Laplacian of Gaussian Edge Detection using Canny Method, Study notes of Computer Science

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Lecture-

Edge Detection: LG, Canny

Edge Detection

Edge Detectors

• Gradient operators: Sobel, Prewit, Robert• Laplacian of Gaussian (Marr-Hildreth)
• Gradient of Gaussian (Canny)• Facet Model Based Edge Detector

(Haralick)

Laplacian of Gaussian EdgeDetector

• Generate a mask for LG for a given• Apply mask to the image
• Detect zerocrossings– Scan along each row, record an edge point at
  • Repeat above step along each columnthe location of zerocrossing.

s

Zerocrossings

• Four cases of zerocrossings :{+,-}, {+,0,-},{-,+}, {-,0,+}
• Slope of zerocrossing {a, -b} is |a+b|.• To detect zerocrossing apply threshold to

the slope. If the slope is above somethreshold, then that point is an edge point.

Gaussian

g ( x )= e^2 -^ o^ x^22

x -3 -2 -1^0123 g(x)

Standarddeviation

2-D Gaussian( 22 2 2 )

g ( x , y )= e - x^ o^ +^ y

s = 2

Separability of Gaussian h ( x , y )= f ( x , y )* g ( x , y )

h ( x , y )=( f ( x , y )* g ( x ))* g ( y )

Requires n^2 multiplications for a n by n mask, for each pixel.

This requires 2n multiplications for a n by n mask, for each pixel.

Separability of Laplacian ofGaussian

h ( x , y )= f ( x , y )D^2 g ( x , y ) h ( x , y )=( f ( x , y ) gxx ( x ))* g ( y )+( f ( x , y )* gyy ( y ))* g ( x )

Requires n^2 multiplications for a n by n mask, for each pixel.

This requires 4n multiplications for a n by n mask, for each pixel.

Separability

Decomposition of LG into four 1-D convolutions

• Convolve the image with a second derivative of Gaussianmask • Convolve the resultant image from step (1) by a Gaussian g (^) yy ( y )along each column mask• Convolve the original image with a Gaussian mask, g(x) along each row. Call the resultant image I x. g ( y ) along each column•Convolve the resultant image from step (3) by a secondderivative of Gaussian mask along each row. Call the resultant image •Add I x (^) and I y. I y^. g^ xx ( x^ )

Non-maxima Suppression

• Suppress the pixels which are not localmaxima. M ( x , y )= ÔÓÔÌÏ M ( 0 x , y ) ififMM (( xx , otherwise , yy ))>> MM (( xx ¢,¢¢, y ¢ y )¢¢&)

Quantization in Eight PossibleDirections

xy direction(magnitude fx , fy )Gradient==q=(tan fx Vector^2 - +^1 fffy^2 )

Hysteresis Thresholding

Gradientmagnitude Highlow

Hysteresis Thresholding

• Scan the image from left to right, top-bottom. If
  • The gradient magnitude at a pixel is above ahigh threshold declare that as an edge point
  • Then recursively consider thepixel.• If the gradient magnitude is above the low threshold neighbors of this declare that as an edge pixel.

Connected Component

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Sequential

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