Digital Image Processing UNIT-2.ppt

Chapter 2
IMAGE ENHANCEMENT IN THE
SPATIAL DOMAIN

Outline
 Background
 Basic Gray-level transformation
 Histogram Processing
 Arithmetic-Logic Operation
 Basics of Spatial Filtering
 Smoothing Spatial Filters
 Sharpening Spatial Filters
 Combining Spatial Enhancement Methods
 Fuzzy techniques*

 Image enhancement approaches fall into two
broad categories: spatial domain methods
and frequency domain methods.
 The term spatial domain refers to the image
plane itself.
 g(x,y)= T[f(x,y)] , T is an operator on f, defined
over some neighborhood of f(x,y)
Background

Size of Neighborhood
 Point processing
 Larger neighborhood: mask (kernel,
template, window) processing

Gray-level Transformation
Contrast stretching thresholding

Basic Gray Level Transformation
 Image negatives: s =L-1-r
 Log transformation: s =clog(1+r)
 Power-law transformation: s=cr

Gamma Correction (I)
 Cathode ray tube (CRT) devices have an intensity-to-voltage
response that is a power function, with exponents varying
from 1.8 to 2.5.

cr
s 

cr
s 

Power-Law Transformation (I)

cr
s 

Power-Law Transformation (II)

cr
s 

Piece-wise Linear Transformation
 Contrast stretching
 Gray-level slicing
(Figure 3.11,12)
 Bit-plane slicing
(Figures 3.13-15)

Histogram Processing
 The histogram of a digital image with gray-levels in
the range [0,L-1] is a discrete function h(rk)=nk
where rk is the kth gray level and nk is the number of
pixels in the image having gray level rk
 Normalized histogram: p(rk)=nk/MN.
 Easy to compute, good for real-time image
processing.

Histogram Transformation

 T(r) is a monotonically increasing function

1
0
),
( 


 L
r
r
T
s
1
0
for
1
)
(
0 




 L
r
L
r
T

Histogram Equalization
 What if we take the transformation T to be:
 It can be shown that ps(s)=1/(L-1)
 Example 3.4 (p.125)

 d
p
L
r
T
s
r
r )
(
)
1
(
)
(
0





 Example 3.5 (p.126)
Histogram Equalization: Discrete
Case

 






k
j
j
k
j
j
r
k
k n
MN
L
r
p
L
r
T
s
0
0
1
)
(
)
1
(
)
(

Histogram Equalization: Examples

Histogram Statistics
 N-th moment of r about its mean:
)
(
)
(
)
(
1
0
i
L
i
n
i
n r
p
m
r
r 






Arithmetic Operations
 Image Subtraction
 Image Averaging

Basics of Spatial Filtering
•Mask, convolution
kernels
•Odd sizes

Spatial Correlation and
Convolution

 







b
b
t
a
a
s
t
y
s
x
f
t
s
w
y
x
f
y
x
w )
,
(
)
,
(
)
,
(
)
,
(

 







b
b
t
a
a
s
t
y
s
x
f
t
s
w
y
x
f
y
x
w )
,
(
)
,
(
)
,
(
)
,
(
Correlation
Convolution

Smoothing Spatial Filters
 Smoothing linear filters: averaging filters, low-pass
filters
 Box filter
 Weighted average
 Order-statistics filters:
 Median-filter: removing salt-and-pepper noise
 Max filter
 Min filter

Sharpening Spatial Filters
 Foundation:
)
(
2
)
1
(
)
1
(
)
(
)
1
(
2
2
x
f
x
f
x
f
x
f
x
f
x
f
x
f













The Laplacian
 Development of the method:
)
,
(
4
)
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
)
,
(
2
)
1
,
(
)
1
,
(
)
,
(
2
)
,
1
(
)
,
1
(
2
2
2
2
2
2
2
2
2
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
f
y
x
f
y
x
f
y
x
f
y
f
y
x
f
y
x
f
y
x
f
x
f
y
f
x
f
f
































Combining Spatial Enhancement
Methods
(a) original (b) Laplacian, (c) a+b, (d) Sobel of (a)
(a) (b) (c) (d)

-39-
gray-level image histogram
 Represents the relative frequency of occurrence of the various gray levels
in the image
 For each gray level, count the number of pixels having that level
 Can group nearby levels to form a big bin & count #pixels in it

-40-
interpretations of histogram
 if pixel values are i.i.d random
variables  histogram is an
estimate of the probability
distribution of the r.v.
 “unbalanced” histograms do not
fully utilize the dynamic range
 Low contrast image: narrow
luminance range
 Under-exposed image:
concentrating on the dark side
 Over-exposed image:
concentrating on the bright side
 “balanced” histogram gives more
pleasant look and reveals rich
details

-41-
contrast stretching

)
(
 T

2

1

2

1

0 L-1
1
s
2
s
3
s
L-1
Stretch the over-concentrated gray-levels
Piece-wise linear function, where the slope in the
stretching region is greater than 1.

-42-
… in practice
 intuition about a “good” image:
 a “uniform” histogram spanning a large variety of gray tones
 can we figure out a stretching function automatically?

-43-
histogram equalization
 goal: map the each luminance level to a new value such that the output
image has approximately uniform distribution of gray levels
 two desired properties
 monotonic (non-decreasing) function: no value reversals
 [0,1][0.1] : the output range being the same as the input range
pdf
cdf
o
1
1
o
1
1

-44-
histogram equalization
 make
 show
o
1
1

-45-
implementing histogram
equalization
1
-
L
...,
0,
i
for
)
(
)
(
)
( 1
0





L
i
i
i
i
u
x
n
x
n
x
p




u
x
i
u
i
x
p
L
v
0
)
(
)
1
(
)
(
' v
round
v 



u
x
i
u
i
x
p
v )
( Rounding or
Uniform
quantization
u v v’
pu(xi)
 Only depend on
the input image
histogram
 Fast to implement
 For u in discrete
prob. distribution,
the output v will
be approximately
uniform
compute
histogram
equalize
round the
output




u
x
i
i
x
n
MN
L
v
0
)
(
1
or

-46-
a toy example




u
x
i
u
i
x
p
L
v
0
)
(
)
1
(

-47-
a toy example
1.33
3.08
4.55
5.67
6.23
6.65
6.86
7.00
1
3
5
6
6
7
7
7

-48-
histogram equalization example

-49-
contrast-stretching vs. histogram equalization
 function form
 reversible? loss of information?
 input/output?
 automatic/interactive?
input gray level u
output
gray
level
v
a b
o




-50-
histogram matching
 Histogram matching/specification
 Want output v with specified p.d.f. pV(v)
 Use a uniformly distributed random vairable W as
an intermediate step
• W = FU(u) = FV(v)  V = F-1
V (FU(u) )
 Approximation in the intermediate step needed
for discrete r.v.
• W1 = FU(u) , W2 = FV(v) 
take v s.t. its w2 is equal to or just above w1

-51-
histogram matching example

-52-
local histogram processing
 problem: global spatial processing not always desirable
 solution: apply point-operations to a pixel neighborhood
with a sliding window

-53-
spatial filtering in image
neighborhoods

-54-
kernel operator / filter masks
Spatial
Filtering
f g
(.)
(.) w
TN 
 

 




a
a
i
b
b
j
j
n
i
m
f
j
i
w
n
m
g )
,
(
)
,
(
)
,
(
N
n
M
m




1
1
kernel

-55-
Smoothing: Image Averaging
Low-pass filter, leads to
softened edges
smoothing
operator

-56-
spatial averaging can suppress
noise
 image with iid noise y(m,n) = x(m,n) + N(m,n)
 averaging
v(m,n) = (1/Nw)  x(m-k, n-l) + (1/Nw)  N(m-k, n-l)
• Nw: number of pixels in the averaging window
 Noise variance reduced by a factor of Nw
 SNR improved by a factor of Nw
 Window size is limited to avoid excessive blurring
UMCP
ENEE408G
Slides
(created
by
M.Wu
&
R.Liu
©
2002)

-57-
smoothing operator of different sizes
3x3
5x5
15x15
9x9
35x35
original

-58-
directional smoothing
 Problems with simple spatial averaging mask
 Edges get blurred
 Improvement
 Restrict smoothing to along edge direction
 Avoid filtering across edges
 Directional smoothing
 Compute spatial average along several directions
 Take the result from the direction giving the smallest changes before
& after filtering
 Other solutions
 Use more explicit edge detection and adapt filtering accordingly

W
UMCP
ENEE408G
Slides
(created
by
M.Wu
&
R.Liu
©
2002)

-59-
non-linear smoothing operator
 Median filtering
 median value  over a small window of size Nw
 nonlinear
• median{ x(m) + y(m) }  median{x(m)} + median{y(m)}
 odd window size is commonly used
• 3x3, 5x5, 7x7
• 5-pixel “+”-shaped window
 for even-sized windows take the average of two middle values as
output
 Other order statistics: min, max, x-percentile …

-60-
median filter example
iid noise
more at lecture 7, “image restoration”
 Median filtering
 resilient to statistical outliers
 incurs less blurring
 simple to implement

-61-
image derivative and sharpening

-62-
edge and the first derivative
 Edge: pixel locations of abrupt luminance
change
 Spatial luminance gradient vector
 a vector consists of partial derivatives
along two orthogonal directions
 gradient gives the direction with
highest rate of luminance changes
 Representing edge: edge intensity +
directions
 Detection Methods
 prepare edge examples (templates) of different
intensities and directions, then find the best match
 measure transitions along 2 orthogonal directions

-63-
edge detection operators
Robert’s
operato
r
Sobel’s
operato
r

























y
f
x
f
G
G
f
y
x
Image gradient:
y
x G
G
f 



-64-
edge detection example
http://flickr.com/photos/reneemarie11/97326485/
Roberts
Sobel

-65-
second derivative in 2D
Image
Laplacian:

-66-
laplacian of roman ruins
http://flickr.com/photos/starfish235/388557119/

-67-
unsharp masking
 Unsharp masking is an image manipulation technique for increasing the
apparent sharpness of photographic images.
 The "unsharp" of the name derives from the fact that the technique uses
a blurred, or "unsharp", positive to create a "mask" of the original image.
The unsharped mask is then combined with the negative, creating a
resulting image sharper than the original.
 Steps
 Blur the image

Subtract the blurred
version from the
original (this is called
the mask)

Add the “mask” to
the original

-68-
high-boost filtering
Avg.
+
+
-
)
,
(
)
,
(
)
,
( y
x
f
y
x
Af
y
x
f lp
hb 

Unsharp mask:
high-boost with A=1

Spring 2012
Meetings 5 and 6, 7:20PM-
10PM
FUZZY SET, MEMBERSHIP FUCTION, OPERATIONS
Figure 1. Input output membership functions.
Digital Image Processing, 3rd
Ed, by R.C. Gonzalez, Richard, Prentice Hull 2007

Spring 2012
10PM
Fuzzy rule based system
Figure 2. The basic steps of a fuzzy rule based system
Digital Image Processing, 3rd

Spring 2012
10PM
Fuzzy Algorithms
The calculation complexity of the five steps approach,
introduced above, includes fuzzifycation and defuzzifycation
procedures, which are very time consuming. To speed up the
algorithms a multi-variable single output function is often
employed. As variables, this function may use multiple
membership function. The results shown on the next slide
are produced by using such a function which includes three
membership functions, for:
- dark, gray and white colors.

Spring 2012
10PM
Fuzzy Image Enhancement- Results
Figure 3. Fuzzy contrast enhancement using a single output multi-variable
function, which includes dark, bright and gray color membership
functions. Digital Image Processing, 3rd

Digital Image Processing UNIT-2.ppt