SEMINAR SERIES ON ADVANCED MEDICAL IMAGE PROCESSING2 FUNDAMENTAL CONCEPTS

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SEMINAR SERIES ONADVANCED MEDICAL IMAGE
PROCESSING(2) FUNDAMENTAL CONCEPTS

• LIXU GU
• Robarts Research Institute
• London, Ontario, Canada
• July 10, 2002

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Road Map

• Thresholding
• Connected Component Labeling
• Region Property
• Neighbourhood Operators
• Noise Reduction

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Threshold?
oops ...
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Thresholding

• Thresholding is a simple, non-contextual,
  efficient segmentation technique
• Usually refer to Intensity Thresholding
• Classify pixels (voxels) into two categories
• Create a binary image (binarisation)
• Thresholding can employ either a fixed or an
  adaptive threshold value
• A variety of techniques have been devised to
  automatically choose a threshold, but no one is
  robust

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Threshold Value vs. Histogram

• Thresholding usually involves analyzing the
  histogram
• Different image features give rise to distinct
  features in a histogram (Bimodel)
• In general the histogram peaks corresponding to
  two features will overlap

• An example of a threshold value is the mean
  intensity value

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Fixed Thresholding

• Fixed or Global threshold the threshold value is
  held constant throughout the image
• Fixed Threshold is in the form of (T is the
  threshold)

Reverse Threshold
Normal Threshold

• A variation which uses two thresholds to define a
  range of intensity values

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Fixed Thresholding

• The success of thresholding depends critically on
  the selection of an appropriate threshold
• An Example

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Single Threshold (Example)
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Double Thresholds (Example)
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Isodata Algorithm

• This iterative threshold selection technique was
  developed by Ridler and Calvard
• The algorithm works as
• Select an initial threshold T0 (e.g. the mean
  intensity)
• Partition the image into two groups (R1 and R2)
  using the T0
• Calculate the mean intensity values ?1 and ?2 of
  the partitions R1 and R2.
• Select a new threshold Ti (?1 ?2 )/2
• Repeat steps 2-4 until Ti Ti-1

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Optimal Threslding

• Histogram shape can be useful in locating the
  threshold. However it is not reliable for
  threshold selection when peaks are not clearly
  resolved
• Optimal thresholding a criterion function is
  devised that yields some measure of separation
  between regions
• A criterion function is calculated for each
  intensity and that which maximizes/minimizes this
  function is chosen as the threshold

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Otsus Methold

• Otsus thresholding method is based on selecting
  the lowest point between two classes (peaks).
• Frequency and Mean value
• Frequency N total pixel number
• Mean ni number of pixels in
  level i
• Analysis of variance (variancestandard
  deviation2)
• Total variance

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Otsus Methold

• between-classes variance (?b2 ) The variation of
  the mean values for each class from the overall
  intensity mean of all pixels
• ?b2 ?0 (?0 - ?t ) 2 ?1(?1 - ?t ) 2,
• Substituting ?t ?0 ?0 ?1?1, we get
• ?b2 ?0?1(?1 - ?0 )2
• ?0, ?1, ?0, ?1 stands for the frequencies and
  mean values of two classes, respectively.

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Otsus Method

• The criterion function involves between-classes
  variance to the total variance is defined as
• ? ?b2 / ?t2
• All possible thresholds are evaluated in this
  way, and the one that maximizes ? is chosen as
  the optimal threshold

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Entropy Method

• Entropy is served as a measure of information
  content
• A threshold level t separates the whole
  information into two classes, and the entropy
  associated with them is
• Optimal threshold is the one maximize
• H Hb Hw

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Comparing Threshold Value
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Other Algorithms

• Moment Preservation Algorithm Optimal threshold
  makes the original and thresholded images have
  the closest moments
• Minimum Error Algorithm The criterion function
  is minimized to find the best fit between the
  statistical model (normal distribution) and the
  histogram.

• Triangle Algorithm
• Maximized distance d indicate the optimal
  threshold

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Adaptive Thresholding

• Adaptive thresholding is also called local (or
  regional) thresholding
• Employ more than one threshold value.
• Works when the background intensity level is not
  constant and the object varies within the image.
• Examines the relationships between intensities of
  neighboring pixels to adapt the threshod
  according to the intensity statistics of
  different regions.
• Difficulties of thresholding
• Poor image contrast, Spatial non-uniformities,
  Ambiguity

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Threshold In VTK/ITK

• VTK
• Fixed thresholding (double thresholds acceptable)
  
• vtkImageThreshold output ? image
• vtkThreshold output ? unstructured grid
• vtkThresholdPoint output ? polygonal data
• ITK
• Fixed thresholding (double threshold acceptable)
• itkBinaryThresholdImageFilter output ?
  image
• itkBinaryThresholdImageFunction output ?
  Boolean
• Optimal thresholding
• itkOtsuThresholdImageCalculator output ? image

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Connected Component Labeling

• Pixel Connectivity
• for a pixel P(x,y)
• 4-neighbours N4(P) (x1,y), (x-1,y),
  (x,y1),(x,y-1)
• 8-neighbours N8(P) N4(P) ?(x1,y1),
  (x1,y-1),
• (x-1,y1),(x-1,y-1)
• 4(8)-connected two pixels within N4(P) or N8(P)
  
• Connected components labeling
• groups the pixels in an image into components
  based on pixel connectivity
• Labels components with a gray level
• or a color (color labeling)

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Labeling Algorithm

• Connected component labeling works by scanning an
  image, pixel-by-pixel (from top to bottom and
  left to right) in order to identify connected
  pixel regions
• Intensity criterion(IC) the same set of
  intensity values ( 1 for a binary image a value
  range for a gray level image)
• Labeling algirithm for a pixel P satisfy IC
• Step 1 (First Scan)
• If all neighbors do not satisfy IC, assign a new
  label to P
• if only one neighbor satisfy IC, assign its label
  to P
• if one or more of the neighbors satisfy IC,
  assign one of the labels to P and make a note of
  the equivalences.

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Labeling Algorithm

• Step2 (Resolve equivalence) The equivalent
  label pairs are sorted into equivalence classes
  by a equivalence resolve algorithm ( e.g.
  Floyd-Warshall algorithm) and a unique label is
  assigned to each class
• Step3 (Second scan) Each label is replaced by
  the label assigned to its equivalence classes

• Examples

banner S Labels banner C Labels
00000 00000 1 2
1 2 1 1
33333 1 4 1
5 4 1 3
66666 44444
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Region Property

• Region Properties are important features for the
  region analysis (or measurement) after the
  regions have been labeled (segmented)
• Region Properties
• Perimeter and Area
• Center, Radius and Diameter
• Centroid Moments and Orientation
• Extreme Points and Curvature
• Intensity Properties

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Region Property

• Perimeter and Area
• Perimeter The length of the contour of a
  connected component (region).
• calculated from the chain-code of the contour,or
• estimated by the number of pixels on the contour.
• Area The number of unit squares contained.
• Picks formula A(P) nI nB/2-1
• nI , nB number of interior points or the
  points lie on borders, respectively.

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Region Property

• Center, Radius and Diameter
• Eccentricity of a point P in F is the maximum of
  distance d(p,q) for all points q?F
• ecc(p) max d(p,q) q?F
• Center The set of points P of least eccentricity
  
• Radius The value of the least eccentricity
  d(p,p)
• Diameter The value of the greatest
  eccentricity d(p,q)

F
P
q
P
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Region Property

• Centroid Moments and Orientation
• Centroid Given F, a set of n connected pixels
  (xi,yi), we can define a centroid c as
• Moments The discrete (k,l)-order central moment
  is defined as
• Orientation Orientation is defined here as an
  angle ?

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Region Property

• Extreme Points and Curvature
• Extreme Points
• Topmost left 5. Bottommost right
• Topmost right 6. Bottommost left
• Rightmost top 7. Leftmost bottom
• Rightmost bottom 8. Leftmost top
• Curvature is a very important region property
  for measuring contours. TBD
• Intensity Properties
• Gray level mean and variance, intensity-weighted
  centroid (gravity center) and moments.

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8
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7
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5
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Neighbourhood Operators

• Different domain
• Numerical Operations Addition, Subtraction,
  Multiply, division, minima, maxima, convolution
  
• Symbolic Opeartions AND, OR, NOT, or
  table-look-up operations.
• Recursive or not
• Non-recursive Operations Region-Growing
  operator, Region-shrinking operator
• Recursive OperationsThinning operator, distance
  operator

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Convolution VS. Correlation

• Given an image f(x,y) and a kernel w(a,b)
• Convolution
• Correlation
• Difference Index is different.
• Kernel asymmetric convolution flips the kernel.

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Image Convolution
Image Convolution

• Convolution includes three steps
• Position the center of the kernel at the first
  pixel of an image and flip the kernel.
• Multiply and sum.
• Slide to next pixel.Repeat steps 2 and 3 until
  the whole image is processed
• Demos

Sobel
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Noise Reduction

• Noise reduction is also called noise cleaning or
  smoothing.
• Coherence vs. incoherence
• Replace incoherent pixel values by values more
  specially coherent which are based on some or all
  the pixels in an appropriate neighbourhood.
• Two categories
• Convolution based Gaussian smoothing, median
  filter
• Morphology based Opening, Closing (TBD)

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Gaussian Smoothing

• Gaussian kernel
• The effect of Gaussian smoothing is to blur an
  image. The degree of smoothing is determined by
  the standard deviation ? of the Gaussian
• The Gaussian outputs a weighted average' of each
  pixel's neighbourhood, with the average weighted
  more towards the value of the central pixels

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Median Filter

• Mean Filter The idea of mean filtering is simply
  to replace each pixel value in an image with the
  mean (average') value of its neighbours,
  including itself.

• Median Filter Instead of simply replacing the
  pixel value with the mean of neighbouring pixel
  values, replaces it with the median of those
  sorted values.

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Gaussian vs. Median
Gaussian ( ? 2.5)
Median ( 5 X 5)
Source
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VTK/ITK

• VTK
• vtkImageConvolve()
• vtkImageGaussianSmooth()
• vtkImageMedian3D()
• vtkImageLogic()
• vtkImageMathematics()
• ITK
• itkMeanImageFilter
• itkMedianImageFilter
• itkDiscreteGaussianImageFilter

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Discussion

• Seminar Series notes
• http//www.imaging.robarts.ca/coders/
• or
• http//www.imaging.robarts.ca/lgu/research_E.html