Fuzzy Logic Edge Detection

CSE 6369 - Reasoning with Uncertainty

Joel Martin

Dec 8, 2015

"A New Fuzzy Approach for Edge Detection"

Yasar Becerikli and Tayfun M. Karan

Proceedings of the 8th International Workshop on Artificial Neural Networks, IWANN 2005

PDF

Background: Fuzzy Logic Models

  • Mamdani
    • Crisp output value using defuzzification
    • More intuitive and human meaningful
  • Sugeno (Takagi-Sugeno-Kang)
    • Crisp output value using weighted averages
    • Less intuitive but more efficient (no output membership functions)

Background: Edge Detection

  • Core component of most image and video analysis algorithms especially feature dectection and extraction.
  • Edge: Boundary or abrupt change in an image. Points (line segment) where image brightness changes sharply.
  • Commonly a 3x3 kernel is convolved with the source image to produce a image representing detected edges.

Image Example

0.1 0.2 0.1 0.8 0.8 0.2
0.3 0.2 0.1 0.8 0.7 0.1
0.3 0.2 0.8 0.7 0.8 0.1
0.3 0.2 0.7 0.8 0.9 0.2

Edge Detection: Sobel Kernel

  • Horizontal:
    -1 -2 -1
    0 0 0
    1 2 1
  • Vertical:
    -1 0 1
    -2 0 2
    -1 0 1

Edge Detection: Prewitt Kernel

  • Horizontal:
    -1 -1 -1
    0 0 0
    1 1 1
  • Vertical:
    -1 0 1
    -1 0 1
    -1 0 1

Vertical Prewitt Mask Example

0.1 0.2 * -1 0.1 * 0 0.8 * 1 0.8 0.2
0.3 0.2 * -1 0.1 * 0 0.8 * 1 0.7 0.1
0.3 0.2 * -1 0.8 * 0 0.7 * 1 0.8 0.1
0.3 0.2 0.7 0.8 0.9 0.2

0.2*-1 + 0.2*-1 + 0.2*-1
+ 0.8*1 + 0.8*1 + 0.8*1
= 1.7

Vertical Prewitt Mask Example

0.4 -0.2 1.2 1.3 -1.3 -1.5
0.6 0.2 1.7 1.3 -1.9 -2.3
0.6 0.7 1.7 0.8 -1.9 -2.4
0.4 0.9 1.1 0.2 -1.2 -1.7

Left edge threshold: ≥ 1.0

Right edge threshold: ≤ -1.0

Gradients in Each Direction

Assign labels to 3x3 grid:

z1 z2 z3
z4 z5 z6
z7 z8 z9

Calculate gradients:

  • ∇D1 = (z5 − z1)2 + (z9 − z5)2 (NW/SE Diagonal)
  • ∇D2 = (z5 − z2)2 + (z8 − z5)2 (Vertical)
  • ∇D3 = (z5 − z3)2 + (z7 − z5)2 (NE/SW Diagonal)
  • ∇D4 = (z5 − z4)2 + (z6 − z5)2 (Horizontal)

Gradients in Each Direction

Approximation with absolute values (for efficiency):

  • ∇D1 = |z5 − z1| + |z9 − z5| (NW/SE Diagonal)
  • ∇D2 = |z5 − z2| + |z8 − z5| (Vertical)
  • ∇D3 = |z5 − z3| + |z7 − z5| (NE/SW Diagonal)
  • ∇D4 = |z5 − z4| + |z6 − z5| (Horizontal)

Fuzzy Logic Rules

Fuzzy Membership Sets

Directional Components

Final Edge Value

y = y̅1 + y̅2 + y̅3 + y̅4

Results / Comparison

Original Prewitt Fuzzy

Conclusion

  • Similar quality and computation efficiency to traditional convolution edge detection methods.
  • Added flexibility compared to traditional convolution methods.
  • Simple to implement (understandable)

Questions?

These notes: http://kanaka.github.io/fuzzy-edge