1. 2.3.3 Line and curve detection
Lines and curves are important features in computer
vision because they define the contours of objects in
the image.
Take the result of edge detection as input, see if we
can link up the edge pixels and detect the existence
of lines and curves.

2. Line detection - Hough transform (HT)
A line y = mx + n can be represented by a parameter
pair (m, n) which is one point in the parameter space.
You can rewrite the line equation as n = -mx + y. For
a point p = [x, y]T, m and n can vary and form a line in
the parameter space, representing all possible lines
passing through p.

3. A line containing N edge points p1, …, pN is identified
in the parameter space by the intersection of N lines
associated with p1, …, pN.
m
p2 n
p1
n’
image m’
parameter space

4. As m and n vary, the corresponding entry in the
parameter space will be increased by 1. Ideally, all
entries covered by the lines in the parameter space will
have the count of 1 except that the entry (m’, n’) has the
count of N.
Therefore, HT is a voting algorithm. To detect line is
to search for peak in the parameter space.
There is a problem using the parameter space
formed by m, n!

5. [0, 0] j
θ 360°
ρ
.
θ .
.
i 10°
0°
0 10 ... M2 + N2
ρ
discrete image I[i, j] parameter space
In implementation, we usually adopt the polar
representation ρ = j cosθ - i sinθ.

6. Line detection algorithm:
• input edge detection result (M x N binary image)
• quantize ρ and θ to create the parameter space,
ρ ∈[0 , M 2 + N 2 ],θ ∈ [ 0 ,2π ]
• initialize all entries of the parameter space to zero
• for each edge point [i, j], increment by 1 each entry
that satisfies the polar representation, e.g. for a
specific quantized value of θ, find the closest
quantized value of ρ
• find the local maxima (ρ, θ), each has the count >
user-defined threshold τ

7. Hough transform can be implemented using the
MATLAB function hough.
Advantages:
• It is possible to detect the line even some points are
missing.
• False edge points are unlikely to contribute to the
same set of parameters and generate a false peak in
the parameter space.

8. Disadvantages:
• You can generalize HT to detect curve y = f(x, a),
where a = [a1, …, aP]. The parameter space is P-
dimensional. However, search time increases rapidly
with the number of parameters.
• Non-target shapes can produce spurious peaks in
parameter space, e.g. low-curvature curves can be
detected as lines.

9. Another approach to detect line is to fit edge pixels to a
line model - model fitting approach
For a generic line ax + by + c = 0, or , find the
parameter vector [a, b, c]T which results in a line going
as near as possible to each edge point. A solution is
reached when the distance between the line and all the
edge points is minimum – least squares problem.

10. Ellipse fitting
Many objects contain circular shapes, which usually
appear as ellipses in image. Therefore, ellipse
detectors are useful tools for computer vision.
According to the model fitting approach, find the best
ellipse that can fit the edge points.

11. Assume that the set of edge points pi = [xi, yi]T,
i = 1, ...,N belong to a single arc of ellipse
xTa = ax2+bxy+cy2+dx+ey+f = 0
where x = [x2, xy, y2, x, y, 1]T and a = [a, b, c, d, e, f]T.
Find the parameter vector a, associated to the ellipse
which fits p1…pN best in the least squares sense
N 2
min ∑ x i a
T
a
i =1

12. To avoid the trivial solution a = 0 and force the
solution to be an ellipse
0 0 − 2 0 0 0
0 1 0 0 0 0
 
2 T

T −2 0 0 0 0 0
b − 4 ac = −1 = a Ca = a  a
0 0 0 0 0 0
0 0 0 0 0 0
 
0
 0 0 0 0 0 
where C is called the constraint matrix. C is rank-
deficient.

13. The problem becomes min a T X T Xa = min a T Sa
a a
where X is called the design matrix
 x2 x y y2 x y 1
 1 1 1 1 1 1 
 x2 x y y2 x y 1
X = 2 2 2 2 2 2 
 L 
 2 
xN x y y2 x y 1
 N N N N N 
and S is called the scatter matrix S = XTX.

14. Using Lagrange multipliers, the problem
(generalized eigenvalue problem) can be solved by
Sa = λCa.
Ellipse detection algorithm:
• normalize the edge points
• build X
• compute S
• build C
• compute eigenvalues, a is the eigenvector
corresponding to the only negative λ
Use the MATLAB function eig to compute eigenvalues
and eigenvectors.

15. Original image Edge detection result
Ellipse detection result

16. 2.3.4 Color
Humans use color information to distinguish objects.

17. Color model
Color can be represented in 3 bytes – one byte for each
of Red, Green and Blue (RGB). The encoding of an
arbitrary color in the visible spectrum can be made by
combining the encoding of three primary colors RGB.
RGB color model is additive and corresponds well to
monitors.
(255, 0, 0) red
(0, 255, 0) green
(255, 255, 0) yellow
(0, 0, 0) black
(255, 255, 255) white

18. Some computer vision algorithms can perform better
using normalized colors.
R
normalized red r=
R+G+B
G
normalized green g =
R+G+B
normalized blue B
b=
R+G+B

19. Another color model is called Hue-Saturation-Intensity
(HSI). It has the advantage that color information
(chromaticity) represented by H and S, is separated from
intensity. Hue describes the tone of the color. Saturation
provides a measure of its purity.
R+G+B 2R − G − B
I= cosH =
3 2 (R − G) 2 + (R − B)(G − B)
3
S = 1− min(R, G, B)
R+G+B
Normalize RGB components to 1. Subtract H from
360° when B/I > G/I. H is not defined when S is zero.
S is not defined when I is zero.

20. Conversion between RGB and HSI can be implemented
using the MATLAB functions rgb2hsv and hsv2rgb.
An RGB color image in MATLAB corresponds to a
3D matrix of dimensions M x N x 3.
image = imread(filename);
[height, width, color] = size(image);

21. Histogram
A histogram counts the number of pixels of each kind
(e.g. grey level, color).
Create a histogram by reading the image pixels one by
one and incrementing the appropriate bin of the
histogram. For color image, you can create 3
histograms, one for each of the 3 color components,
e.g. RGB. Each histogram may have 256 bins.

23. Histogram can be used to determine how similar a test
image T to a reference image R.
Assume both histograms hT and hR have K bins.
K
intersection = ∑ min(h T [i], h R [i])
i =1
K
∑ min(h T [i], h R [i])
match = i =1
K
∑h
i =1
R [i]
The match value indicates how much color content of
the reference image is present in the test image. It is
relatively invariant to translation, rotation and scale
changes of image.

24. Sometimes, you may want to compute a dissimilarity
measure
K
distance = ∑ h T [i] − h R [i]
i =1
Noise filtering techniques and edge detectors can be
extended to color images under the componentwise
paradigm.

25. 2.3.5 Texture
Texture feature can be a powerful descriptor of an
image. It gives us information about the spatial
arrangement of the colors or intensities in an image.
Same histograms – 50% black, 50% white

26. Texture is commonly found in natural scenes and man-
made objects.
However, there is not a universally agreed upon
definition of texture.

27. There are two main approaches to describe texture
properties:
Structural approach – texture is a set of primitive texture
elements (texels) in some regular or repeated
relationship.
Statistical approach – texture is a quantitative measure
of the arrangement of colors or intensities in a region.
The first approach can work well for man-made, regular
patterns. The second approach is more general and easier
to compute and is used more often in practice.

28. Edgeness
The number of edge pixels in a given region indicates
the busyness of that region.
p
Edgeness per unit area =
N
where |p| is the number of edge pixels in a region of N
pixels.

29. To include both gradient magnitude and gradient
orientation
Edge-based histogram = (hmag, horient)
where hmag is the normalized histogram of gradient
magnitude of that region, and horient is the normalized
histogram of gradient orientation of that region.

30. Co-occurrence matrices
A co-occurrence matrix is a 2D array C in which both
the rows and the columns represent a set of image
values (intensities, colors). The value Cd[i, j] indicates
how many times value i co-occurs with value j in some
designated spatial relationship. The spatial relationship
is represented by a vector d = (dr, dc), indicating the
displacement of the pixel having value of j from the
pixel having value of i by dr rows and dc columns.

31. j
1 1 0 0 0 1 2
1 1 0 0 0 4 0 2
0 0 2 2 i 1 2 2 0
0 0 2 2 2 0 0 2
image I C(0,1)
It is common to normalize the co-occurrence matrix
and so each entry can be considered as a probability.
C d [i, j]
N d [i, j] =
∑∑ C d [i, j]
i j

32. Numeric features can be computed from the co-
occurrence matrix that can be used to represent the
texture more compactly.
Energy = ∑∑ N d [i, j]
2
i j
Entropy = −∑∑ N d [i, j]log 2 N d [i, j]
i j
Contrast = ∑∑ (i - j) 2 N d [i, j]
i j
N d [i, j]
Homogeneity = ∑∑
i j 1+ i - j

33. ∑∑ (i − µ )(j − µ )N
i j
i j d [i, j]
Correlation =
σiσ j
where µi, µj are the means and σi, σj are the standard
deviations of the row and column sums
N d [i] = ∑ N d [i, j]
j
N d [j] = ∑ N d [i, j]
i

34. Summary
♦ Hough transform
♦ ellipse fitting
♦ color models
♦ histogram
♦ texture features