sections 3.1-3.3 digital image processing gonzales and woods irina rabaev intensity transformations
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Sections 3.1-3.3Digital Image Processing
Gonzales and Woods
Irina Rabaev
Intensity Transformations
Representing digital image
value f(x,y) at each x, y is called intensity level or gray level
Intensity Transformations and Filters
g(x,y)=T[f(x,y)]
f(x,y) – input image,g(x,y) – output imageT is an operator on f defined over a neighborhood of point (x,y)
Intensity Transformation1 x 1 is the smallest possible neighborhood.In this case g depends only on value of f at
a single point (x,y) and we call T an intensity (gray-level mapping) transformation and write
s = T(r) where r and s denotes respectively the
intensity of g and f at any point (x, y).
Some Intensity Transformation Functions
Image NegativesDenote [0, L-1] intensity levels of the image.
Image negative is obtained by s= L-1-r
Log Transformations
s = clog(1+r), c – const, r ≥ 0Maps a narrow range of low intensity values in the input into a
wider range of
output levels. The opposite is true for higher values of input levels.
Power–Law (Gamma) transformation
s = crγ, c,γ –positive constantscurve the grayscale components either to brighten the intensity
(when γ< 1) or darken the intensity (when γ > 1).
Power –Law (Gamma) transformation
Power –Law (Gamma) transformation
Contrast stretchingContrast stretching is a process that expands the range of intensity levels in a image so that it spans the full intensity range of the recording medium or display device.Contrast-stretching transformations increase the contrast between the darks and the lights
Thresholding function
Intensity-level slicingHighlighting a specific range of gray levels in an image
Histogram processing
The histogram of a digital image with gray levels in the range [0, L-1] is a
discretefunction h(rk)=nk , where rk is the kth
gray level and nk is the number of pixels in
the image having gray level rk. It is common practice to normalize a histogram by dividing each of its values
by the total number of pixels in the image, denoted by the product MN.
Thus, a normalized histogram is given by h(rk)=nk/MN
The sum of all components of anormalized histogram is equal to 1.
Histogram Equalization
Histogram equalization can be used to improve the visual appearance of an image.
Histogram equalization automatically determines a transformation function that produce and output image that has a near uniform histogram
Histogram EqualizationLet rk, k[0..L-1] be intensity levels and let
p(rk) be its normalized histogram function.The intensity transformation function for
histogram equalization is
k
jj
k
jjrkk
LknMN
L
rpLrTs
0
0
1,...,2,1,0,1
)()1()(
Histogram Equalization - Example
Let f be an image with size 64x64 pixels and L=8 and let f has the intensity distribution as shown in the table
rknkp r(rk
)=nk/MN
07900.19
210230.25
18500.21
36560.16
43290.08
52450.06
61220.03
7810.02
.00.7,86.6,65.6,23.6,67.5,55.4
08.3))()((7)(7)(
33.1)(7)(7)(
765432
10
1
011
0
0
000
ssssss
rprprprTs
rprprTs
rrj
jr
rj
jr
round the values to the nearest integer
Local histogram Processing
Define a neighborhood and move its center from pixel to pixel. At each location, the histogram of the points in the neighborhood is computed and histogram equalization transformation is obtained.
Using Histogram Statistics for Image Enhancement
The intensity variance:
Denote: ri – intencity value in the range [0, L-1],p(i) - histogram component corresponding to value ri .
Using Histogram Statistics for Image EnhancementLet (x, y) be the coordinates of a pixel in an image, and let Sxy
denote a
neighborhood (subimage) of specified size, centered at (x, y).
The mean value of the pixels in this neighborhood is given by
where is the histogram of the pixels in region Sxy.
The variance of the pixels in the neighborhood is given by
)(1
0i
L
iSiS rprmxyxy
xySp
)()(1
0
22i
L
iSSiS rpmrxyxyxy
Using Histogram Statistics for Image Enhancement
Tungsten filament
Using Histogram Statistics for Image Enhancement
Using Histogram Statistics for Image Enhancement