edge detection - stanford university

Digital Image Processing: Bernd Girod, © 2013 Stanford University -- Edge Detection 1

Edge detection

n Gradient-based edge operatorsl Prewittl Sobell Roberts

n Laplacian zero-crossingsn Canny edge detectorn Hough transform for detection of straight linesn Circle Hough Transform


Gradient-based edge detection

n Idea (continous-space): local gradient magnitude indicates edge strength

n Digital image: use finite differencesto approximatederivatives

n Edge templates

difference -1 1( )

central difference -1 0[ ] 1( )

Prewitt−1 0 1−1 0[ ] 1−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Sobel−1 0 1−2 0[ ] 2−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

grad f x, y( )( ) = ∂ f x, y( )

∂x

⎛

⎝⎜

⎞

⎠⎟

2

+∂ f x, y( )

∂y

⎛

⎝⎜

⎞

⎠⎟

2


Practical edge detectors

n Edges can have any orientationn Typical edge detection scheme uses K=2 edge templatesn Some use K>2

s x,y[ ]

e1 x, yéë ùû

e2 x, yéë ùû

eK x, yéë ùû

Combination,e.g.,

ek x, yéë ùû2

kå

orMaxk ek x, yéë ùû

edgemagnitude

(edge orientation)

M x, yéë ùû

a x, yéë ùû


Gradient filters (K=2)

Prewitt−1 0 1−1 0[ ] 1−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1 −1 −10 0[ ] 01 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Sobel−1 0 1−2 0[ ] 2

−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1 −2 −10 0[ ] 0

1 2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Roberts 0[ ] 1-1 0

æ

èç

ö

ø÷

1[ ] 00 -1

æ

èç

ö

ø÷Central

Difference

0 0 0−1 0[ ] 10 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

0 −1 00 0[ ] 00 1 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟


Kirsch operator (K=8)

Kirsch +5 +5 +5−3 0[ ] −3−3 −3 −3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−3 +5 +5−3 0[ ] +5−3 −3 −3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−3 −3 +5−3 0[ ] +5−3 −3 +5

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−3 −3 −3−3 0[ ] +5−3 +5 +5

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−3 −3 −3−3 0[ ] −3+5 +5 +5

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−3 −3 −3+5 0[ ] −3+5 +5 −3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+5 −3 −3+5 0[ ] −3+5 −3 −3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+5 +5 −3+5 0[ ] −3−3 −3 −3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟


Magnitude of image filtered with

(log display)


(log display)

Prewitt operator example

−1 0 1−1 0[ ] 1−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1 −1 −10 0[ ] 01 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Original1024x710


Prewitt operator example (cont.)

Sum of squared horizontal and

vertical gradients(log display)

threshold = 900 threshold = 4500 threshold = 7200


Sobel operator example

Sum of squared horizontal and

vertical gradients(log display)



Roberts operator example

Original1024x710


(log display)


(log display)

1[ ] 00 −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

0[ ] 1−1 0

⎛

⎝⎜⎜

⎞

⎠⎟⎟


Roberts operator example (cont.)

Sum of squared diagonal gradients

(log display)



Edge orientationCentralDifference

0 0 0−1 0[ ] 10 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

0 −1 00 0[ ] 00 1 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Gradient scatter plot

x-component

y-co

mpo

nent

x-componenty-

com

pone

nt

Roberts

0[ ] 1

−1 0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1[ ] 0

0 −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟


Edge orientation

Sobel

−1 0 1−2 0[ ] 2

−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1 −2 −10 0[ ] 0

1 2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Prewitt

−1 0 1−1 0[ ] 1−1 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1 −1 −10 0[ ] 01 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x-componenty-

com

pone

nt

x-component

y-co

mpo

nent



Edge orientation

5x5 “consistent”gradient operator

[Ando, 2000]

x-component

y-co

mpo

nent


-0.0604 -0.1632 0 0.1632 0.0604-0.4286 -1.1335 0 1.1335 0.4286-0.7448 -1.9612 0[ ] 1.9612 0.7448-0.4286 -1.1335 0 1.1335 0.4286-0.0604 -0.1632 0 0.1632 0.0604

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

-0.0604 -0.4286 -0.7448 -0.4286 -0.0604-0.1632 -1.1335 -1.9612 -1.1335 -0.1632

0 0 0[ ] 0 00.1632 1.1335 1.9612 1.1335 0.1632 0.0604 0.4286 0.7448 0.4286 0.0604

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟


Gradient consistency problem

Knowngray

value

Calculate this valueusing gradient field

Same result,regardless of path?


Laplacian operator

n Detect edges by considering second derivative

n Isotropic (rotationally invariant) operatorn Zero-crossings mark edge location

∇2 f x, y( ) = ∂2 f x, y( )

∂x2 +∂2 f x, y( )

∂y2

Edge profile f(x)

Detectzero crossing

f’(x) f”(x)


Approximations of Laplacian operator by 3x3 filter

0 1 01 −4⎡⎣ ⎤⎦ 1

0 1 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

1 1 11 −8⎡⎣ ⎤⎦ 1

1 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

-20

2

-20

20

2

4

6

8

wxwy

|H|

-20

2

-20

20

5

10

15

wxwy

|H|


Zero crossings of Laplacian

n Sensitive to very fine detail and noise è blur image firstn Responds equally to strong and weak edges

è suppress zero-crossings with low gradient magnitude


Laplacian of Gaussian

n Filtering of image with Gaussian and Laplacian operators can be combined into convolution with Laplacian of Gaussian (LoG) operator

LoG x, y( ) = − 1πσ 4

1− x2 + y2

2σ 2

⎛

⎝⎜

⎞

⎠⎟e

−x2+y2

2σ 2

2s =

-4-2

02

4

-4-2

02

4-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

XY -4-2

02

4

-4-2

02

4-40

-30

-20

-10

0

10

XY


Discrete approximation of Laplacian of Gaussian

2s =

-20

2

-20

20

50

100

150

200

wxwy

|H|

0 0 1 2 2 2 1 0 0

0 2 3 5 5 5 3 2 0

1 3 5 3 0 3 5 3 1

2 5 3 -12 -23 -12 3 5 2

2 5 0 -23 -40 -23 0 5 2

2 5 3 -12 -23 -12 3 5 2

1 3 5 3 0 3 5 3 1

0 2 3 5 5 5 3 2 0

0 0 1 2 2 2 1 0 0


Zero crossings of LoG

2 2s = 4 2s = 8 2s =2s =


Zero crossings of LoG – gradient-based threshold

2 2s = 4 2s = 8 2s =2s =


Canny edge detector

1. Smooth image with a Gaussian filter2. Approximate gradient magnitude and angle (use Sobel, Prewitt . . .)

3. Apply nonmaxima suppression to gradient magnitude4. Double thresholding to detect strong and weak edge pixels5. Reject weak edge pixels not connected with strong edge pixels

[Canny, IEEE Trans. PAMI, 1986]

M x, y⎡⎣ ⎤⎦ ≈∂f∂x

⎛

⎝⎜

⎞

⎠⎟

2

+∂f∂y⎛

⎝⎜

⎞

⎠⎟

2

α x, y⎡⎣ ⎤⎦ ≈ tan−1 ∂f

∂y∂f∂x

⎛

⎝⎜

⎞

⎠⎟


Canny nonmaxima suppression

n Quantize edge normal to one of four directions:horizontal, -45o, vertical, +45o

n If M[x,y] is smaller than either of its neighbors in edge normal direction è suppress; else keep.



Canny thresholding and suppression of weak edges

n Double-thresholding of gradient magnitude

n Typical setting: n Region labeling of edge pixelsn Reject regions without strong edge pixels

Strong edge: M x, y⎡⎣ ⎤⎦ ≥θhigh

Weak edge: θhigh > M x, y⎡⎣ ⎤⎦ ≥θ low

θhigh θ low = 2...3



Canny edge detector

2s = 2 2s = 4 2s =


Hough transform

n Problem: fit a straight line (or curve) to a set of edge pixelsn Hough transform (1962): generalized template matching techniquen Consider detection of straight lines y = mx + c

x

y

m

c

edge pixel


Hough transform (cont.)

n Subdivide (m,c) plane into discrete “bins,” initialize all bin counts by 0n Draw a line in the parameter space [m,c] for each edge pixel [x,y] and increment

bin counts along line.n Detect peak(s) in [m,c] plane

x

y

m

c

detect peak


Hough transform (cont.)

n Alternative parameterization avoids infinite-slope problem

n Similar to Radon transform

p2

xcosq + ysinq = r

x

y

r

qedge pixel

r

p-

-p2 q


Hough transform example

Original image

200 400 600 800

100

200

300

400

500

600

700

800



200 400 600 800

100

200

300

400

500

600

700

800

Original image



Paper (3264x2448)

Global thresholding

-50 0 50

-4000

-2000

0

2000

4000 0

500

1000

1500

q

r

De-skewed Paper

-81.8 deg

-50 0 50

-4000

-2000

0

2000

4000 0

500

1000

1500

-81.8 deg


Circle Hough Transform

n Find circles of fixed radius r

n Equivalent to convolution (template matching) with a circle

x

y

0x

edge pixel

0y

Circle center


Circle Hough Transform for unknown radius

n 3-d Hough transform for parametersn 2-d Hough transform aided by edge orientation è “spokes” in parameter space

x

y

0x

edge pixel

0y

Circle center

x0 , y0 ,r( )


Example: circle detection by Hough transform

Original coins image

Prewitt edge detection Detected circles

edge detection - stanford university

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