SEMINAR SERIES ON ADVANCED MEDICAL IMAGE PROCESSING7 Mathematical Morphology I

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SEMINAR SERIES ONADVANCED MEDICAL IMAGE
PROCESSING(7) Mathematical Morphology (I)

• LIXU GU
• Robarts Research Institute
• London, Ontario, Canada
• September 6, 2002

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Road Map

• Mathematical Morphology
• Binary Morphology
• Binary Operations Dilation, Erosion, Opening,
  Closing, Hit-and-Miss
• Shape Feature Detection
• Pattern Spectrum
• Recursive Dilation and Erosion
• Distance Transform and Skeleton
• Applications

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Mathematical Morphology

• A methodology for the quantitative analysis of
  spatial structures which was initiated by
  G.Matheron and J.Serra at Paris School of Mines.
• It aims at the analyzing the shape and the forms
  of the objects.
• Its mathematical origins stem from set theory,
  topology, lattice algebra, random functions,
  stochastic geometry, etc.
• Extremely useful, not yet often used

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Uses of Mathematical Morphology

• image enhancement
• image segmentation
• image restoration
• edge detection
• texture analysis
• feature generation
• skeletonization

• shape analysis
• image compression
• component analysis
• curve filling
• general thinning
• feature detection
• noise reduction

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Reference

• Homepage
• Center of Mathematical Morphology
  (http//cmm.ensmp.fr/index_eng.html) at Ecole des
  Mines de Paris.
• Morphology Digest (http//www.cwi.nl/projects/morp
  hology) edited by Henk Heijmans, Centre for
  Mathematics and Computer Science, Amsterdam, The
  Netherlands
• Book
• Image analysis and mathematical morphology by
  J. Serra (call  TA1632.S47  v.2 1988 )

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Structuring Element

• Structuring element (SE) is also called the
  kernel, but I reserve this term for the similar
  objects used in convolutions
• Origin the SE is typically translated to each
  pixel position in the image based on the origin.

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Dilation

• Binary Dilation also called Minkowski addition.
  An image F dilated by a SE K is defined as
• It can be regarded as an expansion operation.

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Properties of Dilation

• Commutative
• Associative
• Translation Invariance
• Increasing
• Decomposition
• Multi-Dilations

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Erosion

• Binary Erosion also called Minkowski
  subtraction. An image F eroded by a SE K is
  defined as
• It can be regarded as an shrinking operation

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Properties of Erosion

• Non-Commutative
• Non-Inverses
• Translation Invariance
• Increasing in A
• Decreasing in B
• Decomposition

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Opening

• Binary Opening An image F opened by a SE K is
  defined as
• It can remove the small regions which are smaller
  than the structuring element

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Properties of Opening

• Translation
• Antiextensivity
• Increasing monotonicity
• Idempotence

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Closing

• Binary Closing An image F closed by a SE K is
  defined as
• It can fill the small holes which are smaller
  than the structuring element

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Properties of Closing

• Translation
• Extensivity
• Increasing monotonicity
• Idempotence

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Hit-and-Miss

• Hit-and-miss A morphological shape detector.
• F ? K (F ? K1) ? (Fc ? K2)c,
• K1 ? K2 ?, K1 ?K, K2 ? K
• can be used to look for particular patterns of
  foreground and background pixels in an image

?

K1
K2
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Shape Feature Detection

• Binary Opening is a powerful shape detector by
  using different structuring elements
• Example1 Distinguish circles and lines

? r3X9KSquare
? r5Kdisk
? r9X3KSquare
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Shape Feature Detection

• Example2 Decompose a printed circuit board in
  its main parts.

Detecting holes
Detecting square islands
Detecting circle islands
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Shape Feature Detection
Detecting Rectangular islands
Detecting Thin connections
Detecting Thick connections
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Operators In Software

• VTK
• vtkImageDilateErode3D()
• vtkImageOpenClose3D()
• ITK
• itkBinaryDilateImageFilter()
• itkBinaryErodeImageFilter()
• MATLAB
• mmdil(), mmero(), mmopen(),mmclose(), mmse2hmt()

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Pattern Spectrum

• Pattern Spectrum is known as granulometric size
  density. It is employed to measure the size
  distribution of an object.
• Pattern spectrum PSrik(F) of a set F in terms of
  SE rik is defined as

Where, Card(F) denotes the cardinality of set F
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Pattern Spectrum

• Pattern spectrum not only can detect the size of
  parts in an image, but also can analyze the
  shapes of them.

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Pattern Spectrum

• Another example size analysis

Organ mass
Pattern Spectrum analysis
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Recursive Dilation

• Recursive Dilation is defined as
• where, i is defined as scalar factor and K as
  its base.
• Recursive Dilation is employed to compose SE
  series in the same shape but different sizes.

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Recursive Dilation
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Recursive Erosion

• Recursive Erosion is also called successive
  erosion which is defined as
• When performing recursive erosions of an object,
  its components are progressively shrunk until
  completely disappeared.
• Useful for distance transform and segmentation

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Recursive Erosion
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Distance Transform

• The distance transform is an operator normally
  only applied to binary images.
• The result of the transform is a greylevel image
  showing the distance to the closest boundary from
  each point.

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Distance Metrics

• Euclidean Distance
• City Block Distance (N8)
• Chessboard Distance (N4)

Chessboard metric
Original
Euclidean metric
City block metric
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Distance Transform

• Perform multiple recursive erosions with a
  suitable SE until all foreground regions of the
  image have been eroded away.
• Label each pixel with the number of erosions that
  had been performed before it disappeared, then
  get the distance transform result.
• Suitable SE for different distance metrics
• A square SE gives the chessboard distance
  transform
• A cross shaped SE gives the city block distance
  transform
• A disc shaped SE gives the Euclidean distance
  transform.

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Distance Transform
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Skeleton

• Skeleton is a process for reducing foreground
  regions in a binary image
• preserves the topology (extent and connectivity)
  of the original region while throwing away most
  of the original foreground pixels

• locus of centers of bi-tangent circles that fit
  entirely within the foreground region

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Skeleton

• Skeleton subset Si(F) is defined as
• where n is the largest value of i before the set
  Si(F) becomes empty. SE K is chosen to
  approximate a disc
• Skeleton is then the union of the skeleton
  subsets

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Skeleton

• Reconstruction the original object can be
  reconstructed by given knowledge of the skeleton
  subsets Si(F), the SE K, and i
• Examples of skeleton

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Skeleton
Skeleton

• Examples of skeleton (continue)

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VTK/ITK/MATLAB

• VTK
• vtkImageEuclideanDistance ()
• vtkImageCityBlockDistance ()
• vtkImageSkeleton2D ()
• ITK
• itkEuclideanDistance()
• itkDistanceMetric()
• Matlab
• mmpatspec - Pattern spectrum
• mmdist - Distance transform
• mmskelm - Morphological skeleton
• mmskelmrec - Morphological skeleton reconstruction

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Application 1

• Detect the teeth of a gear

Subtraction
labeling
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Application 2 Grid identification from Biochip
image
Pattern Spectrum Spot size 5 (pixel)
Origin
Otsu threshold
enthopy threshold
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Application 2 Grid identification from Biochip
image
Morphological noise reduction
Grid identified with noise
Grid identified without noise
Grid identification final result
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Application 3

• Segment vertibra and ribs

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Discussion