Digital Image Processing
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Digital Image Processing
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The Hit-or-Miss Transformation
The morphological hit-or-miss transformation is a basic tool for shape detection.
Consider the set A from Figure 9.12 consisting of three shapes (subsets) denoted
C, D, and E. The objective is to locate one of the shapes, say, D.
Let the origin of each shape be located at its center of gravity. Let D be enclosed
by a small window, W. The local background of D with respect to W is defined
as the set difference (W-D) (Figure 9.12(b)). Figure 9.12(c) shows the
complement of A. Fig. 9.12(d) shows the erosion of A by D. Figure 9.12(e)
shows the erosion of the complement of A by the local background set (W-D).
From Figures 9.12(d) and (e) we can see that the set of location for which D
exactly fits inside A is the intersection of the erosion of A by D and the erosion
of Ac by (W-D) as shown in Figure 9.12(f).
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If B denotes the set composed of D and its background, the match (or the set of
matches) of B in A, denoted A B is:
( ) ( )c
A B A D A W D
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We can generalize the notation by letting B = (B1, B2), where B1 is the set
formed from elements of B associated with an object and B2 is the set of
elements of B associated with the corresponding background (B1=D, B2=W-D)
in the preceding example).
1 2( )
cA B A B A B
The set A B contains all the (origin) points at which, simultaneously, B1 found
a match (“hit”) in A and B2 found a match in Ac. Taking into account the
definition and properties of erosion we can rewrite the above relation as:
1 2( ) ( )A B A B A B
The above three equations for A B are referred as the morphological hit-or-
miss transform.
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Some Basic Morphological Algorithms
When dealing with binary images, one of the principal applications of
morphology is in extracting image components that are useful in the
representation and the description of shape. We consider morphological
algorithms for extracting boundaries, connected components, the convex hull,
and the skeleton of a region.
The images are shown graphically with 1s shaded and 0s in white.
Boundary Extraction
The boundary of a set A, denoted β(A), can be obtained by first eroding A by B
and then performing the set difference between A and its erosion.
( )A A A B
where B is a suitable structuring element.
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Digital Image Processing
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Digital Image Processing
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Filling Holes
A hole may be defined as a background region surrounded by a connected
border of foreground pixels. We present an algorithm based on set dilation,
complementation, and intersection for filling holes in an image.
Let A denote a set whose elements are 8-connected boundaries, each boundary
enclosing a background region (i.e. a hole). Given a point in each hole, the
objective is to fill all the holes with 1s.
We form an array, X0, of 0s (the same size as the array containing A), except at
the location in X0 corresponding to the given point in each hole, which is set to
1. The following procedure fills all the holes with 1s:
1, 1,2,3, ...
c
k kX X B A k
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where B is the symmetric structuring element in Figure 9.15(c). The algorithm
terminates at iteration step k if Xk=Xk-1. The set Xk then contains all the filled
holes. The set union of Xk and A contains all the filled holes and their
boundaries.
Digital Image Processing
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Digital Image Processing
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Extraction of Connected Components
Extraction of connected components from binary images is important in many
automated image analysis applications.
Let A be a set containing one or more connected components. Form an array
X0 (of the same size as the array containing A) whose elements are 0s
(background values), except at each location known to correspond to a point in
each connected component in A, which we set to 1 (foreground value). The
objective is to start with X0 and find all the connected components.
The procedure that accomplishes this task is the following:
1( ) , 1,2,3, ...
k kX X B A k
where B is a suitable structuring element. The procedure terminates when Xk =
Xk-1, with Xk containing all connected components of the input image.
Digital Image Processing
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Digital Image Processing
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Digital Image Processing
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Figure 9.18(a) shows an X-ray image of a chicken breast that contains bone
fragment. It is of considerable interest to be able to detect such objects in
processed food befor packing and/or shiping. In this case, the density of the
bones is such that their normal intensity values are different from the
background. This makes extraction of the bones from the background a simple
matter by using a single threshold. The result is the binary image in Figure
9.18(b). We can erode the thresholded image so that only objects of
„significant” size remain. In this example, we define as significant any object
that remains after erosion with a 5×5 structuring elemnt of 1s. The result of
erosion is shown in Figure 9.18(c). The next step is to analyse the objects that
remain. We identify these objects by extracting the connected components in the
image. There are a total of 15 connected components, with four of them being of
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dominant size. This is enough to determine that significant undesirable objects
are containd in the original image.
Thinning, thickening
( ) ( )cA B A A B A A B - thinning
1 2{ , , , }nB B B B
Bi is the rotated version of Bi-1
1 2{ } ( (( ) ) )i nA B A B B B
Thickening
( )A B A A B
1 2{ } ( (( ) ) )i nA B A B B B
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Digital Image Processing
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Digital Image Processing
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Digital Image Processing
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Digital Image Processing
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Color Image Processing Color Image Processing
Color Image Processing
Color is very important characteristic of an image that in most cases
simplifies object identification and extraction form a scene. Human eye can
discern thousands of color shades and intensities and only two dozen shades of
gray.
Color image processing is divided in 2 major areas: full-color (images acquired
with a full-color sensor) and pseudo-color (gray images for which color is
assigned) processing.
The colors that humans can perceive in an object are determinde by the nature
of the light reflected from the object.Visible light is composed of a relatively
narrow band of frequencies in the electromagnetic spectrum (390nm to750nm).
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A body that reflects light that is balanced in all visible wavelengths appears
white to the observer. A body that favors reflectance in a limited range of the
visible spectrum exhibits some shades of color.
For example, blue objects reflect light with wavelengths from 450 to 475 nm,
while absorbing most of the energy of other wavelengths.
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How to characterize light? If the light is achromatic (void of color) its only
attribute is its intensity (or amount) – determined by levels of gray (black-grays-
white).
Chromatic light spans the electromagnetic spectrum from approximately 400 to
720 nm. Three basic quantities are used to describe the quality of a chromatic
light source: radiance, luminance, and brightness.
- Radiance is the total amount of energy that flows from the light source
(usually measured in watts).
- Luminance (measured in lumens – lm) gives a measure of the amount of
energy an observer percieves from a light source. For example, the light
emitted from a source operating in the infrared region of the spectrum could
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have significant energy (radiance), but an observer would hardly perceive it
(the luminance is almost zero).
- Brightness is a subjective descriptor, that cannot be measured, it embodies
the achromatic notion of intensity and is a factor describing color sensation.
Cones are the sensors in the eye responsible for color vision. It has been
established that the 6 to 7 million cones in the human eye can be devided into
three principal sensing categories, corresponding roughly to red, green, and blue.
Approximately 65% of all cones are sensitive to red light, 33% are sensitive to
green light, an only about 2% are sensitive to blue (but the blue cones are the
most sensitive).
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Due to these absorbtion characteristics of the human eye, colors are seen as
variable combinations of the so-called primary colors : red (R), green (G), and
blue (B).
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For the purpose of standardization, the CIE (Commission Internationale de
l’Eclairage) designated in 1931 the following specific wavelength values to the
three primary colors: blue= 435.8 nm, green = 546.1 nm, and red=700 nm. The
CIE standards correspond only approximately with experimental data.
These three standard primary colors, when mixed in various intensity
proportions, can produce all visible colors.
The primary colors can be added to produce the secondary colors of light –
magenta (red+blue), cyan (green+blue), and yellow (red+green). Mixing the
three primaries, or a secondary with its opposite primary color in the right
intensities produces white light.
We must differentiate between the primary colors of light and the primary
colors of pigments. A primary color for pigments is one that substracts or absorb
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a primary color of light and reflects or transmits the other two. Therefore, the
primary colors of pigments are magenta, cyan, and yellow, and the secondary
colors are red, green, and blue.
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The characteristics usually used to distinguish one color from another are
brightness, hue, and saturation.
Brightness embodies the achromatic notion of intensity. Hue is an attribute
associated with the dominant wavelength in a mixture of light waves. Hue
represents dominat color as percieved by an observer (when we call an object to
be red, orange or yellow we refer to its hue). Saturation refers to the relative
purity or the amount of white light mixed with a hue. The pure spectrum colors
are fully saturated. Color such as pink (red+white) and lavender (violet+white)
are less saturated, with the degree of saturation being inversely proportional to
the amount of white light added.
Hue and saturation taken together are called chromaticity, and therefore a color
may be characterized by its brightness and chromaticity.
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The amounts of red, green, and blue needed to form any particular color are
called the tristimulus values and are denoted X, Y and Z, respectively. A color is
specified by its trichromatic coefficients, defined as:
Xx
X Y Z
Yy
X Y Z
Zz
X Y Z
1x y z
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For any wavelength of light in the visible spectrum, the tristimulus values
needed to produce the color coresponding to that wavelength can be obtained
from the existing curves or tables.
Another approach for specifying colors is to use the CIE chromaticity
diagram, which shows color compositin as a function of x (red) and y (green); z
(blue) is obtained from relation z = 1-x-y.
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The positions of the various spectrum colors (from violet at 380 nm to red at
780 nm) are indicated around the boundary of the tongue-shaped chromaticity
diagram.
The chromaticity diagram is useful for color mixing because a straight-line
segment joining any two points in the diagram defines all the different color
variation that can be obtained by combining these two colors. This procedure
can be extended to three colors: to triangle determined by the three color-points
on the diagram embodies all the possible colors that can be obtained by mixing
the three colors.
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Digital Image Processing
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Color Models
A color model (color space or color system) is a specification of a coordinate
system and a subspace within that system where each color is represented by a
single point.
http://www.colorcube.com/articles/models/model.htm
Most color models in use today are oriented either toward hardware (color
monitors or printers) or toward applications where color manipulation is a goal.
The most commonly used hardware-oriented model is RGB (red-green-blue) –
for color monitors, color video cameras.
The CMY (cyan-magenta-yellow) and CMYK (cyan-magenta-yellow-black)
models are in use for color printing.
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The HSI (hue-saturation-intensity) model corespond with the way humans
describe and interpret colors. The HSI model has the advantage that it decoupes
the color and gray-scale information in an image, making it suitable for using
the gray-scale image processing techniques.
The RGB Color Model
In the RGB model, each color appears decomposed in its primary color
components: red, green, blue. This model is based on a Cartesian coordinate
system. The color subspace of interest is the unit cube (Figure 6.7), in which the
primary and the seconadary colors are at the corners; black is at the origin, and
white is at the corner farthest from the origin.
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The gray scale (point of equal RGB values) extends from black to white along
the line joining these two points. The different colors in this model are points on
or inside the cube, and are defined by vectors extending from the origin.
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Images represented in the RGB color model consist of three component images,
one for each primary color. The number of bits used to represent each pixel in
RGB space is called the pixel depth. Consider an RGB image in which each of
the red, green, and blue images are an 8-bit image. In this case, each RGB color
pixel has a depth of 24 bits. The term full-color image is used often to denote a
24-bit RGB color image. The total number of colors in a 24-bit RGB image is
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3
82 16.777.216
A convenient way to view these colors is to generate color planes (faces or cross
sections of the cube).
A color image can be acquired by using three filters, sensitive to red, green, and
blue.
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Because of the variety of systems in use, it is of considerable interest to have a
subset of colors that are likely to be reproduced faithfully, resonably
independently of viewer hardware capabilities. This subset of colors is called the
set of safe RGB colors, or the set of all-systems-safe colors. In Internet
applications, they are called safe Web colors or safe browser colors.
We assume that 256 colors is the minimum number of colors that can be
reproduced faithfully by any system. Forty of these 256 colors are known to be
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processed differently by varoius operating system. We have 216 colors that are
common to most systems, and are the safe colors, especially in Internet
applications. Each of the 216 safe colors has a RGB representation with:
, , 0,51,102,153,204,255R G B
We have (6)3=216 possible color values. It is costumary to express these values
in the hexagonal number system.
Each safe color is formed from three of the two digit hex numbers from the
above table. For example purest red if FF0000. The values 000000 and FFFFFF
represent black and white respectively.
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Figure 6.10(a) shows the 216 safe colors, organized in descending RGB values.
Figure 6.10(b) shows the hex codes for all the possible gray colors in the 216
safe color system.
Figure 6.11 shows the RGB safe-color cube.
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http://www.techbomb.com/websafe/
The CMY and CMYK Color Models
Cyan, magenta, and yellow are the secondary colors of light but the primary
color of pigments. For example, when a surface coated with yellow pigment is
illuminated with white light, no blue light is reflected from the surface. Yellow
substracts blue light from reflected white light (which is composed of equal
amounts of red, green, and blue light).
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Most devices that deposit color pigments on paper, such as color printers and
copiers, require CMY data input and perform RGB to CMY conversion.
Assuming that the color values were normalized to range [0,1], this conversion
is:
1
1
1
C R
M G
Y B
From this equation we can easily deduce, that pure cyan does not reflect red,
pure magenta does not reflect green, and pure yellow does not reflect blue.
Equal amount of pigments primary, cyan, magenta, and yellow should produce
black. In practice, combining these colors for printing produces a muddy-
looking black. In order to produce true black (which is the predominant color in
printing), a fourth color, black, is added, giving rise to the CMYK color model.
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The HSI Color Model
The RGB, CMY, and other similar color models are not well suited for
describing colors in terms that are practical for human interpretation.
We (humans) describe a color by its hue, saturation and brightness. Hue is a
color attribute that describes a pure color, saturation gives a measure of the
degree to which a pure color is diluted by white light and brightness is a
subjective descriptor that embodies the achromatic notion of intensity.
The HSI (hue, saturation, intensity) color model, decouples the intensity
component from the color information (hue and saturation) in a color image.
What is the link between the RGB color model and HSI color model? Consider
again the RGB unit cube. The intensity axis is the line joining the black and the
white vertices. Consider a color point in the RGB cube. Let P be a plane
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perpedicular to the intensity axis and containing the color point. The intersection
of this plane with the intensity axis gives us the intensity of the color point. The
saturation (purity) of the considered color point increases as a function of
distance from the intensity axis (the saturation of the point on the intensity axis
is zero).
In order to determine how hue can be linked to a given RGB point, consider a
plane defined by black, white and cyan. The intensity axis is also included in
this plane. The intersection of this plane with the RGB-cube is a triangle. All
point contained in this triangle would have the same hue (i.e. cyan).
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The HSI space is represented by a vertical intensity axis and the locus of color
points that lie on planes perpedicular to this axis. As the planes move up and
down the intensity axis, the boundary defined by the intersection of this plane
with the faces of the cube have either triangular or hexagonal shape.
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In the plane shown in Figure 6.13(a) primary colors are separated by 120º. The
secondary colors are 60º from the primaries. The hue of the point is determined
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by an angle from some reference point. Usually (but not always) an angle of 0º
from the red axis designates 0 hue, and the hue increases countercloclwise from
there. The saturation (distance from the vertical axis) is the length of the vector
from the origin to the point. The origin is defined by the intersection of the
color plane with the vertical intensity axis.
Converting colors from RGB to HSI
if
if 360
B GH
B G
2
( ) ( )arccos
2 ( ) ( )( )
R G R B
R G R B G B
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31 min{ , , }S R G B
R G B
1
3I R G B
It is assumed that the RGB values have been normalized to the range [0,1] and
that angle θ is measured with respect to the red axis of the HSI space in Figure
6.13. Hue can be normalized to the range [0,1] by dividing it to 360º. The other
two HSI components are in this range if the RGB values are in the interval [0,1].
R=100, G=150, B=200 H=210º, S=1/3, I=150/255=0.588
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Converting colors from HSI to RGB
Given values of HSI we now want to find the corresponding RGB values in the
same range.
RG sector (0º ≤ H < 120º)
(1 )
cos1
cos(60 )
3 ( )
B I S
S HR I
H
G I R B
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GB sector (120º ≤ H < 240º)
(1 )
cos120 , 1
cos(60 )
3 ( )
R I S
S HH H G I
H
B I R G
BR sector (120º ≤ H < 240º)
(1 )
cos240 , 1
cos(60 )
3 ( )
G I S
S HH H B I
H
R I G B
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Lab color space
The effectiveness of such transformations is judged ultimately in print. The
transformations are developed and evaluated on monitors. It is necessary to have
a high degree of consistency between the monitors and the output devices. This
is best accomplished with a device-independent color model that relates the
color gamut of the monitors and output devices, as well as any other devices
being used, to one another. The model of choice for many color management
systems (CMS) is the CIE L*a*b* model, also called CIELAB. The L*a*b*
color components are given by the following equations:
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* 116 16W
YL h
Y
* 500W W
X Ya h h
X Y
* 200W W
Y Zb h h
Y Z
3 0.008856
( ) 167.787 0.008856
116
q q
h qq q
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, ,W W W
X Y Z are reference tristimulus values – typically the white of a
perfectly reflecting diffuser under CIE standard D65 illumination
( 0.3127 , 0.33290 , 1x y z x y ).
The L*a*b* color space is colorimetric (i.e. colors perceived as matching are
encoded identically), perceptually uniform (i.e. color differences among various
hues are perceived uniformly), and device independent. Like the HSI system, the
L*a*b* system is an excellent decoupler of intensity (represented by lightness
L*) and color (represented by a* for red minus green and b* for green minus
blue), making it useful in both image manipulation (tone and contrast editing)
and image compression applications.
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Other color spaces
YIQ – for the NTSC (National Television System Committee) television system
in US
Y – luminance
I (in-phase), Q (quadrature) – chrominance
YUV – for the PAL (Phase Alternation Line) and SECAM (Séquentiel Couleur
à Mémoire) television system in Europe
(I, Q) – obtained by rotating (U,V)
YCbCr – digital video transmission
More about color spaces in: Andreas Koschan, Mongi Abidi, Digital Color
Image Processing, Wiley, 2008
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Color Difference
RGB, CMY, Lab – Euclidean distance
HSI - F1=(H1, S1, I1), F2=(H2, S2, I2)
HSI
if
if
2 2
1 2 1 2
2 2
1 2 1 2
1 2 1 2
1 2 1 2
( , ) ( ) ( ) ,
2 cos
2
d F F I C I I I
C S S S S
H H H H
H H H H
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Pseudo-color Image Processing
Pseudo-color (also called false color) image processing consists of assigning
colors to gray values based on a specified criterion. The main use of
pseudo-color is for human visualization and interpretation of gray-scale events
in an image or sequence of images.
Intensity (Density) Slicing
If an image is viewed as a 3-D function, the method can be described as one of
placing planes parallel to the coordinate plane of the image; each plane then
“slices” the function in the area of intersection.
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The plane at ( , )i
f x y l slices the image function into two levels. If a
different color is assigned to each side of the plane, any pixel whose intensity
level is above the plane will be coded with one color and any pixel below the
plane will be coded with other color. Levels that lie on the plane itself may be
arbitrarily assigned one of the two colors. The result is a two color image whose
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relative appearance can be controlled by moving the slicing plane up and down
the intensity axis.
Let [0, L-1] represent the gray scale, let l0 represent black (f(x,y)=0) and level
lL-1 represent white (f(x,y)=L-1). Suppose that P planes perpendicular to the
intensity axis are defined at levels l1, l2, …, lP , 0<P<L-1. The P planes partition
the gray scale into P+1 intervals, V1, V2, …, VP+1. Intensity to color assignments
are made according to the relation:
if ( , ) ( , )k k
f x y c f x y V .
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Measurements of rainfall levels with ground-based sensors are difficult and
expensive, and total rainfall figures are even more difficult to obtain because a
significant portion of precipitations occurs over the ocean. One way to obtain
these figures is to use a satellite. The TRMM (Tropical Rainfall Measuring
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Mission) satellite utilizes, among others, three sensors specially designed to
detect rain: a precipitation radar, a microwave imager, and a visible and infrared
scanner. The results from the various rain sensors are processed, resulting in
estimates of average rainfall over a given time period in the area monitored by
the sensors. From these estimates, it is not difficult to generate gray-scale
images whose intensity values correspond directly to rainfall, with each pixel
representing a physical land area whose size depends on the resolution of the
sensors.
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Basics of Full-Color Image Processing
3 4
( , ) ( , )
: / , ( , ) ( , ) ( , )
( , ) ( , )
( , )
( , )( , )
( , )
( , )
R
G
B
c x y R x y
f D f x y c c x y G x y
c x y B x y
C x y
M x yf x y c
Y x y
K x y
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Color Transformations
- processing the components of a color image within the context of a single-
color model
( , ) ( , )g x y T f x y
1 2( , , ..., ) , 1,2, ..., ( ( , ) , ( , ) )
i i ns T r r r i n f x y r g x y s
ri, si are the color components of f(x, y) and g(x, y), n is the number of color
components, and {T1, T2,…, Tn} is a set of transformations or color mapping
functions that operate on ri to produce si. (n=3 or n=4)
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In theory, any transformation can be performed in any color model. In practice,
some operations are better suited to specific color models.
Suppose we wish to modify the intensity of a color image, using
( , ) ( , ) , 0 1g x y k f x y k
In the HSI color space, this can be done with:
s1= r1 , s2= r2 , s3=k r3
In the RGB/CMY color model all components must be transformed
s1= kr1 , s2= kr2 , s3=kr3 (RGB)
si = kri+(1-k) , i=1,2,3 (CMY)
Although the HSI transformation involves the fewest number of operations, the
costs for converting an RGB or CMY(K) image to the HSI color space are much
bigger than the transformations.
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Color Complements
The hues directly opposite one another on the above color circle are called
complements (analogous to the gray-scale negatives).
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Unlike the intensity transformation, the RGB complement transformation
functions used in this example do not have straightforward HSI space
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equivalent. The saturation component of the complement cannot be computed
from the saturation component of the input image alone.
Color Slicing
Highlighting a specific range of colors in an image is useful for separating
objects from their surroundings. The basic idea is either to:
- display the colors of interest so they stand out from the background
- use the region defined by the colors as a mask for further processing.
One of the simplest ways to “slice” a color image is to map the colors outside
some range of interest to a neutral color. If the colors of interest are enclosed
by a cube (or hypercube, if n>3) of width W and centered at a prototypical
(e.g. average) color with components 1 2, , ...,
na a a the set of
transformations is:
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if
otherwise
0.5 , 1, 1,2, ...,2
j j
i
i
Wr a j n
s i n
r
These transformations highlight the colors around the prototype by forcing all
other colors to the midpoint of the reference color space (an arbitrarily chosen
neutral point).
For the RGB color space, for example, a suitable neutral point is middle gray or
color (0.5, 0.5, 0.5).
If a sphere is used to specify the colors of interest, the transformations are:
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if
otherwise
2 2
0
1
0.5 ( ), 1,2, ...,
n
j j
ji
i
r a Rs i n
r
where R0 is the radius of the enclosing sphere and 1 2, , ...,
na a a are the
components of its center.
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Histogram Processing
It is not advisable to histogram equalize the components of a color image
independently. This can produce wrong colors. A more logical approach is to
spread the color intensity uniformly, leaving the colors (e.g., hues) unchanged.
The HSI color space is ideally suited for this type of approach.
The unprocessed image contains a large number of dark colors that reduce the
median intensity to 0.36. Histogram equalizing the intensity component, without
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altering the hue and saturation produced image Figure 6.37(c). The image is
brighter. Figure 6.37(d) was obtained by increasing also the saturation
component.
Color Image Smoothing
Let Sxy denote a neighborhood centered at (x,y) in an RGB color image. The
average of the RGB component vectors in this neighborhood is:
( , )
1( , ) ( , )
xys t S
c x y c s tK
( , )
( , )
( , )
1( , )
1( , ) ( , )
1( , )
xy
xy
xy
s t S
s t S
s t S
R s tK
c x y G s tK
B s tK
Digital Image Processing
Week 4
Digital Image Processing
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Digital Image Processing
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Color Image Sharpening
2( , ) ( , ) ( , )g x y f x y c f x y
2
2 2
2
( , )
( , ) ( , )
( , )
R x y
c x y G x y
B x y
Digital Image Processing
Week 4
Image Compression
Image compression is the art and science of reducing the
amount of data required to represent an image.
Consider a two-hour standard definition (SD) television
movie using 720480 24 bit pixel arrays. A digital movie is
a sequence of video frames in which each frame is a full-color
still image. Because video players must display the frames
sequentially at rates 30 fps (frames per second), SD digital
video must be accessed at:
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frames pixels bytesbytes/sec
sec frame pixel30 (720 480) 3 31.104.000
and a two-hour movie consist of
bytes sechours bytes( GB)
sec hour
2 1131.104.000 (60 ) 2 2.24 10 224
To put a two-hour movie on a DVD, each frame must be
compressed by a factor of 26.3(on average). The compression
must be even higher for high definition (HD) television where
image resolution reach 1920108024 bit/image.
Digital Image Processing
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Fundamentals
The term data compression refers to the process of reducing
the amount of data required to represent a given quantity of
information. Data and information are not the same thing;
data are the means by which information is expressed.
Because various amounts of data can be used to represent the
same amount of information, representations that contain
irrelevant or repeated information are said to contain
redundant data.
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Let b and b′ denote the number of bits in two representations
of the same information, the relative data redundancy R of
the representation with b bits is:
11R
C
Where C, commonly called the compression ratio is
bC
b
If C=10 - the larger representation has 10 bits of data for
every 1 bit of data in the smaller representation. The
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corresponding relative data redundancy of the larger
representation is 0.9 (R=0.9), indicating that 90% of its data
is redundant.
In the context of digital image compression, b usually is the
number of bits needed to represent an image as a 2-D array of
intensity values. Two-dimensional intensity arrays (far from
optimal) suffer from three principal types of data
redundancies:
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1. Coding redundancy. A code is a system of symbols
(letters, numbers, bits…) used to represent a body of
information or set of events. Each piece of information
or event is assigned a sequence of code symbols, called a
code word. The number of symbols in each code word is
its length. The 8-bit codes that are used to represent the
intensities in most 2-D intensity arrays contain more bits
than are needed to represent the intensities.
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2. Spatial and temporal redundancy. Because the pixels
of most 2-D intensity arrays are correlated spatially (i.e.
each pixel is similar to or dependent on neighboring
pixels), information is unnecessarily replicated in the
representations of correlated pixels. In a video sequence,
temporally correlated pixels (i.e., those similar to or
dependent on pixels in nearby frames) also duplicate
information.
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3. Irrelevant information. Most 2-D intensity arrays
contain information that is ignored by the human eye.
This information is redundant in the sense that it is not
used.
Compression is achieved when one or more redundancy is
reduced or eliminated.
Digital Image Processing
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Digital Image Processing
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Code Redundancy
Assume that a discrete random variable rk in the interval
[0, L-1] is used to represent the intensities of an MN image
and that each rk occurs with probability p(rk).
( ) , 0,1,2,..., 1k
k
np r k L
M N
Where L is the number of intensity values, and nk is the
number of times that the k-th intensity appears in the image.
If the number of bits used to represent each value of rk is l(rk),
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then the average number of bits required to represent each
pixel is:
1
0
( ) ( )L
avg k k
k
L l r p r
The total number of bits required to represent an MN image
is avg
MNL . If the intensities are represented using a natural
m-bit fixed-length code, avg
L m .
Consider image in Figure 8.1 (a) (M=N=256) and the coding
Table 8.1.
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For Code 1 8avg
L . For Code 2 we have:
bits0.25 2 0.47 1 0.25 3 0.03 3 1.81avg
L
The total number of bits needed to represent the entire image
is 256 256 1.81 118.621avg
MNL .
Digital Image Processing
Week 4
256 256 8 84.42
256 256 1.81 1.81C
1 11 1 0.774
4.42R
C
Thus, 77.4% of the data in the original 8-bit 2-D intensity
array is redundant.
The compression achieved by code 2 results from assigning
fewer bits to the more probable intensity values than to the
less probable ones, thus resulting a variable-length code. The
best fixed-length code that can be assigned to the intensities
Digital Image Processing
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of the image in Figure 8.1(a) is the natural 2-bit counting
sequence {00, 01, 10, 11} but the resulting compression is
only C=8/2=4:1 which is about 10% less than the 4.42:1
compression of the variable-length code.
Coding redundancy is present when the codes assigned to a
set of events (such as intensity values) do not take full
advantage of the probabilities of the events. Coding
redundancy is almost always present when the intensities of
an image are represented using a natural binary code. Most
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images are composed of objects that have a regular and
somewhat predictable morphology (shape) and reflectance,
and are sampled so that the objects being depicted are much
larger than the picture elements. For most images, certain
intensities are more probable than others (that is, the
histograms of most images are not uniform). A natural binary
encoding assigns the same number of bits to both the most
and least probable values, failing to minimize the value of
avgL and resulting in coding redundancy.
Digital Image Processing
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Digital Image Processing
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Spatial and Temporal Redundancy
Consider the computer-generated image in Figure 8.1(b). In
the corresponding 2-D intensity array:
All 256 intensities are equally probable
Because the intensity of each line was selected randomly,
its pixels are independent of one another in the vertical
direction
Digital Image Processing
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Because the pixels along each line are identical, they are
maximally correlated (completely dependent on one
another) in the horizontal direction.
The first observation tells us that this image cannot be
compressed by variable-length coding alone. Observation 2
and 3 reveal a significant spatial redundancy that can be
eliminated, for instance, by representing this image as a
sequence of run-length pairs, where each run-length pair
specifies the start of a new intensity and the number of
Digital Image Processing
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consecutive pixels that have that intensity. A run-length based
representation compresses the original 2-D, 8-bit intensity
array by
256 256 8128 :1
256 2 8C
Each 256-pixel line of the original representation is replaced
by a single 8-bit intensity value and length 256 in the
run-length representation.
In most images, pixels are correlated spatially (in both x and
y) and in time (in case of video sequences). Because most
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pixels’ intensities can be predicted reasonably well from
neighboring intensities, the information carried by a single
pixel is small. Much of its visual contribution is redundant in
the sense that can be inferred from its neighbors. To reduce
the redundancy associated with spatially and temporally
correlated pixels, a 2-D intensity array must be transformed
into a more efficient but usually ‘non-visual’ representation.
Transformations of this type are called mappings. A mapping
is said to be reversible if the pixels of the original 2-D
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intensity array can be reconstructed without error from the
transformed data set; otherwise the mapping is said to be
irreversible.
Irrelevant Information
One of the simplest ways to compress a set of data is to
remove superfluous data from the set. In the context of DIP,
information ignored by the system which uses the image
(human eye, computer programs) are obvious candidates for
omission. Thus, the computer-generated image in Figure
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8.1(c), because it appears to be a homogeneous field of grey,
can be represented by its average intensity alone – a single
8-bit value. The original 2562568 bit intensity array is
reduced to a single byte; the resulting compression is
65.536:1.
Digital Image Processing
Week 4
Figure 8.3(a) shows the histogram of the image in Figure
8.1(c) – there are some intensity values (125 through 131)
actually present. The human visual system averages these
intensities, perceives only the average value, and ignores the
small changes in intensity that are present in this case.
Digital Image Processing
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Figure 8.3(b), a histogram equalized version of the image in
Figure 8.1 (c), makes the intensity changes visible and reveals
two previously undetected regions of constant intensity.
If the image in Figure 8.1 (c) is represented by its average
value alone, this ‘invisible’ structure is lost.
The kind of redundancy can be eliminated because the
information itself is not essential for normal visual processing
and/or the intended use of the image. Because its omission
results in a loss of quantitative information, its removal is
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commonly referred to as quantization (mapping of a broad
range of input values to a limited number of output values).
Because information is lost, quantization is an irreversible
operation.