3-D Computer Vision
CSc 83020
Feature Detection and Grouping
Finding Corners
CSc 83020 3-D Computer Vision – Ioannis Stamos
What Is a Corner?
Large gradients in more than one direction.
CSc 83020 3-D Computer Vision – Ioannis Stamos
A Simple Corner Detector
  I x2
C
 I x I y
I I
I
Find eigenvalues of C
x y
2
y
1



2 
If the smaller eigenvalue is above a threshold then we have a corner.
CSc 83020 3-D Computer Vision – Ioannis Stamos
A Simple Corner Detector
  I x2
C
 I x I y
I I
I
Iy  0
 a 0
C

0
0


a 0 
C

0
a


Find eigenvalues of C
x y
2
y
1



2 
If the smaller eigenvalue is above a threshold then we have a corner.
CSc 83020 3-D Computer Vision – Ioannis Stamos
A Simple Corner Detector
CSc 83020 3-D Computer Vision – Ioannis Stamos
Sobel Operator
-1 -2 -1
S1= 0 0 0
1 2 1
Edge Magnitude =
-1
-2
-1
S2 =
2
0
0
0
1
2
1
2
S1 + S2
Edge Direction = tan-1
S1
S2
CSc 83020 3-D Computer Vision – Ioannis Stamos
Sobel in Matlab
CSc 83020 3-D Computer Vision – Ioannis Stamos
Canny Edge Detector
CSc 83020 3-D Computer Vision – Ioannis Stamos
Comparison
Sobel
CSc 83020 3-D Computer Vision – Ioannis Stamos
Canny
Canny Edge Detection
Steps:
1.
2.
Apply derivative of Gaussian
Non-maximum suppression

3.
Thin multi-pixel wide “ridges” down to single
pixel width
Linking and thresholding


Low, high edge-strength thresholds
Accept all edges over low threshold that are
connected to edge over high threshold
CSc 83020 3-D Computer Vision – Ioannis Stamos
Non-Maximum Supression
Non-maximum suppression:
Select the single maximum point across the width
of an edge.
CSc 83020 3-D Computer Vision – Ioannis Stamos
Linking to the Next Edge Point
Assume the marked
point q is an edge
point.
Take the normal to
the gradient at that
point and use this to
predict continuation
points (either r or p).
CSc 83020 3-D Computer Vision – Ioannis Stamos
Edge Hysteresis



Hysteresis: A lag or momentum factor
Idea: Maintain two thresholds khigh and klow
 Use khigh to find strong edges to start edge
chain
 Use klow to find weak edges which continue
edge chain
Typical ratio of thresholds is roughly
khigh / klow = 2
CSc 83020 3-D Computer Vision – Ioannis Stamos
Canny Edge Detection (Example)
gap is gone
Strong +
connected
weak edges
Original
image
Strong
edges
only
Weak
edges
courtesy of G. Loy
CSc 83020 3-D Computer Vision – Ioannis Stamos
Canny Edge Detection (Example)
Using Matlab with default thresholds
CSc 83020 3-D Computer Vision – Ioannis Stamos
Bridge Example Again
edge(im,’canny’)
CSc 83020 3-D Computer Vision – Ioannis Stamos
Summary: Canny Edge
Detection

Most commonly used method
Traces edges, accommodates variations in
contrast
Not a linear filter!

Problems with corners


CSc 83020 3-D Computer Vision – Ioannis Stamos
Towards Global Features
Local versus global
CSc 83020 3-D Computer Vision – Ioannis Stamos
From Edges to Lines
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Lines



Assume edge
detection
Each pixel is either
edge or not
How do we find the
line?
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Lines



Assume edge
detection
Each pixel is either
edge or not
How do we find the
line?
CSc 83020 3-D Computer Vision – Ioannis Stamos
Least Squares Fitting of Lines
y  mx  c
E
 yi  mxi  c2
i
N
xi , yi 
Minimize E
E
0
m
E
0
c
yi  mxi  c
CSc 83020 3-D Computer Vision – Ioannis Stamos
Least Squares Fitting of Lines
E
 yi  mxi  c2
i
N
Minimize E
E
0
m
E
0
c
Problem: E must be formulated carefully!
CSc 83020 3-D Computer Vision – Ioannis Stamos
Least Squares Fitting of Lines
  x sin   y cos  0
E
  xi sin  yi cos 2
N
i
xi , yi 
Minimize E
E
0

E
0

  xi sin   yi cos
CSc 83020 3-D Computer Vision – Ioannis Stamos
Hough Transform
b  m x  y
y
x
CSc 83020 3-D Computer Vision – Ioannis Stamos
Hough Transform: Quantization
y
b
x
Detecting Lines by finding maxima / clustering in parameter space
CSc 83020 3-D Computer Vision – Ioannis Stamos
m
m
Hough Transform: Algorithm

For each image point, determine




most likely line parameters b,m (direction of
gradient)
strength (magnitude of gradient)
Increment parameter counter by strength
value
Cluster in parameter space, pick local maxima
CSc 83020 3-D Computer Vision – Ioannis Stamos
Hough Transform: Results
Image
Edge detection
Hough Transform
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Lines Using the Hough
Transform
xi , yi 
  xi cos  yi sin 
CSc 83020 3-D Computer Vision – Ioannis Stamos
Algorithm




Discretize the parameter spaces ρ and θ.
Create Accumulator array A(1..R,1..T).
Set A(k,h)=0 for all k and h.
For each image edge E(i,j)=1

For h=1…T
ρ =i cosθd(h)+j sinθd (h)
 Find index k: ρd is closest to ρ
 Increment A(h,k) by one.


Find all local maxima (kp, hp) such that A (kp, hp)>τ
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Lines Using the Hough
Transform
Strong local peaks
correspond to lines.
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Lines Using the Hough
Transform
Resolution Issues
…
CSc 83020 3-D Computer Vision – Ioannis Stamos
From
Forsyth and
Ponce
CSc 83020 3-D Computer Vision – Ioannis Stamos
Summary Hough Transform

Smart counting




Local evidence for global features
Organized in a table
Careful with parameterization!
Problem: Curse of dimensionality


Works great for simple features with 3
unknowns
Will fail for complex objects
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Circles by Hough Transform
y
b0
r
a0
Equation of Circle:
(xi,yi)
x
( xi  a0 )2  ( yi  b0 )2  r 2
CSc 83020 3-D Computer Vision – Ioannis Stamos
Finding Circles by Hough Transform
y
b0
r
a0
(xi,yi)
x
( xi  a0 )2  ( yi  b0 )2  r 2
Equation of Circle:
If radius r is known:
Circles!
b
y
(xi,yi)
x
83020 3-D
Computer Vision – Ioannis Stamos
AccumulatorCSc
array
A(a,b)
a
Finding Circles by Hough Transform
y
b0
r
(xi,yi)
a0
x
If r is not known
Use accumulator array A(a,b,r)
For each (xi,yi) increment A(a,b,r) such that
( xi  a )2  ( yi  b )2  r2
CSc 83020 3-D Computer Vision – Ioannis Stamos
Using Gradient Information
Can save lot of computations!
Given: location (xi,yi)
y
Edge direction φi
x
CSc 83020 3-D Computer Vision – Ioannis Stamos
Using Gradient Information
Can save lot of computations!
Given: location (xi,yi)
y
Edge direction φi
x
Assume r is known:
(xi,yi)
y
φi
a=x-rcosφ
b=y-rsinφ
Need to increment
(a,b)
only one point
CSc 83020 3-D Computer Vision
– Ioannis
Stamos
x in
Accumulator
Array.
Hough Transform for Curves




Curve y=f(x,a)
a=[a1, … , ap] the parameters of the curve.
Limitation: size of parameter space wrt # of
parameters.
Solution: variable-resolution parameter space.
CSc 83020 3-D Computer Vision – Ioannis Stamos
Hough Transform






Pattern Matching.
More efficient than template matching.
Handles occlusion.
Finds all instances of pattern.
Handling inaccurate edge locations?
Drawbacks?
CSc 83020 3-D Computer Vision – Ioannis Stamos