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					         MATHEMATICAL MORPHOLOGY

I.    INTRODUCTION

II.   BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY
                   INTRODUCTION

Mathematical morphology

   • Self-sufficient framework for image processing and
     analysis, created at the École des Mines
     (Fontainebleau) in 70’s by Jean Serra, Georges
     Mathéron, from studies in science materials

   • Conceptually simple operations combined to define
     others more and more complex and powerful

   • Simple because operations often have geometrical
     meaning

   • Powerful for image analysis
                INTRODUCTION

• Binary and grey-level images seen as sets



                           X

                           Xc
                                X = { (x, y, z) , z  f (x,y) }

                 f (x,y)
                INTRODUCTION

• Operations defined as interaction of images with a
  special set, the structuring element
         MATHEMATICAL MORPHOLOGY

I.    INTRODUCTION

II.   BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY
                BINARY MORPHOLOGY

1. Erosion and dilation
2. Common structuring elements
3. Opening, closing
4. Properties
5. Hit-or-miss
6. Thinning, thickenning
7. Other useful transforms :
   i.   Contour
   ii. Convex-hull
   iii. Skeleton
   iv. Geodesic influence zones
                   BINARY MORPHOLOGY

Notation
                                                                             x
                                                     -2 -1 0 1    2

                                                -2
                                                -1
                                                 0
                                                 1
                                                 2
                                B
                                            y
                          A special set :
                          the structuring        Origin at center in this
           X                                     case, but not necessarily
                          element
 No necessarily compact                          centered nor symmetric
 nor filled
                 BINARY MORPHOLOGY

Dilation : x = (x1,x2) such that if we center B on them,
           then the so translated B intersects X.




          X                                        difference



          B
                  BINARY MORPHOLOGY

Dilation : x = (x1,x2) such that if we center B on them,
           then the so translated B intersects X.

How to formulate this definition ?

1) Literal translation




 2) Better : from Minkowski’s sum of sets
               BINARY MORPHOLOGY

Minkowski’s sum of sets :




                            l




                                   l
             BINARY MORPHOLOGY

Dilation :




                      l




                                 Dilation
               BINARY MORPHOLOGY

Dilation is not the Minkowski’s sum




                                      l
BINARY MORPHOLOGY




                l




            l



   b b bb           l
                BINARY MORPHOLOGY

Dilation with other structuring elements
                BINARY MORPHOLOGY

Dilation with other structuring elements
                BINARY MORPHOLOGY

Erosion : x = (x1,x2) such that if we center B on them,
          then the so translated B is contained in X.




                                                    difference
                 BINARY MORPHOLOGY

Erosion : x = (x1,x2) such that if we center B on them,
          then the so translated B is contained in X.

How to formulate this definition ?

1) Literal translation




2) Better : from Minkowski’s substraction of sets
BINARY MORPHOLOGY
               BINARY MORPHOLOGY

Erosion with other structuring elements
               BINARY MORPHOLOGY

Erosion with other structuring elements
                                          Did not belong to X
                   BINARY MORPHOLOGY

Common structuring elements shapes
                                        = origin


                                                x

                                          y


            disk           circle



   segments 1 pixel wide

                                       Note :
   points
            BINARY MORPHOLOGY

Problem :
BINARY MORPHOLOGY
                BINARY MORPHOLOGY

Problem :
                        <d/2




                        d/2
            d
                 BINARY MORPHOLOGY

Implementation : very low computational cost




                      0
                      1 (or >0)




    Logical or
                BINARY MORPHOLOGY

Implementation : very low computational cost




                      0
                      1




  Logical and
               BINARY MORPHOLOGY

Opening :
            also




                   Supresses :                    difference
                      • small islands
                      • ithsmus (narrow unions)
                      • narrow caps
               BINARY MORPHOLOGY

Opening with other structuring elements
                   BINARY MORPHOLOGY

Closing :
            also




                      Supresses :
                         • small lakes (holes)
                         • channels (narrow separations)
                         • narrow bays
               BINARY MORPHOLOGY

Closing with other structuring elements
                BINARY MORPHOLOGY

Application : shape smoothing and noise filtering
               BINARY MORPHOLOGY

Application : segmentation of microstructures (Matlab Help)


                                               disk radius 6

                                                original
                                                negated
                                                threshold
                                                closing
                                                opening
                                                and with
                                                threshold
                BINARY MORPHOLOGY

Properties

   • all of them are increasing :




   • opening and closing are idempotent :
            BINARY MORPHOLOGY

• dilation and closing are extensive
  erosion and opening are anti-extensive :
             BINARY MORPHOLOGY
• duality of erosion-dilation, opening-closing,...
            BINARY MORPHOLOGY
• structuring elements decomposition




  operations with big structuring elements can be done
  by a succession of operations with small s.e’s
                BINARY MORPHOLOGY

Hit-or-miss :




Bi-phase structuring element




                        “Hit” part   “Miss” part
                         (white)       (black)
                BINARY MORPHOLOGY

Looks for pixel configurations :




                                   background
                                   foreground
                                   doesn’t matter
             BINARY MORPHOLOGY


isolated points at
4 connectivity
                BINARY MORPHOLOGY

Thinning :



Thickenning :


Depending on the structuring elements (actually, series
of them), very different results can be achieved :

   • Prunning
   • Skeletons
   • Zone of influence
   • Convex hull
   • ...
                BINARY MORPHOLOGY

 Prunning at 4 connectivity : remove end points by a
 sequence of thinnings




1 iteration =
        BINARY MORPHOLOGY

                      1st iteration




2nd iteration     3rd iteration: idempotence
               BINARY MORPHOLOGY

What does the following sequence ?

      background            doesn’t
                            matter
               foreground
                BINARY MORPHOLOGY

1. Erosion and dilation
2. Common structuring elements
3. Opening, closing
4. Properties
5. Hit-or-miss
6. Thinning, thickenning
7. Other useful transforms :
   i.   Contour
   ii. Convex-hull
   iii. Skeleton
   iv. Geodesic influence zones
                BINARY MORPHOLOGY

i. Contours of binary regions :
BINARY MORPHOLOGY

          8-connectivity         4-connectivity
          contour                contour




      4-connectivity       8-connectivity

    Important for perimeter computation.
                BINARY MORPHOLOGY

ii. Convex hull : union of thickenings, each up to idempotence
BINARY MORPHOLOGY
                  BINARY MORPHOLOGY

iii. Skeleton :

       Maximal disk : disk centered at x, Dx, such that
       Dx  X and no other Dy contains it .

       Skeleton : union of centers of maximal disks.
          BINARY MORPHOLOGY

Problems :

• Instability : infinitessimal variations in the border of X
  cause large deviations of the skeleton

• not necessarily connex even though X connex




• good approximations provided by thinning with
  special series of structuring elements
            BINARY MORPHOLOGY




1st
iteration
                BINARY MORPHOLOGY




result of
1st iteration



                         2nd iteration reaches
                         idempotence
BINARY MORPHOLOGY



                    20 iterations
                    thinning




                    40 iterations
                    thickening
               BINARY MORPHOLOGY

Application : skeletonization for OCR by graph matching
                    BINARY MORPHOLOGY

     Application : skeletonization for OCR by graph matching




 Hit-or-Miss




and 3 rotations
                BINARY MORPHOLOGY

iv. Geodesic zones of influence :

      X set of n connex components {Xi}, i=1..n .
      The zone of influence of Xi , Z(Xi) , is the set of
      points closer to some point of Xi than to a point of
      any other component. Also, Voronoi partition.
      Dual to skeleton.
      BINARY MORPHOLOGY
                  thr     erosion
                             7x7




GZI               and     opening
                              5x5
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
       BINARY MORPHOLOGY
 thr             erosion
                    7x7




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