SEMINAR SERIES ON ADVANCED MEDICAL IMAGE PROCESSING EDGE DETECTION

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SEMINAR SERIES ONADVANCED MEDICAL IMAGE
PROCESSING EDGE DETECTION

• MINGYUE DING, PH. D.
• Robarts Research Institute
• London, Ontario, Canada
• July 25, 2002

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CONTENTS

• What is an edge?
• Gradient, curvature and edge detection
• Line detection-Hough transform technique
• Edge representation-B-Spline fitting technique

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WHAT IS AN EDGE?

• An edge is a set of connected pixels that lie on
  the boundary between two regions
• The pixels on an edge are called edge points
• Position orientation of edge
• Gray level discontinuity across an edge

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WHY A GRAY LEVEL DISCONTINUITY HAPPENED?

• Different colors, brightness, textures, material,
  tissues,
• Different normal directions of surfaces
• Different illuminance

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DIFFERENT EDGES
A
Different color
Different brightness
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DIFFERENT EDGES
Different texture
Different surfaces
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UNIQUENESS OF EDGE

• Most of edges are unique in space, i.e., its
  position and orientation keep the same in the
  space when viewing from different points
• Non-unique edge-Limb edge

A

C

A
B
C
D


O1
D
B
Image 2
Image 1
O2
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TYPES OF EDGES

• Gray level profile
  derivatives
• Step edge
• Ramp edge
• Peak edge

1st
2nd
1st
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DIFFERENT TYPES OF EDGES IN A CT IMAGE
Ramp edge
Step edge
Peak edge
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DEFINITION OF GRADIENT

• A point is defined as an edge point if its 2-D
  first or second -order derivative is greater than
  a specified threshold.
• Gradient of digital image, , is defined
  by a vector

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GRADIENT CALCULATATION

• Gradient at an edge point, (x,y), can also be
  interpreted as a complex number with its
  magnitude determined by
• and the direction determined by

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EDGE OPERATORS

• The partial derivatives in x and y, ,
• can be estimated using different ways
• Roberts operator
• Prewitt operator
• Sobel operator

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GRADIENT MASKS

• All operators can be performed by the
  convolution using different masks.
• Roberts operator masks

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GRADIENT OPERATORS

• Prewitt operator masks
• Sobel operator masks

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CURVATURE

• Curvature is defined as the rate of change of
  edge direction.
• Suppose is an edge, its
  curvature, , is defined as

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CURVATURE

• The curvature at A is larger than B

B
R1
R2
A
edge
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LINE DETECTION

• Line is the most often used and important edge in
  an image
• A lot of edges are straight lines
• Any edge can be considered as piecewise line edge
• How to detect a straight line in an image?
• To find the straight line that has the maximum
  number of points passing through it

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NEEDLE DETECTION IN US IMAGE
Binary image
2D US gray level image
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HOUGH TRANSFORM

• In 1967, Hough proposed to detect line in
  parameter space, called Hough transform (HT)
  technique

B
y
b
b-xnayn
(x1,y1)


(x2,y2)
yAxB

A
b-x2ay2
b-x1ay1
Cluster
(xn,yn)

x
a
Parameter space
Image space
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LINE-POINT DUALITY OF HOUGH TRANSFORM

• Image space point, line, line detection
• Parameter space line, point,cluster detection
• Accumulator array
• Quantilization of slope and intercept of line
• Thus, the problem of line detection in image
  space becomes a cluster detection in parameter
  space

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COMPUTATIONAL COMPLEXITY OF THE HT

• Directly perform in (a,b) space
• 
  (1)
• where
• is the size of binary image
• are the possible numbers of a and
  b
• is the number of addition operations
  needed at each point in the binary image
• is the fraction of points in the binary
  image

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PROBLEMS IN THE HT

• If
  
• 
  
  
• 
  (2)
  Problems
• Too slow (570s on a computer of 108 adds/s)
• Both a and b are unbounded when the line is
  parallel to y-axis

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STANDARD HOUGH TRANSFORM (SHT)

• In 1972, Duda Hart proposed a Standard HT
  (SHT). They used the polar coordinate equation of
  a straight line

Y
(3)
X
O
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STEPS OF ACCUMULATOR ARRAY CALCULATATION IN SHT
1. Search a point in the binary image 2. Vary the
possible values of from 0 to 3. Calculate
from Eq. (3) 4. Increase by
1 5. If we find a new point in the binary image,
go to Step 1 otherwise stop.
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COMPUTATIONAL COMPLEXITY OF THE SHT

• 
  (4)
• If we use the same values of the parameters in
  (2), the computational complexity of the SHT is
• It is Nb 512 times faster than HT, but still
  too slower for a real-time processing.

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MOTIVATIONS OF REAL-TIME HOUGH TRANSFORM (RTHT)

• Use the smallest image containing the line to
  replace the original image
• Use Look-up Table for the operations of sin and
  cos
• Two-stage searching strategy

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SMALLEST IMAGE CONTAINING LINE
M
M
Smallest image
N

• Original image

N
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TWO-STAGE SEARCHING STRATEGY

• 1. At the coarse stage, we use lower resolution
  image to detect the
  line where is the original image,
  is the magnifying factor,
  
  
• 
  .
• 2. At the fine stage, we use with the
  approximate line direction determined at the
  coarse stage.

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DETERMINATION OF OPTIMAL IMAGE RESOLUTION

• The computational complexity at the coarse stage
  is
• 
  (5)
• The computational complexity at the fine stage is
• 
  (6)
• The total computational complexity of the RTHT is
• 
  (7)

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DETERMINATION OF OPTIMAL IMAGE RESOLUTION

• By deriving CRTHT in Eq. (7), we determine

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COMPUTATIONAL COMPLEXITY OF THE RTHT

• Substituting the value of into Eq. (7)
• If we use the same values in (2), the
  computational complexity of the RTHT is

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REAL-TIME NEEDLE TRACKING
Patient biopsy
Agar phantom
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EDGE REPRESENTATION

• Chain code easy but sensitive
  to noise
• Edge fitting
• More robust and continuous
• B-spline is a powerful and popular model used in
  graphics and image processing

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B-SPLINE FITTING
What is spline? A spline is a thin flexible
strip made of bamboo or steel that can be bent to
pass through or near given points in the plane In
B-spline, B means Basic
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DEFINITION OF B-SPLINE

• Mathematically splines are piecewise polynomials
  with pieces that are smoothly connected together.
• 
  
• 
  (8)
• (N1
  convolution)

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PROBLEMS IN B-SPLINE

• Need to determine the coefficients for every
  Basic functions
• Difficult to determine the control points
• Difficult to let the B-spline passing through the
  given points accurately.

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CARDINAL-SPLINE

• 
  
• 
  (9)
• where is called the n-order
  cardinal-spline function and it can be calculated
  from the n-order B-spline function

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BASIC FUNCTIONS
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EXAMPLES




Cardinal-spline
Control points
Cardinal-spline used in initializing a prostate
contour
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EXAMPLES

CONTOUR FITTING USING B-SPLINE
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3D RECONSTRUCTION RESULTS
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EDGE IN VTK/ITK

• VTK
• Edge operator
• vtkEdgePoint, vtkImageGradient,
• vtkImageGradientMagnitude,
  vtkImageSobel2D,
• vtkImageSobel3D,
• B-spline
• vtkSpline, vtkCardinalSpline.
• ITK
• Edge operator
• itkSobelEdgeDetectionImageFilter,
  itkSobelOperator
• itkGradientMagnitudeImageFilter
• itkGradientToMagnitudeImageFilter
• itkGradientVectorFlowImageFilter
• B-spline itkBSplineInterpolationWeightFunction
  
• itkBSplineInterpolateImageFunction

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