The document discusses local and non-metric similarities between images. It outlines that local similarities are needed to compare images and detect small differences. The document proposes using a local dissimilarity map (LDM) approach to quantify local differences between images. The LDM approach involves calculating a distance measure, such as the Hausdorff distance, within a sliding window over the images to identify localized regions of difference. This localized comparison approach could be useful for applications like analyzing ancient printings.

1. Local and Non-Metric Similarities
Between Images
-
Why, How and What for ?
Frédéric Morain-Nicolier
CReSTIC - URCA - Troyes
2012.09.25
1

2. Outlines
• Local similarities
• Non metric similarities
• Conclusion :
• Local non-metric similarities ?
• solutions
• open problems
2

3. Local Similarities
3

4. Local Similarities,
Why ?
4

5. 3D Acquisition
?
[Troyes Library - Early Rennaissance Collection
- Bibliothèque Bleue]
5

6. 3D Acquisition
[Thanks to Le2I - Le Creusot Team]
6

7. 3D Acquisition : Economical Solution
7

8. 3D Acquisition : An Example
ANALYSIS AND CONSERVATION OF ANCIENT WOODEN STAMPS
ANALYSIS AND CONSERVATION OF ANCIENT WOODEN STAMPS 115
7 Range image (‘Pisces’ stamp) computed from projec-
6 3D View of acquired points of ‘Pisces’ stamp: point tion of point cloud
cloud acquired with Minolta scanner
Else t5Mzk*s (M.m and the neigh-
a stamp, the printing zones are high elevation ones;
7 Range image (‘Pisces’ stamp) computed
bourhood is in a low zone)
6 3D View of acquired points of ‘Pisces’range imagespoint
the high grey-level pixels in the stamp: have to tion of point cloud
End if
be binarized as black (pixel50). The non-printing If pixel.t Then pixel50
cloud acquired with Minolta scanner
zones must therefore be binarized as white (pixel51). Else pixel51
Stamp 3D model Range Image
A modified Niblack’s algorithm is used to binarize End if Else t5Mzk*s (M.m an
range images.11,12 The main task is to adapt the End Do
a stamp, the printing zones are high elevation ones;
threshold over the image. The threshold is deter- bourhood is in a low zone
The local threshold computing can be summarized
the high grey-level pixels from the local meanimageslocal standard
mined in the range and the have to End if
by equation (1)
deviation, computed on a restricted neighbourhood t~max(M,m)zk às (1)
be binarized as blackeach pixel.
for
(pixel50). The non-printing If pixel.t Then pixel50
By modifying the k parameter, it is possible to
zones must therefore be binarized as white (pixel51).
Define: Else pixel51
simulate the inking and printing process for various
m the local mean computed on a [w6w]
8
A modified Niblack’s algorithm is used to binarize
neighbourhood End if
conditions (ink quantity, paper quality or humidity,
11,12 ink fluidity, exerted pressure etc.). Figure 8 shows the

9. 3D Acquisition ➙ Virtual Printing
RVATION OF ANCIENT WOODEN STAMPS 115
local threshold
7 Range image (‘Pisces’ stamp) computed from projec-
point tion of point cloud
Else t5Mzk*s (M.m and the neigh-
ones; bourhood is in a low zone)
ave to End if
inting If pixel.t Then pixel50
el51). Else pixel51
narize End if
pt the End Do
deter- The local threshold computing can be summarized
ndard by equation (1)
rhood t~max(M,m)zk às (1)
By modifying the k parameter, it is possible to
simulate the inking and printing process for various
w6w]
conditions (ink quantity, paper quality or humidity,
ink fluidity, exerted pressure etc.). Figure 8 shows the
mplete
same range image as that used in Fig. 7, binarized for
k50.1–0.8. Note the ‘inking’ variations produced by
d on a
the k value modification.
3.3 Comparison between virtual and real stamping
In order to test the proposed method, the results of 9
virtual printing were compared with real ones. For

10. 3D : Virtual vs. Real Fidelity ?
Virtual Resolution ! Real
10

11. A Local Comparison is needed !
11

12. A Local Comparison is Needed !
12

13. A Local Comparison is Needed !
small diffs
scattered diffs
big localised diffs 13

14. A Local Comparison is Needed !
14

15. Local Similarities,
How ?
15

16. Local Dissimilarity Map (LDM)
Φ( , )
Which measure ?
Which size ?
e 7.4 – Comparaison de deux groupes de lettres. L’image de la CDL indique clai
CDL indique cla
lisations des dissimilarités. La comparaison est également quantifiée.
quantifiée.
LDM
on de deux groupes de lettres. L’image de la CDL indique clairement
n de deux L’image de la CDL indique clairement
milarités. La comparaison est également16quantifiée.
milarités. également quantifiée.

17. Local Dissimilarity Map
Which measure
between small
images ?
• MSE - PSNR?
➡ pixel to pixel diffs
A
dA,B (p) = |A(p) B(p)|
|A(p) B(p)|
➡ low information
and
hard to interpret B
17

18. Local Dissimilarity Map
Which measure
between small
images ? DH(A, B) = max (h(A, B), h(B, A))
✓ ◆
• Binary image = set of h(A, B) = max min d(a, b)
pixels (foreground) a2A b2B
h(B, A)
➡ Hausdorff distance DH(A, B)
[Huttenlocher 1993]
DH(A, Tv A) = kvk
➡ numerous variations, A
including partial HD
ieme
B
hK (A, B) = Ka2A d(a, B)
18

19. Local Dissimilarity Map
Which size for the sliding
window ?
➡ adaptative
➡ must encompass
the «diffs» but no
more
A B
(a) (a) (b) (b) (c
➡ increase until the
Figure 7.2 – Illustration de la notion de dissimilarité locale. Les Les imag
Figure 7.2 – Illustration de la notion de dissimilarité locale. image A
measure equals itsdissimilaires. Les Les fenêtrescentrepetite et AetetABetcar carc), petites.perme
dissimilaires. fenêtres de taille
de taille petitemoyenne (en (en ne permetten
la dissimilarité locale au centre des des images
la dissimilarité locale au images
moyenne
B trop
c), ne
trop petites.
theoretical max ⇒ stopping criterion
Nous pouvons donner uneune idée générale. les pixels situés dans la
Nous pouvons donner idée générale. Si Si les pixels situés dans
partiennent à des des traits grossiers,fenêtre doitdoit avoir une taille su⇥
partiennent à traits grossiers, la la fenêtre avoir une taille su⇥san
écarts résultant de la comparaison de ces ces traits. les traits sont fins,
écarts résultant de la comparaison de traits. Si Si les traits sont fi
fenêtre soitsoit trop grande. Dans le cas contraire, des écarts qui sont plu
fenêtre trop grande. Dans le cas contraire, des écarts qui ne ne sont
19

20. rmax = max r|DHW (p,r) (A, B) = r . (7.27)
r>0
Local Dissimilarity Map
En pratique ce théorème indique que tant que la distance de Hausdor locale est égale au
rayon de la fenêtre, la mesure optimale n’est pas atteinte.
In the Hausdorff distancedissimilarités locales
7.1.4.4 Définition de la carte des
case :
La carte des dissimilarités locales est définie maintenant aisément. Elle regroupe l’ensemble
des mesures de dissimilarités locales réalisés pour di érentes positions. L’algorithme général
lorsque la distance locale est basée sur la distance de Hausdor est le suivant :
Algorithme 7.1 Algorithme itératif de calcul de la carte des dissimilarités locales (CDL)
entre deux images binaires A et B. W (p, n) désigne la fenêtre carrée centrée au pixel p et de
rayon n.
Pour chaque pixel p, faire
1. n ⇤ 1
2. tant que DHW (p,n) (A, B) = n et n ⇥ DH(A, B), faire
n⇤n+1
3. CDLA,B (p) = DHW (p,n 1) (A, B) =n 1
84
20

21. Local Dissimilarity Map
With the Hausdorff distance : fast computation
CDLA,B (p) = |A(p) B(p)| max (d(p, A), d(p, B))
Distance transform (or function) based
TDA (p) = d(p, A)
(distance to the nearest foreground pixel : very fast with a chamfer distance)
⇒ linear expression (binary images) :
CDLA,B = A.TDB + B.TDA
[Baudrier PhD - Pattern Recognition 2008]
21

22. Local Dissimilarity Map : Toy Examples
CDLa,b
Figure 7.3 – Comportement de la CDL avec des motifs simp
comparer. d est la CDL entre a et b, e est la CDL entre b et c e
CDL de
le niveau b,c gris est foncé, plus grande est la valeur locale de la
FigureFigure 7.3 – Comportement de la CDL avec des simples ; a,b,c sont lessont les images à
7.3 – Comportement de la CDL avec des motifs motifs simples ; a,b,c images à
comparer. d est la d est la CDL et b, e est b, e est la CDL entreet fet cCDL la CDL entre Plus c. Plus
comparer. CDL entre a entre a et la CDL entre b et c b la et f entre a et c. a et
le niveau de gris est foncé, plus grande est la valeur locale de la mesure.mesure.
le niveau de gris est foncé, plus grande est la valeur locale de la
Cet algorithme est coûteux en temps de calcul car itératif. En
Figure 7.3 – Comportement de la CDL avec des motifs simples ; a,b,c sont le
est en O(m4 ) pour deux images composées deet f⇥ m pixels. No
comparer. d est la CDL entre a et b, e est la CDL entre b et c m la CDL entre
A quantified and localized information
le niveau de gris est distance de grandeEn elaet, la complexité de mesure.
la foncé, plus Hausdor est utilisée dans de mesure locale de dis
locale la la calcul
Cet algorithme est coûteuxcoûteux en temps de calcul car itératif.valeur et, la complexité de calcul
en temps de calcul car itératif. est
7.3 – Comportement de algorithme est des motifs simples ; a,b,c sont les imagese à
Cet la CDL avec très rapide. En
. d est laest en entreen et b, 4 ) est la deux images composées de m ⇥ m pixels.c. Plus avons montré que lorsque
4 ) pour deux images composées de m ⇥ m pixels. Nous avons montré que lorsque
CDL O(m a O(m e pour CDL entre b et c et f la CDL entre a et Nous
est
de gris est distancedistance deest estvaleur locale Théorème mesure locale dedes dissimilaritéscalcul peut être
la foncé,la de grande Hausdor est dansde la mesure.
plus Hausdor la utilisée utilisée dans la locale La dissimilarité, le calcul peut être entre deux
la mesure 7.8. de carte dissimilarité, le locales
22

23. Local Dissimilarity Map : Toy Examples
➡ structural informations
23

24. 3.7. G´n´ralisation aux images en niveaux de gris
e e 85
Local Dissimilarity Map : Toy Examples
➡How to save time during holidays ?
24

25. Local Similarities,
What for ?
25

26. Ancients Printings
É. Baudrier et al. al. / Pattern Recognition 41 (2008) 1461 – 1478
É. Baudrier et / Pattern Recognition 41 (2008) 1461 – 1478
et 1471
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É. É. Baudrier al. al.Pattern Recognition 41 41 (2008) 1461 – 1478
Baudrier et / / Pattern Recognition (2008) 1461 – 1478 14711471
É. Baudrier et al. al. / Pattern Recognition 41 (2008) 1461 – 1478
É. Baudrier et / Pattern Recognition 41 (2008) 1461 – 1478 1471
1471
É. É. Baudrier al. al.Pattern Recognition 41 41 (2008) 1461 – 1478
Baudrier et et / / Pattern Recognition (2008) 1461 – 1478 14711471
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27. Ancients Printings
É. Baudrier et al. / Pattern Recognition 41 (2008) 1461 – 1478 1471
É. Baudrier et al. / Pattern Recognition 41 (2008) 1461 – 1478
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18 Fig. 8. Medieval impressions and their LDMaps. Here are four medieval impressions. Imp. 1, Imp. 2 and Imp. 3 illustrate the same scene with a di
1616 and their LDMaps. Here are four medieval impressions. Imp. 1, Imp. 2 and Imp. 3 illustrate the same scene with a
Fig. 8. Medieval impressions16 16
16
16 kind 8. Medieval impressions Imp. their LDMaps. Here aare four medieval 14
Comparaison d’illustrations 12
14
14
Figure 20 – 14 Imp. their LDMaps. Here aare four 16 anciennes.Imp. 1, Imp. 22 and Imp. 3illustrate the same16 scene with a
Fig. of grass and helmets in and 3. Imp. 4 illustrates distinct scene. impressions. Imp. 1, images a, b3et c représentent la a di
7.9 14 and 3. Imp. 4 illustrates distinct medieval impressions. Les Imp. and Imp. illustrate the same with m scene
kind 8. Medieval impressions
Fig. of grass and helmets in 14in Imp. 3. Imp. 4 illustrates a distinct 16
scene. 20 16
kindFigure 18CDL Imp.(f) CDL d’illustrations anciennes. Les images a, b et cde scènes dissimila
of grass and 7.9 –12 Comparaison ; (g) CDLscene. (h) CDL . La comparaison représentent la m
14
14 kind of grass and helmets
20 12 12 20
18 14
scène. (e)18 10 helmets12 ; 3. Imp. 4 illustrates a distinct14
in
12 scene. ; 10
14 18 14
12
12 a,b
10 a,c a,d 10 16 c,d 12
10
10 scène. (e)methods are (f) CDLa,c ; (g)réparties; sur CDLc,d .the measure result is h).image comparaison
produit 14
16
CDLa,b ; importantes ones12a,d (h) whether La comparaison de scènes dissimila
8
10
8 8 16
comparisondes valeurscompared with the CDLobtained 8 toute l’image (en g et an La (case of the LD
16 10 12 14
14
12
10
comparison methods86are compared with the ones 10obtained 6 whether LSDMap) or aresultvalue (case of(case ofof the L
comparisonThe five classification methods réparties sur toute l’image (en result is h).image comparaison
14
ones obtained6
methods are compared with the are the follo-
8 10 whether the measure realet is an image 10 HD and LD
12 the the measure an (case the its
88
produit 10 valeurs importantes
manually. des 12 and
10 faible measure g
12
scènes methods66 produit with the ones 8obtained 4 and the LSDMap) or a real value ou très8the HD and i
12
manually. The10 five 64classification des valeurs importanteswhetherLSDMap) or a resultvalue(case of 8(case of is itsL
comparison similaires classificationmethods are the follo-4 La the
en the In the first case,(en is an (case of method the
the nombrereal e) image the HD and
the classification localisées
6 are compared methods are the follo- 8 10
6 wing ones: The five
manually. 4 and
ations). a
44 scènes similaires produit methods are the follo-2 2 ations). Inin Sectioncase, In theclassificationmethod is is
manually. The
wing ones:
wing ones: 6
8 five42classification des valeurs importantes enthe LSDMap) or a real valueou très6localisées
8
2 4 6 6 and 88 faible nombre (en e) (case of the HD and 6
described the first case, the classification 4 an empirica
ations).
66 first 6.2. the second case, method a
2 6 20 4
2 wing ones: 44 based on the LDMap,
• our method 02 4 4 0
0 describedforthe first case,In the second case, an empirica
ations). Inin each class C In and C
Section 6.2. the classification method is
described Section 6.2. sim the second is computed from
44
tribution case, an empiri
0 0 2
In and Cdissim 2 0
2 dissim
0 • ourthe so-calledbased on the LDMap,
• our method Local the LDMap,
method22 based on Simple Difference Map (LSDMap) us-
0 22 2
describedfor eachthe modes of the dissim isis computed fr
in each class Csim and C empirical distribution
Section 6.2. sim the second case, an empiri
tribution set. different C
Medieval impressions and their LDMaps. Here are four medieval on the LDMap, Imp. 2 and Imp. 30illustrate the same 0 learning a As class
tribution 0 computed fro
• and are four illustrateLocal SimpleImp. theasimple difference locallythe same0 scene set. eachthe modes of the empiricalcomputed f
ourImp. the medievalthe Simple with 1, Imp.Map (LSDMap) us-
• the the
• so-called
edieval impressions and their LDMaps.2Here ing so-called based impressions. Imp. 1, different (LSDMap) us-
impressions.
method00 Localmap, but Difference 2Map Imp. 316
distance same sceneDifference and
0illustrate
scene with
tribution with As the easy of the dissim 16 distribution
learning for a different Csim and C empirical distributi
class is
16
ure andimpressions. Imp. Imp.Imp.d’illustrations anciennes.(F, G)imagesdifferencelocally learning ladefined, anmodesand efficient classification me
f medieval impressions. Imp. 1, Imp. 2 and Imp.33 illustrate the same scene with a different b 16 c représentent
grass 7.9 – Comparaisonillustrates
4 20
medieval helmets Imp. 3. 3. 1, 4 illustrates a a distinctscene.
in Imp. scene. with
20Local Simple Difference Map a,
quite well
20
Les simple ∩ W − Get W |,us- learning set.mêmeanmodesand efficient classification m
• ing instead ofcthe HD: but with la = |F difference locally 20
ons anciennes. Les images a, bdistance map, H SD W the même (LSDMap)
ass and helmets in
t scene.
scene.
Imp. the ing the distance map, but with the simple
so-called
distinct
the et 18 représentent
18
18 14
14
∩ quite well defined, an easy method. empirical distributi
quite maximum likelihood and the
the
18
18
18 As the easy of efficient 14
is 18 well defined, 14
classification
16
e. (e) CDLa,b ; (f) CDLa,c ;insteadglobal the map,SD W (F, G)La=comparaisonW |, scènes16 well defined, an easymethod.
(g) CDL16 HD: H but with . G) |F difference W de
• instead of ; (h) CDL (F, simple W − G 12 |,
18
16
ingthe of theHD,HD: H SD Wc,dthe = |F ∩∩ W − G 12locally
the distance16
a,d 16
∩ ∩
16 16dissimilaires
quite maximum likelihood and efficient 14
16
is 14 maximum likelihood method.
is the
the
16
16
12
classification
12
La,d ; methods are c,d . La comparaison14HD, H SD W (F, G) =measure result1014 an image (case of the LDMap
(h) CDL compared withinsteadglobal HD: 14de scènes dissimilaires
16
arisondes valeurs importantes theones the sur whether the |F ∩ W g 14 W |, La comparaison likelihood method. 14 14
duitmethods are comparedmeasuretherépartiesimage touteof the LDMap − Get12h). imageis 12 maximum de
son
• of 14
• the global HD,
PHD,14 obtained
12
12 l’image 27 result10 an
(en ∩is 14
14
the
12 of the LDMap
10
12
10
12
rties sur toute l’image (enresult isobtained comparaison de
obtained whether the with the g et 10 La (case
ones h). an whether the measure 12is
8 (case
12
12
6.3.2. Results
10 8

28. Ancients Printings
• LDM classification
• similar
• dissimilar
➡ SVM
Bonne classification (en %) CDL DPP DH PHD MHD
pour Csim 98 90 60 83 77
pour Cdissim 97 92 75 81 83
Table 7.1 – Performances en classification d’images similaires issues de la base d’impressions
anciennes. CDL = carte des dissimilarités locales, DPP : di érence pixel à pixel, DH : distance
de Hausdor , PHD : distance de Hausdor partielle, MHD : distance de Hausdor modifiée
(voir 7.2.1.4).
[Baudrier PhD]
28

29. Tumor Evolution
t1 t2 t3
(a) (b) (c)
? ?
Fig. 2. Segmented MRI.
7.12 – Un exemple de la segmentation d’une tumeur. Seule une coupe d’u
représentée, à trois dates di érentes.
values in (f) do not reflect et al. 2007] in this case. As
[Nicolier the similarity
a short conclusion, the LDMap is a useful tool for non-
29

30. (S1) Coupes de la segmentation 1
A. Segmentation Method R ESULTS
III.
using SVM with RBF kernel.
(j)
Tumor Evolution
A. MRI images are acquired on a 1.5T GE (General
Segmentation Method
Electric Co.) machine using an axial1.5T IR (Inversion
on a 3D Slices 18 to 23 (a-f) of the Local Distance Volu
MRI images are acquired Fig. 5. GE (General
Recuperation) machine using an axialan axial FSE (Fast
Electric Co.) T1-weighted sequence, 3D IR (Inversion
Spin Echo) T2-weighted, ansequence, anPD-weighted 3) and 2 (fig 4). The distances are absolute
volumes 1 (fig se-
Recuperation) T1-weighted axial FSE axial FSE (Fast
quence and T2-weighted, an axial FSE examination, we distance histogram (logarithmic scale in g
Spin Echo) an axial FLAIR. eq. one PD-weighted se-
For (3). (j) is the (a) (b) (c)
have 24and an axial FLAIR.signals with a voxel size
quence slices of the four colormap of images (a-f).
For one examination, we
of 0.47 ⇥slices ⇥ 5.5 mm3. signals with a and all size
0.47 of the four All the slices voxel the (a) (b) (c)
have 24
examinations are⇥ 5.5 mm3. All SPM software. all the
of 0.47 ⇥ 0.47 registrated using the slices and
examinations are registrated using SPM software. using
We use the first examination for training SVM
RBF kernelthe first examination for training SVM using
We use [10]. The training set was obtained from one
(a) slice(b) using mouse to(c)
(c) RBF by choose ten pixels into the tumour
kernel [10]. The training set was obtained from one (d) (e) (f)
(a) and (b) outside. We perform theten pixels into the tumour
sliceten using mouse to choose first segmentation of this
by (c)
(c)
volume by usingWe perform the first segmentation of this Fig. 4. Segmentation of the de (e) volume (slices from(f) to 23, a-f)
and (b) outside. the SVM model obtained. So, we build using SVM with RBF kernel. second segmentation 18
(d) Coupes
(S2) la 2
(c) t1 (a) ten (c)hundred points into the tumour
automatically about one model obtained. So, we build Fig. 4. Segmentation of the second volume (slices from 18 to 23, a-f)
volume by using the SVM
his case. As and outside from all one hundred slices. into retraining a using SVM with RBF kernel.
automatically about the tumoral points We the tumour
second SVM fromuse it fortumoral slices. Wesegmentation (a) (b)
ol for non-
his case. As and outside and all the perform a second retraining a
for improve the first it for perform this last SVM model, As a true distance is used to compute the LDV, the
). It is non-
for thus
ol case. As (d) second SVM and use result. With a second segmentation
(e) (f)
his we perform the segmentationWith given scalar are true physical distance in mm. The given
). It is non-
thus (d) for improve the first result. of others examinations. At
Fig. 3. Segmentation of the first volume (slices from (f) to 23, a-f) this last SVM model,
(e) 18 As a true distance is used to compute the LDV, the
ol for using SVM with RBF kernel. eachperform the segmentation of others examinations. At given scalar are true physical there are mm. more low
we segmentation of examination, we use this 2 steps
distances histogram indicates distance inmuch The given
). It is thus (d) Coupesprocess for improve of examination, we use this 2 steps distances than (b) distances. This a coherent fact as non-
Fig. 3. Segmentation of thede (e) segmentation(f) to 23, Fig 2 contains an example
(S1) each la segmentation the 1
first volume (slices from 18 result. a-f) (a) distances histogram indicates there are much more low
high (c)
using SVM with RBF kernel. zero LDV values aredistances. in theaintersection of as non-
Fig. 3. Segmentation of thede la obtained segmentation. All the nine slices of two
the for (slices from result. a-f)
process segmentation18 distances high obtained This coherent fact the two
(S1) Coupesof first volumeimprove the 1 to 23, Fig 2 contains an example volumes. than intersection is locally filled with increasing
segmented volumes are given All the and slices of two zero LDVThe are obtained in the intersection of the two
using SVM with RBF kernel. the obtained segmentation. in figs 3 nine 4.
of values (d) (e)
E (General values, starting from zero locally filledmaximum local
segmented volumesDetails are given in figs 3 and 4. volumes. The intersection is up to the with increasing
E (Inversion B. Implementation distance starting from volumes. to the maximum 7. Us Fig.
(General
l FSE (Fast Fig. 6. a new mageJ is 15.56mm. This represent the higher straight distance
values, between the zero the The maximum tumor hasvo
A three-dimensional view of up LDV between re local
distance
E (Inversion
(General B. The computation DetailsLDV is done with
Implementation of the distance between the volumes. The maximum has progresse distance
leighted se-
FSE (Fast
R (Inversion
2.
plugin computation2GHz opteron, donecomparison mageJ between the two volumes. the higher straight directed dista
The [6]. With a of the LDV is the with a new of the is 15.56mm. This represent
(d) (e) (f) distance
t
weighted we 2
ination, se-
l voxel (Fast
FSE
(a) (b) ⇥ 512 ⇥ 9 volumes is done in 39 seconds.
plugin [6].
(c) The proposed Local Distance Volume can be used to
two 512 With a 2GHz opteron, the comparison of the between the two volumes. and 4). (b) is
(j)
ination, size
we (a)
(VDM) - coupes duprecisely the variations between two volumes. S
track more volume des dissimilarités can be used to
locales entre
weighted these-
and all size two (b) ⇥ (c)
C. Results 512 ⇥ 9 volumes is done in 39 seconds.
512 The proposed Local Distance Volume
voxel we
ination,
Fig. 5. Slices 18 to 23 (a-f) of the
The Hausdorff Distance in a window (eq. (3)) is defined as
track more precisely the variations between two2 volumes. ds
C.The LDV is computed between Figure 1 and – Exemple de volume des dissimilarités (j) is the distance histogram (l
volumes 7.13 2. The the maximum of two directed distance. In(3)) is Les deux
locales.
are.
and all size (a) (b) (c) volumes 1 (fig 3) and (fig 4). The
voxel the
SVM using
Results The Hausdorff Distance in a window (eq. the present case
eq. (3). defined as
are.
and all one results LDVpresented in betweenS2)the z-resolution is L’histogramme indique la useful information.(a-f). (A, B)
are is computed fig 5. As sont 1 and 2. The the maximum of two directed distance.of imagespresent case(e
comparées. the directed distances carry répartition des distances
clearly seen (by high negative distances). colormap hW
d from the
Local Dissimilary Volume
The volumes In the
SVM using
are. tumour slightly greather than the xfigy 5. As the (with a ratio of carry the information on voxels sont expriméesW (A, B)
results are presented in resolutions z-resolution is the directedniveaux de gris, present in vol. 1h ennot in
Les distances, traduites par des distances carry useful information. and mm.
(j)
the
d from one
SVMof this
using (d) 11.7), the obtained distance depends only(with a ratiothe vol. 2. SymmetricallyonW (B, A) carry in vol. 1 and not on
(e) (f) information h voxels Volume the information in
slightly greather than the x y resolutions lightly on of carry the (a-f) of the Local Distance present between
ationtumour
the Fig. 5. Slices 18 to 23
z-axis information. distance dependsobtained distance is voxels present indistances andabsolute accordinginformation B)
Only when the only lightly vol. 2. (fig 4). The vol. hWare not carry the to hW (A, on
2 (B, A) in vol. 1. So
o, from this
d we build
one (d) Coupesgreathersegmentation the z-information has beenon the(fig 3) and 2 Symmetricallytumor has regressed and h (B, A)
Fig. 4. Segmentation of the11.7), the obtained from(f) to 23, a-f)
(S2) de la 2 volumes 1
the of
ationtumour
the tumour using SVM with RBF kernel.
(e) than 11.7mm, 18
second volume (slices
Fig. 4. Segmentation of the de (e) volume (slices froma18 to 23, a-f)
IV. CONCLUSION
z-axis information. Only when the obtained distance (j) is the distance histogram the 2 and notinin vol. 1. the hW (A, B)
taken indicates where (logarithmic scale gray) and So W
eq. (3). is voxels present in vol.
ation build
o, weof this (d) Coupesinto account. Fig 6 is (f) z-information has been taken images (a-f). where the tumor has regressedillustrated(B, fig.
(S2) second segmentation 2
la than 11.7mm, the
greather colormap of
three-dimensional representation where the tumor has progressed. This is and hW in A)
indicates
retraining a
the tumour using SVM with RBF kernel.
o, we build of the LDV. Fig is 18 to 23, a-f)
Fig. 4. Segmentation of theinto account. (slices6from a three-dimensional representation
second volume (a) (b) (c) 97
7 and fig. 8. The has progressed. This central occlusion is
where the tumor augmentation of the is illustrated in fig.
egmentation
retraining a
VM tumour
the model, using SVM with RBF kernel. the LDV.
of
As a true distance is used to compute the LDV, the A distance measure between volumes has
30
7 and fig. 8. The augmentation of the central occlusion is

31. Binary Pattern Localization
x
y
P I(x,y)
I(x,y)
P
CDLP,I(x,y)
I
31

32. Binary Pattern Localization
Local dissimilarities aggregation :
XX
MDGI,P = CDLI,P (k, l)
k l
with CDLI,P = I.TDP + P.TDI
) DI,P = TD2 P +I TD2
I P sum of two
oriented measures
Chamfer score [Borgefors,
1988]: how much
I looks like à P?
32

33. Binary Pattern Localization : Example
Fig. 2.Chamfer score MDG
In (c), image response by Borgefors chamfer matcher. – In (d), image
obtained with symmetric LDM-matcher. – A good match with the reference
pattern is reported by low values (with dark gray levels).
ce pattern, the ideal location in (a)
33

34. Binary Pattern Localization : Example
I
[Morain-Nicolier et al. 2009]
34

35. ('%+*
"#$
Brain Internal Structures Segmentation
!
(',-*(','*
!
! !
! !
!
!
/01!! !
!!
"#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"%$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"&$!
"#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"%$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"&$!
"#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"%$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"&$!
"#$! !"%$! !"&$ !
'()*+,!-./!"0$!12!3*405!#+0(5!066,0+05%,!"#$!%7+7508!"%$!09(08!"&$!:0)(;;08!
"#$!
"#$! !"%$!
!"%$! !"&$
!"&$ ! !
'()*+,!-./!"0$!12!3*405!#+0(5!066,0+05%,!"#$!%7+7508!"%$!09(08!"&$!:0)(;;08!
'()*+,!-./!"0$!12!3*405!#+0(5!066,0+05%,!"#$!%7+7508!"%$!09(08!"&$!:0)(;;08!
35

36. Brain Internal Structures Segmentation
caudate
putamen putamen
thalamus
"#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"%$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"&$!
36

37. Brain Internal Structures Segmentation
!
"#. !
! !
!
!
!
!
! Deformation ?
!
!
!
Atlas %/'! "#$!
Test Volume
%+'
! !
%&'! !"($! "#. !! )*+ %/'! !%+'
37
!"#$! !!!

38. Brain Internal Structures Segmentation
Optimal transformation :
regularization
T ⇤ = argminT 2 {Esim (B, A T ) + Ereg (T )}
similarity measure
Classic solution :
1 2
Esim (B, A T ) = kB A Tk
2
Displacement field :
(A T B)(p)
u(p) = 2 rB(p)
(A T B)2 (p) + krB(p)k
38

39. representation of the corresponding structure in the deformed deformed shape map shapeaccording to the following p
representation of the corresponding structure in the unified unified MðpÞ map MðpÞ according to the
8
atlas after the transformation T. Under the constraint of the shape the shape > Fs ðpÞ
atlas after the transformation T. Under the constraint of 8
< i > Fsif Fsi ðpÞ Z0 Fsi ðpÞ Z0
< i
ðpÞ if
similarity term, the optimal transform would leadwould lead to the final maxðF ðpÞÞ maxðF ðpÞ o0if and ðpÞ o0F ðpÞÞj
similarity term, the optimal transform to the final
Brain Internal Structures Segmentation
MðpÞ ¼ MðpÞs¼
i
if Fssi ðpÞÞ
i
Fsi jmaxð and
si
segmented structure shape as closer as as closer asatlas.in the atlas. Therefore >
segmented structure shape that in the that Therefore :0 >
: 0 if F ðpÞ o0if and ðpÞ o0F ðpÞÞj
si Fsi jmaxð and
si
the above overall costoverall cost function can be modified as
the above function can be modified as
where e is the threshold, ethreshold, e ¼ Fsi ðpÞÞg. Ther
E ¼ Esim ðB; A3TÞ¼ Esim ðB; A3TÞ þ Ereg ðB; A3TÞ þ Eshape ðFS ðAÞ; Fshape ðFSþ Ereg ðTÞ
E þ Ereg ðTÞ ¼ Eintensity ðTÞ ¼ Eintensity ðB; A3TÞ þ ESSD
where e is the ¼ mini fmaxp ð mini fmaxp ðF
SSD SSD SSD S ðA3TÞÞ ðAÞ; FS ðA3TÞÞ þ Ereg ðTÞ
in Eqs. (7),in Eqs. (7), (8)should be should be rep
(8) and (10) and (10) replaced by M
ð8Þ ð8Þ
implementation procedure. procedure.
implementation
By extending the originalthe original Demons registration[25],
By extending Demons registration algorithm algorithm [25],
Shape constraint introduction :
an optimal an optimal solution can be obtained by the alternating strategy.
solution can be obtained by the alternating strategy.
The displacement vectors related to the intensity and the shape at
The displacement vectors related to the intensity and the shape at
3.2. Topology correction strategy
3.2. Topology correction strategy
the point p theinterest regions are regions are
of point p of interest
As is mentioned, mentioned, topology preservatio
As is topology preservation of a defor
ðA3TðpÞÀBðpÞÞðA3TðpÞÀBðpÞÞ is important in registration-segmentation method
is important in registration-segmentati
uintensity ðpÞ ¼ À
uintensity ðpÞ ¼ À 2 rBðpÞ rBðpÞ ð9Þ ð9Þ
8 ðA3TðpÞÀBðpÞÞ 22
ðA3TðpÞÀBðpÞÞ þ JrBðpÞJ þ JrBðpÞJ 2 brain case.brain case. bijectivity bijectivity and
Although Although and smoothing
adopted in optimizing the cost function like Eq. (4) l
adopted in optimizing the cost function o
>0
<
ushape ðpÞ ¼ À shape ðpÞ ¼ À
u
ðFS ðA3TðpÞÞÀFSSðA3TðpÞÞÀFS ðAðpÞÞÞ
ð F ðAðpÞÞÞ
2 2 2
p⇥C helpful for preventing topology change, it ischange, it
rFS ðAðpÞÞ 2 rFS ðAðpÞÞ helpful for preventing topology hard to b
ðFS ðA3TðpÞÞÀFSSðA3TðpÞÞÀFrFS ðAðpÞÞJ rFS ðAðpÞÞJ
ðF ðAðpÞÞÞ þJ S ðAðpÞÞÞ þJ theory. The topology preservation problem of t
theory. The topology preservation pro
S (p) = d(p, C) p⇥S
ð10Þ algorithm has been investigated in recent years [3
ð10Þ algorithm has been investigated in rece
cases without topology preservation using this metho
cases without topology preservation usin
>
:
The combined displacement vector is vector is
The combined displacement found in some published reliable experiments [3
found in some published reliable exp
d(p, C)
uðpÞ ¼ ð1ÀbuðpÞ ¼ ð1ÀbÞuintensityðpÞ þ bushape ðpÞ
Þuintensity ðpÞ þ bushape p ⇥ ¬S
ð11Þ experiments. By optimizing optimizing the cost fu
ð11Þ experiments. By the cost function over
diffeomorphism, a diffeomorphic Demons algorith
diffeomorphism, a diffeomorphic Demo
posed [36,37]. In this paper, a different simple topolog
posed [36,37]. In this paper, a different sim
method is proposedis proposed based onfield vector
method based on the vector the analys
Let T ¼ ðX; Y; ZÞ T ¼ ðX; Y; ZÞ deformation field, whe
Let denote the denote the deformatio
the new point p of point p ðx; y; zÞ afte
the new position of position ðx; y; zÞ after deformati
1 β1 β
0.5 0.5
− + − +
x
x0 x 0 x 0 x0 x 0 x 0
Fig. 1. An example of the shape representationrepresentation (a) (b) its shape (b) its shape
Fig. 1. An example of the shape (a) the putamen the putamen
S
distance representationrepresentation map.
distance map.
S Fig. 2. The function parameter b.
Fig. 2. The function of the balance of the balance
39