1. Pixel-Based Processing
ECE 847: Digital Image Processing
Stan Birchfield
Clemson University

2. detect objects, classify fruit,
Challenge ?
original image
detect objects, classify fruit,
find banana stem
Steps:
binarize / threshold
clean up binary image
find regions corresponding to foreground objects
compute properties of regions
classify regions based on properties
S. Birchfield, Clemson Univ., ECE 847,

3. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

4. What is a digital image? 75 81 83 96 94 62 74 76 87
100 80 89 93 53 67 86 90
105 60 77 99
115
S. Birchfield, Clemson Univ., ECE 847,

5. Accessing pixels in an image
Timing Test (640 x 480 image, get/set each pixel using 2.8 GHz P4)
C / C++
Method 1: 1D array val = p[ y * width + x];
Method 2: 2D array val = p[y][x];
Method 3: pointers val = *p++;
6.6 ms
5.6 ms
0.5 ms
(even faster with SIMD operations)
Matlab
Method 1: 2D array val = im(y,x);
Method 2: parallelize im = im + constant;
21.4 ms
2.5 ms
S. Birchfield, Clemson Univ., ECE 847,

6. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

7. Types of image transformations
Graylevel transforms I’(x,y)  f( I(x,y) ) (arithmetic, logical, thresholding, histogram equalization, …)
Geometric transforms I’(x,y)  f( I(x’,y’) ) (flip, flop, rotate, scale, …)
Area-based transforms I’(x,y)  f( I(x,y), I(x+1,y+1), … ) (morphological operators, convolution)
Global transforms I’(x,y)  f( I(x’,y’), x’,y’ ) (Fourier transform, wavelet transform) A
S. Birchfield, Clemson Univ., ECE 847,

8. identity inverse contrast threshold enhancement
Graylevel transforms
identity inverse contrast threshold enhancement
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9. Geometric transforms
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10. Histograms image histogram Throws away all spatial information!
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11. Histogram equalization
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12. Running sum Example: [ 6 3 9 7 4 2]  [6 9 18 25 29 31]
Step 1: Initialize 1D array
int hist[256]
...
Step 2: Compute running sum
sum[0] = 0
for g = 1 to 255, sum[g] = sum[g-1] + hist[g]
Example:
[ ]  [ ]
S. Birchfield, Clemson Univ., ECE 847,

13. Histogram equalization
pdf
Steps:
Compute normalized histogram (PDF) int hist[256] = 0…0 float norm[256] for x, hist[ I(x) ]++ for g = 0 to 255, norm[g] = hist[g] / npixels
Compute cumulative histogram (CDF) cum[0] = norm[0] for g = 1 to 255, cum[g] = cum[g-1] + norm[g]
Transform for x, I(x) = 255*cum[ I(x) ]
equal area
cdf
equal spacing
new gray level
running sum
old gray level
S. Birchfield, Clemson Univ., ECE 847,

14. Histogram equalization *
PDF p
desired
PDF p’
equal area
equal area a a’
a’+d
new gray level k’
CDF
By definition,
d { q
d { a’
equal
spacing
new gray level k’
old gray level k
where p is the PDF and q is the CDF
S. Birchfield, Clemson Univ., ECE 847,

15. Histogram equalization *p
desired
PDF
PDF
equal area p’
1/(L-1) a b
L-1 a’
a’+d
L-1
old gray level k
new gray level k’
scaled CDF
L-1
q·(L-1)
d { a’
new gray level k’
L-1
old gray level k
S. Birchfield, Clemson Univ., ECE 847,

16. Histogram equalization *p
desired
PDF
PDF
equal area p’
1/(L-1) a1 b1 a2 b2
a1’
a1’+d
a2’
a2’+d
L-1
L-1
new gray level k’
scaled CDF
d {
L-1
a2’
q·(L-1)
d {
a1’
new gray level k’
L-1
old gray level k
S. Birchfield, Clemson Univ., ECE 847,

17. Integral image Integral image is a 2D running sum S(x,y) = SS I(x,y)
To compute, S(x,y) = I(x,y) S(x-1,y-1) + S(x-1,y) + S(x,y-1)
To use, V(l,t,r,b) = S(l,t) + S(r,b) - S(l,b) - S(r,t) Returns sum of values inside rectangle
Note: Sum of values in any rectangle can be computed in constant time!
S. Birchfield, Clemson Univ., ECE 847,

18. Thresholding Valley separates light from dark How to find it?
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19. A simple thresholding algorithm
Repeat until convergence:
T  ½ (m1 + m2)
T. Ridler and S. Calvard, Picture Thresholding Using an Iterative Selection Method, IEEE Transactions on Systems, Man and Cybernetics, 8(8): , August 1978.
S. Birchfield, Clemson Univ., ECE 847,

20. Ridler-Calvard algorithm
S. Birchfield, Clemson Univ., ECE 847,

21. percentage of pixels in first group variance of pixels in first group
Otsu’s method
Threshold t splits image into two groups
Calculate within-group variance of each group
Search over all t to minimize total within-group variance; moment calculations make search efficient
percentage of pixels in first group
variance of pixels in first group
S. Birchfield, Clemson Univ., ECE 847,

22. Otsu’s method
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23. Local Entropy Thresholding (LET) *
Compute co-occurrence matrix
Threshold s divides matrix into four quadrants
Pick s that maximizes HBB(s)+HFF(s)
Uses spatial information
S. Birchfield, Clemson Univ., ECE 847,

24. Otsu vs. LET * image histogram Otsu LET
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25. Adaptive thresholding
background
image
histogram
from Gonzalez and Woods, p. 597
S. Birchfield, Clemson Univ., ECE 847,

26. Double thresholding (hysteresis)
graylevel
threshold too high:
misses part of object
object 1
object 2
threshold too low:
captures noise
noise
pixel
Algorithm:
Threshold with high value
Threshold with low value, retaining only the pixels that are contiguous with existing ones
S. Birchfield, Clemson Univ., ECE 847,

27. Double thresholding example
1. Use high threshold
to get seed pixel
low threshold
high threshold
3. Repeat
2. Floodfill
using seed pixel
S. Birchfield, Clemson Univ., ECE 847,

28. Double thresholding example
image
low threshold retains some background
combined using floodfill on low threshold with seeds from high threshold
high threshold removes some foreground
S. Birchfield, Clemson Univ., ECE 847,

29. We will explain Floodfill later
Double thresholding
We will explain Floodfill later
S. Birchfield, Clemson Univ., ECE 847,

30. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

31. Binary image as a set =
S. Birchfield, Clemson Univ., ECE 847,

32. Set operations Fundamental operators Other operators DeMorgan’s Laws
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33. Minkowski operators Minkowski addition: Minkowski subtraction:
S. Birchfield, Clemson Univ., ECE 847,

34. Add each element of B to each element of A1
Minkowski addition
Add each element of B to each element of A1
S. Birchfield, Clemson Univ., ECE 847,

35. Minkowski subtraction
Intersection of translating
A2 to every point in B
Two examples:
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36. Minkowski addition: “Center-out” approach
(“center-out” is more intuitive)
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37. Minkowski subtraction: “Center-out” approach
(“center-out” is more intuitive)
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38. Morphological operations
morphology – study of form or shape
Mathematical morphology – grew out of research at Fontainebleau in 1964
We will deal only with binary images
Two fundamental operations: Dilation and Erosion
(same as Mink. add.)
(same as Mink. sub. after reflection)
B is the structuring element (SE)
S. Birchfield, Clemson Univ., ECE 847,

39. Dilation Algorithm (center-out): Looks like nonlinear convolution
Flip B horizontally and vertically (but often B is symmetric, so nothing is changed)
Move B around
Anywhere they intersect, turn center pixel on
Looks like nonlinear convolution
S. Birchfield, Clemson Univ., ECE 847,

40. Erosion Algorithm (center-out): Duality b/w dilation and erosion:
Move B around
If there is not complete overlap, turn center pixel off
Duality b/w dilation
and erosion:
S. Birchfield, Clemson Univ., ECE 847,

41. Common structuring elements
Usually SE is either B or B8
Greatly simplifies algorithm (and erosion is same as Min. sub.)
S. Birchfield, Clemson Univ., ECE 847,

42. Dilation example Let B be 3x3 with all 1s. for y B A for x
if ( in(x-1, y) || in(x-1, y-1) || in(x-1, y+1) || in(x+1, y+1) )
out(x, y) = 1
else
out(x, y) = 0 B A
S. Birchfield, Clemson Univ., ECE 847,

43. Erosion example Let B be 3x3 with all 1s. for y B A for x
if ( in(x-1, y) && in(x-1, y-1) && in(x-1, y+1) && in(x+1, y+1) )
out(x, y) = 1
else
out(x, y) = 0 B A
S. Birchfield, Clemson Univ., ECE 847,

44. Why does dilation require flipping the structuring element?
Simple example: A B
Without flipping
With flipping  
Result is flipped!
Fixed!
S. Birchfield, Clemson Univ., ECE 847,

45. Closing and Opening Dilation fills gaps. Erosion removes noise
Combine them:
First dilate, then erode (closing)
First erode, then dilate (opening)
Closing/opening retain object size
Repeated applications do nothing
Duality between closing and opening
S. Birchfield, Clemson Univ., ECE 847,

46. Closing: dilate, then erode Opening: erode, then dilate
Closing and Opening
Closing: dilate, then erode
Opening: erode, then dilate
S. Birchfield, Clemson Univ., ECE 847,

47. Morphology Example thresholded difference open then close erode dilate
S. Birchfield, Clemson Univ., ECE 847,

48. Accessing out-of-bounds pixels
If kernel is near boundary of image, some pixels of kernel will be out of bounds (OOB)
What to do?
Do not use OOB pixels (change kernel size near boundary)
Do not compute output for pixels near boundary
Extend image function:
zero pad – OOB pixels have value of zero
replicate – OOB pixels have value of closest pixel
reflect – pixel values are extended by mirror reflection at border
wrap – pixel values are extended by repeating
extrapolate – pixel values are extended by extrapolating function near boundary
No solution is best
Problem also occurs with convolution
S. Birchfield, Clemson Univ., ECE 847,

49. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

50. Binary images Each pixel is ON or OFF
We will say that foreground pixels are ON, background pixels are OFF
In computer, each pixel has value of 0 or 1.
We adopt convention that 0=OFF, 1=ON.
Usually 0=black and 1=white, but sometimes display is reversed
S. Birchfield, Clemson Univ., ECE 847,

51. Neighbors 4-neighbors N4 8-neighbors N8 diagonal neighbors ND
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52. Adjacency 4-adjacency: 8-adjacency: m-adjacency: or and Example: m 8 4q p q or
and
Example: m 8 4
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53. Adjacency
Use different connectedness for foreground and background to avoid inconsistency
foreground: D8
background: D4
dumbbell
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54. Connectivity, Regions and Boundaries
path from p to q is sequence of adjacent pixels
p and q are connected in subset S if path exists between them containing only pixels in S
connected component of p is set of pixels in S that are connected to p
S is a region if it is a connected set
boundary of S is the set of pixel in S that have one or more neighbors not in S
S. Birchfield, Clemson Univ., ECE 847,

55. Boundary definitions binary image outer region boundary outer hole
complete
boundary
inner
region
boundary
inner
hole
boundary
inner
complete
boundary
S. Birchfield, Clemson Univ., ECE 847,

56. Floodfill Fill a region with a new color Start with a seed point
All pixels are colored if they have the same color as the seed and are connected to the seed via pixels that have the same color as the seed
seed pixel
S. Birchfield, Clemson Univ., ECE 847,

57. Floodfill * Repeated applications of dilation can fill a region
image A
structuring
element B
complement Ac
... but terribly inefficient
S. Birchfield, Clemson Univ., ECE 847,

58. Floodfill Efficient implementation uses a stack called the frontier.
Function image = Floodfill(seed_point,
image,
new_color)
old_color = image(seed_point)
if (old_color == new_color) return
frontier.Push(seed_point)
image(seed_point) = newcolor
while (NOT frontier.IsEmpty())
q = frontier.Pop();
for each neighbor r of q,
if image(r) == old_color,
frontier.Push(r)
image(r) = new_color
return image
frontier:
S. Birchfield, Clemson Univ., ECE 847,

59. Floodfill
S. Birchfield, Clemson Univ., ECE 847,

60. Floodfill This code performs the same logic,
but does not change the pixels of the input image
S. Birchfield, Clemson Univ., ECE 847,

61. Revisit: Double thresholding
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62. Connected components
Recall that connected component of p is set of pixels in S that are connected to p
Can solve the problem by repeated applications of Floodfill – assigning a new label to each new region
Slightly more efficient implementation involves two passes through the image (union-find)
S. Birchfield, Clemson Univ., ECE 847,

63. Connected components using floodfill
S. Birchfield, Clemson Univ., ECE 847,

64. Classical union-find connected component labeling algorithm
scan left-to-right, top-to-bottom U L P
I(P) == I(U) ?
4-neighbor
algorithm:
YES NO
C(P)=C(U) and set Equiv(U,L)
C(P)=C(L)
C(P)=C(U)
C(P)=n++
I(P) == I(L) ?
4-neighbor
mask:
8-neighbor
mask:
S. Birchfield, Clemson Univ., ECE 847,

65. Classical connected components: first pass1 1 1 1 1 2 2 2 2 3 3 3 2 4 3 5 5 2 4
image
components 1 2 3 4 5
equivalence table
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66. Classical connected components: traverse table1 1 1 1 1 2 2 2 2 3 3 3 2 4 3 5 5 2 4
image
components 1 2 3 4 5
equivalence table
S. Birchfield, Clemson Univ., ECE 847,

67. Classical connected components: second pass1 1 1 1 1 3 3 3 4 3 4
image
components 1 2 3 4 5
equivalence table
S. Birchfield, Clemson Univ., ECE 847,

68. Another example
from

69. Connected components Function labels = ConnectedComponents(image)
// first pass
for y=0 to height-1,
for x=0 to width-1,
pix = image(x,y)
if pix == image(x-1,y) && pix == image(x,y-1)
labels(x,y) = labels(x,y-1)
SetEquivalence(labels(x-1,y), labels(x,y-1))
else if pix == image(x-1,y)
labels(x,y) = labels(x-1,y)
else if pix == image(x,y-1)
else
labels(x,y) = next_label++
// second pass
labels(x,y) = GetEquivalentLabel( labels(x,y) )
S. Birchfield, Clemson Univ., ECE 847,

70. Connected components Function SetEquivalence(a, b)
aa = GetEquivalentLabel(a)
bb = GetEquivalentLabel(b)
if aa > bb
equiv[aa] = bb
else if aa < bb
equiv[bb] = aa
Function b = GetEquivalentLabel(a)
if a == equiv[a] return a
else
equiv[a] = GetEquivalentLabel(equiv[a])
return equiv[a]
S. Birchfield, Clemson Univ., ECE 847,

71. S. Birchfield, Clemson Univ. , ECE 847, http://www. ces. clemson

72. Connected components quantized image components
S. Birchfield, Clemson Univ., ECE 847,

73. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

74. Perimeter Perimeter of binary region:
Dilate, then subtract
Erode, then subtract
(Which is better?)
This only gives the pixels as binary image
To get ordered list of contiguous pixels, use Wall Follow algorithm
S. Birchfield, Clemson Univ., ECE 847,

75. Wall following Algorithm: Start at ON pixel
Move in any direction until end of region reached
Turn right
Repeat:
If pixel to the left is ON,
Turn left and move forward
Else if pixel in front is OFF,
Else
move forward
- This traverses clockwise, using 4-connectedness
(other versions are possible)
- Can be used to compute perimeter
S. Birchfield, Clemson Univ., ECE 847,

76. Distance functions D is a distance function (metric) if
Common distance functions:
shape of each?
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77. Other distances Mahalanobis (weight differently along each axis)
Hausdorff (distance b/w point sets) A and B are subsets of S
S. Birchfield, Clemson Univ., ECE 847,

78. Length along path Freeman formula c = 0 (overestimates)
# of isothetic moves
# of diagonal moves
Freeman formula
c = 0
(overestimates)
Pythagorean theorem
c = 1
(underestimates)
Kimura’s method
c = ½ (good compromise)
3 methods compared:
Freeman:
L = 16.5
No=8
Pythagorean:
L = 15.2
Example:
Kimura:
L = 15.7
Nd=6
circle with radius = 10
* isothetic (blue)
* diagonal (orange)
True arc length:
L = 10*p/2 = 15.7
Nd=6
S. Birchfield, Clemson Univ., ECE 847,

79. Chamfering How to approximate “correct” distance (Euclidean)?
Along a path,
Let m be number of isothetic moves (horizontal or vertical)
Let n be number of diagonal moves
Define chamfer distance between p and q as dab = min {am+bn} over all possible paths
Chamfer distance is metric if a, b > 0
If b > 2a, then diagonal moves are ignored
Special cases:
a=1, b=infinity (Manhattan)
a=1, b=1 (chessboard)
a=1, b=sqrt(2) (another reasonable choice)
a=1, b=1.351 (best approximation to Euclidean)
a=3, b=4 (best integer approximation, scaled by factor of 3)
[G. Borgefors, Distance transformations in digital images, CVGIP, 1986]
S. Birchfield, Clemson Univ., ECE 847,

80. Chamfering (cont.)
Chamfering is process of computing distance to each foreground pixel
Two-pass algorithm
Traverse top-to-bottom, left-to-right
Traverse bottom-to-top, right-to-left
In each case, compute minimum distance
foreground pixels L: R: L: R:
4-connectivity
8-connectivity
S. Birchfield, Clemson Univ., ECE 847,

81. “? :” means “if-else” (C conditional expression operator)
Chamfering (cont.)
Algorithm (for D4 using Manhattan distance):
Function labels = Chamfer(image)
// first pass
for y=0 to height-1,
for x=0 to width-1,
d(x,y) = image(x,y) ? : min(infinity, 1+d(x-1,y), 1+d(x,y-1))
// second pass
for y=height-1 to 0 step -1,
for x=width-1 to 0 step -1,
d(x,y) = image(x,y) ? : min(d(x,y), 1+d(x+1,y), 1+d(x,y+1))
Note:
“? :” means “if-else” (C conditional expression operator)
S. Birchfield, Clemson Univ., ECE 847,

82. Uses of chamfering
Find center of biggest part of shape (need to compute signed distance)
Shape matching
Useful for watershed thresholding (we will see this later)
centroid
largest chamfer
value
S. Birchfield, Clemson Univ., ECE 847,

83. Outline Pixel-based image transformations (thresholding, …)
Removing noise from binary image (morphological operators)
Labeling regions (floodfill, connected components)
Computing distance in an image (chamfer distance, …)
Region properties (area and boundary properties)
S. Birchfield, Clemson Univ., ECE 847,

84. } Descriptors We will focus on regions Region Boundary Euler number
convex hull
skeleton
moments
compactness
eccentricity (elongatedness)
Boundary
chain code
signature
r-q plot
s-y plot
boundary segments
shape number
Fourier descriptors
We will focus
on regions
S. Birchfield, Clemson Univ., ECE 847,

85. Euler number E = C- H 1-1=0 1-2=-1 1-0=1 2-0=2 Measure of topology
components
holes
1-1=0
1-2=-1
1-0=1
2-0=2
Measure of topology
S. Birchfield, Clemson Univ., ECE 847,

86. Euler number Euler number computed as (N1-N2-2N3)4, where
N1 is the number of 2x2 patterns with 1 in either corner, 0 elsewhere (4 possible patterns)
N2 is the number of 2x2 patterns with 0 in either corner , 0 elsewhere (4 possible patterns)
N3 is the number of 2x2 checkerboard patterns (2 possible patterns)
Explained in W.K. Pratt, Digital Image Processing, 1991
S. Birchfield, Clemson Univ., ECE 847,

87. Convex hull Convex – any line connecting two points in R is also in R
Convex hull – the minimal convex set containing R
convex
not convex
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88. should skeleton touch boundary?
Skeletons
should skeleton touch boundary?
Point on skeleton is equidistant to >= 2 boundary points (without crossing)
Or, equivalently,
where wavefronts (grassfires) meet (shock graph)
centers of maximal balls
Blum’s Medial Axis Transform (MAT)
Skeleton is highly sensitive to noise
S. Birchfield, Clemson Univ., ECE 847,

89. Skeleton examples
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90. Skeletonization by thinning
First compute MAT
Then delete points such that
do not remove endpoints
do not break connectivity
do not excessively erode region
S. Birchfield, Clemson Univ., ECE 847,

91. Moments in 1D The pth moment of a function f: The pth central moment:
What are ?
centroid
(area, 0, variance, skew, kurtosis) m2 is also moment of inertia about axis passing through centroid
S. Birchfield, Clemson Univ., ECE 847,

92. Moments in 2D The pqth moment of a 2D region: The pqth central moment:
can be
grayscale or
binary (more typical)
(area)
(variance in x)
(variance in y)
S. Birchfield, Clemson Univ., ECE 847,

93. Moments in 2D (cont.) The pqth normalized central moment:
Hu moments (derived from these) are invariant to
translation
rotation
scaling
skew (at least one of them is)
S. Birchfield, Clemson Univ., ECE 847,

94. Moments example Simple 5x5 binary image: m00 = m10 = m01 = m11 = m20 =
12.0
23.0
21.0
37.0
65.0
63.0
m00 =
m10 =
m01 =
m11 =
m20 =
m02 =
12.0
0.0
-3.3
20.9
26.3
S. Birchfield, Clemson Univ., ECE 847,

95. (and normalized / Hu) are computed without touching the pixels again!
Computing moments
One pass through image
to compute
regular moments
Central moments
(and normalized / Hu) are computed without
touching the pixels again!
(Typically, f is a binarized image containing a single non-zero region)
S. Birchfield, Clemson Univ., ECE 847,

96. Projections Project image onto x, y, or diagonal
Projection offers compact representation of image
First-order moments of image are equal to first-order moments of projection
Second-order moments require diagonal projection
S. Birchfield, Clemson Univ., ECE 847,

97. Compactness Define compactness as
Related to eccentricity (elongatedness)
Define compactness as
(area = # of pixels,
perimeter = length along boundary)
compactness = 1
compactness < 1
(Note: discretization effects of computing perimeter are severe)
Alternative: Use
S. Birchfield, Clemson Univ., ECE 847,

98. Eccentricity Approximate region with ellipse eccentricity = a / a =
Formula given by PCA
S. Birchfield, Clemson Univ., ECE 847,

99. Principal components analysis (PCA)
Given n points in d dimensional space
Goals:
fit ellipsoid to data
reduce dimensionality
Ellipsoid specified by
center (given by mean of data)
orientation of axes (given by eigenvectors of covariance matrix of data)
length of axes (given by square root of eigenvalues of covariance matrix of data)
S. Birchfield, Clemson Univ., ECE 847,

100. PCA (cont.) n points in d dimensional space mean covariance
eigenvalues and eigenvectors
matrix form
factorize covariance matrix
S. Birchfield, Clemson Univ., ECE 847,

101. PCA (cont.) reconstruct x linear transformation
but elements of y are decreasingly important
project to k-dimensional space
linear transformation
data uncorrelated!
vectors ei form an orthonormal basis for space
S. Birchfield, Clemson Univ., ECE 847,

102. PCA (cont.)
S. Birchfield, Clemson Univ., ECE 847,

103. PCA example Simple 5x5 binary image: m = C = C = PLPT = major axis =
minor axis =
(2.0, 2.0)
(1/9) * [ ]
[-10 12]
[ ] [ ] [ ]
[ ] [ ] [ ]
e1√l1 = [ ]T
e2√l2 = [ ]T e1 e2 l1 l2
S. Birchfield, Clemson Univ., ECE 847,

104. Best fitting ellipse The elliptical binary region in image
bounded by the ellipse
has the following second-order moments
To be an ellipse,
(which is always
true for a
covariance matrix)  
Note:
S. Birchfield, Clemson Univ., ECE 847,

105. Eccentricity and Direction
Covariance matrix of region
Eigenvalues solve det(C-lI)=0)
Eccentricity ranges from 0 (circle) to 1 (line)
Direction
Eigenvectors
Hint: Use atan2(y,x),
passing numerator
and denominator
as separate arguments
(clockwise angle from positive x-axis)
S. Birchfield, Clemson Univ., ECE 847,

106. } Eigenfaces mean face eigenvectors
N x N image is a point in an N2 dimensional space
N2 x N2 Covariance matrix captures relationships
between face images in ensemble
Turk and Pentland, 1991
S. Birchfield, Clemson Univ., ECE 847,

107. Fruit classification (high threshold) original image (low threshold)
Note: double threshold
and connected components
can be computed simultaneously:
scan high threshold image
if pixel is 1, then floodfill using low threshold image
(connected components)
(double threshold)
mij
mij
(moments)
(classification)
(banana stem)
S. Birchfield, Clemson Univ., ECE 847,

108. Sample results fruit1.pgm fruit2.pgm
S. Birchfield, Clemson Univ., ECE 847,