Ultimate grain filter

Abstract

This work introduces a residual operator called ultimate grain filter which is a powerful image operator based on numerical residues. With a multi-scale approach, the ultimate grain filter analyzes an image under a series of grain filters of increasing grain sizes. Thus, contrasted objects can be detected if a relevant residue is generated when they are filtered out by one of these grain filters. We also present an efficient algorithm for ultimate grain filter computation by using a structure called tree of shapes. In fact, since the result of a given grain filter can be obtained by pruning the corresponding tree and reconstructing it, we show that the result of the ultimate grain filter (which is based on numerical residues from a family of grain filters) can be obtained by the computation of the difference (remaining nodes) of the corresponding pruned trees. Finally, we propose the use of ultimate grain filter to extract contrasted objects using a priori knowledge of an application.

Full text

ULTIMATE GRAIN FILTER
Wonder Alexandre Luz Alves and Ronaldo Fumio Hashimoto
Department of Computer Science, Institute of Mathematics and Statistics,
University of S˜
ao Paulo, S˜
ao Paulo, Brazil
ABSTRACT
This work introduces a residual operator called ultimate grain filter
which is a powerful image operator based on numerical residues.
With a multi-scale approach, the ultimate grain filter analyzes an
image under a series of grain filters of increasing grain sizes. Thus,
contrasted objects can be detected if a relevant residue is generated
when they are filtered out by one of these grain filters. We also
present an efficient algorithm for ultimate grain filter computation
by using a structure called tree of shapes. In fact, since the result
of a given grain filter can be obtained by pruning the corresponding
tree and reconstructing it, we show that the result of the ultimate
grain filter (which is based on numerical residues from a family of
grain filters) can be obtained by the computation of the difference
(remaining nodes) of the corresponding pruned trees. Finally, we
propose the use of ultimate grain filter to extract contrasted objects
using a priori knowledge of an application.
Index Terms—residual morphological operator; ultimate grain
filter; grain filter; tree of shapes; connected operator.
1. INTRODUCTION
An operator in Mathematical Morphology (MM) can be seen as a
mapping between complete lattices [1]. In particular, mappings on
the set of all gray level images F(D)defined on domain D ⊂ Z2
are of special interest in MM and they are called image operators,
i.e., ψ:F(D)→ F(D). In this study we focus on a special type
of operator known as residual operator and is defined as the differ-
ence (pixel-by-pixel) between two operators, say ψand ϕ, applied
to a given image f∈ F(D), i.e., r(f) = ψ(f)−ϕ(f). In the
literature, it is possible to find many applications of residual oper-
ators in Image Processing: morphological gradients (which are the
residue of a dilation δand an erosion ε, i.e., r(f) = δ(f)−ε(f)),
white top-hats (which are the residue of the identity operator id
and an opening γ, i.e., r(f) = id(f)−γ(f)); and black top-hats
(which are the residue of a closing φand the identity operator, i.e.,
r(f) = φ(f)−id(f)) are examples of residual operators. Residual
operators have been extended to extract residues of two families (in-
dexed by a set I) of operators, applied to an image, {ψi(f) : i∈I}
and {ϕi(f) : i∈I}as the supremum of residual operators, i.e.,
r(f) = supi∈I{ri(f) : ri(f) = ψi(f)−ϕi(f)}[2]. For ex-
ample, the ultimate opening operator is defined as the supremum of
residual operators when ψiand ϕiare consecutive openings, i.e.,
ri(f) = γi(f)−γi+1(f)[2].
These residual operators have been extended to extract residues
of families of connected operators [3, 4, 5, 6, 7]. One of the ad-
vantages is that the connected operators do not create new contours.
Thus, contrasted objects are detected when they are filtered out by a
connected operator, generating an important connected residue. At
first, Retornaz and Marcotegui [3] proposed the ultimate attribute
opening (UAO) and by duality ultimate attribute closing (UAC) for
scene text localization. Then, Fabrizio and Marcotegui [8] devel-
oped an efficient algorithm for computing UAO using component
tree. Later, Hern´
andez and Marcotegui [5] added shape information
to extract residues from UAO. Differently from these works [3, 5, 8],
Meyer [4] proposed to extract residues from a family of levelings
obtained by using erosions and dilations as markers. The obtained
levelings produce positive and negative residues (which are not the
case for the UAO and UAC). In this way, Meyer [4], in contrast to
Beucher [2], treats separately the positive and negative residues.
Given the above considerations, in this work, we propose a new
residual operator called ultimate grain filter (UGF). A powerful
residual operator constructed through a multi-scale approach. It pro-
vides the same treatment to both poles of the image, which allows
us, differently from UAO (resp. UAC), to assume no hypotheses
about the contrast of objects and this is the main advantage of UGF
in compasison to the UAO (resp. UAC). We also show that the
family of grain filters is represented by a family of trees that satisfies
a scale-space based on levelings. Lastly, we use these trees to build
an efficient algorithm for computing UGF.
The remainder of this paper is structured as follows. For the sake
of completeness, Sections 2 and 3 briefly recall some definitions and
properties of image representation by tree structures and grain fil-
ters. In particular, Section 3 provides the first result of this work.
The other contributions of this paper are given in Section 4, where
we introduce the UGF and present an efficient algorithm for its com-
putation as well experimental results. Finally, Section 5 concludes
this work and presents some future research direction.
2. PRELIMINARIES
For any µ∈Zwe define Xµ(f) = {p∈ D :f(p)< µ}and
Xµ(f) = {p∈ D :f(p)≥µ}as the lower and upper level sets at
value µ, respectively. From these sets we deduce two other sets L(f)
and U(f)composed of the connected components (CCs) of the lower
and upper level sets of f, i.e., L(f) = {C∈ CC4(Xµ(f)) : µ∈Z}
and U(f) = {C∈ CC8(Xµ(f)) : µ∈Z}, where CC4(X)and
CC8(X)are sets of 4and 8connected CCs of X, respectively. Then,
the pairs ordered consisting of the families of CCs of lower and upper
level sets and the usual set inclusion relation, i.e., (L(f),⊆)and
(U(f),⊆), induce two dual trees [9]. They can be represented by
non-redundant data structures known as min-tree and max-tree [10].
Combining this pair of dual component trees, min-tree and max-
tree, into a single tree, we have the tree of shapes [9, 11]. Note that,
if C∈ L(f)then the CCs of D\Care in U(f)(or vice versa, C∈
U(f)⇒ CC4(D\C)⊆ L(f)). Then, let P(D)denote the powerset
of Dand let sat :P(D)→ P(D)be the operator of saturation [12]
(or filling holes) and SAT (f) = {sat(C) : C∈ L(f)∪ U (f)}
be the family of CCs of the upper and lower level sets with holes
filled. The elements of SAT (f), called shapes, are nested by an
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inclusion relation and thus the pair (SAT (f),⊆), induces the tree
of shapes [9, 12].
Tree of shapes, and also component trees, are complete repre-
sentation of images which can be represented by a compact and non-
redundant data structure [12, 13] so that a pixel p∈ D which is asso-
ciated to the smallest shape of the tree containing it, by parenthood
relationship, is also associated to all its ancestors shapes. Then, we
denote by SC (Tf, p)the smallest shape containing pin tree Tf.
3. GRAIN FILTERS AND LEVELINGS
Grain filters are powerful connected filters that have as characteristic
the simplification of the image by regional extrema. The first kind of
grain filters has been introduced by L. Vicent [14] in order to remove
small CCs of level sets called grain, i.e.,
∀x∈ D,[γκ
λ(f)](x) = sup
C∈L0:x∈C
{inf
y∈C{f(y)}},
∀x∈ D,[φκ
λ(f)](x) = inf
C∈U0:x∈C{sup
y∈C
{f(y)}},
where L0and U0are subsets of L(f)and U(f), respectively, such
that its elements have area κgreater than λ. These operators (that
do not commute, i.e., γκ
λ(φκ
λ(f)) 6=φκ
λ(γκ
λ(f))), are opening (and
closing respectively), the grains are the CCs of level sets. A second
kind of grain filters was introduced by Masnou [15] and their prop-
erties were studied by V. Caselles and P. Monasse [12, 16]. In this
filter, the grains are “shapes” and defined as: ∀x∈ D,
[νκ
λ(f)](x) = sup
B∈Bλ
{inf
y∈x+B{f(y)}},
where Bλis family of sets such that B∈ Bλsatisfies: (i)Bis
connected and closed; (ii) 0 ∈sat(B);(iii)κ(sat(B)) ≥λ;(iv)
0/∈B⇒κ(CC(D\B , 0)) ≤λ[17].
If we store the information of the area of CCs or “shapes” in
each node of the tree, then we can efficiently obtain these operators
by pruning. For example, if the chosen tree is the max-tree (resp.
min-tree) then the reconstruction of the pruned tree is an area open-
ing (resp. closing) whereas if it is the tree of shapes, the resulting
operator is a grain filter [12]. In fact, many different kinds of in-
formation can be stored in the nodes of the trees, and we call them
attributes. Formally, an attribute is a function κ:P(D)→Rand it
can represent node information such as area, volume, height, width,
circularity. Some attributes can be increasing, i.e., if A, B ∈ P(D)
and A⊆B, then κ(A)≤κ(B). If the attributes are increasing then
the attribute values also follow a well-defined ordering in these trees
due to the hierarchy of the level sets. Thanks to this property, we can
perform connected filtering (extensive, anti-extensive and self-dual)
simply pruning the tree nodes whose attribute value κis lower than
a given threshold λfollowed by reconstruction of the pruned tree.
Remembering that the connected filters represent a wide class
of filters in which F. Meyer [18, 19, 20, 21] extensively studied their
specializations. One of these specializations, known as levelings, are
powerful simplifying filters that preserves order and do not create
new structures (regional extrema and contours).
Definition 1. An operator ψ:F(D)→ F (D)is said leveling, if for
any f∈ F(D)the following relation is valid for all pairs of neigh-
boring pixels, i.e., ∀(p, q)neighbors: [ψ(f)](p)>[ψ(f)](q)⇒
f(p)≥[ψ(f)](p)and [ψ(f)](q)≥f(q).
Levelings are increasing operators and constrain the values
of neighboring pixels, since f(p)≥[ψ(f)](p)>[ψ(f)](q)≥
f(q)⇒f(p)> f(q). Moreover, it is possible to show that
if ψ(f)is the result of filtering using increasing attributes from
the max-tree, min-tree or tree of shapes built from an image f,
then ψ(f)is a leveling of f. In other words, attribute openings
(resp. closings) and grain filters are levelings. In the case of the
max-tree (resp. min-tree) the resulting operator is anti-extensive
(resp. extensive), that is, ψ(f)≤f(resp. ψ(f)≥f). Further-
more, ψ(f)is a leveling of fsince ψ(f)≤f(resp. ψ(f)≥f)
and for any pair of neighboring pixels (p, q)the following condi-
tion holds: [ψ(f)](p)>[ψ(f)](q)⇒[ψ(f)](q) = f(q)(resp.
[ψ(f)](p)>[ψ(f)](q)⇒[ψ(f)](p) = f(p)).
Now, in the case of tree of shapes, it is needed to know how
are the relations between neighboring pixels in the nodes of the tree.
Note that, in the tree there are two types of connectivity. In this
sense, Proposition 1 and 2 help us understand how the neighboring
pixels are related in the tree.
Proposition 1. Let Tfbe the tree of shapes of an image f. Then,
∀(p, q)neighbors such that f(p)6=f(q), we have: either SC(Tf, p)
and SC (Tf, q)are comparable; or SC(Tf, p)∩ SC(Tf, q ) = ∅.
Proposition 2. If A, B ∈ SAT (f)such that B⊆Aand Aand B
are of different types (lower or upper). Then, for any pixel in B, its
neighborhood pixels are in a node rooted in the subtree of node A.
Theorem 1. Let fbe an image and consider the tree Tf(max-tree,
min-tree or tree of shapes) that represents f. Consider that the tree
was filtered by an increasing attribute and this pruned tree was re-
constructed into an image fλ. Then, fλis a leveling of f.
Hint of the proof. The proof for the component tree is trivial. Now,
to prove for the tree of shapes, we need just to check if for all (p, q)
neighbors, the definition of leveling is valid. Let us consider two
cases, where f(p) = f(q)and where f(p)6=f(q). In the first case,
fλmeets the definition of leveling by vacuity. In the second case, we
have SC(Tf, p)6=SC(Tf, q)and the pruning in Tfcan: either (1)
preserve both nodes; or (2) eliminate both nodes; or (3) eliminate
one of the two nodes. In such cases, Propositions 1 and 2 help to
prove that the definition of leveling is satisfied.
Levelings can be nested to create a scale-space decomposition of
an image [20]. Suppose Tfis the tree that represents an image fand
fλis the result of a filtering by increasing attribute in Tfby a thresh-
old λ(i.e., a pruning in Tf). Denote by Tfλthis pruned version of
Tf. Likewise, consider the tree Tfλ+1 pruned obtained by a filtering
of an increasing attribute in Tfλby a threshold λ+ 1. Hence, by
Theo. 1, we have that fλ+1 is a leveling of fλand fλis a leveling of
f. Then, from Def. 1, we have that, for all pairs of neighboring pix-
els (p, q),fλ+1 (p)> fλ+1(q)⇒fλ(p)≥fλ+1 (p)> fλ+1(q)≥
fλ(q)⇒f(p)≥fλ(p)≥fλ+1(p)and fλ+1 (q)≥fλ(q)≥f(q),
showing that fλ+1 is also a leveling of f. This shows that the tree
generates a family of levelings that further simplifies the image f,
thus constituting a morphological scale-space [4, 12, 20]. Fig. 1
shows some examples of images presented in a scale-space of level-
ings obtained by pruning the min-tree, max-tree and tree of shapes.
4. ULTIMATE GRAIN FILTER
Remembering, a residual operator is defined as the difference be-
tween two operators, say ψand φ, applied to a given image f∈
F(D), i.e., ∀p∈ D,[r(f)](p) = [ψ(f)](p)−[φ(f)](p). So we
say that if r(p)>0then r(p)is a positive residue; if r(p)<0
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min-tree
area = 0 area = 491 area = 1966 area = 9953
max-tree
area = 0 area = 491 area = 1966 area = 9953
tree of shapes
area = 0 area = 491 area = 1966 area = 9953
Fig. 1. Examples of images presented in a scale-space of levelings
obtained by pruning the min-tree, max-tree and tree of shapes, re-
spectively.
then r(p)is negative residue; otherwise, r(p)is a null residue. Dif-
ferently from S. Beucher [2] that defined the extended residual op-
erator as supremum of either positive residual operators or negative
residual operators, in this work, we will follow Meyer [4] and con-
sider the extended residual operator as supremum of positive and
negative residual operators, i.e., r(f) = supλ∈I{rλ(f) : rλ(f) =
r+
λ(f)∨r−
λ(f)}where r+
λ(f) = [ψλ(f)−φλ(f)]∨0and r−
λ(f) =
[φλ(f)−ψλ(f)] ∨0.
As original contribution of this work, we present in this section a
residual operator, called ultimate grain filter (UGF), where the prim-
itives are consecutive grain filters, say ψλand ψλ+1, of an indexed
family {ψλ:λ∈I}of grain filters of increasing sizes, i.e.,
R±(f) = R+(f)∨ R−(f),
R+(f) = sup
λ∈I
{r+
λ(f) : r+
λ(f) = [ψλ(f)−ψλ+1(f)] ∨0}and
R−(f) = sup
λ∈I
{r−
λ(f) : r−
λ(f) = [ψλ+1(f)−ψλ(f)] ∨0}.
Note that, if ψλ=γκ
λ(resp. ψλ=φκ
λ) then R±is the UAO
(resp. UAC). We call the operator R±of UGF only when ψλ=νκ
λ.
The UGF is a powerful operator based on numerical residues. With
a multi-scale approach, the UGF analyzes an image under a series of
grain filters of increasing sizes. Contrasted objects are detected when
they are filtered out by a grain filter, generating important residues.
Fig. 2 shows some examples of applications of UAO, UAC and UGF
for extraction of contrasts in a given image. The UGFs, as grain
input image fUAO of fUAC of fUGF of f
Fig. 2. Examples of extraction of contrast.
filters, are operators that provide the same treatment to both poles
of the image (see Fig. 2 how UGF extracted both back and white
contrasts), which allow us, differently from UAOs (resp. UACs),
not to assume any hypotheses about the contrast of objects and this
is the main advantage of UGFs in comparison to the UAOs (resp.
UACs). Moreover, it is possible to show that the UGFs are connected
operators, i.e., preserve contours of the input image.
4.1. Associated information
It can be associated to each pixel x∈ D a set of extracted attributes
of a node C, containing x, that generated the maximum non-null
residue in rλfor x. In this regard, a set of associated information
can be extracted at the time of computation of the UGF. For example,
a function that associates to each pixel the index that produced the
maximum non-null residue [2], i.e.,
q±(x) = q+(x), if R+(x)>R−(x),
q−(x), otherwise,
where q+(x) = max{λ+ 1 : [r+
λ(f)](x) = [R+(f)](x)>0}and
q−(x) = max{λ+ 1 : [r−
λ(f)](x)=[R−(f)](x)>0}. Thus,
SC (Tfq(x), x)is the node containing xthat produced the maximum
non-null residue. This information may be very useful to perform an
analysis of the extraction of residues.
4.2. Algorithm
As the family of grain filters of a image fcan be obtained by prun-
ing its corresponding tree Tf, then two consecutive grain filters, say
fλand fλ+1, can be obtained by pruning Tfby thresholds I(λ)and
I(λ+ 1), respectively. Note that, I:I→Ris a monotone function
that associated the family of indices to the values of indices. There-
fore, we can use the nodes of Tfλthat are not in Tfλ+1 to compute
the residue between fλand fλ+1. Then, let Nr(λ)the set of nodes
removed from Tfλto build the pruned tree Tfλ+1. Therefore, we
can calculate the positive (resp. negative) residues of the pixels of a
given node C∈ N r(λ)by
r+
λ(C) =













[level(C)−level(parent(C)) ∨0]
if parent(C)/∈ N r(λ),
[level(C)−level(parent(C)) ∨0]
+r+
λ(parent(C)),
otherwise,
where level(C)is the gray level associated to node Cand
parent(C)is the parent node of C. Therefore, given an index
λ∈I={0,1, ..., IMAX }we need to know what are the nodes that
are in Nr(λ)and this leads us to think about the family of primitives
of the UGF. For example, the set of all the possible primitives with
the attribute κ, is given by Nr(λ) = {C∈ Tfλ:κ(C)≤ I(λ)}.
The UGF often underestimates the contrast of blurred ob-
jects in the image, since contours of objects are composed of
small gradual transitions [6] of gray levels. Thus, the contrast
of an object is divided into several transitions and the greatest
value of the residue obtained from UGF is inevitably smaller
than the real contrast of object. One way to reduce this discom-
fort is to simplify the family of primitives such that the transi-
tions of the gray levels between two consecutive primitives are
more abrupt. So, suppose that C∈ N r(λ)and parent(C)/∈
Nr(λ). Then, we have κ(parent(C)) −κ(C)>0since
κ(C)≤ I(λ)and κ(parent(C)) >I(λ). Therefore, if
κ(parent(C)) −κ(C)> , then we can add nodes in Nr(λ)
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corresponding small gradual transitions of node C. Thus, we have
Nr(λ) = Nr(λ)∪ {C∈[A, B] : A∈ N r(λ),parent(A)/∈
Nr(λ), B is the first ancestor of A such that κ(parent(C)) −
κ(C)> }. An adaptive way to find is to use extinction val-
ues [22]. We also can use shape recognition techniques to choose
nodes of Nr(λ)based on a criterion of similarity Ω[5]. This al-
lows us to extract only the residues of nodes that satisfy the criterion
Ω, i.e., Nr,Ω(λ) = {C∈ N r(λ) : Csatisfies the criterion Ω}.
Now, we present Alg. 1 to compute the UGF. This algorithm is
based on the algorithm presented by Fabrizio and Marcotegui [8] to
compute the UAO using max-tree and the concept of gradual tran-
sition shown in [6]. This algorithm uses a top-down approach: the
residue of the parent node is calculated; it is then propagated to the
child nodes. Every child node compares its residue with the residue
of the parent node and keeps the maximum of these two values in
R. At the same time, it can also store the index that generated the
maximum residue. After this step, each child node becomes a parent
node and the process is repeated.
Algorithm 1: Compute the ultimate grain filter.
1void UGF(node C)begin
2if C∈ N r,Ω(κ(C)) then
3r+[C] = level(C)−parent(level(C)) ∨0;
r−[C] = parent(level(C)) −level(C)∨0;
4if parent(C)∈ N r,Ω(κ(C)) then
5r+[C] = r+[C] + r+[parent(C)];
r−[C] = r−[C] + r−[parent(C)];
6R+[C] = R+[parent(C)] ∨r+[C];
7R−[C] = R−[parent(C)] ∨r−[C];
// Here: compute q+and q−
8foreach node Schild of Cdo
9UGF (S);
Alg. 1 needs as input a tree (max-tree, min-tree or tree of
shapes). These trees can be built by simple algorithms based union-
find proposed in [13, 23] with complexity O(|D| log(|D|)). Once
the tree is built (see algorithms in [10, 13]), we execute the UGF
function, for all child nodes of the root node. Since the test in
line 4can be done in constant time and UGF function visits each
node of the tree exactly once and the number of nodes of the tree
is at most the number of pixels in the image, UGF is executed with
time complexity O(|D|). Then, we can reconstruct the output of
UGF (by ∀p∈ D,R±[p] = R+[SC(T, p)] ∨R−[S C(T, p)])
and associated information in O(|D|)time complexity. There-
fore, to extract the UGF, we build the tree with time complexity
O(|D| log(|D|)) [10, 13] and compute the residues with time com-
plexity O(|D|). Thus, the time complexity of the proposed algorithm
is O(|D| log(|D|)).
4.3. Example of application
Contrast extraction, through numerical residues, has been used suc-
cessfully in various applications in a preprocessing step such as text
location [3, 24], segmentation of building facades [5], and restora-
tion of historical documents [4]. In this sense, we present a simple
example of application (in problem of extraction of objects) that evi-
dences its potential for future applications. It is worth noting that for
the success of the application is very important to select the primi-
tives based on the knowledge of the application domain. So, let us
consider the problem of text location and cell segmentation, where
objects are characters and cells respectively. Thus, we added shape
information, by using Ω, based on heuristics of text location [7, 24]
and cell segmentation, respectively. The results are shown in Fig. 3,
where (3a)and (3d)are the input images; (3b)and (3e)are the re-
sults of UGF; and (3c)and (3f)are labeled images of the associated
function q±. In Fig. 4, (4a)is the input image and (4b)and (4c)are
labeled images of the associated function q±. Watch a demo of UGF
at http://score.ime.usp.br/wonder/icip2014.
(a) (b) (c)
(d) (e) (f)
Fig. 3. Example of extraction of residues applied to text location
using height (bounding box) attribute and heuristics for the problem
of text location to select the family of primitives.
(a) (b) (c)
Fig. 4. Application of the UGF with circular shape information.
5. CONCLUSION
In this work, we introduced a new residual operator called ultimate
grain filter (UGF) based on numerical residues. A powerful residual
operator constructed through a multi-scale approach. It provides the
same treatment to both poles of the image, which allows us, differ-
ently from the UAO (resp. UAC), not to assume hypotheses about
the contrast of objects and this is the main advantage of UGF in
comparison to the UAO (resp. UAC). We also showed that the fam-
ily of grain filters is represented by a family of trees that satisfies a
scale-space based on levelings. These trees are used to build an effi-
cient algorithm for computing UGF. Extraction of contrast by means
of numerical residues has been used successfully in various appli-
cations, such as text location, segmentation of building facades and
binarization of historical documents. Experimental results applied
to text location show the potentiality of using UGF to real image
processing problems.
Acknowledgements
We would like to thank the financial support from CAPES, CNPq,
FAPESP (grant #2011/50761-2) and NAP eScience-PRP-USP.
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6. REFERENCES
[1] Jean Serra, Image Analysis and Mathematical Morphology,
vol. 2: Theoretical Advances, Academic Press, London, 1988.
[2] Serge Beucher, “Numerical residues,” Image Vision Comput.,
vol. 25, no. 4, pp. 405–415, Apr. 2007.
[3] Thomas Retornaz and Beatriz Marcotegui, “Scene text lo-
calization based on the ultimate opening,” in Proceedings of
the 8th International Symposium on Mathematical Morphol-
ogy and its Applications to Image and Signal Processing, 2007,
vol. 1 of ISMM ’08, pp. 177–188.
[4] Fernand Meyer, “Levelings and flat zone morphology,” Pattern
Recognition, International Conference on, vol. 0, pp. 1570–
1573, 2010.
[5] Jorge Hernandez and Beatriz Marcotegui, “Shape ultimate at-
tribute opening,” Image and Vision Computing, vol. 29, no. 8,
pp. 533 – 545, 2011.
[6] Beatriz Marcotegui, Jorge Hernandez, and Thomas Retornaz,
“Ultimate opening and gradual transitions,” in Mathematical
Morphology and Its Applications to Image and Signal Process-
ing, vol. 6671, pp. 166–177. 2011.
[7] W.A.L. Alves, A. Morimitsu, J.S. Castro, and R.F. Hashimoto,
“Extraction of numerical residues in families of levelings,” in
Graphics, Patterns and Images (SIBGRAPI), 2013 26th SIB-
GRAPI - Conference on, 2013, pp. 349–356.
[8] Jonathan Fabrizio and Beatriz Marcotegui, “Fast implemen-
tation of the ultimate opening,” in Proceedings of the 9th In-
ternational Symposium on Mathematical Morphology, 2009,
ISMM ’09, pp. 272–281.
[9] Vicent Caselles, Enric Meinhardt, and Pascal Monasse, “Con-
structing the tree of shapes of an image by fusion of the trees
of connected components of upper and lower level sets,” Posi-
tivity, vol. 12, no. 1, pp. 55–73, 2008.
[10] Edwin Carlinet and Thierry Graud, “A comparison of many
max-tree computation algorithms,” in Mathematical Mor-
phology and Its Applications to Signal and Image Processing,
vol. 7883 of Lecture Notes in Computer Science, pp. 73–85.
Springer Berlin Heidelberg, 2013.
[11] Pascal Monasse and Frederic Guichard, “Fast computation of
a contrast-invariant image representation,” IEEE Transactions
on Image Processing, vol. 9, no. 5, pp. 860–872, 2000.
[12] Vicent Caselles and Pascal Monasse, “Geometric description
of images as topographic maps,” Lecture Notes in Mathemat-
ics, 2010.
[13] Thierry Graud, Edwin Carlinet, Sbastien Crozet, and Laurent
Najman, “A quasi-linear algorithm to compute the tree of
shapes of nd images,” in Mathematical Morphology and Its
Applications to Signal and Image Processing, vol. 7883 of Lec-
ture Notes in Computer Science, pp. 98–110. Springer Berlin
Heidelberg, 2013.
[14] Luc Vincent, “Morphological area openings and closings for
grey-scale images,” in Proc. of the Workshop Shape in Picture.
1992, pp. 197–208, Springer.
[15] Simon Masnou and Jean-Michel Morel, “Image restoration
involving connectedness,” in In Proceedings of the 6 th Inter-
national Workshop on Digital Image Processing and Computer
Graphics. 1998, pp. 334–6, SPIE.
[16] Vicent Caselles and Pascal Monasse, “Grain filters,” Journal of
Mathematical Imaging and Vision, vol. 17, no. 3, pp. 249–270,
2002.
[17] Pascal Monasse, Morphological representation of digital im-
ages and application to registration, Ph.D. thesis, Universit´
e
Paris IX-Dauphine, Paris, France, June 2000.
[18] Fernand Meyer, “From connected operators to levelings,” in
Proceedings of the 4th International Symposium on Mathemat-
ical Morphology and its Applications to Image and Signal Pro-
cessing, 1998, pp. 191–198.
[19] Fernand Meyer, “The levelings,” in Proceedings of the 4th In-
ternational Symposium on Mathematical Morphology and its
Applications to Image and Signal Processing, 1998, ISMM
’98, pp. 199–206.
[20] Fernand Meyer and Petros Maragos, “Nonlinear scale-space
representation with morphological levelings,” Journal of Vi-
sual Communication and Image Representation, vol. 11, no. 2,
pp. 245 – 265, 2000.
[21] Fernand Meyer, “Levelings, image simplification filters for
segmentation,” Journal of Mathematical Imaging and Vision,
vol. 20, no. 1-2, pp. 59–72, 2004.
[22] Alexandre Gonalves Silva and Roberto de Alencar Lotufo,
“Efficient computation of new extinction values from extended
component tree,” Pattern Recognition Letters, vol. 32, no. 1,
pp. 79 – 90, 2011.
[23] Ch Berger, T. Geraud, R. Levillain, N. Widynski, A. Baillard,
and E. Bertin, “Effective component tree computation with ap-
plication to pattern recognition in astronomical imaging,” in
Image Processing, 2007. ICIP 2007. IEEE International Con-
ference on, 2007, vol. 4, pp. IV – 41–IV – 44.
[24] W.A.L. Alves and R.F. Hashimoto, “Text regions extracted
from scene images by ultimate attribute opening and decision
tree classification,” in Proceedings of the 23rd SIBGRAPI Con-
ference on Graphics, Patterns and Images, 2010, pp. 360–367.
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