Post on 13-Dec-2015
Edge Detection and Geometric Primitive Extraction
Jinxiang Chai
Outline
Edge detection
Geometric primitive extraction
Edge Detection
What are edges in this image?
Edge Detection
What are edges in this image?
Origin of Edges
• Edges are caused by a variety of factors
depth discontinuity
surface color discontinuity
illumination discontinuity
surface normal discontinuity
Edge Detection
One of the most important vision problems - really easy for human - really difficult for computers - fundamental for computer vision (object recognition,
3D reconstruction, etc.)
How to tell a pixel is on the edge?
Characterizing edges• An edge is a place of rapid change in
the image intensity function
imageintensity function
(along horizontal scanline) first derivative
edges are related to derivative
Gradient
A vector variable - Direction of the maximum growth of the function
- Magnitude of the growth
- Perpendicular to the edge direction
2
)/
/tan(
),(
),(
22
xf
yfa
y
f
x
fyxf
yy
fxx
fyxf
),( yxf
Gradients
How to Calculate Gradient?
How to compute ?x
f
Intensity
Pixel column
The Good ole’ Taylor Series
Subtracting the second from the first we obtain
...)(''2
1)(')()( 2 xfhxhfxfhxf
...)(''2
1)(')()( 2 xfhxhfxfhxf
or…
)(2
)()()(' 2hO
h
hxfhxfxf
Discrete Gradient Estimation
For discrete functions, we can use the first order approximation of the gradient
where h corresponds to the step size
h
hxfhxfxf
2
)()()('
Discrete Gradient Estimation
For discrete functions, we can use the first order approximation of the gradient
where h corresponds to the step size
For our purposes, h corresponds to the width of 1 pixel =>
h
hxfhxfxf
2
)()()('
2
)1,()1,(),(2
),1(),1(),(
yxIyxI
y
yxI
yxIyxI
x
yxI
Discrete Gradient Estimation
So how can we compute the image gradient efficiently?
- Using our good old friend convolution!
1
0
1
*
101*
Iy
I
Ix
I
- Dropped off the “divide by 2” for speed considerations.- This only scales the gradient.
Taking the Discrete Derivative
] 1 0 1[
abs()
Basic Edge Detection Step 1
INPUT IMAGE
1) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
But isn’t edge detection susceptible to noise?
Basic Edge Detection Steps 1-2
INPUT IMAGE
2) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
1) NoiseSmoothing
16/
121
242
121
Discrete Gradient EstimationRemember that the gradient is a vector and we have calculated the coefficients in the x and y directions at each point in the image
After convolving, we get the magnitude of the gradient from at each point (pixel) from
In practice, we often sum the absolute values of the components for computational efficiency
22
),(
y
I
x
IyxG
Basic Edge Detection (cont’d)
INPUT IMAGE
1) NoiseSmoothing
2) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
2
122 ),(),(
|),(|
y
yxI
x
yxIyxI
“GRADIENT” IMAGE
16/
121
242
121
Basic Edge Detection (cont’d)
INPUT IMAGE
1) NoiseSmoothing
2) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
2
122 ),(),(
|),(|
y
yxI
x
yxIyxI
“GRADIENT” IMAGE
16/
121
242
121
What does the gradient image mean?
Basic Edge Detection (cont’d)
INPUT IMAGE
1) NoiseSmoothing
2) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
2
122 ),(),(
|),(|
y
yxI
x
yxIyxI
“GRADIENT” IMAGE
16/
121
242
121
What does the gradient image mean?
- Magnitude of intensity changes around each pixel
Discrete Gradient EstimationSo how do we segment the edges from the rest of the scene?
- Thresholding!
The Effects of ThresholdingWhile edge features are independent of illumination, the edge strength is not!
Results from threshold values of 50 and 100
Basic Edge Detection Summary
INPUT IMAGE
1) NoiseSmoothing
EDGE IMAGE
2) EdgeEnhancement
Horizontal [-1 0 1]
Vertical [-1 0 1]T
),( yxI
x
yxI
),(
y
yxI
),(
2
122 ),(),(
|),(|
y
yxI
x
yxIyxI
“GRADIENT” IMAGE
3)Threshold
16/
121
242
121
The effects of Filtering Noise
Threshold20
Gaussian SmoothingUnsmoothed Edges
Threshold50
Sobel Edge DetectionOne of the dominant edge detection schemes uses the Sobel
operators
Convolving each of these with the original image generates horizontal and vertical gradient images that are combined as before
101
202
101
VSobel
121
000
121
HSobel
Sobel Edge Detection
One of the dominant edge detection schemes uses the Sobel operators
101
202
101
VSobel
121
000
121
HSobel
- Can be approximated as a derivative of Gaussian
- First Gaussian smoothing and then compute derivatives
Ix
G
x
IG
)(
Sobel Edge Detection
One of the dominant edge detection schemes uses the Sobel operators
101
202
101
VSobel
121
000
121
HSobel
- Can be approximated as a derivative of Gaussian
- First Gaussian smoothing and then compute derivatives
Ix
G
x
IG
)( Why?
Sobel Edge Detection
One of the dominant edge detection schemes uses the Sobel operators
101
202
101
VSobel
121
000
121
HSobel
- Can be approximated as a derivative of Gaussian
- First Gaussian smoothing and then compute derivatives
- In practice we may still need to smooth for noise
Ix
G
x
IG
)(
Robert and Prewitt Edge Detectors
The Prewitt is similar to the Sobel, but uses a different kernel
Roberts was an early edge detector kernel
101
101
101
VP
111
000
111
HP
11
111R
11
112R
Sobel Edge Detector
Java Applet
Second Derivative Edge Detector
Sobel Operator can produce thick edges; ideally we are looking for infinitely thin boundaries
Second Derivative Edge Detector
Sobel Operator can produce thick edges; ideally we are looking for infinitely thin boundaries
An alternative approach is to look for local extrema in the first derivative
A peak in the first derivative corresponds what in the second derivative?
Second Derivative Edge Detector
Sobel Operator can produce thick edges; ideally we are looking for infinitely thin boundaries
An alternative approach is to look for local extrema in the first derivative
A peak in the first derivative corresponds what in the second derivative?
Localization with Laplacian
An equivalent measure of the second derivative in 2D is the Laplacian:
Numerically, we approximate the Laplacian using the following filtering kernel:
Zeros crossings of the filter corresponds to positions of maximum gradient.
- can be used for detecting edges - might be sensitive to noise - need to filter the image
Laplacian of Gaussian
In the LoG, there are two methods which are mathematically equivalent:
- Convolve the image with a gaussian smoothing filter and compute the Laplacian of the result
- Convolve the image with the linear filter that is the Laplacian of the Gaussian filter
The Laplacian of Gaussian Kernel
),()()),(( 22 yxIGyxIG
2
22
24
222222 2
yx
eyx
x
G
x
GG
The Laplacian of Gaussian Filter Kernel
0.1
4.1
To avoid detection of insignificant edges, only the zero crossings whose corresponding first derivative is above some threshold, are selected as edge point.
Laplacian Edge Detector
Java applet: Click here
Edge Detection
• Matlab functions for edge detection: - BW = edge(I,'sobel') % Sobel detector
- BW = edge(I,'prewitt') % Prewitt detector
- BW = edge(I,'roberts') % Robert detector
- BW = edge(I,'log') % Laplacian of Gaussian detector
- BW = edge(I,'canny') % canny detector
Outline
Edge detection
Geometric primitive extraction
Finding straight lines
• One solution: try many possible lines and see how many points each line passes through
• Hough transform provides a fast way to do this
Hough transform• An early type of voting scheme
• General outline: – Discretize parameter space into bins– For each feature point in the image, put a vote in every bin in the
parameter space that could have generated this point– Find bins that have the most votes
P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959
Image space Hough parameter space
Parameter space representation
• A line in the image corresponds to a point in Hough space
Image space Hough parameter space
Source: S. Seitz
Parameter space representation• What does a point (x0, y0) in the image
space map to in the Hough space?
Image space Hough parameter space
Parameter space representation• What does a point (x0, y0) in the image
space map to in the Hough space?– Answer: the solutions of b = –x0m + y0
– This is a line in Hough spaceImage space Hough parameter space
Parameter space representation• Where is the line that contains both (x0,
y0) and (x1, y1)?
Image space Hough parameter space
(x0, y0)
(x1, y1)
b = –x1m + y1
Parameter space representation• Where is the line that contains both (x0,
y0) and (x1, y1)?– It is the intersection of the lines b = –x0m + y0 and
b = –x1m + y1
Image space Hough parameter space
(x0, y0)
(x1, y1)
b = –x1m + y1
• Problems with the (m,b) space:– Unbounded parameter domain– Vertical lines require infinite m
Parameter space representation
• Problems with the (m,b) space:– Unbounded parameter domain– Vertical lines require infinite m
• Alternative: polar representation
Parameter space representation
sincos yx
Each point will add a sinusoid in the (,) parameter space
Algorithm outline• Initialize accumulator H
to all zeros• For each edge point (x,y)
in the imageFor θ = 0 to 180 ρ = x cos θ + y sin θ H(θ, ρ) = H(θ, ρ) + 1
endend
• Find the value(s) of (θ, ρ) where H(θ, ρ) is a local maximum– The detected line in the image is given by
ρ = x cos θ + y sin θ
ρ
θ
features votes
Basic illustration
Square Circle
Other shapes
Several lines
A more complicated image
http://ostatic.com/files/images/ss_hough.jpg
features votes
Effect of noise
features votes
Effect of noise
• Peak gets fuzzy and hard to locate
Dealing with noise• Choose a good grid / discretization
– Too coarse: large votes obtained when too many different lines correspond to a single bucket
– Too fine: miss lines because some points that are not exactly collinear cast votes for different buckets
• Increment neighboring bins (smoothing in accumulator array)
• Try to get rid of irrelevant features – Take only edge points with significant gradient
magnitude
Incorporating image gradients• Recall: when we detect an
edge point, we also know its gradient direction
• But this means that the line is uniquely determined!
• Modified Hough transform:
• For each edge point (x,y) θ = gradient orientation at (x,y)ρ = x cos θ + y sin θH(θ, ρ) = H(θ, ρ) + 1
end
Line detection
• Matlab functions for line detection:
[H, theta, rho] = hough(BW)
Hough transform for Other Shapes
• How many dimensions will the parameter space have?
Hough transform for Other Shapes
• How many dimensions will the parameter space have?
(x0,y0, r) (x0,y0, a,b,theta)
Things to remember
• Edge detector - First order method: smooth-
>gradients->thresholding
- Second order method: zero crossings for LOG
• Hough transform = points vote for shape parameters