Crosssection topology Michel Couprie

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Cross-section topologyMichel Couprie

• Contributors
• G. Bertrand
• M. Couprie
• J.C. Everat (PhD)
• F.N. Bezerra (PhD)

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Functions of 2 variables
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On hills and dales
Cayley 1859, Maxwell 1870
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BINARY IMAGES (SETS)
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Topology preservation - notion of simple point

• A topology-preserving transformation preserves
  the connected components of both X and X

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Simple point
Set X (black points)

• Definition (2D) A point p is simple (for X) if
  its modification (addition to X, withdrawal from
  X) does not change the number of connected
  components of X and X

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Simple point local characterization ?
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Simple point local characterization ?
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Connectivity numbers

• T(p)number of Connected Components of (X - p)
  N(p) where N(p)8-neighborhood of p
• T(p)number of Connected Components of (X - p)
  N(p)
• Characterization of simple points (local) p is
  simple iff T(p) 1 and T(p) 1

U
U
T2,T2
T1,T1
T1,T0
T0,T1
Isolated point
Interior point
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Homotopy

• We say that X and Y are homotopic (they have the
  same topology) if Y may be obtained from X by
  sequential addition or deletion of simple points

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Homotopy illustration
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GRAYSCALE IMAGES (FUNCTIONS)
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Homotopy of functions

• Basic idea consider the topology of each
  cross-section (threshold) of a function
• Given a function F (Z2 Z) and k in Z, we
  define the cross-section Fk as the set of points
  p of Z2 such that F(p) ? k
• We say that two functions F and G are homotopic
  if, for every k in Z, Fk and Gk are homotopic (in
  the binary sense)Beucher 1990

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Homotopy of functions
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F(x,y)
G(x,y)
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Homotopy of functions
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F(x,y)
G(x,y)
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Homotopy of functions
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F(x,y)
G(x,y)
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Destructible point Bertrand 1997

• Definition a point p is destructible (for F) if
  it is simple for Fk, with k F(p)
• Property p is destructible iff its value may be
  lowered by one without changing the topology of
  any cross-section
• Definition a point p is constructible (for F) if
  it is destructible for -F (duality)

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Destructible point examples
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F(x,y)
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F3
F2
F1
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Destructible point examples
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F(x,y)
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F3
F2
F1
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Destructible point examples
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F(x,y)
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F3
F2
F1
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Destructible point counter-examples
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F(x,y)
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F3
F2
F1
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Destructible point counter-examples
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F(x,y)
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F3
F2
F1
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Connectivity numbers

• N(p) q in N(p), F(q) ? F(p)
• T(p) number of Conn. Comp. of N(p) .
• N--(p) q in N(p), F(q) lt F(p)
• T--(p) number of Conn. Comp. of N--(p) .
• N, T, N-, T- similar
• If an adjacency relation (eg. 4) is chosen for
  T, T, then the other adjacency (8) must be
  used for T-, T--

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Destructible point local characterization

• The point p is destructible if and only if
  T(p) 1 and T--(p) 1

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T 1
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T-- 1
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T-- 1
destructible
non-destructible
non-destructible
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Classification of pointsBertrand 97

• The local configuration of a point p corresponds
  to exactly one of the eleven following cases

• well (T- 0)
• minimal constructible(T T- 1, T-- 0)
• minimal convergent(T gt 1, T-- 0)
• constructible divergent (T T- 1, T-- gt 1)

• peak (T 0)
• maximal destructible(T T-- 1, T 0)
• maximal divergent(T-- gt 1, T 0)
• destructible convergent (T T-- 1, T gt 1)

• interior (T T-- 0)
• simple side (T T-- T T- 1)
• saddle (T gt 1, T-- gt 1)

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Classification of points examples
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Simple side
Saddle
Interior
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Peak
Maximaldestructible
Maximaldivergent
Destructibleconvergent
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Grayscale skeletons

• We say that G is a skeleton of F if G may be
  obtained from F by sequential lowering of
  destructible points
• If G is a skeleton of F and if G contains no
  destructible point, then we say that G is an
  ultimate skeleton of F

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Ultimate skeleton 1D example
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Ultimate skeleton illustration
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Ultimate skeleton illustration
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Ultimate skeleton 2D example
Original image F
Ultimate skeleton G of F
Regional minima of F (white)
Regional minima of G (white)
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Thinness

• In 1D, the set of non-minimal points of an
  ultimate skeleton is  thin  (a set X is thin if
  it contains no interior point). Is it always true
  in 2D ?
• The answer is no, as shown by the following
  counter-examples.

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Thinness
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Thinness
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Basic algorithm
Basic ultimate grayscale thinning(F) Repeat until
stability Select a destructible point p for F
F(p) F(p) 1 Inefficient O(n.g), where
n is the number of pixels g is the maximum gray
level
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Lowest is best

• The central point is destructible
  it can thus be lowered down to 5
  without changing the topology.
• It can obviously be lowered more
• down to 3 (since there is no value between 6 and
  3 in the neighborhood)
• down to 1 (we can check that once at level 3, the
  point is still destructible)

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Two special values

• If p is destructible, we define
• ?-(p)highest value strictly lower than F(p)
  in the neighborhood of p
• ?-(p)lowest value down to which F(p) can
  be lowered without changing the topology

Here ?-(p)3 ?-(p)1
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Better but not yet good

• If we replace F(p) F(p) 1 in the basic
  algorithm by F(p) ?-(p), we get a faster
  algorithm. But its complexity is still bad. Let
  us show why

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Fast algorithm
Fast ultimate grayscale thinning(F) Repeat until
stability Select a destructible point p for F
of minimal graylevel F(p)
?-(p) - Can be efficiently implemented thanks to
a hierarchical queue- Execution time roughly
proportional to n (number of pixels)
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Complexity analysis open problem
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Non-homotopic operators

• Topology preservation strong restriction

over segmentation

• Our goal Change topology in a controlled way

regional minima ? ? segmented regions
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Altering the topology

• Control over topology modification.
• Criteria
• Local contrast notion of ?-skeleton
• Regional contrast regularization
• Size topological filtering
• Topology crest restoration

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?-destructible point
not ?-destructible
?-destructible
?
?
?
Illustration (1D profile of a 2D image)
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?-destructible point

• Definition
• Let X be a set of points, we define
  F-(X)minF(p), p in X
• Let ? be a positive integer
• A destructible point p is ?-destructible
• A k-divergent point p is ?-destructible if at
  least k-1 connected components ci (i1,,k-1) of
  N--(p) are such that F(p) - F-(ci) ? ?

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?-skeleton examples
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Topological filtering
C reconstruction of B under A
B thinning peak deletion

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Topological filtering (cont.)
Original image
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Topological filtering (cont.)
Homotopic thinning (n steps)
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Topological filtering (cont.)
Peak deletion
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Topological filtering (cont.)
Homotopic reconstruction
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Topological filtering (cont.)
Final result
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Topological filtering (cont.)

• Comparison with other approaches (median filter,
  morphological filters) better preservation of
  thin and elongated features

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Crest restoration

• Motivation

Thinning thresholding
Gradient
Original image
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Crest restoration (cont.)

• p is a separating point if
• there is k such that T(p, Fk)2
• p is extensible if
• p is a separating point, and
• p is a constructible or saddle point, and
• there is a point q in its neighborhood that is an
  end point or an isolated point for Fk, with
  kF(p)1

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Crest restoration (cont.)

• p is a separating point if
• there is k such that T(p, Fk)2
• p is extensible if
• p is a separating point, and
• p is a constructible or saddle point, and
• there is a point q in its neighborhood that is an
  end point or an isolated point for Fk, with
  kF(p)1

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Crest restoration (cont.)
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Crest Restoration result
Significant crests have been highlighted (in
green)
Before crest restoration
After crest restoration
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Crest restoration (cont.)
Gradient
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Crest restoration (cont.)
Thinning crest restoration thresholding (1
parameter)
Thinning hysteresis thresholding (2 parameters)
Thinning hysteresis thresholding (2 parameters)
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Conclusion

• Strict preservation of both topological and
  grayscale information
• Combining topology-preserving and
  topology-altering operators
• Control based on several criteria (contrast,
  size, topology)

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Perspectives

• Study of complexity
• Extension to 3D
• Topology in orders (G. Bertrand)

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References

• G. Bertrand, J. C. Everat and M. Couprie "Image
  segmentation through operators based upon
  topology", Journal of Electronic Imaging, Vol. 6,
  No. 4, pp. 395-405, 1997.
• M. Couprie, F.N. Bezerra, Gilles Bertrand
  "Topological operators for grayscale image
  processing", Journal of Electronic Imaging,
  Vol. 10, No. 4, pp. 1003-1015, 2001.
• www.esiee.fr/coupriem/Sdi/publis.html