Edge Detection

• An edge is a place of rapid change in the image intensity function (or grey-level function), which manifests as boundaries between physically distinct regions.
  • i.e. a peak in the derivative of the image function

Convolution based Edge Detection>

see Convolution

• Edge detection is same as differentiation a.k.a. derivative filter
• -
• gradient direction is perpendicular to the edge

Derivative of Gaussian convolution kernel

see Gaussian Distribution

• For noisy images -
  • scale selection is important
    • edges may be noise
    • Location of edges at coarse scale can direct the search for finer-scale edges
    • edges can persist across scales, allows fusion across scales
  • Perform Image smoothing before derivative filter to reduce effect of noise
• Derivative of Gaussian filter in 2 dimensions, along x
• derivatives can be along x and y directions (2 filters) → requires atleast 2 convolutions
• allows selection of scale of edge detection
  • through σ

Kernels

Prewitt Kernel

• vertical edges, detecting edges across x direction, smooths along y direction
  • -1 0 1
  • -1 0 1
  • -1 0 1
• horizontal edges, detecting edges across y direction, smooths along x direction
  • -1 -1 -1
  • 0 0 0
  • 1 1 1

Sobel Kernel

• vertical edges, detecting edges in x direction, weighted smoothing along y direction
  • -1 0 1
  • -2 0 2
  • -1 0 1
• horizontal edges, detecting edges in y direction, weighted smoothing along x direction
  • -1 -2 -1
  • 0 0 0
  • 1 2 1
• 2 filters: Derivative images I’_x and I’ _y
  

Laplacian of Gaussian Edge Detector

a.k.a. Marr–Hildreth Edge Detector a.k.a. Mexican hat wavelet

• second derivative of gaussian kernel
• 
• 2nd Derivative of Gaussian filter in 2 dimensions
• 
• detect zero crossings → where the 2nd derivative crosses zero → indicates an edge
• convolution kernel has a negative center and a positive surrounding
• again, scale selection is done through σ
• isotropic response
  • one filter (all directions)
    • no edge direction
  • indirect edge magnitude
• kernel examples

 2  -1  2 
-1  -4 -1  
 2  -1  2

1   1   1
1  -8   1
1   1   1

Canny Edge Detector

• best attributes of both the gradient operator and the Laplacian operator
• Multi stage algorithm
  1. Gaussian Smoothing to select scale and smoothen
  2. edge detection
     • convolve image intensity function with derivative of Gaussian (Sobel operator)
     • Edge magnitude and direction detected
  3. Compute Laplacian of Gaussian along the Gradient Direction at each pixel
     • zero crossings → edge location
     • less effected by pixels that are unrelated to the edge
  4. thinning: suppress non maxima of derivative
     • non maxima derivatives across an edge
     • sub pixel precision: detect precise edge locations
  5. Track using hysteresis thresholds
     • Finalize the detection of edges by suppressing all the other edges that are weak and not connected to strong edges.
     • declare each pixel as being an edge or not
     • single threshold: if the edge magnitude is less than some value T it is not an edge
     • 

Difference of Gaussians Edge Detector

• a fast approximation of the Laplacian of Gaussians
• subtraction of one Gaussian-Blurred (Gaussian Smoothing) version of an image from another, less blurred image
• for scales of gaussians sσ and σ →
• 
• DoG = (s-1) NLoG

Edge illusions

Cannot be detected using edge detectors

Herring Illusion

horizontal lines are actually parallel

Cafe Wall Illusion

horizontal lines are actually parallel