TOPOLOGICAL GRADIENT OPERATORS FOR EDGE DETECTION

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TOPOLOGICAL GRADIENT OPERATORS FOR EDGE DETECTION

• Hakan G. Senel
• Anadolu University, Turkey

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Outline

• Edge Detection
• Importance of Gradient Estimation
• A Better Gradient Estimation?
• Less smoothing
• Larger gradient kernel
• Problems of larger kernels
• Fuzzy Set Representation of Images and Fuzzy
  Image Topology
• Degree of Connectedness Map (DOCM)
• Properties of DOCM
• Fuzzy Topology Based Gradient Estimation
• Results
• Conclusions

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Edge Detection

• Edge detection is the process of localizing pixel
  intensity transitions i.e. step edges
• Typical algorithm
• Compute directional derivatives (gradient
  kernels Sobel, Prewitt, etc)
• Find direction and magnitude
• Threshold the magnitude (global or adaptive)
• Operate on the binary image

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Gradient Based Edge Detection

• Common misconceptions about gradient based edge
  detection
• Small kernel is necessary
• Step edges must be detected
• Smoothing is a must
• Single size kernel is enough

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Why Small Gradient Kernels?

• Small kernels are used
• to avoid localization problems
• to diminish the effects of near objects
• to detect step edges
• to increase computational performance

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Small Kernel Sizes

• Small Kernels Require smoothing. Why?

Derivative Filter
F sampling frequency, 2k1 filter length
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Small Kernel Sizes

• To determine derivative direction, we have two
  alternatives.
• Derivatives along x and y axes

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Large Gradient Kernels

• Yield better derivative approximation than small
  kernels
• Better smoothing along the direction
  perpendicular to the derivative directions
• Better directional estimate

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Large Kernels

• have problems
• Smearing effect
• Large response area around the edge
• High computational requirements

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How Can We Integrate Larger Kernels?

• Objects that are not connected to the center
  pixel have to removed
• Kernel response around an edge should be limited

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Solution
TOPOLOGICAL GRADIENT KERNELS
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Images as Fuzzy Sets

• Image is scaled between 0.0 and 1.0 and converted
  to fuzzy set
• Each pixel value is converted to degree of
  membership within the set of bright objects
• If applied to reverse image, each value becomes
  degree of membership within the set of dark
  objects

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Fuzzy Topology
Strength of a path weakest link
Degree of connectedness strength of the
strongest path
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Degree of Connectedness Map

• In a neighborhood, degree of connectedness is
  assigned to its pixel
• Visualization of how pixels are connected to the
  center pixel
• Depicts the relationship between the center pixel
  and other pixels

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Degree of Connectedness Map
DOCM
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Degree of Connectedness Map
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Degree of Connectedness Map
Let I be an image and p(x,y) denotes a pixel.
I(x,y) is the intensity value of the image at
point (x,y). W is a (2n1)x(2n1) observation
window. Origin is denoted by (xo,yo). Degree of
connectedness for bright objects is
Degree of connectedness for dark objects is
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Topological Gradients
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Topological Gradient Results
Slowly varying edge
TG for bright objects
magnitude
TG for dark objects
Conventional Sobel gradient
x
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Topological Gradient Results
Slowly varying edge
magnitude
Conventional gradient
x
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Real Images
Topological Gradient 9x9 in 11x11
Conventional Gradient 9x9
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Real Images
Topological Gradient 9x9 in 11x11
Conventional Gradient 9x9
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Results
A 1000x90 synthetic image that contains a ramp
edge is formed.
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Results

• BR Blur rate is defined to assess the performance
  of TG

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Results

• R is defined as the resillience against noise on
  flat image areas

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Results
BLUR RATES AND RESILLIENCE FOR DIFFERENT NOISE
LEVELS Topo. Sobel 11x11 Conventional Noise
in 13x13 DOCM 11x11 Sobel s BR R BR
R 0 0.000 0.00 0.099 0.00 5 0.164 19.31 0.
176 47.16 10 0.203 38.88 0.216 95.55 15 0.235 5
7.65 0.251 142.98
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Conclusions

• This paper presents a method to facilitate the
  use of large gradient kernels by
• Limiting the gradient response area around the
  edge
• Smoothing due to more samples to compute gradient
• at the expense of computational requirements.