edge filters

• References :

  • Done Reading
  • Reading
    • https://nptel.ac.in/courses/106/106/106106224/ [IIT Madras Week 2 part 1]
    • Cornell CV lecture on Edge
    • https://szeliski.org/Book/ [Book Chapter 3]
    • https://www.wikiwand.com/en/Sobel%5Foperator
    • Joseph Redmon Lectures
    • https://evergreenllc2020.medium.com/fundamentals-of-image-gradients-and-edge-detection-b093662ade1b
  • To read:

• Questions :

• Criteria for a good edge detector
  • Good detection
    • find all real edges, ignoring noise or other artifacts
  • Good Localization
    • detect edges as close as possible to true edges
  • Single Response
    • return one point only for each true edge point

from skimage.filters import roberts, sobel, scharr, prewitt, farid
roberts_img = roberts(img)
sobel_img = sobel(img)
scharr_img = scharr(img)
prewitt_img = prewitt(img)
farid_img = farid(img)

Robert, Scharr, Prewitt, Farid

• The x-coordinate in figure ref:edge-filter-kernel is defined as increasing in the “right”-direction, and the y-coordinate is defined as increasing in the “down”-direction. At each point in the image, the resulting gradient approximations can be combined to give the gradient magnitude, using: [wikipedia]

  \( {M} = \sqrt{ ({M}_{x})^{2} + ({M}_{y})^{2}} \)

• Using this information, we can also calculate the gradient’s direction(orientation):

  \( \Theta = \arctan{\frac{M_{y}}{M_{x}}} \)

where, for example, \(\Theta\) is 0 for a vertical edge which is lighter on the right side.

Sobel

• Computes the image gradient in the x direction

• Computes the image gradient in the y direction

Canny edge

Algorithm

• Smooth image (only want “real” edges, not noise)
• Calculate gradient direction and magnitude
• Non-maximum suppression perpendicular to edge
  • check if pixel is local maximum along gradient problem
• Threshold into strong, weak, no edge
  • Define two thresholds: low and high
  • hysteresis:
    • Use the high threshold to start edge curves and the low threshold to continue them
  • if gradient at pixel > ‘High’ = ’edge pixel’
  • if gradient at pixel < ‘Low’ = ’non edge pixel’
  • if gradient at pixel >= ‘Low’ and <= ‘High’ = ’edge pixel’ iff,
    • it is connected to an ’edge pixel’ directly or via pixels between ‘Low’ and ‘High’
• Connect together components
• Tunable: Sigma, thresholds
• Canny edge pipeline

• Effect of \(\sigma\) in Cannyu Edge Detector
  • The choice of σ (Gaussian kernel spread/size) depends on desired behavior
    • large σ detects large-scale edges
    • small σ detects fine edges

Python implementation

canny_edge = cv2.Canny(img, 50, 80)
# auto canny
sigma = 0.3
median = np.median(img)
lower = int(max(0, (1.0 - sigma) * median))
upper = int(min(255, (1.0 + sigma) * median))
auto_canny = cv2.Canny(img, lower, upper)