Intensity Transformations and
Spatial Filtering: WHICH ONE LOOKS BETTER?
3.1
Intensity Transformations and
Spatial Filtering: WHICH ONE LOOKS BETTER?
3.2
1
Intensity Transformations and
Spatial Filtering
• Goal: Image enhancement seeks
– tto improve
i
the
th visual
i l appearance off an image,
i
or
– convert it to a form suited for analysis by a human or a
machine.
• Image enhancement does not, however,
– seek to restore the image, nor
– increase its information contents
• Peculiarity:
– actually, there is some evidence which suggests that a distorted image
can be more pleasing than a perfect image!
3.3
Information Content
• Suppose a source, e.g. an image generates a discrete
set of independent messages (grey levels) rk, with
probabilities pk, k=1,...,L.
• Then the information associated with rk is defined as
Ik =-log2 pk bits
• Since the sum of pk’s is 1 and each pk ≤ 1, Ik is nonnegative.
g
• This definition also implies that the information
conveyed is large when an unlikely message is
generated.
3.4
2
Intensity Transformations and
Spatial Filtering
Major Problem in Image Enhancement:
the
h llackk off a generall standard
d d off image
i
quality
lit
makes it very difficult to evaluate the performance
of different IE schemes.
Thus, Image Enhancement algorithms are mostly
application-dependent, subjective and often adhoc
hoc.
Therefore, mostly subjective criteria are used in
evaluating image enhancement algorithms.
3.5
Subjective Criteria Used in Image Enhancement
A. Goodness scale: how good an image is
Overall goodness
scale
l
Group goodness scale
Excellent (5)
Best (7)
Good (4)
Well above average (6)
Fair (3)
Slightly above average (5)
Poor (2)
Average (4)
Unsatisfactory (1)
Slightly below average (3)
B. Impairment scale: how bad the
degradation is in an image
• not noticeable (1)
• just noticeable (2)
• definitely noticeable but only slight
impairment (3)
• impairment acceptable (4)
• somewhat objectionable (5)
• definitely objectionable (6)
• extremely objectionable (7)
Well below average (2)
Worst (1)
the numbers in parenthesis indicate a numerical
weight attached to the rating.
3.6
3
Intensity Transformations and
Spatial Filtering
Spacial Domain:
g(x,y) = T[f(x,y)]
f(x,y) is the original image, g(x,y) is the output image and
T[.] is an operator on f defined over a neighborhood of
(x,y).
Special case: neighborhood size is 1 pixel Æ T[.] is called
3.7
intensity or mapping transformation function.
Lack of contrast
3.8
4
Intensity Transformations and
Spatial Filtering
Contrast Stretching
Poor contrast is the most common defect in images and is
causedd by
b reduced
d d and/or
d/ nonlinear
li
amplitude
lit d range or
poor lighting conditions.
A typical contrast stretching transformation is shown
below (examples are given later):
v
vc
γ
vb
β
va
α
u
0
L
a
b
3.9
Intensity Transformations and
Spatial Filtering
Contrast Stretching
A special case of contrast stretching is illustrated above
(bi-level output) and is called thresholding.
3.10
5
Intensity Transformations and
Spatial Filtering
Basic Grey Level Transformations
s = T[r]
3.11
Intensity Transformations and
Spatial Filtering
Image Negative
s = L -1 - r
r is the input grey level, s is the output grey level and
L-1 is the maximum value of r.
3.12
6
Intensity Transformations and
Spatial Filtering
Log Transformation
s = c log(1+r)
c is a constant and r ≥ 0.
This transformation maps a narrow range of low gray-level
input values into a wider range of output levels.
USE: Expand values of dark pixels in an image while
3.13
compressing higher level values. Inverse log will do the opposite
Intensity Transformations and
Spatial Filtering
Power-Law transformations
s = crγ (c and γ are positive constants)
3.14
7
Intensity Transformations and
Spatial Filtering
Power-law Transformation: Gamma Correction
3.15
Intensity Transformations and
Spatial Filtering
Power-Law Transformation for MR Image Enhancement
Magnetic
Resonance (MR)
image of a fractured
human spine
3.16
8
Intensity Transformations and
Spatial Filtering
Power-Law Transformation for Aerial Image
Original aerial image
has a washed-out
appearance, i.e.
compression of
grey levels is needed.
3.17
Intensity Transformations and
Spatial Filtering
Piecewise-Linear Transformations
1. Contrast
S
Stretching
hi
a scanning electron
microscope image
of pollen magnified
700 times.
3.18
9
Intensity Transformations and
Spatial Filtering
Piecewise-Linear Transformations
2. Level slicingg
an input
image
result after
applying
transformation
in (a).
Applications: enhancing features, e.g. masses of water in satellite imagery
3.19
and enhancing flaws in X-ray images.
Intensity Transformations and
Spatial Filtering
Piecewise-Linear Transformations, another example
10
Intensity Transformations and
Spatial Filtering
Piecewise-Linear Transformations
•3.
3. Bit-plane
Bit plane Slicing
3.21
Intensity Transformations and
Spatial Filtering
3. Bit-plane
Bit plane Slicing: example 1:
3.22
11
Intensity Transformations and
Spatial Filtering
3.23
3. Bit-plane Slicing: Example 2:
12
Intensity Transformations and
Spatial Filtering
Histogram Processing
Basic image types:
dark
light
low-contrast
high-contrast
3.25
Intensity Transformations and
Spatial Filtering
Histogram Processing
Histogram processing re-scales an image so that
the enhanced image histogram follows some
desired form.
The modification can take on many forms:
– histogram equalization
equalization, or
– histogram shaping
•
e.g. exponential or hyperbolic histogram
3.26
13
Intensity Transformations and
Spatial Filtering
Histogram Equalization
• To transfer the gray levels so that the
histogram of the resulting image is equalized
to be a constant:
• The purpose:
– to equally use all available gray levels;
• This figure shows that for any given mapping
function
between the input and
output images, the following holds:
• i.e., the number of pixels mapped from x to
y is unchanged.
3.27
Intensity Transformations and
Spatial Filtering
Histogram Equalization
To equalize the histogram of the input image, we let p(y) be a constant. Assume
that the gray levels are in the range of 0 and 1 (0<x<1, 0<y<1). Then we
have:
i.e., the mapping function for histogram equalization is:
where
is the cumulative probability distribution of the input image, which is
monotonically non-decreasing function.
3.28
14
Intensity Transformations and
Spatial Filtering
Histogram Equalization
•
Histogram equalization is
based on the following idea:
– If p(x) is large, y=f(x)
has a steep slope, dy
will be wide, causing
p(y) to be low;
– If p(x) is small, y=f(x)
has a shallow slope
slope, dy
will be narrow, causing
p(y) to be high.
3.29
Intensity Transformations and
Spatial Filtering
Histogram Equalization
For discrete gray levels, the gray level of the input takes
one of the discrete values;; and the continuous
mapping function:
becomes discrete:
where Pi is the probability for the gray level of any
given pixel to be (0<i<L):
3.30
15
Intensity Transformations and
Spatial Filtering
Histogram Equalization
•
The resulting function is in the range
converted to the gray levels
ways:
and it needs to be
by one of the two following
•
where
is the floor, or the integer part of a real number x, and adding
0.5 is for proper rounding. Note that while both conversions map
to the highest gray level L - 1, the second conversion also maps
to 0
to stretch the gray levels of the output image to occupy the entire dynamic
range
.
3.31
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Example: Assume the images have pixels in 8 gray levels. The following table shows
the equalization process corresponding to the two conversion methods above:
0/7
790
0.19
0.19
1/7
0.19
0.19
0/7
0.19
0.19
1/7
1023
0.25
0.44
3/7
0.25
0.44
2/7
0.25
0.44
2/7
850
0.21
0.65
5/7
0.21
0.65
4/7
0.21
0.65
3/7
656
0.16
0.81
6/7
5/7
0.16
0.81
4/7
329
0.08
0.89
6/7
6/7
0.08
0.89
5/7
245
0.06
0.95
7/7
6/7
122
0.03
0.98
7/7
7/7
81
0.02
1.00
7/7
0.11
1.00
0.24
0.89
7/7
7/7
0.11
1.00
7/7
y1j=[y’(L-1)+0.5], y2j=[(y’-y’min)/(1-y’min) (L-1) +0.5],
3.32
16
Intensity
Transformations and
Spatial Filtering
Histogram
Equalization
3.33
Intensity Transformations and
Spatial Filtering
Histogram Equalization
• In the following example, the
histogram of a given image is
equalized.
li d Although
Alth
h the
th
resulting histogram may not
look constant, but the
cumulative histogram is a exact
linear ramp indicating that the
density histogram is indeed
equalized. The density
histogram is not guaranteed to
be a constant because the pixels
of the same gray level cannot be
separated to satisfy a constant
distribution.
3.34
17
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Programming Hints:
Find histogram of given image:
Build lookup table:
Image Mapping:
3.35
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Histogram equalization produces an output image by point rescaling such that the histogram of the new image is uniform.
uniform
Assume an image R of size M=NxN.
• Let
H R ( j) =
# pixels with value rj
M
• where j=1,2, ..., J. HR(j) is just the fractional number of pixels
whose amplitude is quantized to reconstruction level rj.
We want to produce an enhanced image S whose histogram
(normalized) is
H S (i ) =
1
K
i = 1,2,..., K
i.e. a histogram that is as flat as possible.
3.36
18
Intensity Transformations and
Spatial Filtering
Histogram Equalization
The scaling algorithm:
1 compute the
1.
h average value
l off the
h histogram.
hi
2. starting at the lowest gray level of the original,
combine pixels in the quantization bands until the
sum is closest to the average.
3. Rescale all of these pixels to the first reconstruction
l l at the
level
h midpoint
id i off the
h enhanced
h
d image
i
first
fi
quantization band.
4. Repeat for higher gray level values.
3.37
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Remark:
1 Hi
1.
Histogram equalization
li i works
k best
b on images
i
with
ih
details “hidden” in dark regions.
2. Good quality originals are usually degraded when their
histograms are equalized.
3. Other histogram modifications are possible,
ex. a useful one is called histogram hyperbolization.
3.38
19
Intensity Transformations and
Spatial Filtering
Histogram Equalization
4. However, a normalized image histogram is just an approximation of the
image probability density function! Therefore, histogram modification is
nothing
thi but
b t modification
difi ti off the
th underlying
d l i pdf!
df!
E.g., in histogram equalization, the enhanced image is desired to have a uniform
p.d.f. That is, if pr(r) is the pdf of the original image r, then the pdf ps(s) of
the enhanced image s is uniform.
This is not a difficult probability problem:
Consider a random variable (RV) R with pdf pR(r) and cumulative distribution
function (cdf) PR(r).
Find a transformation T(.) such that the new RV S=T(R) is uniform, i.e., pS(s) is
a uniform density.
3.39
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Stop/Resume
Claim: S=T(R)= PR(r) will do it.
Proof:
r
S = PR (r ) = ∫ p R (t ) dt
0
Let's find the cdf of S (PS(s)).
PS ( s ) = Pr ob[ PR (r ) ≤ s ] = Pr ob[ R ≤ PR−1 ( s )]
for 0 ≤ s ≤ 1.
=∫
PR−1 ( s )
0
pR (t ) dt = PR ( PR−1 ( s)) = s
3.40
20
Intensity Transformations and
Spatial Filtering
Histogram Equalization
In discrete case,
r
S = PR (r ) = ∫ pR (t ) dt
0
becomes
k
k
nj
j =0
j =0
n
sk = T (rk ) = ∑ pR (rj ) =∑
(*)
, , ,
for k=0,1,...,L-1.
3.41
Intensity Transformations and
Spatial Filtering
Histogram Equalization
any transformation such as plotted above can be used as a
transformation provided that
• it is single valued and monotonically increasing, and
• it takes values in [0,1] for inputs in [0,1].
3.42
21
Intensity Transformations and
Spatial Filtering
Histogram Equalization
3.43
Intensity Transformations and
Spatial Filtering
Histogram Equalization
k
k
nj
j =0
j =0
n
sk = T (rk ) = ∑ pR (rj ) =∑
(*)
3.44
22
Intensity Transformations and
Spatial Filtering
3.45
Intensity Transformations and
Spatial Filtering
Histogram Equalization
Photo of the Mars moon Phobos and its higtogram
g g
3.46
23
Intensity Transformations and
Spatial Filtering
Histogram Equalization
3.47
Intensity Transformations and
Spatial Filtering
3.48
24
Intensity Transformations and
Spatial Filtering
Localized Histogram Equalization
3.49
25
Intensity Transformations and
Spatial Filtering
3.51
Intensity Transformations and
Spatial Filtering
mS ( x , y )
σ S ( x, y )
(1, E )
⎧E. f (x, y) if mS( x, y) < k0mG and k1σG < σS( x, y) < k2σG ⎫
g(x, y) = ⎨
⎬
otherwise
⎩ f (x, y)
⎭
3.52
26
Intensity Transformations and
Spatial Filtering
⎧E. f (x, y) if mS( x, y) < k0mG and k1σG < σS( x, y) < k2σG ⎫
g(x, y) = ⎨
⎬
otherwise
⎩ f (x, y)
⎭
3.53
Intensity Transformations and
Spatial Filtering
Enhancement using logic operations
AND
OR
3.54
27
Intensity Transformations and
Spatial Filtering
Enhancement using arithmetic operations
3.55
Note: Fig. 3.14(a) is the original picture in 3rd ed.
28
Intensity Transformations and
Spatial Filtering
Enhancement using arithmetic operations
3.57
Intensity Transformations and
Spatial Filtering
Enhancement using spatial averaging operations
When images are displayed (or printed), they
often have suffered from noise and
interferences from several sources including:
– electrical sensor noise,
– photographic grain noise, and
– channel errors.
These noise
Th
i effects
ff t can usually
ll be
b removedd by
b
simple ad hoc “noise-cleaning” techniques
applied to local neighborhoods of input pixels.
3.58
29
Intensity Transformations and
Spatial Filtering
Enhancement using spacial averaging operations
Consider a noisy image:
g ( x, y ) = f ( x, y ) + η ( x, y )
where the second term is noise which is uncorrelated
with the input and has zero mean. Then, averaging K
different noisy images:
g (x, y) =
1
K
K
∑
i=1
g i(x, y)
produces an output image with
E[ g ( x, y )] = f ( x, y ) and σ g2 =
1 2
ση
K
3.59
Intensity Transformations and
Spatial Filtering
Enhancement using averaging operations
original
i i l
result of
aver 8 noisy
images
noisy image,
image N(0,64
N(0 642)
16 noisy images
128 noisy images
64 noisy
images
3.60
30
Intensity Transformations and
Spatial Filtering
K=8
K=16
K=64
notice how the noise
variance is decreasing
with
i h increasing
i
i K.
K
K=128
3.61
Intensity Transformations and
Spatial Filtering
Each pixel u(m,n) is
replaced by a
weighted average of
its neighbourhood
pixels
3.62
31
Intensity Transformations and
Spatial Filtering
Linear Filtering
M
g ( x, y ) =
N
∑ ∑ w(m, n) f ( x + m, y + n)
m=− M n=− N
M
N
∑ ∑ w(m, n)
m=− M n=− N
where g(x,y) is the output image and f(x,y) is the
input image. In the mask above, M=N=1.
3.63
Intensity Transformations and
Spatial Filtering
Spacial averaging operations
Consider again a noisy image:
g ( x, y ) = f ( x, y ) + η ini ( x, y )
where the second term is noise which is uncorrelated with the input and has zero
mean. Let’s apply a local averaging filter (all weights are equal) with size
K=(2M+1)x(2N+1):
g ( x, y ) =
1
K
⎡1
⎤
∑ ∑ g ( x, y ) = ⎢ K ∑ ∑ f ( x, y ) ⎥ + η
⎣
( x , y )∈W
( x , y )∈W
⎦
out
( x, y )
produces an output image with
2
σ out
=
1 2
σ in
K
Therefore, if the input is constant over W, the SNR has improved by a factor of
K!!
3.64
32
Intensity Transformations and
Spatial Filtering
3.65
Intensity Transformations and
Spatial Filtering
original
3 x 3 mask
5 x 5 mask
9 x 9 mask
15 x 15 mask
35 x 35 mask
3.66
33
Intensity Transformations and
Spatial Filtering
Linear Filtering Example
Image from the hubble space telescope in orbit around the Earth.
Here, we want to blur the image in order to see large objects.
3.67
Intensity Transformations and
Spatial Filtering
Q: What happen when important details must be
preserved or the noise is non-Gaussian?
non Gaussian?
A: may consider a number of techniques:
– median and order statistics filtering,
– sharpening spacial filters
• directional
di ti l filtering,
filt i
– unsharp masking
– hybrid combinations
3.68
34
Intensity Transformations and
Spatial Filtering
Nonlinear Filtering
Median or order statistics filters may perform much better in
the presence of non-Gaussian noise (Nonlinear Signal
Processing course!)
3.69
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters
(subtract pixels from right to left)
(subtract 1st derivative again
from right to left)
3.70
35
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters
First Order Derivative
Second Order Derivatives
• First order derivative
– produces thicker edges
– has stronger response to grey-level steps
• Second order derivative
– has a much stronger response to details
– pproduces a double response
p
at stepp changes
g in ggrey
y level
– has a stronger response to a line than to a step and to a point than to a
line.
• Conclusion
– Second derivative is more useful to enhance image details
3.71
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters: Laplacian
• Isotropic 2nd order derivative (Laplacian)
∇2 f =
• In digital form:
∂2 f
∂2 f
+
∂2 x2 ∂2 y2
∂2 f
= f ( x + 1, y ) + f ( x − 1, y ) − 2 f ( x, y )
∂2x2
• in the x-direction and in the y-direction:
∂2 f
= f ( x, y + 1) + f ( x, y − 1) − 2 f ( x, y )
∂2 y2
• 2-D Laplacian:
∇2 f =
∂2 f
∂2 f
+ 2 2
2 2
∂ x
∂ y
• this can be implemented using the mask in the next
slide
3.72
36
Intensity Transformations and
Spatial Filtering
isotropic for
rotations in
increments of 90°
* 45°
negatives of the above two masks
3.73
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters: Laplacian
• Enhancement using Laplacian:
⎧ f ( x, y ) − ∇ 2 f ( x, y ) if the mask center coeff is negative
g ( x, y ) = ⎨
2
⎩ f ( x, y ) + ∇ f ( x, y ) if the mask center coeff is positive
3.74
37
Intensity Transformations and
Spatial Filtering
image of the north
pole of the moon
result of
filtering with
the 45o mask
Laplacian image
scaled for display
purpose
enhanced
image
3.75
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters: Laplacian
• The previous expression :
⎧ f ( x, y ) − ∇ 2 f ( x, y ) if the mask center coeff is negative
g ( x, y ) = ⎨
2
⎩ f ( x, y ) + ∇ f ( x, y ) if the mask center coeff is positive
• The above two operations can be combined and thus
simplified:
g(x, y) = 5 f (x, y) −[ f (x +1, y) + f (x −1, y) + f (x, y +1) + f (x, y −1)]
• This operation can be implemented using the mask in
the next slide.
3.76
38
Intensity Transformations and
Spatial Filtering
90°
45°
results for the 90°
SEM iimage
results for the 45°
Notice how much
sharper it is!
3.77
Intensity Transformations and
Spatial Filtering
Laplacian with high-boost filtering
gives better results if the original image is darker than
desired see example in next slide.
desired,
slide
3.78
39
Intensity Transformations and
Spatial Filtering
-1 -1 -1
-1 8 -1
-1 -1 -1
b
-11 -11 -11
-1 9.7 -1
-1 -1 -1
-1 -1 -1
-1 9 -1
-1 -1 -1
3.79
Intensity Transformations and
Spatial Filtering
Sharpening Spacial Filters: First Derivative, the Gradient
• First derivatives are implemented using the magnitude
of the gradient
gradient. The gradient is given by:
⎡ ∂f ⎤
⎡G x ⎤ ⎢ ∂x ⎥
∇f = ⎢ ⎥ = ⎢ ∂f ⎥
⎣⎢G y ⎦⎥ ⎢ ⎥
⎣⎢ ∂y ⎦⎥
• The magnitude of the gradient is given by:
⎡⎛ ∂f ⎞ 2 ⎛ ∂f ⎞ 2 ⎤
∇f = mag (∇f ) = [G x2 + G y2 ]1 / 2 = ⎢⎜ ⎟ + ⎜⎜ ⎟⎟ ⎥
⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦
1/ 2
• note that components of the gradient are linear operators, but the magnitude
is not! Also the partial derivatives are not isotropic, but the magnitude is!
• The magnitude can be (for implementation reasons) approximated by:
∇f ≈ G x + G y
3.80
40
Intensity Transformations and
Spatial Filtering
f(x,y)
}
Roberts cross-gradient
operator (2x2)
∇f ≈| z 9 − z 5 | + | z 8 − z 6 |
}
Gy
Sobel operators (3x3)
Gx
3.81
Intensity Transformations and
Spatial Filtering
3.82
41
original
original+
Laplacian
Laplacian of origginal
Intensity Transformations and
Spatial Filtering
Combining spatial enhancement methods
•Single technique may not
produce desirable results
•Must thus devise a strategy
for the given application at hand
This application:
nuclear whole body scan
want to detect deseases, e.g.
bone infection and tumors
Sobel of orig
Strategy:
•use Laplacian to highlight
details,
•gradient to enhance edges,
•grey-level trans. to
increase dynamic range
3.83
Intensity Transformations and
Spatial Filtering
smoothed
Sobel
orig + mask
Mask obtained by:
(orig + Lap) x (smoothed Sobel)
Fi l result
Final
l
power law to
orig. + mask
3.84
42
Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com
Chapter 3
Intensity Transformations & Spatial Filtering
Self-study: Section 3.8. Using Fuzzy Techniques for Intensity
Transformations and Spatial Filtering.
© 1992–2008 R. C. Gonzalez & R. E. Woods
43