Edge Detection

Today’s readings

Cipolla and Gee

supplemental: Forsyth, chapter 9

Watt, 10.3-10.4

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

Look for zero-crossings of

2D edge detection filters

is the Laplacian operator:

The Canny edge detector

original image (Lena)

The Canny edge detector

Effect of Gaussian kernel width

Edge detection by subtraction

Gaussian - subtraction filter

An edge is not a line...

Finding lines in an image

Option 1:

Search for the object at every possible position in the image

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

Summary

Things to take away from this lecture

What is an edge and where does it come from

Edge detection by differentiation

Image gradients

continuous and discrete

filters (e.g., Sobel operator)

Effects of noise on gradients

Derivative theorem of convolution

Derivative of Gaussian (DoG) operator

Laplacian operator

Laplacian of Gaussian (LoG)

Canny edge detector (basic idea)

Effects of varying sigma parameter

Approximating an LoG by subtraction

Hough Transform


Today’s readings
	Cipolla and Gee
		supplemental: Forsyth, chapter 9
	Watt, 10.3-10.4


	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


	Option 1:
		Search for the object at every possible position in the image
		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


Things to take away from this lecture
	What is an edge and where does it come from
	Edge detection by differentiation
	Image gradients
		continuous and discrete
		filters (e.g., Sobel operator)
	Effects of noise on gradients
	Derivative theorem of convolution
	Derivative of Gaussian (DoG) operator
	Laplacian operator
		Laplacian of Gaussian (LoG)
	Canny edge detector (basic idea)
		Effects of varying sigma parameter
	Approximating an LoG by subtraction
	Hough Transform