Chapter 5: Edge Detection

Edge DetectionINEL 6088 - Fall 2011 - M. Toledo

Tuesday, September 13, 11

2

ideal

~real


Edge point – point at which the local intensity changes significantly

Edge fragment – edge (point) and orientation

Edge detector – produces a set of edges (points or fragments)

Edge linking – orders a list of edges

Contour – list of edges

Edge following – searching the (input) image to determine contours


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Real image

Edge detector produces:• correct edges• false edges (false positives)• can miss edges (false negatives)


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Edge Detection using Gradient

G[f(x,y)]=[Gx Gy]T

Gx = δf/δx ≈ f[i,j+1]-f[i,j] « gradient at [i, j+½] Gy = δf/δy ≈ f[i+1,j]-f[i,j] « gradient at [i+½, j]

To get the gradient at the same point [i+½, j+½], use:

Simple way


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Steps:• filtering• enhancement• detection• sometimes, localization

Roberts: P5 = |P5 - P9| + |P8 – P6|

i

j

Sobel: C= mult. by 2

Prewitt: Sobel with c=1


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Filtered with 7x7 gaussian filter


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unfiltered


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Filtered, noisy


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Unfiltered, noisy


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Edge Detectors that use the Second Derivative

Change in sign

indicates edgeTwo ways:

• Laplacian• Second directional derivative


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Laplacian

Combined to get

Also used:


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Second directional derivative

n Second derivative computed in the direction of the gradient

n Implement using the formula:


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Laplacian of Gaussian (LoG)

Problem with Laplacian and second-derivative-operator: • very sensitive to noise• Small peaks in first derivative produce zero-crossing in the second derivative.

Solution: Filter out noise before edge enhancement

n Smoothing: Gaussian smoothingn Enhancement: Second-derivative edge enhancementn Detection: zero-crossing in the second derivative with a corresponding

large peak (i.e. above some threshold) in the first derivativen If desired, use linear interpolation to locate the edge with sub-pixel

resolution


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Mexican-hatOperator

The output of the LoG operator, h(x,y) is given by:

h(x, y) = r2[g(x, y) ? f(x, y)]

h(x, y) = [r2g(x, y)] ? f(x, y)

r2g(x, y) =

x

2+ y

2 � 2�

2

�

4exp

✓�x

2+ y

2

2�

2

◆


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Two ways of doing LoG:• Gaussian smoothing followed by laplacian• Convolution of image with a linear filter that is the laplacian of a gaussian filter

To obtain real edges, it might be necessary to combine information from filters of different sizes. The problem of combining edges obtained from different size operators still remains.


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LoG ResultsSmoothing causes blurring large σ: better noise filtering but more blurring – can cause edge merging

By applying filters of different sizes and analyzing the results, better edge detection can be accomplished.


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An alternative approach is to fit a function to the image and then detect

the edges in the function…


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Facet model: fit a function only in the local neighborhood of each pixel.


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Example: bicubic polynomial


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Masks for computing the coefficients of the bicubic approximation.


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The angle may be chosen to be the angle of the approximating

plane:

sin ✓ =

k3pk22 + k23

cos ✓ =

k2pk22 + k23

f

00✓ (x0, y0) = 2(3k7 cos

2✓ + 2k8 sin ✓ cos ✓ + k9 sin

2✓)x0

+2(k8 cos2✓ + 2k9 sin ✓ cos ✓ + 3k10 sin

2✓)y0

+2(k4 cos2✓ + k5 sin ✓ cos ✓ + k6 sin

2✓)

f

0✓(x, y) =

@f

@x

cos ✓ +

@f

@y

sin ✓

f

00✓ (x, y) =

@

2f

@x

2cos

2✓ + 2

@

2f

@x@y

cos ✓ sin ✓ +

@

2f

@y

2sin

2✓


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There is an edge at (x0,y0) if for some ρ, | ρ |< ρ0 where ρ0 is the length of the side of a pixel,

fθ”(x0,y0; ρ) = 0 and

fθ’(x0,y0; ρ) ≠ 0

x0 = ⇢ cos ✓

y0 = ⇢ sin ✓

f

00✓ (x0, y0) = 6(k10 sin

3✓ + k9 sin

2✓ cos ✓ + k8 sin ✓ cos

2✓ + k7 cos

3✓)⇢

+2(k6 sin2✓ + k5 sin ✓ cos ✓ + k4 cos

2✓)

= A⇢+B


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Canny Edge detectionn Let I[i,j] be the image. First use Gaussian smoothing:

S[i,j]=G[i,j;σ]*I[i,j]n Find approx. gradient usin 2×2 first-diff. approx.

q x direction:

P[i,j]=(S[i,j+1]-S[i,j]+S[i+1,j+1]-S[i+1,j])/2q y direction:

Q[i,j]=(S[i,j]-S[i+1,j]+S[i,j+1]-S[i+1,j+1])/2


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Canny edge detection (cont)

n Magnitude and orientation: rectangular to polar conversion:q M[I,j]=√(P[i,j]2+Q[i,j]2) q θ[i,j] = arctan(Q[i,j]/P[i,j])

n Non-maxima Suppressionq Thins the ridges of gradient magnitude to one pixelq Passes a 3x3 neighborhood across the magnitude

array M[I,j] and replace pixels with 0 if not greater than neighbors


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Possible gradient orientations are partitioned into the following sectors for

non-maxima suppression

876

504

321

Replace pixel 0 with value 0 if it is not bigger than neighboring pixels in the (quantized) direction of the

gradient


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Canny Edge detection (cont)

n Thresholdingq Apply two threshold t1 and t2, with t2 = 2*t1q This produces two threshold images T1 & T2q T2 will have fewer edges, but might have gapsq When the end of a contour in T2 is reached, look into

T1 at the 8-neighboring locations to see if there are further edge points that can be linked to the contour.


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Test Image (256x256)


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Canny with 7x7 Gaussian smoothing + gradient approx + NMS


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Canny with 31x31 Gaussian smoothing


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Edge Detector Performance Evaluationn Probability of false edgesn Probability of missing edgesn Edge angle estimation errorn Mean square distance between estimated and real edgesn Tolerance to distorted edges


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Pratt´s Figure of Merit

detected edges

ideal edges

distance betweenactual and ideal

edges

constant used to penalize displaced edges

FM =

1

max(IA, II)

IAX

i=1

1

1 + ↵d2i


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Sequential Methods

Edge following: scan image looking for strong edge extend the edge in the proper direction by looking at neighboring edges link if directions are compatible look at large neighborhood to fill in missing edges


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Chapter 5: Edge Detection