Extraction of morphologic characteristics from discrete scalar field

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• Extraction of morphologic characteristics from
  discrete scalar field

Luca Chiarabini
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General Idea

• Our domain is a set of discrete rappresentations
  of scalar fields f,

f ?2 ? ?

• A mountains or hills surface, are example of
  real scalar fields

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General Idea

• We obtain a discrete rappresentation of a scalar
  field f if we approximate every portion of f (for
  example) with a triangular plane serface

• We call such kind of rappresentation
  Triangulation of Scalar Fild f

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General Idea

• A very interesting new research field is the
  study of a images rappresentation semantic

• The semantic of the scalar field rappresentation
  is the set of Morphological important region

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General Idea

• Let ? a scalar fields f triangulation. We say
  that a triangle t?? is,

Non morphologically important if t represent a
portion of plane surface of f
morphologically important if t represent a
portion of crest (of a mountain) of f
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General Idea

• In the litterature the canonical approaches to
  morfological study of surfaces rappresentation,
  are

• Study of Gaussian curvature of surface

• Application of Morse Theory

• Watershed Transform

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Goal

• To study Watershed Transform on Morse functions

• To study Watershed Transform on triangulation of
  scalar field

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Morse Theory

• Let f a C2-differenziable function defined on
  plane ?2 in ?.

We say that f is a Morse function if, for all
critical point x of f, x is non-degenere
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Morse Theory

• Let f ?2 ? ? a Morse function. We say that
  ? I ? ?2 I? ? , is a Integral curve if

• Im(? ) ? Dom( f )

• ? connect two critical points

• ? follow the steepest descent way of f , that is

?t?I. ? '(t) ?(f(? (t)))
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Morse Theory

• Let f ?2 ? ? a Morse function

• For every critical point x of f we consider the
  set of integral lines that start from x we get
  a stable decomposition of domain of f

• In the same way, for every critical point x, we
  consider the set of integral line that end in x
  we get a unstable decomposition of domain of f

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Morse Theory

• The boundary of stable and unstable decomposition
  constitute a critical net

• The critical net is composed from the union of
  integral line that link a minima point to saddle
  point, and integral line that link a saddle point
  to maxima point

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Watershed Transform

• The goal of this tecnique is recognize objects
  boundary in digital images

• We can see an image as a topographic relief

So, applay the Watershed Transform to topographic
relief means extract reliefs ridge lines
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Watershed Transform

• Topographic relief

• We punched the regional minima and

and flood the entire relief in a lake

• When the water in two distinct basin is nearly to
  merg a dam in built to prevent the merging

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Watershed Transform
Theorem
Let f a Morse function. Then the Watershed
Lines of f is equal to the subset of integral
lines of critical net such that link a saddle
point to maximum point
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Watershed Transform

• There are two canonical approaches in order to
  extract the Watershed line from digital image

Simulation of immersion
Steepest descent
We flood the image from bottom. For every
waters level (driven by pixels gray level) we
calculate the Watershed lines (that is, the
pixels equidistant from two distinct basin).
Watershed lines is equal to the set of pixels
from wich exist two paths of steepest descent
that end in two differnt minima regions.
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Watershed Transform

• The basins and the Watershed lines extracted from
  digital image, depend to pixels adiacency

4-adjacency
8- adjacency

• Let I a digital image (4/8 - adjacency). The
  Watershed lines extracted from I by simulation of
  immersion or steepest descent can be not equal.

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Watershed Transform (simulation of immersion)

• Toy digital image, 4-adjacency

We identify two distinct basins
Waters level 0
Watershed pixels, because these pixels are
equidistant from two distinct basins build in the
previous step
This pixel is merged with blue basin
Waters level 1
Waters level 2
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Watershed Transform(simulation of immersion)

• The flooding process end when the hight of the
  water is equal to the maximum grey level of the
  image (in this case, 3)

Waters level 3

• Let n the number of pixel of the image. The
  algorithms

• work in O(nlog(n)) steps (in the wrost case)

• use O(n) memory unit (1unit1byte)

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Watershed Transform (steepest descent)

• Basic idea

If from a pixel p exists two differents steepest
descent way with the same lenght that arrived to
two different regional minima, we say that p is
Watershed pixel. Watershed lines is the set of
Watershed pixels
this pixel belong to blue basin
Watersheds pixel
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Watershed Transform(steepest descent)

• In this case are detected the same set of
  Watershed pixels as simulation of immersion
  approach

• Let n the number of image pixels. The algorithms

• work in O(n) steps (in the wrost case)

• use O(n) memory resources

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Watershed Transform

• Sometimes this two approaches return different
  result on the same image

(The Watersehed pixel are shadowed)
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Watershed on Triangulated Surface

• We will see

• The meaning of Watershed Transform on
  triangulated surface

• Rapresentation of triangulated surface

• Which kind of segmentation algorithms to use

• Policy in order to face the sovrasegmentation
  problem

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Watershed on Triangulated Surface

• We want to segment the triangulated surface into
  several regions ( minima basins ), where each
  region is a connected component of triangle

• The set of common edges between two adjacent
  regions compose the Watershed lines

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Watershed on Triangulated Surface

• Rappresentation of a domain

Dual
Direct
Mesh
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Watershed on Triangulated Surface

• The kind of segmentation policy depend on the
  kind of domain rappresentation we choosed

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Watershed on Triangulated Surface

• Segmentation algorithms

• Give a different label to every minima node /
  regional minima

• Follow steepest descent way (from every node)
  until we meet a minima node / regional minima /
  labelled node

• We label all nodes of the followed path with the
  same label of the node that we met

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Watershed on Triangulated Surface

• We build a subdivision of the input Mesh in
  connected component of triangles (using a
  labelled direct rappresentation)

Obtaining
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Watershed on Triangulated Surface

• Normally in a discrete rappresentation of
  terrains the number of insignificant regional
  minima is too big. So arise the
  Sovrasegmentation problem.

• Its necessary use region merging policy

Idea the region less deep than a prefixed
threshold are merged with an adiacency region
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Watershed on Triangulated Surface

• The segmentation algorithm works in O(n) steps,
  being n the number of Mesh nodes.

• The merging algorithm works in O(r) steps, being
  r the number of regions created (r is always less
  than the number of triangles of the Mesh)

• We can represent a triangulated surface as a
  planar graph, where the number of arcs and faces
  (edges and triangles) is linear in the number of
  nodes (vertexes)

• So, the temporal and spatial complexity of the
  method that we propose is linear in the number of
  vertexes of the Mesh.

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Conclusion

• Equivalence between Watershed lines and a subset
  critical net lines extracted from a Morse function

• Develop and analisys of a Watershed algorithms on
  triangular Mesh

Future Work

• Extension of Morse Theory on Piecewise-Linear
  functions

• Extension of watershed transform on
  Piecewise-Linear functions

• Relation between Watershed line and Critical net
  extracted from Piecewise-Linear functions