[CS5670] Lecture 2: Edge detection

Edge detection

• Convert a 2D image into a set of curves
• Extracts salient features of the scene
• More compact than pixels

 

Origin of edges

 

• Edges are caused by a variety of factors

 

 

Images as functions

• Edges look like steep cliffs

 

 

Characterizing edges

• An edge is a place of rapid change in the image intensity function

 

Image derivatives

• How can we differentiate a digital image $F[x,y]$

 

1. Option 1: reconstruct a continuous image, $f$, then compute the derivative
2. Option 2: take discrete derivative (finite difference)

 

$$\frac{\partial f}{\partial x} [x,y] \approx F[x+1, y] - F[x,y]$$

 

• How would you implement this as a linear filter?

 

 

Image gradient

 

• The gradient of an image: $$\bigtriangledown f = [ \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} ]$$

 

• The gradient points in the direction of most rapid increase in intensity

 

 

• The edge strength is given by the gradient magnitude:

 

$$||\bigtriangledown f|| = \sqrt{(\frac{\partial f}{\partial x})^2 (\frac{\partial f}{\partial y}})^2$$

 

• The gradient direction is given by:

 

$$\theta = tan^{-1} (\frac{\partial f}{\partial x} / \frac{\partial f}{\partial y})$$

 

• How does this relate to the direction of the edge?

 

 

Effects of noise

 

• Where is the edge?

 

Solution: smooth first

 

 

To find edges, look for peaks in $\frac{d}{dx} (f \star h)$

 

 

 

Associative property of convolution

 

Differentiation is convolution, and convolution is associative:

 

$$\frac{d}{dx} (f \star h) = f \star \frac{d}{dx} h$$

 

This saves us one operation: $f$

 

 

 

The 1D Gaussian and its derivatives

 

$$G_{\sigma}(x) = \frac{1}{\sqrt{2\pi} \sigma} e ^ -({\frac{x^2}{2\sigma^2}})$$

 

$$G'_{\sigma}(x) = \frac{d}{dx} G_{\sigma}(x) = - \frac{1}{\sigma} (\frac{x}{\sigma}) G_{\sigma} (x)$$

 

2D edge detection filters

 

 

$$h_{\sigma}(u,v) = \frac{1}{2\pi \sigma^2} e ^ -({\frac{u^2+v^2}{2\sigma^2}})$$

 

 

$$\frac{\partial}{\partial x} h_{\sigma} (u,v)$$

 

Derivative of Gaussian filter

 

 

The sobel operator

 

• Common approximation of derivative of Gaussian

 

• The standard definition of the Sobel operator omits the 1/8 term
  • doesn’t make a difference for edge detection
  • the 1/8 term is needed to get the right gradient magnitude

 

Sobel operator: example

 

 

 

Example

 

original

smoothed gradient magnitude

 

• Which method is the best way to compute the derivative of an image?

 

 

Finding edges

 

thesholding

 

Get Orientation at Each Pixel

 

• Get orientation (below, threshold at minimum gradient magnitude)

 

 

 

Non-maximum supression

 

• Check if pixel is local maximum along gradient direction
  • requires interpolating pixels p and r

 

Before Non-max Suppression

 

 

After Non-max Suppression

 

 

Thresholding edges

 

• Still some noise
• Only want strong edges
• 2 thresholds, 3 cases
  • R > T: strong edge
  • R < T but R > t: weak edge
  • R < t: no edge
• Why two thresholds?

 

 

Connecting edges

 

• Strong edges are edges!
• Weak edges are edges if they connect to strong
• Look in some neighborhood (usually 8 closest)

 

 

Canny edge detector

1. Filter image with derivative of Gaussian
2. Find magnitude and orientation of gradient
3. Non maximum suppression
4. Linking and thresholding (hysteresis)

• Define two thresholds: low and high
• Use the high threshold to start edge curves and the low threshold to continue them

 

Reference: Lecture 1: Images and image filtering