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Carpools

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]?

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

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 spread/size)

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


	Mailing list: csep576@cs.washington.edu
			you should have received messages
	Project 1 out today (due in two weeks)
	Carpools


	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


	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