Mathematical Morphology I: Structures

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Mathematica l Morphology IStructures
June 10-17 2007
GEONET VI
Image analysis Complete lattices
Remarquable elements Types of lattices
Case of numerical functions Lattices of
opérateurs
Jean Serra, ESIEE
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Image Processing (I)

• gt Image processing may be divided into three
  classes of questions, namely Codification,
  Extraction of characteristics and Segmentation.
• 1) Codification
• Codification concerns the representation of the
  images. It comprises

Numerical Image
Analog Image
Acquisition Transformation of analog images
into numerical ones. Compression Modification
of image representation. Synthesis Generation
of an image from a more symbolic representation.
Acquisition
Numerical Image n 1
Numerical Image n 2
Compression
Numerical Image
Numbers
Synthesis
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Image Processing (II)

• 2) Extraction of Characteristics
• The aim here is to improve image quality or to
  exhibit some of its features. This includes in
  particular measurements, noise reduction, and
  filtering.
• 3) Segmentation
• Segmentation consists in partitioning the
  images into zones which are homogeneous according
  to a given criterion.

Image2
Image1
Image Tranformations
Numbers
Pixels
Regions
Segmentation
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Four Approaches in Image Processing
Geometric Space
Abstract Spaces

• Linear
• Convolution,
• Fourier ,Wavelets
• Tomography
• Kriging, Splines

• Statistical
• Multivariate Analysis
• Neural Nets
• Stereology

Linear

• Morphological
• Morph. Filtering
• Hierarchies
• (e.g. Granulometry)
• Random Sets
• Watersheds

• Syntactical
• semantic approaches
• Grammars
• Indexation

Non linear
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Definitions of Mathematical Morphology
Mathematics
Physics
Lattice theory for objects or operators in
continuous or discrete spaces Topological and
stochastic models.

• Signal analysis techniques based on set theory
  aiming at the study of relations between physical
  and structural properties.

Engineering
Signal Processing
Algorithms and software / hardware tools for
developing signal processing applications.
Nonlinear signal processing technique based on
minimum and maximum operations.
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Basic Structures

• Mathematical morphology
• The basic structure is a complete lattice i.e. a
  set o such that
• 1) o is provided with a partial ordering, i.e. a
  relation with
• A A
• A B, B A Þ A B
• A B, B C Þ A C
• 2) For each family of elements XiÎP, there
  exists in o
• a greatest lower bound Xi, called infimum ( or
  inf.)
• a smallest upper bound Xi, called supremum (
  or sup.)

Linear signal processing The basic structure
in linear signal processing is the vector space
i.e. a set of vectors V and a set of scalars K
such that 1) - K is a field - V is a
commutative group 2) - There exists a
multiplicative law between scalars and vectors.
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Basic Operations

• Mathematical Morphology
• Since the Lattice structure lies on the ordering
  relation, on the sup and the inf, the basic
  operations are those which preserve these
  fundamental laws, namely
• Ordering Preservation
• XYÞY(X)Y(Y)Û increasingness
• Commutation under Supremum.
• Y (Xi ) Y (Xi ) Û Dilation
• Commutation under Infimum
• Y (Xi ) Y (Xi ) Û Erosion

Linear Signal Processing Since the structure is
that of a vector space, whose fundamental laws
are addition and scalar product, then The basic
operations are those which preserve these
laws, i.e. which commute under them Y(ª li
fi ) ª li Y (fi ) The resulting operator is
called convolution.

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Examples of Lattices
Lattice of subsets P(E) of a set E The partial
ordering is defined by the inclusion law Sup
Inf ? Extremes E, Æ
Lattices of real or integer numbers This total
ordering is given by the succession of the
values Sup (usual sense) Inf Extremes
-T, T

• Lattice of convex sets
• The order is defined by the inclusion law
• Sup Convex hull of the union
• Inf Intersection

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Lattices of Functions
_
_

• If E is an arbitrary set, and if T designates R,
  Z one of their closed subsets, then the
  functions f E T generate in turn a new
  lattice, denoted by TE, for the product ordering
• f g iff f(x)
  g(x) for all x Î E ,
• where sup and inf derive directly from those of
  T, i.e.
• ( fi )(x) fi (x) ( fi )(x)
  fi (x) .
• By convention, the same symbol 0 stands for the
  minimum in T and in TE .
• In TE, the pulse functions
  
  
  
  
  kx,t (y) t when x y
  kx,t (y) 0 when x ¹ y
  
• are sup-generators, i.e. any f E T is
  a supremum of pulses.
• The approach extends directly to the products
  of T type lattices, i.e. to multivariate
  functions ( e.g. color images, motion).

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Lattices of Partitions

• Definition A partition of space E is a mapping
  D E Ds(E) such that
• (i) " x Î E , x Î D(x)
• (ii) " (x, y) Î E,
• either D(x) D(y)
• or D(x) ? D(y) Æ
• The partitions of E form a lattice g for the
  ordering according to which D D' when each
  class of D is included in a class of D'. The
  largest element of g is E itself, and the
  smallest one is the pulverizing of E into all its
  points.

The sup of the two types of cells is the
pentagon where their boundaries coincide. The
inf, simpler, is obtained by intersecting the
cells.
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Atoms , Co-primes and Complement

• A subset L of lattice L is called a sub-lattice
  if it closed under and and contains the two
  extremes 0 and m of L.
• A lattice L is complemented when for every a Î L
  , there exists one b Î L at least such that
• a b m
  a b 0 .
• A non zero element a of a lattice L is an atom
  if
• x a L x 0
  or x a .
• An element x Î L is said to be a co-prime when
• x a b L x a or x b
  .
• Moreover, element x Î L is strongly co-prime
  when for any (finite or not) family bi , iÎI
• x bi , iÎI L x b
  for some b Î bi .

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Sup-generators Distributivity

• A lattice L is sup-generated when it has a subset
  X, called a sup-generator, such that every a Î L
  is the supremum of the elements of X that it
  majorates
• a x Î X , x a
• When the sup-generators are co-prime (resp.
  atomic), then lattice L is said to be co-prime
  (resp. atomic) .
• Lattice L is distributive if, for all a , y ,
  z Î L
• a ( y z ) ( a y ) ( a z ) or,
  equivalently
• a ( y z ) ( a y ) ( a z ) .
• When theses conditions extend to infinity,
  lattice L is infinite distributive
• a ( yi , i Î I ) ( a yi ) , i Î
  I
• a ( yi , i Î I ) ( a yi ) , i Î
  I
• ( NB the two conditions are no longer
  equivalent !)

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Characterisation of s(E) Lattices

• Theorem ( G.Matheron) The four following
  statements are equivalent
• L is complemented and generated by the class Q
  of its co-primes
• L is atomic ( of class Qa) and generated by
  the class Qs of its strong co-primes
• L is isomorphic to a s(E) type lattice
• L is isomorphic to lattice s(Q) .
• When they are satisfied, L is infinite
  distributive and we have
• Q Qa Qs .
• Other lattices
• The function lattice TE is infinite distributive
  but not complemented. The pulses are
  sup-generating co-primes, and even strong
  co-primes when T is discrete (T finite, T Z ),
  but they are not atoms .
• The lattice g of the partitions is sup-generated,
  but neither distributive nor complemented.

_
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Notion of duality

• Examples of involution
• Lattice of subsets of a set The involution is
  the complement. It translates to the classical
  notion of foreground and background
• Lattice of real functions bounded by 0,M The
  involution is the reflection with respect to M/2.

The two laws of Sup. and Inf. play a symmetrical
role. Each involution (c) that permutes them
generates a duality. More precisely, Definition
Two operators y and y are dual with respect to
the involution (c) when y ( Xc ) y (X) c
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Self duality

• Mathematical morphology
• The fundamental duality between Sup. and Inf.
  translates to all morphological tools.
• In general, morphological operations go by pair
  and correspond to each other by duality as
  examples erosion and dilation, opening and
  closing.
• However, operators may also be
• - self-dual, i.e.
• y ( Xc ) y (X) c ( e.g. morph. centre)
• - or invariant under duality, i.e.
• y ( Xc ) y (X) (e.g. boundary set in Rn )

• Linear processing
• The convolution operation is self dual, that is
  dual of itself
• f (-g) - (f g)
• This means that positive or negative (bright and
  dark) components are processed in a symmetrical
  way.

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Input Output Comparison

• Extensivity anti-extensivity A transformation
  is extensive if its output is always greater than
  its input. By duality, it is anti-extensivity
  when the output is always smaller than the input.
• Extensivity X Í Y (X)
  anti-extensivity X Ê Y (X) X Í E
• Set
  (extensivity) Function
  (extensivity)
• Idempotence A transformation is idempotent if
  its output is invariant with respect to the
  transformation itself
• Idempotence Y Y (X) Y (X)

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Lattices of Operators

• With every lattice L is associated the class L'
  of the operations a LL . Now, L' turns out to
  be a lattice where a b (in
  L' ) O a(A)
  b(A) for all A Î L (ai) (A)
  ai (A) (ai) (A)
  ai (A) ( in L' ) ( in L )
  ( in L' ) ( in L )
• for example, The mappings which are
• - increasing , - or extensive , -
  or anti-extensive ,
• over L are each a sub-lattice of L'
• More generally, we shall meet lattices for
• - openings - filters
  - activity etc...

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Notion of Residues in Morphology

• The theory of morphological filters has
  highlighted the increasing and idempotence
  properties, as well as the ordering rules between
  transformations.
• There is a family of transformations which
  studies the difference between two (or many)
  basic transformations. Their common basis relies
  on the notion of difference also called residue.

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Classification of residues

• The residues that are used in practice can be
  classified in three groups
• 1) Residues of two primitives
• 2) Residues of two family of primitives
• 3) Residues relying on "hit or miss"
  transformations

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Magnification and Operators

• It is often necessary to modify the working
  scale, i.e. to perform a
• space similitude X lX. Here two situations can
  be distinguished
• 1) Transformations which commute under
  magnification
• y (X) (1 /l) y (l X)
• Examples boundary, skeleton, centre of gravity.
• 2) Family of transformations that are compatible
  with magnification
• y1 ( X) (1 / l) yl (l X)
• Examples Granulometries, FAS, associated with
  similar structuring
• elements B(l) lB. We then have yB (X) (1/l)
  ylB (l X).
• N.B. - Family yB is globally invariant under
  magnification
• - Mutatis mutandis, the above notions are
  also valid for functions
• - They model Multi-resolution
  Decomposition ( i.e. pyramids ).

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Curves and Measurements

• Every Image processing ends either by a new image
  (e.g. filtering) or by a measurement, i.e. by
  numbers.
• Measurements
• The simplest and the most often used measurement
  is the labelling of presence / absence. The
  second one is Lebesgue measure, or its digital
  versions. There exist a few other ones,
  topological or metrical.
• Families depending on a positive parameter
• In this case, the two usual representations
  consist in
• either associating a measurement with each
  transformation of the family and obtaining a
  curve depending on parameter l. Examples
  histogram, size distributions
• or considering the family of transformations as
  the sections of a numerical function. Example
  distance function.

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References

• On Roots
• Mathematical morphology has two main roots
  lattice theory and random geometry. It was
  created by G. Matheron and J.Serra in 1964 and
  known by their three basic publications
• MAT75 mainly deals with sets (topological
  framework, random sets, boolean models,
  convexity, granulometry, representation of
  increasing mapping by dilations),
• SER82 concentrates on translation invariant
  mapping (extension to functions, discrete
  morphology, thinning/thickening, combination of
  operators) and
• SER88 enlarges the approach the lattice
  framework (dilation, theory of morphological
  filtering, connectivity, skeletons, boolean
  functions).This lattice approach was pursued by
  H.Heijmans and Ch. Ronse HEI90RON91.
• In addition, there exists three
  excellent treatises on the subject COS89
  SCH93 HEI 94 and SER01. Instructive
  overviews of mathematical morphology can be found
  in SER87,HAR87,GIA88,DOU92b and SER97.