Lect0 1 digital imaging processing part 1.ppt

chapter 1

1. CS589-04 Digital Image ProcessingCS589-04 Digital Image Processing
Lecture 1 Lecture 1
Introduction & FundamentalsIntroduction & Fundamentals
Spring 2008Spring 2008

2. Weeks 1 & 2 2
Introduction to the courseIntroduction to the course
► Article Reading and Project
Medical image analysis (MRI/PET/CT/X-ray tumor
detection/classification)
Face, fingerprint, and other object recognition
Image and/or video compression
Image segmentation and/or denoising
Digital image/video watermarking/steganography
and detection
Whatever you’re interested …

3. Weeks 1 & 2 3
Introduction to the courseIntroduction to the course
► Evaluation of article reading and project
Report
Article reading
— Submit a survey of the articles you read and the list of the
articles
Project
— Submit an article including introduction, methods,
experiments, results, and conclusions
— Submit the project code, the readme document, and some
testing samples (images, videos, etc.) for validation
Presentation

4. Weeks 1 & 2 4
Journals & Conferences Journals & Conferences
in Image Processingin Image Processing
► Journals:
— IEEE T IMAGE PROCESSING
— IEEE T MEDICAL IMAGING
— INTL J COMP. VISION
— IEEE T PATTERN ANALYSIS MACHINE INTELLIGENCE
— PATTERN RECOGNITION
— COMP. VISION AND IMAGE UNDERSTANDING
— IMAGE AND VISION COMPUTING
… …
► Conferences:
— CVPR: Comp. Vision and Pattern Recognition
— ICCV: Intl Conf on Computer Vision
— ACM Multimedia
— ICIP
— SPIE
— ECCV: European Conf on Computer Vision
— CAIP: Intl Conf on Comp. Analysis of Images and Patterns
… …

5. ►What is an image?What is an image?
►A digital image can be considered as a A digital image can be considered as a
discrete representation of data possessing discrete representation of data possessing
both both spatial (layout) spatial (layout) and and intensity (color) intensity (color)
information. information.
►consider treating an image as a consider treating an image as a
multidimensional signalmultidimensional signal..
Weeks 1 & 2 5

6. Weeks 1 & 2 6
IntroductionIntroduction
► What is Digital Image Processing?

Digital Image
— a two-dimensional function
x and y are spatial coordinates
The amplitude of f is called intensity or gray level at the point (x, y)
Digital Image Processing
— process digital images by means of computer, it covers low-, mid-, and high-level
processes
low-level: inputs and outputs are images
mid-level: outputs are attributes extracted from input images
high-level: an ensemble of recognition of individual objects
Pixel
— the elements of a digital image
( , )f x y

7. Weeks 1 & 2 7
Origins of Digital Image Origins of Digital Image
ProcessingProcessing


Sent by submarine cable
between London and
New York, the
transportation time was
reduced to less than
three hours from more
than a week

8. Weeks 1 & 2 8
Origins of Digital Image Origins of Digital Image
ProcessingProcessing

9. Weeks 1 & 2 9
Sources for ImagesSources for Images
► Electromagnetic (EM) energy spectrumElectromagnetic (EM) energy spectrum
► AcousticAcoustic
► UltrasonicUltrasonic
► ElectronicElectronic
► Synthetic images produced by computerSynthetic images produced by computer

10. Weeks 1 & 2 10
Electromagnetic (EM) energy spectrumElectromagnetic (EM) energy spectrum
Major uses
Gamma-ray imaging: nuclear medicine and astronomical observations
X-rays: medical diagnostics, industry, and astronomy, etc.
Ultraviolet: lithography, industrial inspection, microscopy, lasers, biological imaging,
and astronomical observations
Visible and infrared bands: light microscopy, astronomy, remote sensing, industry,
and law enforcement
Microwave band: radar
Radio band: medicine (such as MRI) and astronomy

11. Weeks 1 & 2 11
Examples: Gama-Ray Imaging Examples: Gama-Ray Imaging

12. Weeks 1 & 2 12
Examples: X-Ray Imaging Examples: X-Ray Imaging

13. Weeks 1 & 2 13
Examples: Ultraviolet Imaging Examples: Ultraviolet Imaging

14. Weeks 1 & 2 14
Examples: Light Microscopy Imaging Examples: Light Microscopy Imaging

15. Weeks 1 & 2 15
Examples: Visual and Infrared Imaging Examples: Visual and Infrared Imaging

16. Weeks 1 & 2 16
Examples: Visual and Infrared Imaging Examples: Visual and Infrared Imaging

17. Weeks 1 & 2 17
Examples: Infrared Satellite Imaging Examples: Infrared Satellite Imaging


USA 1993USA 2003

18. Weeks 1 & 2 18
Examples: Infrared Satellite Imaging Examples: Infrared Satellite Imaging

19. Weeks 1 & 2 19
Examples: Automated Visual InspectionExamples: Automated Visual Inspection

20. Weeks 1 & 2 20
Examples: Automated Visual InspectionExamples: Automated Visual Inspection


The area in which
the imaging
system detected
the plate
Results of
automated
reading of the
plate content
by the system

21. Weeks 1 & 2 21
Example of Radar ImageExample of Radar Image

22. Weeks 1 & 2 22
Examples: MRI (Radio Band)Examples: MRI (Radio Band)

23. Weeks 1 & 2 23
Examples: Ultrasound ImagingExamples: Ultrasound Imaging

24. Weeks 1 & 2 24
Fundamental Steps in DIPFundamental Steps in DIP


Result is more
suitable than
the original
Improving the
appearance
Extracting image
components
Partition an image
into its constituent
parts or objects
Represent image for
computer processing

25. Weeks 1 & 2 25
Light and EM SpectrumLight and EM Spectrum
c , : Planck's constant.E h h

26. Weeks 1 & 2 26
Light and EM SpectrumLight and EM Spectrum
►The colors that humans perceive in an object The colors that humans perceive in an object
are determined by the nature of the light are determined by the nature of the light
reflected from the object.reflected from the object.

e.g. green objects reflect light with wavelengths
primarily in the 500 to 570 nm range while absorbing
most of the energy at other wavelength

27. Weeks 1 & 2 27
Light and EM SpectrumLight and EM Spectrum
►Monochromatic light: void of color
Intensity is the only attribute, from black to white
Monochromatic images are referred to as gray-scale
images
►Chromatic light bands: 0.43 to 0.79 um
The quality of a chromatic light source:
Radiance: total amount of energy
Luminance (lm): the amount of energy an observer perceives
from a light source
Brightness: a subjective descriptor of light perception that is
impossible to measure. It embodies the achromatic notion of
intensity and one of the key factors in describing color sensation.

28. Weeks 1 & 2 28
Image AcquisitionImage Acquisition
Transform
illumination
energy into
digital images

29. Weeks 1 & 2 29
Image Acquisition Using a Single SensorImage Acquisition Using a Single Sensor

30. Weeks 1 & 2 30
Image Acquisition Using Sensor StripsImage Acquisition Using Sensor Strips

31. Weeks 1 & 2 31
Image Acquisition ProcessImage Acquisition Process

32. Weeks 1 & 2 32
A Simple Image Formation ModelA Simple Image Formation Model
( , ) ( , ) ( , )
( , ): intensity at the point ( , )
( , ): illumination at the point ( , )
(the amount of source illumination incident on the scene)
( , ): reflectance/transmissivity
f x y i x y r x y
f x y x y
i x y x y
r x y
 
at the point ( , )
(the amount of illumination reflected/transmitted by the object)
where 0 < ( , ) < and 0 < ( , ) < 1
x y
i x y r x y

33. Weeks 1 & 2 33
Some Typical Ranges of illuminationSome Typical Ranges of illumination
► IlluminationIllumination
Lumen — A unit of light flow or luminous flux
Lumen per square meter (lm/m
2
) — The metric unit of measure
for illuminance of a surface
On a clear day, the sun may produce in excess of 90,000 lm/m
2
of
illumination on the surface of the Earth
On a cloudy day, the sun may produce less than 10,000 lm/m
2
of
illumination on the surface of the Earth
On a clear evening, the moon yields about 0.1 lm/m
2
of illumination
The typical illumination level in a commercial office is about 1000 lm/m
2

34. Weeks 1 & 2 34
Some Typical Ranges of ReflectanceSome Typical Ranges of Reflectance
► ReflectanceReflectance
0.01 for black velvet
0.65 for stainless steel
0.80 for flat-white wall paint
0.90 for silver-plated metal
0.93 for snow

35. Weeks 1 & 2 35
Image Sampling and QuantizationImage Sampling and Quantization
Digitizing the
coordinate
values
Digitizing the
amplitude
values

36. Weeks 1 & 2 36
Image Sampling and QuantizationImage Sampling and Quantization

37. Weeks 1 & 2 37
Representing Digital ImagesRepresenting Digital Images

38. Weeks 1 & 2 38
Representing Digital ImagesRepresenting Digital Images
►The representation of an M×N numerical
array as

(0,0) (0,1) ... (0, 1)
(1,0) (1,1) ... (1, 1)
( , )
... ... ... ...
( 1,0) ( 1,1) ... ( 1, 1)
f f f N
f f f N
f x y
f M f M f M N
 
 

 
 
 
    

39. Weeks 1 & 2 39
Representing Digital ImagesRepresenting Digital Images
►The representation of an M×N numerical
array as

0,0 0,1 0, 1
1,0 1,1 1, 1
1,0 1,1 1, 1
...
...
... ... ... ...
...
N
N
M M M N
a a a
a a a
A
a a a


   
 
 
 

 
 
 

40. Weeks 1 & 2 40
Representing Digital ImagesRepresenting Digital Images
►The representation of an M×N numerical
array in MATLAB

(1,1) (1,2) ... (1, )
(2,1) (2,2) ... (2, )
( , )
... ... ... ...
( ,1) ( ,2) ... ( , )
f f f N
f f f N
f x y
f M f M f M N
 
 
 
 
 
 

41. Weeks 1 & 2 41
Representing Digital ImagesRepresenting Digital Images
► Discrete intensity interval [0, L-1], L=2
k
► The number b of bits required to store a M × N
digitized image
b = M × N × k

42. Weeks 1 & 2 42
Representing Digital ImagesRepresenting Digital Images

43. Weeks 1 & 2 43
Spatial and Intensity ResolutionSpatial and Intensity Resolution
►Spatial resolution
— A measure of the smallest discernible detail in an
image
— stated with line pairs per unit distance, dots (pixels) per
unit distance, dots per inch (dpi)
►Intensity resolution
— The smallest discernible change in intensity level
— stated with 8 bits, 12 bits, 16 bits, etc.

44. Weeks 1 & 2 44
Spatial and Intensity ResolutionSpatial and Intensity Resolution

45. Weeks 1 & 2 45
Spatial and Intensity ResolutionSpatial and Intensity Resolution

46. Weeks 1 & 2 46
Spatial and Intensity ResolutionSpatial and Intensity Resolution

47. Weeks 1 & 2 47
Image InterpolationImage Interpolation
►Interpolation — Process of using known data
to estimate unknown values
e.g., zooming, shrinking, rotating, and geometric correction
►Interpolation (sometimes called resampling)
— an imaging method to increase (or decrease) the
number of pixels in a digital image.
Some digital cameras use interpolation to produce a larger image
than the sensor captured or to create digital zoom
http://www.dpreview.com/learn/?/key=interpolation

48. Weeks 1 & 2 48
Image Interpolation: Image Interpolation:
Nearest Neighbor InterpolationNearest Neighbor Interpolation
f
1(x
2,y
2) =
f(round(x
2), round(y
2))
=f(x
1,y
1)
f(x
1,y
1)
f
1
(x
3
,y
3
) =
f(round(x
3
), round(y
3
))
=f(x
1
,y
1
)

49. Weeks 1 & 2 49
Image Interpolation: Image Interpolation:
Bilinear InterpolationBilinear Interpolation
2
( , )
(1 )(1 ) (, ) (1 ) ( 1, )
(1 ) (, 1) ( 1, 1)
( ), ( ), , .
f x y
a b f l k a b f l k
a b f l k ab f l k
l floor x k floor y a x l b y k
     
     
     
   
   
(x,y)

50. Weeks 1 & 2 50
Image Interpolation: Image Interpolation:
Bicubic InterpolationBicubic Interpolation
3 3
3
0 0
(, )
i j
ij
i j
f xy axy
 

►The intensity value assigned to point (x,y) is obtained by
the following equation

►The sixteen coefficients are determined by using the
sixteen nearest neighbors.
http://en.wikipedia.org/wiki/Bicubic_interpolation

51. Weeks 1 & 2 51
Examples: InterpolationExamples: Interpolation

52. Weeks 1 & 2 52
Examples: InterpolationExamples: Interpolation

53. Weeks 1 & 2 53
Examples: InterpolationExamples: Interpolation

54. Weeks 1 & 2 54
Examples: InterpolationExamples: Interpolation

55. Weeks 1 & 2 55
Examples: InterpolationExamples: Interpolation

56. Weeks 1 & 2 56
Examples: InterpolationExamples: Interpolation

57. Weeks 1 & 2 57
Examples: InterpolationExamples: Interpolation

58. Weeks 1 & 2 58
Examples: InterpolationExamples: Interpolation

59. Weeks 1 & 2 59
Basic Relationships Between Pixels
►Neighborhood
►Adjacency
►Connectivity
►Paths
►Regions and boundaries

60. Weeks 1 & 2 60
Basic Relationships Between Pixels
►Neighbors of a pixel p at coordinates (x,y)

4-neighbors of p, denoted by N
4
(p):
(x-1, y), (x+1, y), (x,y-1), and (x, y+1).

4 diagonal neighbors of p, denoted by N
D(p):
(x-1, y-1), (x+1, y+1), (x+1,y-1), and (x-1, y+1).

8 neighbors of p, denoted N
8
(p)
N
8(p) = N
4(p) U N
D(p)

61. Weeks 1 & 2 61
Basic Relationships Between Pixels
►Adjacency
Let V be the set of intensity values
4-adjacency: Two pixels p and q with values from V are
4-adjacent if q is in the set N
4
(p).
8-adjacency: Two pixels p and q with values from V are
8-adjacent if q is in the set N
8
(p).

62. Weeks 1 & 2 62
Basic Relationships Between Pixels
►Adjacency
Let V be the set of intensity values
m-adjacency: Two pixels p and q with values from V
are m-adjacent if

(i) q is in the set N
4(p), or
(ii) q is in the set N
D
(p) and the set N
4
(p) ∩ N
4
(p) has no pixels whose
values are from V.

63. Weeks 1 & 2 63
Basic Relationships Between Pixels
►Path

A (digital) path (or curve) from pixel p with coordinates (x
0, y
0) to pixel
q with coordinates (x
n, y
n) is a sequence of distinct pixels with
coordinates

(x
0, y
0), (x
1, y
1), …, (x
n, y
n)
Where (x
i
, y
i
) and (x
i-1
, y
i-1
) are adjacent for 1 i n.
≤ ≤
Here n is the length of the path.

If (x
0, y
0) = (x
n, y
n), the path is closed path.
We can define 4-, 8-, and m-paths based on the type of adjacency
used.

64. Weeks 1 & 2 64
Examples: Adjacency and Path
0 1 1 0 1 1 0 1 10 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 00 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 10 0 1 0 0 1 0 0 1
V = {1, 2}

65. Weeks 1 & 2 65
Examples: Adjacency and Path
0 1 1 0 1 1 0 1 10 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 00 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 10 0 1 0 0 1 0 0 1
V = {1, 2}
8-adjacent

66. Weeks 1 & 2 66
Examples: Adjacency and Path
0 1 1 0 1 1 0 1 10 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 00 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 10 0 1 0 0 1 0 0 1
V = {1, 2}
8-adjacent m-adjacent

67. Weeks 1 & 2 67
Examples: Adjacency and Path
001,11,1 1 11,21,2 1 11,31,3 0 1 1 0 1 1 0 1 1 0 1 1
002,12,1 2 22,22,2 0 02,32,3 0 2 0 0 2 0 0 2 0 0 2 0
003,13,1 0 03,23,2 1 13,33,3 0 0 1 0 0 1 0 0 1 0 0 1
V = {1, 2}
8-adjacent m-adjacent
The 8-path from (1,3) to (3,3):
(i)(1,3), (1,2), (2,2), (3,3)
(ii)(1,3), (2,2), (3,3)
The m-path from (1,3) to (3,3):
(1,3), (1,2), (2,2), (3,3)

68. Weeks 1 & 2 68
Basic Relationships Between Pixels
►Connected in S
Let S represent a subset of pixels in an image. Two
pixels p with coordinates (x
0, y
0) and q with coordinates
(x
n, y
n) are said to be connected in S if there exists a
path
(x
0, y
0), (x
1, y
1), …, (x
n, y
n)
Where
,0 ,( , )
i i
i i n x y S   

69. Weeks 1 & 2 69
Basic Relationships Between Pixels
Let S represent a subset of pixels in an image
►For every pixel p in S, the set of pixels in S that are connected to p
is called a connected component of S.
►If S has only one connected component, then S is called Connected
Set.
►We call R a region of the image if R is a connected set
►Two regions, R
i and R
j are said to be adjacent if their union forms a
connected set.
►Regions that are not to be adjacent are said to be disjoint.

70. Weeks 1 & 2 70
Basic Relationships Between Pixels
►Boundary (or border)
The boundary of the region R is the set of pixels in the region that
have one or more neighbors that are not in R.
If R happens to be an entire image, then its boundary is defined as
the set of pixels in the first and last rows and columns of the image.
►Foreground and background
An image contains K disjoint regions, R
k, k = 1, 2, …, K. Let R
u denote
the union of all the K regions, and let (R
u
)
c
denote its complement.

All the points in R
u
is called foreground;
All the points in (R
u
)
c

is called background.

71. Weeks 1 & 2 71
Question 1
►In the following arrangement of pixels, are the two
regions (of 1s) adjacent? (if 8-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Region 1
Region 2

72. Weeks 1 & 2 72
Question 2
►In the following arrangement of pixels, are the two
parts (of 1s) adjacent? (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Part 1
Part 2

73. Weeks 1 & 2 73
►In the following arrangement of pixels, the two
regions (of 1s) are disjoint (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Region 1
Region 2

74. Weeks 1 & 2 74
►In the following arrangement of pixels, the two
regions (of 1s) are disjoint (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
foreground
background

75. Weeks 1 & 2 75
Question 3
►In the following arrangement of pixels, the circled
point is part of the boundary of the 1-valued pixels if
8-adjacency is used, true or false?
0 0 0 0 0
0 1 1 0 0
0 1 1 0 0
0 1 1 1 0
0 1 1 1 0
0 0 0 0 0

76. Weeks 1 & 2 76
Question 4
►In the following arrangement of pixels, the circled
point is part of the boundary of the 1-valued pixels if
4-adjacency is used, true or false?
0 0 0 0 0
0 1 1 0 0
0 1 1 0 0
0 1 1 1 0
0 1 1 1 0
0 0 0 0 0

77. Weeks 1 & 2 77
Distance Measures
►Given pixels p, q and z with coordinates (x, y), (s, t),
(u, v) respectively, the distance function D has
following properties:
a.D(p, q)
≥
0 [D(p, q) = 0, iff p = q]
b.D(p, q) = D(q, p)
c.D(p, z)
≤
D(p, q) + D(q, z)

78. Weeks 1 & 2 78
Distance Measures
The following are the different Distance measures:
a. Euclidean Distance :
D
e
(p, q) = [(x-s)
2
+ (y-t)
2
]
1/2
b. City Block Distance:
D
4(p, q) = |x-s| + |y-t|
c. Chess Board Distance:
D
8(p, q) = max(|x-s|, |y-t|)

79. Weeks 1 & 2 79
Question 5
►In the following arrangement of pixels, what’s the
value of the chessboard distance between the
circled two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

80. Weeks 1 & 2 80
Question 6
►In the following arrangement of pixels, what’s the
value of the city-block distance between the circled
two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

81. Weeks 1 & 2 81
Question 7
►In the following arrangement of pixels, what’s the
value of the length of the m-path between the
circled two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

82. Weeks 1 & 2 82
Question 8
►In the following arrangement of pixels, what’s the
value of the length of the m-path between the
circled two points?
0 0 0 0 0
0 0 1 1 0
0 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

83. Weeks 1 & 2 83
Introduction to Mathematical Operations in Introduction to Mathematical Operations in
DIPDIP
►Array vs. Matrix Operation

11 12
21 22
b b
B
b b
 

 
 
11 12
21 22
a a
A
a a
 

 
 
11 11 12 21 11 12 12 22
21 11 22 21 21 12 22 22
*
a b a b a b a b
A B
a b a b a b a b
  

 
 
 
11 11 12 12
21 21 22 22
.*
a b a b
A B
a b a b
 

 
 
Array product
Matrix product
Array
product
operator
Matrix
product
operator

84. Weeks 1 & 2 84
Introduction to Mathematical Operations in Introduction to Mathematical Operations in
DIPDIP
►Linear vs. Nonlinear Operation

H is said to be a linear operator;
H is said to be a nonlinear operator if it does not meet the
above qualification.
 ( , ) ( , )H f x y g x y
Additivity
Homogeneity
 
 
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
i i j j
i i j j
i i j j
i i j j
H a f x y a f x y
H a f x y H a f x y
aH f x y a H f x y
ag x y a g x y
  
 
  
 
  
 
 

85. Weeks 1 & 2 85
Arithmetic OperationsArithmetic Operations
►Arithmetic operations between images are array
operations. The four arithmetic operations are
denoted as
s(x,y) = f(x,y) + g(x,y)
d(x,y) = f(x,y) – g(x,y)
p(x,y) = f(x,y) × g(x,y)
v(x,y) = f(x,y) ÷ g(x,y)

86. Weeks 1 & 2 86
Example: Addition of Noisy Images for Noise ReductionExample: Addition of Noisy Images for Noise Reduction
Noiseless image: f(x,y)
Noise: n(x,y) (at every pair of coordinates (x,y), the noise is uncorrelated
and has zero average value)
Corrupted image: g(x,y)
g(x,y) = f(x,y) + n(x,y)
Reducing the noise by adding a set of noisy images,
{g
i
(x,y)}

1
1
( , ) ( , )
K
i
i
g x y g x y
K



87. Weeks 1 & 2 87
Example: Addition of Noisy Images for Noise ReductionExample: Addition of Noisy Images for Noise Reduction

 
 
1
1
1
1
( , ) ( , )
1
( , ) ( , )
1
( , ) ( , )
( , )
K
i
i
K
i
i
K
i
i
E g x y E g x y
K
E f x y n x y
K
f x y E n x y
K
f x y



 
 
 
 
  
 
 
   
 




1
1
( , ) ( , )
K
i
i
g x y g x y
K


2
( , )
1
( , )
1
1
( , )
1
2
2 2
( , )
1
g x y K
g x y
i
K
i
K
n x y
i
K
i
n x y
K
 
 





 

88. Weeks 1 & 2 88
Example: Addition of Noisy Images for Noise ReductionExample: Addition of Noisy Images for Noise Reduction
►In astronomy, imaging under very low light levels
frequently causes sensor noise to render single
images virtually useless for analysis.
►In astronomical observations, similar sensors for
noise reduction by observing the same scene over
long periods of time. Image averaging is then used
to reduce the noise.

89. Weeks 1 & 2 89

90. Weeks 1 & 2 90
An Example of Image Subtraction: Mask Mode An Example of Image Subtraction: Mask Mode
RadiographyRadiography
Mask h(x,y): an X-ray image of a region of a patient’s body
Live images f(x,y): X-ray images captured at TV rates after injection
of the contrast medium
Enhanced detail g(x,y)
g(x,y) = f(x,y) - h(x,y)
The procedure gives a movie showing how the contrast medium
propagates through the various arteries in the area being
observed.

91. Weeks 1 & 2 91

92. Weeks 1 & 2 92
An Example of Image MultiplicationAn Example of Image Multiplication

93. Weeks 1 & 2 93
Set and Logical OperationsSet and Logical Operations

94. Weeks 1 & 2 94
Set and Logical OperationsSet and Logical Operations
► Let A be the elements of a gray-scale image
The elements of A are triplets of the form (x, y, z),
where x and y are spatial coordinates and z denotes the
intensity at the point (x, y).

► The complement of A is denoted A
c


{( , , )|( , , ) }
2 1; is the number of intensity bits used to represent
c
k
A x y K z x y z A
K k z
  
 
{( , , )| ( , )}A x y z z f x y 

95. Weeks 1 & 2 95
Set and Logical OperationsSet and Logical Operations
► The union of two gray-scale images (sets) A and B is
defined as the set
{max( , )| , }
z
A B ab a Ab B   

96. Weeks 1 & 2 96
Set and Logical OperationsSet and Logical Operations

97. Weeks 1 & 2 97
Set and Logical OperationsSet and Logical Operations

98. Weeks 1 & 2 98
Spatial OperationsSpatial Operations
► Single-pixel operations
Alter the values of an image’s pixels based on the intensity.
e.g.,

( )s T z

99. Weeks 1 & 2 99
Spatial OperationsSpatial Operations
► Neighborhood operations
The value of this pixel is determined
by a specified operation involving
the pixels in the input image with
coordinates in S
xy

100. Weeks 1 & 2 100
Spatial OperationsSpatial Operations
► Neighborhood operations

101. Weeks 1 & 2 101
Geometric Spatial TransformationsGeometric Spatial Transformations
► Geometric transformation (rubber-sheet
transformation)
— A spatial transformation of coordinates
— intensity interpolation that assigns intensity values to the spatially
transformed pixels.
► Affine transform
( , ) {( , )}x y T vw
  
11 12
21 22
31 32
0
1 1 0
1
t t
x y v w t t
t t
 
 

 
 
 

102. Weeks 1 & 2 102

103. Weeks 1 & 2 103
Intensity Assignment Intensity Assignment
► Forward Mapping
It’s possible that two or more pixels can be transformed to the
same location in the output image.
► Inverse Mapping
The nearest input pixels to determine the intensity of the output
pixel value.
Inverse mappings are more efficient to implement than forward
mappings.
( , ) {( , )}x y T vw
1
( , ) {( , )}v w T x y



104. Weeks 1 & 2 104
Example: Image Rotation and Intensity Example: Image Rotation and Intensity
InterpolationInterpolation

105. Weeks 1 & 2 105
Image RegistrationImage Registration
► Input and output images are available but the
transformation function is unknown.
Goal: estimate the transformation function and use it to
register the two images.
► One of the principal approaches for image registration
is to use tie points (also called control points)
 The corresponding points are known precisely in the
input and output (reference) images.

106. Weeks 1 & 2 106
Image RegistrationImage Registration
►A simple model based on bilinear approximation:

1 2 3 4
5 6 7 8
Where ( , ) and ( , ) are the coordinates of
tie points in the input and reference images.
x cv cw cvw c
y cv cw cvw c
v w x y
   

   


107. Weeks 1 & 2 107
Image RegistrationImage Registration

108. Weeks 1 & 2 108
Image TransformImage Transform
►A particularly important class of 2-D linear transforms,
denoted T(u, v)

1 1
0 0
( , ) ( , ) ( , , , )
where ( , ) is the input image,
( , , , ) is the ker ,
variables and are the transform variables,
= 0, 1, 2, ..., M-1 and = 0, 1,
M N
x y
T u v f x y r x yu v
f x y
r x yu v forward transformation nel
u v
u v
 
 

..., N-1.

109. Weeks 1 & 2 109
Image TransformImage Transform
►Given T(u, v), the original image f(x, y) can be recoverd
using the inverse tranformation of T(u, v).

1 1
0 0
( , ) ( , ) ( , , , )
where ( , , , ) is the ker ,
= 0, 1, 2, ..., M-1 and = 0, 1, ..., N-1.
M N
u v
f x y T u v s x y u v
s x y u v inversetransformation nel
x y
 
 


110. Weeks 1 & 2 110
Image TransformImage Transform

111. Weeks 1 & 2 111
Example: Image Denoising by Using DCT Example: Image Denoising by Using DCT
TransformTransform

112. Weeks 1 & 2 112
Forward Transform KernelForward Transform Kernel

1 1
0 0
1 2
1 2
( , ) ( , ) ( , , , )
The kernel ( , , , ) is said to be SEPERABLE if
( , , , ) ( , ) ( , )
In addition, the kernel is said to be SYMMETRIC if
( , ) is functionally equal to ( ,
M N
x y
T u v f x y r x y u v
r x y u v
r x y u v r x u r y v
r x u r y v
 
 



1 1
), so that
( , , , ) ( , ) ( , )r x y u v r x u r y u

113. Weeks 1 & 2 113
The Kernels for 2-D Fourier TransformThe Kernels for 2-D Fourier Transform
2 ( / / )
2 ( / / )
The kernel
( , , , )
Where = 1
The kernel
1
( , , , )
j ux M vy N
j ux M vy N
forward
r x yuv e
j
inverse
s x yuv e
MN


 





114. Weeks 1 & 2 114
2-D Fourier Transform2-D Fourier Transform

1 1
2 ( / / )
0 0
1 1
2 ( / / )
0 0
( , ) ( , )
1
( , ) ( , )
M N
j ux M vy N
x y
M N
j ux M vy N
u v
Tuv f x ye
f x y Tuve
MN


 
 
 
 

 





115. Weeks 1 & 2 115
Probabilistic MethodsProbabilistic Methods
Let , 0, 1, 2, ..., -1, denote the values of all possible intensities
in an digital image. The probability, (), of intensity level
occurring in a given image is estimated as

i
k
k
z i L
M N p z
z


( ) ,
where is the number of times that intensity occurs in the image.
k
k
k k
n
p z
MN
n z

1
0
( ) 1
L
k
k
p z



1
0
The mean (average) intensity is given by
= ( )
L
k k
k
m z p z




116. Weeks 1 & 2 116
Probabilistic MethodsProbabilistic Methods
1
2 2
0
The variance of the intensities is given by
= ( ) ( )
L
k k
k
z m p z



th
1
0
The moment of the intensity variable is
( ) = ( ) ( )
L
n
n k k
k
n z
u z z m p z




117. Weeks 1 & 2 117
Example: Comparison of Standard Example: Comparison of Standard
Deviation ValuesDeviation Values
31.614.3 49.2

118. Weeks 1 & 2 118
HomeworkHomework
http://cramer.cs.nmt.edu/~ip/assignments.html