Canny Edge Detector

Sobel Edge Detector

Detecting Edges in Image

Image I

101

202

101

121

000

121

*

*

Idx

d

Idy

d

22

Idy

dI

dx

d

ThresholdEdges

Sobel Edge Detector

Idx

d

Idy

d

I

Sobel Edge Detector

I

22

I

dy

dI

dx

d

100 Threshold

Marr and Hildreth Edge Operator

Smooth by Gaussian

Use Laplacian to find derivatives

IGS * 2

22

2

2

1

yx

eG

Sy

Sx

S 2

2

2

22

Marr and Hildreth Edge Operator

IGIGS ** 222

2

22

22

22

3

2 22

1

yx

eyx

G

Marr and Hildreth Edge Operator

2

22

22

22

3

2 22

1

yx

eyx

G

0.0008 0.0066 0.0215 0.031 0.0215 0.0066 0.00080.0066 0.0438 0.0982 0.108 0.0982 0.0438 0.00660.0215 0.0982 0 -0.242 0 0.0982 0.0215

0.031 0.108 -0.242 -0.7979 -0.242 0.108 0.0310.0215 0.0982 0 -0.242 0 0.0982 0.02150.0066 0.0438 0.0982 0.108 0.0982 0.0438 0.00660.0008 0.0066 0.0215 0.031 0.0215 0.0066 0.0008

X

Y

Marr and Hildreth Edge Operator

Zero CrossingsDetection

I ImageG2*

IG *2 Edge Image

IG *2 Zero Crossings

1

6

3

Quality of an Edge Detector

Robustness to Noise Localization Too Many/Too less Responses

Poor robustness to noise Poor localization Too many responses

True Edge

Canny Edge Detector

Criterion 1: Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives.

Criterion 2: Good Localization: The edges detected must be as close as possible to the true edges.

Single Response Constraint: The detector must return one point only for each edge point.

Canny Edge Detector

Convolution with derivative of Gaussian Non-maximum Suppression Hysteresis Thresholding

Canny Edge Detector

Smooth by Gaussian

Compute x and y derivatives

Compute gradient magnitude and orientation

IGS *2

22

2

2

1

yx

eG

Tyx

T

SSSy

Sx

S

22yx SSS

x

y

S

S1tan

Canny Edge Operator

IGIGS **

T

y

G

x

GG

T

Iy

GI

x

GS

**

Canny Edge Detector

xS

yS

I

Canny Edge Detector

I

22yx SSS

25 ThresholdS

We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?

Non-Maximum Suppression

Non-Maximum Suppression Suppress the pixels in ‘Gradient

Magnitude Image’ which are not local maximum

edge an tonormal direction thealong

in of neighbors theare and Sx,yy,xy,x

otherwise0,,&

,, if,

, yxSyxS

yxSyxSyxS

yxM

yx ,

yx,

yx ,

Non-Maximum Suppression

0

12

3

41420tan41422- :3

41422tan :2

41422tan41420 :1

41420tan41420 :0

.θ.

.θ

.θ.

.θ.-

x

y

S

Sθ tan

Non-Maximum Suppression

22yx SSS M

25ThresholdM

Hysteresis Thresholding

Hysteresis Thresholding

If the gradient at a pixel is above ‘High’, declare it an ‘edge pixel’

If the gradient at a pixel is below ‘Low’, declare it a ‘non-edge-pixel’

If the gradient at a pixel is between ‘Low’ and ‘High’ then declare it an ‘edge pixel’ if and only if it is connected to an ‘edge pixel’ directly or via pixels between ‘Low’ and ‘ High’

Hysteresis Thresholding

M 25ThresholdM

15

35

Low

High