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E.G.M. Petrakis Edge Detection 1
Edge Detection
• Edges: local changes in the image intensity• Edges typically occur on the boundary
between two regions• Important features for image analysis• The changes due to noise are not edges
– real images are very noisy– its very difficult to find edges in noisy images
• The changes due to shading are not edges
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E.G.M. Petrakis Edge Detection 2
I.
line of profile
Original noisy imageII. Edge profile
• The edges are not perfectly sharp
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E.G.M. Petrakis Edge Detection 3
Ideal Edges
step
roof
ramp
line
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E.G.M. Petrakis Edge Detection 4
Noisy Edgesstep
orramp
line or
roof
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E.G.M. Petrakis Edge Detection 5
Edge Detection
• An edge point is a point at the location of a local intensity change
• An edge detector is an algorithm that computes the edges in an image
• A contour is a list of edges or the curve that matches the list of edges
• Edge following is the process of searching the image to determine contours
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E.G.M. Petrakis Edge Detection 6
Edge Results
• The produced set of edges contain – correct edges corresponding to real - useful
information e.g., object boundaries– false edges do not correspond to real edges e.g.,
noise• There are also missing edges
– edge linking may try to recover missing edges– edge following will form contours from
existing and recovered edges
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E.G.M. Petrakis Edge Detection 7
First Derivative Operator
• Gradient: vector denoting the max rate of change of intensity f(x,y):
– direction
– magnitude
jyfi
xff
rrr
ϑϑ
ϑϑ
+=∇
∂∂
∂∂
= −
yf
xf1tanθ
...,max,22
∂∂
∂∂
∂∂
+
∂∂
yf
xf
yf
xf
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E.G.M. Petrakis Edge Detection 8
Gradient
• For any vertical directions (x,y), (x’,y’) and image f(x,y):
( ) ( )yxfyxf ′′∇=∇ ,,rr
′∂∂
′∂∂=′=
∂∂
∂∂= −−
yf
xf
yf
xf 11 tantan θθ
2222
′∂
∂+
′∂∂
=
∂∂
+
∂∂
yf
xf
yf
xf
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E.G.M. Petrakis Edge Detection 9
Digital Approximation of Gradient
• Compute
• Equivalently convolve with
( ) ( ) ( )jifjifjifG xx ,1,, −+=∆≈
( ) ( ) ( )jifjifjifG yy ,1,, +−=∆≈
[ ]11−=xG
−
=1
1yG
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E.G.M. Petrakis Edge Detection 10
Gradient Convolution Masks
• computes edge point at (i+1/2,j+1/2) • computes edge point at (i+1/2,j)
• 2x2 masks: edge point at (i+1/2, j+1/2)
• edge points at center pixel (i,j) use 3x3 masks
−−
=1111
xG
−−
=11
11yG
−
=1
1yG
[ ]11−=xG
−−−
=101202101
41
xG
−−−=
121000121
41
yG
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E.G.M. Petrakis Edge Detection 11
Roberts operator
• Computes the edge at the interpolated point (i+1/2,j+1/2)– magnitude: |Gx| + |Gy|
−
=10
01xG
−=
0110
yG
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E.G.M. Petrakis Edge Detection 12
Sobel Operator
• One the most commonly used edge operators– computes the edge at (i,j)– Smoothing at the same time– places emphasis on pixels closest to (i,j)
−−−
=101202101
41
xG
−−−=
121000121
41
yG
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E.G.M. Petrakis Edge Detection 13
Prewit Operator
• Similar to Sobel but does not place any emphasis on pixels
−−−
=101101101
31
xG
−−−=
111000111
31
yG
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E.G.M. Petrakis Edge Detection 14
Edge operatorswith smoothingfirst
a. Original imageb. Smoothed imagec. Gradient with 1x2,2x1 masks, T=32d. Gradient with 2x2 masks, T=64e. Roberts operator, T=64f. Sobel operator, T=225g. Prewit operator, T=225
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E.G.M. Petrakis Edge Detection 15
a. Original imageb. Simple Gradient with 1x2,2x1 masks, T=64c. Gradient with 2x2 masks, T=64d. Roberts operator, T=64e. Sobel operator, T=225f. Prewitt operator, T=225
Edge operatorswithout smoothing
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E.G.M. Petrakis Edge Detection 16
a. Noisy imageb. Filtered imagec. Simple Gradient with 1x2,2x1 masks,T=32d. Gradient with 2x2 masks, T=64e. Roberts operator, T=64f. Sobel operator, t=225g. Prewitt operator, T=225
Edge operatorswith smoothingon noisy image
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E.G.M. Petrakis Edge Detection 17
a. Noisy imageb. Simple Gradient using 1x2,2x1 masks, T=64c. Gradient using 2x2 masks, T=128d. Roberts operator, T=64e. Sobel operator, T=225f. Prewitt operator, T=225
Edge detectionon noisy imagewithout smoothing
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E.G.M. Petrakis Edge Detection 18
Edge Detection with Edge Patterns
• Comparison (filtering) with 4 or 8 edge masks– each one represents an edge orientation– combine their results– the edge pattern with the max magnitude represents the
magnitude and the orientation of the edge– thresholding is still necessary!
• Optimization: apply 4 of the 8, the other 4 are symmetric • give opposite orientations and negative values of
magnitude • take the absolute value of negative edge strengths
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E.G.M. Petrakis Edge Detection 19
Prewit Edge Patterns
−−−−
111121111
−−−−
111121111
−−−−
111121
111
−−−−
111121111
−
−−−
111121111
−−−−
111121111
−−−
−
111121111
−−−−
111121111 arrow denotes
orientation of edgedetection
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E.G.M. Petrakis Edge Detection 20
Kirsch edge patterns
−−−−−
333305355
−−−−−
335305335
−−−−−
333303
555
−−−−−
355305333
−−−−−
555303333
−−
−−−
553503333
−−−
−−
533503533
−−−−−
333503553 arrow denotes
orientation of edgedetection
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E.G.M. Petrakis Edge Detection 21
Steps in Edge Detection1. Filtering removes noise and improves the performance
of edge detection– there is a trade off between edge strength and noise reduction– filtering smoothes the edge too!!
2. Edge enhancement apply a Gradient operator3. Detection keeps only edge points and eliminates false
edge points– keep points with strong edge content– above a threshold T
4. Localization computes location and orientation of edge
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E.G.M. Petrakis Edge Detection 22
6
3
4
7
4
5
5
6
6
6
93
original image
gradient
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E.G.M. Petrakis Edge Detection 23
7
5
5
6
6
6
9
6
7
6
6
6
9
Detection and Localization
thershold T= 4 keep edges points with strength > 4noise is retained
6
thershold T= 5 keep edges points with strength > 5removes noise but significant parts of the contour are lost
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E.G.M. Petrakis Edge Detection 24
Smoothing & Thresholding
• Smoothing:– high degree of smoothing (large mask, repeated
smoothing) removes noise but blurs the edges– low degree of smoothing retains noise
• Threshold selection: – large thresholds remove noise along with edge
points– low thresholds retain noise
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E.G.M. Petrakis Edge Detection 25
Second Derivative Operators
• Edge points: peaks of the first derivative – use of thresholds – detects too many edge points
• Equivalently, zero’s of the second derivative
• Combine first and second derivatives– locate peak in the first derivative and zero
crossing in the second derivative
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E.G.M. Petrakis Edge Detection 26
Combination of Derivatives
• Edge detection – smoothing by Gaussian– presence of zero crossing in second derivative
– large peak in the first derivative
[ ] 0,2 ≈∇ jifr
[ ] Thresholdjif >∇ ,r
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E.G.M. Petrakis Edge Detection 27
all points in [a,b] are edge points
edge point
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E.G.M. Petrakis Edge Detection 28
Laplacian Operator
• Two dimensional equivalent of the second derivative
• Digital approximation
( ) 2
2
2
22 ,
yf
xfyxf
∂∂
+∂∂
=∇
( ) ( ) ( ) ( )( ) ( )jifjif
jifjifjifjif,41,
1,,1,1,2
−−+++−++≈∇
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E.G.M. Petrakis Edge Detection 29
Proof
( )
],[]1,[2]2,[]),[]1,[(])1,[]2,[(
],[]1,[
],[]1,[2
2
jifjifjifjifjifjifjif
xjif
xjif
xjifjif
xG
xf x
++−+=−+−+−+
=∂
∂−
∂+∂
=∂−+∂
=∂∂
=∂∂
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E.G.M. Petrakis Edge Detection 30
Proof (2)
• Second derivative on X is centered around [i,j+1], – replace j with j-1 to center edge around [i,j]
• Similarly for Y
]1,[],[2]1,[2
2
−+−+=∂∂ jifjifjif
xf
],1[],[2],1[2
2
jifjifjifyf
−+−+=∂∂
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E.G.M. Petrakis Edge Detection 31
Laplacian Mask
• The following mask approximates the Laplacian (both x,y)
−≈∇
010141010
],[2 jif
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E.G.M. Petrakis Edge Detection 32
Response of Laplacian
• Zero’s signal the presence of an edge– trivial zeros (uniform regions) are ignored– strong response to edges, lines, points, noise!
............0
.........01000110011101111
f2∇
→
1231231
122111122011
13221121111111
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E.G.M. Petrakis Edge Detection 33
Response to Step Edges
888888
888888
888888
222222
222222
822
2222
2
2222
2
8
8
88
8
000000
-6-6-6-6-6-6
000000
000000
000000
060
0000
0
6666
6
0
0
00
0f2∇
→
edge lies between 6 and –6
• The edge point (zero crossing location) may need to be approximated to sub-pixel resolution
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E.G.M. Petrakis Edge Detection 34
Response to Ramp Edges
888888
888888
888888
222222
222222
8522
2222
2
5555
5
82
82
8282
82
000000
-3-3-3-3-3-3
000000
000000
000000
0030
3333
3
0000
0
00
00
0000
00f2∇
→
edge
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E.G.M. Petrakis Edge Detection 35
Laplacian of Gaussian
• The second derivative responds strongly to noise
• It is desired to filter the noise before edge detection
• The Laplacian of a Gaussian (LoG) combines Gaussian filtering with Laplacianfor edge detection
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E.G.M. Petrakis Edge Detection 36
One-Step LoG
• Notice that • In practice LoG is
applied in one step by filtering with
• Known also asMexican hat operator
fGfG ∗∇=∗∇ )()( 22
( )
−
−−=∇ 2
2
2
2
42
2exp
211
σσπσrrrG
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E.G.M. Petrakis Edge Detection 37
Discrete Approximation of LoG
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E.G.M. Petrakis Edge Detection 38
LoG Smoothing
• σ controls the degree of smoothing– large σ results in better noise filtering but also – blurs and displaces edges– small σ results in too many noise points and false edges– edge points are better localized using small σ
• The best size of the filter is unknown• Scale space filtering: combine results from
multiple filters with varying – resembles filtering on the retina
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E.G.M. Petrakis Edge Detection 39
LoG filtering with varying σ
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E.G.M. Petrakis Edge Detection 40
DoG Approximation of LoG
• The best engineering approximation of LoGis obtained by taking the Difference of two Gaussians (DoG) with– amplitude ratio 1:1.16
• The same mechanism operates in the retina– many filters with varying σ
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E.G.M. Petrakis Edge Detection 41
DoG: continuous line LoG: dashed line
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E.G.M. Petrakis Edge Detection 42
DoG Filtering on the Retina
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E.G.M. Petrakis Edge Detection 43
“Center-Surround” Filtering
• Filtering by cells in retina resemble the effects of DoG– the photoreceptors are connected to retinal
gaglion cells which send visual data to the brain– “Red” photoreceptors excite “ON center”
gaglion cells (generate neural signals)– “Blue” photoreceptors in its surrounding inhibit
the gaglion cell “OFF surround”