Chapter
Chapter III:
III:
Opening,
Opening, Closing
Closing
Opening and Closing by adjunction
Algebraic Opening and Closing
Top-Hat Transformation
J. Serra
Ecole des Mines de Paris ( 2000 )
Granulometry
Course on Math. Morphology III. 1
Adjunction
Adjunction Opening
Opening and
and Closing
Closing
The problem of an inverse operator
Several different sets may admit a same erosion, or a same dilate. But among
all possible inverses, there exists always a smaller one (a larger one). It is
obtained by composing erosion with the adjoint dilation (or vice versa) .
Erosion
structuring
element
?
The mapping is called adjunction opening,
and is denoted by
γ Β = δΒ εΒ
( general case)
XoB = [(X B) ⊕ Β]
(τ-operators)
By commuting the factors δΒ and εΒ we
obtain the adjunction closing
ϕ Β = εΒ δ Β
(general case),
X•Β
•Β = [X
[ ⊕ Β) B] (τ-operators).
( These operators, due to G. Matheron are sometimes called morphological)
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 2
Properties
Properties of
of Adjunction
Adjunction Opening
Opening and
and Closing
Closing
Increasingness
Adjunction opening and closing are increasing as products of
increasing operations.
(Anti-)extensivity
By doing Y= δΒ(X), and then X = εΒ(Y) in adjunction
δΒ(X) ⊆ Y ⇔ X ⊆ εΒ(Y) , we see that:
δδΒΒεεΒΒ(X)
(X)⊆
⊆XX⊆
⊆εεΒΒδδΒΒ(X)
(X) hence εΒ (δΒ εΒ) ⊆ εΒ ⊆ (εΒ δΒ)εΒ⇒εεΒΒδδΒΒεεΒΒ==εεΒΒ
Idempotence
The erosion of the opening equals the erosion of the set itself.
This results in the idempotence of γ Β and of ϕΒ :
γγΒ γγΒΒ==γγΒΒand,
and,
Β
εΒ (δΒ εΒ) = εΒ ⇒ δΒ εΒ (δΒ εΒ) = δΒ εΒ i.e. by duality
by duality ϕϕΒΒϕϕΒΒ==ϕϕΒΒ
Finally, if εΒ(Y) = εΒ(X), then γ Β(X) = δΒ εΒ(X) = δΒ εΒ(Y) ⊆ Y .
Hence, γ Β is the smallest inverse of erosion εΒ .
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 3
Amending
Amending Effects
Effects of
of Adjunction
Adjunction Opening
Opening
Geometrical interpretation
z ∈ γΒ(X) ⇔ z∈ By and y∈ X B
hence
zz∈∈γ(X)
⇔
γγ(X)
⇔ zz∈∈BByy⊆
⊆XX
γ
• the opened set γΒ(X) is the union of
the structuring elements B(x)
which are included in set X.
• In case of a τ−opening, γΒ(X) is the
zone swept by the structuring
element when it is constrained to
be included in the set.
J. Serra
Structuring element
Ecole des Mines de Paris ( 2000 )
Opening
When B is a disc, the opening amends
the caps, removes the small islands
and opens isthmuses.
Course on Math. Morphology III. 4
Amending
Amending Effects
Effects of
of Adjunction
Adjunction Closing
Closing
Geometrical interpretation
• By taking the complement in the
definition of XoB we see that
v
Structuring element
Closing
v
X•Β
•Β = [(X
[( ⊕ Β) B]
• The τ-closing is the complement
of the domain swept by B as it
misses set X. Note that in most of
the practical cases, set B
is
v
symmetrical, i.e. identical to B.
• Note that when a shift affects both
erosions and dilations, it does not
acts on openings and closings.
J. Serra
When B is a disc, the closing closes the
channels, fills completely the small lakes,
and partly the gulfs.
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 5
Effects
Effects on
on Functions
Functions
• The adjunction opening and
closing create a simpler
function than the original.
They smooth in a nonlinear
way.
• The
opening
(closing)
removes positive (negative)
peaks that are thinner than
the structuring element.
Level
Closing
60
50
Original
40
Opening
30
20
Structuring
element
10
0
• The
opening
(closing)
remains below (above) the
original function.
J. Serra
0
Ecole des Mines de Paris ( 2000 )
20
40
60
80
Sample
Course on Math. Morphology III. 6
Algebraic
Algebraic Opening
Opening and
and Closing
Closing
The three basic properties of adjunction openings δε and closings εδ are also
the axioms for the algebraic notion of an opening and a closing.
Definition : In algebra, any transformation which is:
• increasing, anti-extensive and idempotent is called an (algebraic)
opening,
• increasing, extensive and idempotent is called a (algebraic) closing.
Particular cases :
Here are two very easy ways for creating algebraic openings and closing,
they are often used in practice:
1) Compute various opening (closing) and take their supremum (or the
infimum in case of closings).
2) Use a reconstruction process (see V- 9 and VI - 5)
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 7
Invariant
Invariant Elements
Elements
Let B be the image of lattice L under the algebraic opening γ, i.e. B = γ (L).
Since γ is idempotent, set B generates the family of invariant sets of γ :
b ∈ B ⇔ γ (b) = B .
1/ Classe B is closed under sup. For any family {bj, j ∈ J } ⊆ B, we have
γ (∨bj, j ∈ J) ≥ ∨{γ (bj) , j ∈ J } = ∨( bj, j ∈ J )
by increasingness, and the inverse inequality by anti-extensivity de γ.
Moreover , 0 ∈ B. Note that γ does not commute under supremum.
2/ Therefore, γ is the smallest extension to L of the identity on B, i.e.
γ (x) = ∨ { b : b ∈ B , b ≤ x } ,
x∈L.
[ The right member is an invariant set of γ smaller than x, but also that
contains γ (x). ]
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 8
Representation
Representation of
of openings
openings
Conversely, let B be the class closed under sup. that an arbitrary class B0
generates. With each b ∈ B ∪0 associate dilation
δb(a) = b when a ≥ b
By adjunction, it yields opening
;
γb(x) = ∨ {δΒ(a) , δb(a) ≤ x } =
δb(a) = 0 when not .
b when b ≤ x ,
(2)
0 when not .
[ If b ≤ x, it suffices to take a ≥ b for obtaining a term = b in the ∨ ; if not,
no δb(a) ≠ 0 is ≤ x ] . Then relation (1) becomes
γ (x) = ∨{ δb(a) , b ∈ B }.
Theorem (G. Matheron , J. Serra ) : 1/ With any class B0 ⊆ L corresponds a
unique opening on L that has for invariant sets the b ∈ B0 .
2/ Every algebraic opening γ on lattice L may be represented as the union of
the adjunction openings linked to the invariant sets of γ by relation (2).
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 9
Suprema
Suprema of
of Openings
Openings
Theorem
Theorem::
•• Any
Any supremum
supremum of
of openings
openings
isisstill
stillan
anopening.
opening.
•• Any
Any infimum
infimum of
of closings
closings isis
still
stillaaclosing.
closing.
Application
Application::
For
For creating
creating openings
openings with
with specific
specific
selection
selection properties,
properties, one
one can
can use
use structuring
structuring
elements
elements with
with various
various shapes
shapes and
and take
take their
their
supremum.
supremum.
Opening by
Opening by
sup.
Original
Opening by
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 10
"Top-hat"
"Top-hat" Transformation
Transformation
Goal
• The "top-hat" transformation, due to F. Meyer, aims to suppress
slow trends, therefore to enhance the contrasts of some features
in images, according to size or shape criteria. This operator
serves mainly for numerical functions.
Definition
• The "top-hat" transformation is the residue between the identity
and a (compatible with vertical translation) opening
ρ(f) = f - γ(f) ( functions) ; ρ(X) = X \ γ(X) (sets)
• A dual top-hat can be defined: the residue between a closing
and the identity:
ρ*(f) = ϕ(f) - f ( functions) ; ρ∗(X)
= ϕ(X) \ X (sets)
ρ∗
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 11
Top-hat
Top-hat Properties
Properties
Idempotence
• The top-hat is idempotent (but not increasing). Moreover, when the
original signal is positive the top hat is anti-extensive:
ρ(ρ
ρ(f)) = ρ(f)
f > 0 ⇒ ρ(f) < f
Geometrically speaking, the top-hat reduces to zero the slow trends
of the signal.
Robustness
• If Z stands for the set of points where the opening is strictly smaller
than f, i.e.
Z = { x : (γ f)(x) < f(x) }
and if g is a positive function whose support is included in Z, then
T(g) = g
J. Serra
and
Ecole des Mines de Paris ( 2000 )
T (f+g) = T(f) + T(g)
Course on Math. Morphology III. 12
Use
Use of
of the
the Top-hat
Top-hat
Sets
• The top-hat extracts the objects that
have not been eliminated by the
opening. That is, it removes objects
larger than the structuring element.
Functions
• The top-hat is used to extract
contrasted components with respect
to the background. The basic top-hat
extracts positive components and the
dual top hat the negative ones.
• Typically, top-hats remove the slow
trends, and thus performs a contrast
enhancement.
J. Serra
Ecole des Mines de Paris ( 2000 )
Level
Original
100
80
Opening: γ
60
40
Top hat
20
0
0
20
40
60
80
Sample
Course on Math. Morphology III. 13
An
An Example
Example of
of Top-hat
Top-hat
Comment :The goal is the extraction of the aneurisms ( the small white spots). Top hat c), better
than b) is far from being perfect. Here opening by reconstruction yields a correct solution (VI-7).
Negative image
of the retina.
J. Serra
Top hat by an
hexagon opening
of size 10.
Ecole des Mines de Paris ( 2000 )
Top hat by the sup
of three segments
openings of size 10.
Course on Math. Morphology III. 14
Granulometry:
Granulometry: an
an Intuitive
Intuitive Approach
Approach
•
Granulometry is the study of the size characteristics of sets and of
functions. In physics, granulometries are generally based on sieves ψλ of
increasing meshes λ > 0. Now,
– by applying sieve λ to set X, we obtain the over-sieve ψλ(X)) ⊆ X;
– if Y is another set containing X, the Y-over-sieve, for every λ, is
larger than the X-over-sieve, i.e. X ⊆ Y ⇒ ψλ(X)) ⊆ ψλ(Y)) ;
– if we compare two different meshes λ and µ such that λ ≥ µ, the µover-sieve is larger than the λ−over-sieve, i.e. λ ≥ µ ⇒ ψλ(X)) ⊆
ψµ(X))
– finally, by applying the largest mesh λ to the µ−over-sieve, we obtain
again the λ−over-sieve itself, i.e. ψλ ψµ (X) = ψµ ψλ (X) = ψλ (X)
• Such a description of the physical sieving suggests to resort to openings
for an adequate formalism. The sizes of the structuring elements will play
the role of the the sieves meshes.
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 15
Granulometry:
Granulometry: aa Formal
Formal Approach
Approach
• Matheron Axiomatics defines a granulometry as a family {γλ}
i) of openings depending on a positive parameter λ,
ii) and which decrease as λ increases: λ ≥ µ > 0 ⇒ γλ ≤ γµ .
This second axiom is equivalent to the semi-group where the composition of
two operations is equal to the stronger one, namely
γλ γµ = γµ γλ = γsup(λ,µ
(1)
λ,µ)
λ,µ
⇒ γλ = γλ γλ ≤ (γλ γµ ∨ γµ γλ) ≤ γλ ;
[{ λ ≥ µ > 0 ⇒ γλ ≤ γµ }
conversely, γλ = γµ γλ et γλ ≤ Ι ⇒ γλ ≤ γµ hence semi-group (1).]
• If Bλ et Bµ stand for the invariant elements of γλ and of γµ respectively,
then we easily see that
(1) ⇔ Bλ ⊆ Bµ
• Finally, when γλ's are adjunction openings, i.e. γλ(X)=XoλΒ (with
similar structuring elements), then the granulometry axioms are fulfilled if
and only if the structuring element B is compact and convex.
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 16
An
An Example
Example of
of Granulometry
Granulometry
By duality, the families of closings
{ϕλ , λ>0} increasing in λ generate
anti-granulometries of law
ϕλ ϕµ = ϕµ ϕλ = ϕsup(λ,µ
λ,µ)
λ,µ .
Here, from left to right, closings
by increasing discs.
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 17
Granulometry
Granulometry and
and Measurements
Measurements
• A granulometry is computed
from a pyramid of openings,
or closings, whose each
element is given a size, λ say;
• Value λ is the similarity ratio
holding on the involved
structuring element(s) .
• At the output of each filter,
the area is measured (set
case), or the integral in case of
functions, Mλ say. Then, the
monotonic curve
Fλ = 1 − Mλ / M0
is a Distribution Function .
J. Serra
Measure
open n
Measure
open n-1
Measure
1
Fn
open 2
Measure
open1
n
Measure
initial
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 18
Granulometric
Granulometric Spectrum
Spectrum
One also uses the granulometric spectrum, that is the derivative of the
granulometric distribution function.
Set:
Spectrum
Spectrum
Set:
Set:
Spectrum
Set:
Spectrum
J. Serra
Set:
Set:
Spectrum
Set:
Spectrum
Ecole des Mines de Paris ( 2000 )
Spectrum
Course on Math. Morphology III. 19
References
References
On
OnAdjunction
AdjunctionOpening,
Opening, and
and on
onGranulometry
Granulometry::
•• The
The notion
notion of
of an
an adjunction
adjunction opening
opening (or
(or closing)
closing) and
and that
that of
of aa granulometry
granulometry
were
were introduced
introduced by
by G.Matheron
G.Matheron in
in 1967
1967 {MAT67},
{MAT67}, for
for the
the binary
binary and
and
translation
translation invariant
invariant case.
case. In
In 1975,
1975, {MAT75}
{MAT75} linked
linked them
them with
with algebraic
algebraic
closings
closings in
in E.H.Moore’s
E.H.Moore’s sense
sense (begining
(begining of
of the
the XXth
XXth century)
century) and
and
generalized
generalized the
the notion
notion of
of aa granulometry,
granulometry, but
but still
still in
in the
the set
set case.The
case.The
extension
extension to
to lattices,
lattices, with
with the
the representation
representation theorem,
theorem, isis due
due to
to J.J. Serra
Serra
{SER88.
{SER88. For
For details
details and
and examples
examples about
about granulometric
granulometric spectrum,
spectrum, see
see
{HUN75},{SER82,ch.10},{MAR87a}.For
{HUN75},{SER82,ch.10},{MAR87a}.Foralgebraic
algebraicopenings,
openings,see{RON93}.
see{RON93}.
On
On Gradient
Gradientand
andTop
Tophat
hat mappings
mappings::
•• Morphological
Morphological gradient
gradient appears
appears for
for the
the first
first time
time with
with S.Beucher
S.Beucher and
and
F.Meyer
F.Meyer in
in 1977
1977 {BEU77}
{BEU77} and
and top
top hat
hat operator
operator with
with F.Meyer
F.Meyer in
in 1977
1977
{MEY77},
{MEY77}, followed
followed by
by aa reformulation
reformulation in
in terms
terms of
of function
function mappings
mappings in
in
{SER82,ch.12}.
{SER82,ch.12}. More
More sophisticated
sophisticated versions
versions of
of the
the gradient
gradient can
can be
be found
found in
in
{BEU90},
{BEU90}, and
and top
top hat
hat mappings
mappings with
with non
non flat
flat structuring
structuring elements
elements are
are
proposed
proposedin
in{STE83},
{STE83},(see
(seealso
also{SER88,
{SER88,ch9}).
ch9}).
J. Serra
Ecole des Mines de Paris ( 2000 )
Course on Math. Morphology III. 20