowed HD, but the obtained value interpreta-
tion comes from the local-distance window adaptation in
the algorithm. The distance transform of an image X is an
image where each pixel value is the distance to the nearest
foreground pixel of X.
2.2 LDM for Gray-Level Images
The denition of the fast LDM (def. 3) can be applied
without any modication in the case of gray level images.
The distance transform of each images must be computed.
If in the binary case the distance transform is unique, its
extension for gray level images can be achieved in several
ways. We present here a short review of gray level distance
transform.
Gray weighted distance transform (GWDT) The Gray
Weighted Distance between two pixels is dened in Levi
et al. (1970) and Verbeek et al. (1990) as the smallest
weighted sum of gray level values along the discrete path
between these two points. It corresponds to the surface
area estimation under a curve path. It relies on the followin
cost between two adjacents pixels :
wGWDi =
1
2
(I(ti) + I(ti+1)) jjti ti+1jj; (4)
282

Fig. 1. Images for the tests. First line: computed tomogra-
phy images. Second line: magnetic resonance images.
where jjti ti+1jj is the spatial distance between the two
pixels and I(ti) is the gray value of I for pixel ti.
Weighted distance transform on curved space (WDTOCS)
The path between two points is dened as a n + 1
dimensional path constraint to lie on the hyper-surface
dened by the gray level values. In Toivanen (1996), it
is expressed as the length of the shortest geodesis path
between these two points. It relies on the following cost :
wWDOCSi =
p
(I(ti) + I(ti+1))2 + jjti ti+1jj2: (5)
The main problem with this distance is the inconstency
of the units. By considering gray level values as n + 1
image dimensions, the method mixes spatial and intensity
values. To cope this problem, images values can be scaled
by a coecient, masking the inconstency.
Continuous distance transform (CDT) This distance
transform is based on a generalization of the "white
pixel" and "nd the white nearest neighboring pixel", see
Arlandis et al. (2000). The "white pixel" becomes the
"maximum bright value" and the "nd the white nearest
neighboring pixel" is replaced by "accumulate a maximum
bright value on the neighborhood".
This last denition seems more an "ad-hoc" response to
the addressed problem without straight interpretation.
We prefer the simpler denition of the GWDT and the
WDTOCS.
Each of these distance transforms but the CDT can be
computed in a fast way with a good approximation by a
two-pass algorithm, see Ikonen et al. (2005).
3. LOCAL QUALITY ESTIMATION
In this section, we present some methods designed to
assess local distortions between a coded image and its
reference. First, we recall some results about structural
similarity as proposed by Wang et al. (2004). Then, the
local dissimilarity map for gray-level images is described.
These two metrics are applied to visually compare a
distorted image and its reference.
3.1 SSIM
The Structural Similarity Index (SSIM), proposed by
Wang et al. (2004), is based on the assumption that
the human visual system is highly adapted to extract
structures of objects in a scene. The SSIM is thus based
on the degradation of structural information. The SSIM
index is a full reference metric, it needs a reference image
in order to measure the image quality. The SSIM between
two images A and B is :
SSIM(A;B) =
(2AB + C1)(2AB + C2)
(2A + 
2
B + C1)(
2
A + 
2
B + C2)
; (6)
where A (resp. B) is the mean intensity of A (resp. B),
A (resp. B) is the standard deviation of A intensities
(resp. A), AB is the covariance between A and B inten-
sities. These quantities are computed as follows:
A =
1
mn
m1X
r=0
n1X
c=0
A(r; c); (7)
2A =
1
mn
m1X
r=0
n1X
c=0
(A(r; c) A)
2; (8)
AB =
1
mn
m1X
r=0
n1X
c=0
(A(r; c) A)(B(r; c) B): (9)
C1 = (k1d)2, C2 = (k2d)2 are two small positive constants
to stabilize the division with weak denominator. d is the
dynamic of the pixel-values. k1 = 0:01 and k2 = 0:03 by
default.
The SSIM can be computed locally by measuring local
statistics rather than global ones, see Wang et al. (2004).
Typically A, A and AB are computed in a 8 8 square
sliding window, which moves pixel-by-pixel over entire
image. At each step, the local statistics and SSIM are
calculated within the local window. A SSIM map is thus
obtained, providing local information about structural
similarity between the two images. The choice of the
window radius is decisive. A too small radius will not allow
to catch the structures of the images. A too big value will
smooth the structures and will not allow to mesure small-
size similarities.
3.2 LDM
The local dissimilarity map (LDM) is computed from the
distance transforms of the two compared images :
LDMA;B(x) = jB(x)A(x)jmax(dtA(x);dtB(x)); (10)
Variants of the LDM are obtained from the several solu-
tions to nd a distance transform for gray-level images.
We only retain in this study the gray weighted distance
transform (GWDT) and the weighted distance transform
on curved space (WDTOCS).
3.3 Images
We applied the following metrics to various medical im-
ages modalities : SSIM, LDM with GWD and LDM with
WDOCS. Two modalities were retained : computed to-
mography (CT) and magnetic resonance (MR) images.
Four images were considered for each modality, they are
given in g. 1.
283

Fig. 2. Local quality evaluation: (left) reference image
(CT), (right) JPEG2000 coded image with 0:16 byter-
ate.
Fig. 3. Local quality evaluation: local structural similarity
(SSIM) computed using 77 (left) and 1515 sliding
window.
Fig. 4. Local quality evaluation: local dissimilarity map
with gray weighted distance transform (to the left)
and weighted distance transform on curved space (to
the right).
3.4 Results and analysis
A computed tomography image is taken and coded in
JPEG2000 with a byterate of 0:16. At this high compres-
sion rate, the JPEG200 coding makes some distortions
appear. A zoomed area of the reference and its coded ver-
sion are shown in g. 2. Some parts are clearly distorded,
especially in the border of the structures.
In g. 3 the local SSIM is computed, rst using a 7 
7 sliding window. The SSIM index is a well bounded
measure. Its values range from 1 to 1. The more closer
to 1 is the SSIM, the more similar the images are. In the
rst image of g. 3, the size of the window is clearly not
well chosen as the maximum value is about 0:5. As there
are similar area between coded image and its reference, a
consistent maxima should have been around 1. A second
problem is that lack of readibility of the obtained similarity
image. It is impossible to decide where the distortions
are. A second computation has been runned using a wider
15 15 window. The obtained SSIM image is given to the
left in g. 3. In this image, the maximum value is consistent
with our expectation. The readibility has been improved
as some distortions are emphasized. But due to the wide
size of the window, a smoothing eect is strongly present.
This wide size doesn't allow to catch the small distortions
of the border. This trade-o is a consequence of the global
measure computed to give local information. A large area
is needed by the measure to be signicative. But a large
area produces smoothing eects and doesn't allow to catch
small structures.
In g. 4 the local dissimilarity map is computed using
GWDT and WDTOCS. The two images are very simple to
read and the distortions are clearly localized and quanti-
ed. Small values are related to similar area and distortions
are characterized by the higher values. For the two distance
transforms, the distorted structures are clearly visible. The
previous described trade-o is not present as the LDM
relies internly on an adaptative window. The WDTOCS
produces an even more informative image as the distortion
of the borders are very well detected. These sole images
are a proof of the good quality of the LDM as a distortion
detector between a coded image and its reference.
4. GLOBAL QUALITY ESTIMATION : THE GLOBAL
DISSIMILARITY INDEX
Previous section has shown the excellent visual results
obtained by the local dissimilarity map on gray-level
image. In this section, a global measure is derived from the
local dissimilarity. The obtained index is compared to the
structural similarity index. These measures are computed
on a wide range of coded images with various byterate.
4.1 Global quality from local quality
From the local dissimilarity map computed using gray-
level distance transform, it is needed to compute a scalar.
Classical methods are : compute the sum of the values
or the square root of the sum of the squared values. We
have a strong preference for the second option as it has
been shown its eciency in several studies, by example in
284

Fig. 5. Global quality evaluation: SSIM
Borgefors (1988)]. The following equation is thus retained
for the global dissimilarity index:
GDI(A;B) =
sX
p2A
LDMA;B(p)2 (11)
4.2 Some curves
Firstly, the SSIM is computed between each reference
(8 images : 4 computed tomography and 4 magnetic
resonance) and a wide range of compressed version. Three
compression methods are compared in this study: the well-
known JPEG coding, the JPEG2000 and SPIHT codings,
see Said et al. (1996) for details on SPIHT. For each of
these coding, a wide range of byterates are taken into
account. The results are given in g. 5. Each curve is
obtained from the mean between the SSIM computed
for each modality. The values, at each byterate, of the
SPIHT-SSIM, are very close to 1, proving the eciency
of this coding. The JPEG values are lower than the
JPEG2000 values, indicating the better performances of
the JPEG2000. The sole criticism is the hard readibility
of the behavior for high byterates, say between 0:1 and 8,
the curves are closely intricated around 1.
The same computations are conducted for the global dis-
similarity indexes (produced from GWDT and WDTOCS)
and are given in gs. 6 and 7. The same analysis as
for the SSIM can be made, ranging in eency SPIHT,
JPEG2000 and JPEG (from better to worse). No signi-
cative dierences between GWDT and WDTOCS curves
are seen. However a interesting result is obtained from
the comparison of the JPEG2000 and JPEG curves. For
high byterates, the JPEG curves are under the JPEG200
curves, indicating a better performance for the JPEG
coding. This result is the same as the study of Shiao
et al. (2007). These authors conclude with the fact that
the JPEG2000 outperforms the JPEG coding only at high
compression rates (read low byterates).
5. CONCLUSION AND FUTURE WORKS
In this study, we have proposed a local and global dissim-
ilarity measure. The local measure is a local dissimilar-
ity map computed in the case of gray-level images from
gray-level distance transforms. The computation from the
Fig. 6. Global quality evaluation: GWDT
Fig. 7. Global quality evaluation: WDTOCS
distance transform is thus fast (only two distance trans-
forms are needed, each can be computed in a two pass
algorithm). Excellent visual detection of the distortion
produced by a JPEG2000 coding has been shown, in com-
parison of a locally computed structural similarity index.
A global measure is then derived from the local one, with
a simple formula. This local dissimilarity index have been
tested against images for two modalities, for a wide range
of compression rates, this for eight images. The results are
compared to the global SSIM and are very informatives.
We claim the excellent performance of the local dissimi-
larity map comes from its well dened local behavior, con-
trary of other measures (SSIM, PSNR, etc ...). The good
behavior of the local dissimilarity index comes from the
good performances of the local dissimilarity map. We also
claim it is more consistent to compute a global measure
from a local one, that a local measure from a global one.
Future works following this study will include the use of
a larger images collection. A comparison with subjective
measure and/or task specic measure (e.g. segmentation
of tumoral MRI) is also planned. We also want to obtain
well bounded values, ranging from 0 to 1 by example, in
order to obtain true simularity values.
285

REFERENCES
J. Arlandis, J.C. Perez-Cortes. "The continuos distance
transformation: A generalization of the distance trans-
formation for continuos-valued images". In Pattern
Recognition and Applications, vol. 56 of Frontiers in Ar-
ticial Intelligence and Applications, pages 89-98, 2000
G. Borgefors, "Hierarchical chamfer matching: a paramet-
ric edge matching Algorithm", IEEE Transactions on
Pattern Analysis and Machine Intelligence, vol. 10, n.
6, pp. 849{865, 1988
E. Baudrier, F. Nicolier, G. Millon and S. Ruan, "Binary-
image comparison with local-dissimilarity quantica-
tion", Pattern Recognition, vol. 41, n. 5, pp. 1461{1478,
jan. 2008
P. Cosman, R. Gray, R. Olshen, "Quality evaluation
for compressed medical images : fundamentals", Hand-
book of Medical Imaging, Processing and Analysis,
I.Bankman, Ed. New York : Academic, 2000.
D.P. Huttenlocher, W.J. Rucklidge, "Comparing images
using the hausdor distance", IEEE Transactions on
Pattern Analysis and Machine Intelligence, vol. 15, n.
9, pp. 850{863, 1993
L. Ikonen, P. Toivanen, "Shortest routes on varying height
surfaces using gray-level distance transforms", Image
and Vision Computing, vol. 23, n. 2, pp. 133{141, feb.
2005
G. Levi, U. Montanari, "A gray-weighted skeleton", Infor-
mation and Control, vol. 17, n. 1, pp. 62{91, 1970
F. Morain-Nicolier, S. Lebonvallet, E. Baudrier, S. Ruan,
"Hausdor distance based 3D quantication of brain
tumor evolution from MRI images.," in Conf Proc IEEE
Eng Med Biol Soc, 2007, pp. 5597-5600.
P. Verbeek, B. Verwer, "Shading from shape, the eikonal
equation solved by grey-weighted distance transform",
Pattern Recognition Letters, vol. 11, pp. 681{690, 1990
B. Takacs, "Comparing faces using the modied Hausdor
distance", Pattern Recognition, vol. 31, n. 12, pp. 1873{
1881, 1998
P.J. Toivanen, "New geodesic distance transforms for gray-
scale images", Pattern Recognition Letters, vol. 17, n. 5,
pp. 437{450, 1996
A. Said, W. A. Pearlman, "A new fast and ecient image
codec based upon set partitioning in hierarchical trees",
IEEE Transactions on Circuits and Systems for Video
Technology, vol. 6, pp. 243-250, jun. 1996.
Y.H.Shiao, T.J. Chen, K.S. Chuang, C.H. Lin, C.C.
Chuang, "Quality of compressed medical images", Jour-
nal of Digital Imaging, vol. 20, n. 2, pp. 149{159, 2007
Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, "Im-
age quality assessment: from error visibility to structural
similarity", IEEE Trans. Image Processing, vol. 13, n.
4, pp. 600{612, apr. 2004
286