Edge Detection

Today’s reading

Forsyth, chapters 8, 15.1

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

Edge detection

How can you tell that a pixel is on an edge?

Image gradient

The gradient of an image:

The gradient points in the direction of most rapid change in intensity

The discrete gradient

How can we differentiate a digital image f[x,y]?

Option 1: reconstruct a continuous image, then take gradient

Option 2: take discrete derivative (finite difference)

The Sobel operator

Better approximations of the derivatives exist

The Sobel operators below are very commonly used

Effects of noise

Consider a single row or column of the image

Plotting intensity as a function of position gives a signal

Solution: smooth first

Derivative theorem of convolution

This saves us one operation:

Laplacian of Gaussian

Consider

2D edge detection filters

is the Laplacian operator:

The Canny edge detector

original image (Lena)

The Canny edge detector

Non-maximum suppression

Check if pixel is local maximum along gradient direction

requires checking interpolated pixels p and r

The Canny edge detector

Effect of s (Gaussian kernel size)

Scale space (Witkin 83)

Properties of scale space (w/ Gaussian smoothing)

edge position may shift with increasing scale (s)

two edges may merge with increasing scale

an edge may not split into two with increasing scale

Edge detection by subtraction

Gaussian - image filter

An edge is not a line...

Finding lines in an image

Option 1:

Search for the line at every possible position/orientation

What is the cost of this operation?

Option 2:

Use a voting scheme: Hough transform

Finding lines in an image

Connection between image (x,y) and Hough (m,b) spaces

A line in the image corresponds to a point in Hough space

To go from image space to Hough space:

given a set of points (x,y), find all (m,b) such that y = mx + b

Hough transform algorithm

Typically use a different parameterization

d is the perpendicular distance from the line to the origin

q is the angle this perpendicular makes with the x axis

Why?

Hough transform algorithm

Typically use a different parameterization

d is the perpendicular distance from the line to the origin

q is the angle this perpendicular makes with the x axis

Why?

Basic Hough transform algorithm

Initialize H[d, q]=0

for each edge point I[x,y] in the image

for q = 0 to 180

H[d, q] += 1

Find the value(s) of (d, q) where H[d, q] is maximum

The detected line in the image is given by

What’s the running time (measured in # votes)?

Extensions

Extension 1: Use the image gradient

same

for each edge point I[x,y] in the image

compute unique (d, q) based on image gradient at (x,y)

H[d, q] += 1

same

same

What’s the running time measured in votes?

Extension 2

give more votes for stronger edges

Extension 3

change the sampling of (d, q) to give more/less resolution

Extension 4

The same procedure can be used with circles, squares, or any other shape


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


	The gradient of an image:


	The gradient points in the direction of most rapid change in intensity


	How can we differentiate a digital image f[x,y]?
		Option 1: reconstruct a continuous image, then take gradient
		Option 2: take discrete derivative (finite difference)


	Better approximations of the derivatives exist
		The Sobel operators below are very commonly used


	Consider a single row or column of the image
		Plotting intensity as a function of position gives a signal


	Check if pixel is local maximum along gradient direction
		requires checking interpolated pixels p and r


	Properties of scale space (w/ Gaussian smoothing)
		edge position may shift with increasing scale (s)
		two edges may merge with increasing scale
		an edge may not split into two with increasing scale


	Option 1:
		Search for the line at every possible position/orientation
		What is the cost of this operation?

	Option 2:
		Use a voting scheme: Hough transform


Connection between image (x,y) and Hough (m,b) spaces
	A line in the image corresponds to a point in Hough space
	To go from image space to Hough space:
		given a set of points (x,y), find all (m,b) such that y = mx + b


	Typically use a different parameterization

		d is the perpendicular distance from the line to the origin
		q is the angle this perpendicular makes with the x axis
		Why?


Extension 1: Use the image gradient
	same
	for each edge point I[x,y] in the image
		compute unique (d, q) based on image gradient at (x,y)
			H[d, q] += 1
	same
	same
What’s the running time measured in votes?

Extension 2
	give more votes for stronger edges
Extension 3
	change the sampling of (d, q) to give more/less resolution
Extension 4
	The same procedure can be used with circles, squares, or any other shape