# Image Filtering

Τεχνίτη Νοημοσύνη και Ρομποτική

5 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

82 εμφανίσεις

E.G.M. PetrakisFiltering1
Linear Systems

Many image processing (filtering)
operations are modeled as a linear system
Linear System
h(x,y)
δ(x,y)
dxdy
y
y
x
x
h
y
x
f
y
x
h
y
x
f
y
x
g
∫∫

=

=
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
E.G.M. PetrakisFiltering2
Impulse Response

System’s output to an impulse δ(x,y)
δ(x,y)
t
I
0
E.G.M. PetrakisFiltering3
Space Invariance

g(x,y) remains the same irrespective of the
position of the input pulse

Linear Space Invariance (LSI)
Space Inv. Syst.
δ(x-x0,y-y0)
h(x-x0,y-y0)
LSI System
af1(x,y)+bf2(x,y)
ah1(x,y)+bh2(x,y)
E.G.M. PetrakisFiltering4
Discrete Convolution

The filtered image is described by a discrete
convolution

The filter is described by a n x
m discrete

=
=

=

=
m
l
n
k
l
j
k
i
h
l
k
f
j
i
h
j
i
f
j
i
g
1
1
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
E.G.M. PetrakisFiltering5
Computing Convolution

Invert

Put the mask over each pixel of f(i,j)

For each (i,j) on image
h(i,j)=Ap1+Bp2+Cp3+Dp4+Ep5+Fp6+Gp7+Hp8+Ip9
E.G.M. PetrakisFiltering6
Image Filtering

Images are often corrupted by random
variations in intensity, illumination, or have
poor contrast and can’t be used directly

Filtering: transform pixel intensity values to
reveal certain image characteristics

Enhancement:
improves contrast

Smoothing:
remove noises

Template matching:
detects known patterns
E.G.M. PetrakisFiltering7
Template Matching

Locate the template in the image
E.G.M. PetrakisFiltering8
Computing Template Matching

Match template with image at every pixel

distance 
0
: the template matches the image
at the current location

t(x,y):
template

M,N size of the template
()
(
)
(
)
[]
∑∑
=

=

=
m
x
n
y
y
y
x
x
t
y
x
f
y
x
D
00
2
2
,
,
,
E.G.M. PetrakisFiltering9
()
()
(
)
[]
()
()
()
(
)
∑∑
∑∑
∑∑
∑∑
=

=

=

=

=

=

=

=

+

=

=
M
x
N
y
M
x
N
y
M
x
N
y
M
x
N
y
y
y
x
x
t
y
x
f
y
y
x
x
t
y
x
f
y
y
x
x
t
y
x
f
y
x
D
11
11
2
11
2
11
2
2
,
,
2
,
,
,
,
,
background
constant
correlation:
convolution of
f(x,y) with t(-x,-y)
E.G.M. PetrakisFiltering10
true best match
0
0
0
0
0
0
5
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
0
1
1

6
6
7
8
5
3
3
7
5
3
5
4
1
1
1
1
1
1
1
1
1
template
image
correlation
false match
noise
E.G.M. PetrakisFiltering11
Observations

If the size of f(x,y) is n x
n
and the size
of the template is m x
m the result is
accumulated in a (n-m-1) x
(n+m-1)
matrix

Best match
: maximum value in the
correlation matrix but,

false matches due to noise
E.G.M. PetrakisFiltering12

Sensitive to noise

Sensitive to variations in orientation and scale

Sensitive to non-uniform illumination

Normalized Correlation
(1:image, 2:template):

E
: expected value
()
(
)
(
)(
)
()
(
)
2
1
2
1
2
1
,
q
q
q
E
q
E
q
q
E
y
x
N
σ
σ

=
()
(
)
()
[
]
2
1
2
2
q
E
q
E
q

=
σ
E.G.M. PetrakisFiltering13
Histogram Modification

Images with poor contrast usually contain
unevenly distributed gray values

Histogram Equalization
is a method for
stretching the contrast by uniformly
distributing the gray values

enhances the quality of an image

useful when the image is intended for viewing

not always useful for image processing
E.G.M. PetrakisFiltering14
Example

The original image has very poor contrast

the gray values are in a very small range

The histogram equalized image has better contrast
E.G.M. PetrakisFiltering15
Histogram Equalization Methods

Background Subtraction: subtract the
“background” if it hides useful information

f’(x,y) = f(x,y) –
f
b(x,y)

Static & Dynamic histogram equalization
methods

Histogram scaling
(static)

Statistical scaling (dynamic)
E.G.M. PetrakisFiltering16
Static Histogram Scaling

Scale uniformly entire histogram range:
–[
z
1,zk]:
available range of gray values:

[a,b]:
range of intensity values in image:

scale [a,b] to cover the entire range [z1,zk]

for each z in [a,b] compute

the resulting histogram may have gaps
1
1
)
(
'
z
a
z
a
b
z
z
z
k
+

=
E.G.M. PetrakisFiltering17
Statistical Histogram Scaling

Fills all histogram bins continuously
–p
i
: number of pixels at level zi
input histogram
–q
i
: number of pixels at level zi
output histogram
–k
1= k2=…
: desired number of pixels in histogram bin

Algorithm:
1.
Scan the input histogram from left to right to find
k1
:

all pixels with values z1,z2,…,zk-1
become z1

=

=
<

1
1
1
1
1
1
k
i
i
k
i
i
p
q
p
E.G.M. PetrakisFiltering18
Algorithm (conted)
2.
Scan the input histogram from k1
and to the right to
find k2:

all pixels zk1,zk1+1,…,zk2
become z2

Continue
until the input histogram is exhausted

might also leave gaps in the histogram

=

=
<
+

2
2
1
2
1
1
1
k
i
i
k
i
i
p
q
q
p
E.G.M. PetrakisFiltering19
Noise

Images are corrupted by random
variations
in intensity values called noise due to non-
perfect camera acquisition or environmental
conditions.

Assumptions:

pixel

White noise: The value at a point is
independent on the value at any other point.
E.G.M. PetrakisFiltering20
Common Types of Noise

Salt and pepper noise: random occurrences of both
black and white intensity values

Impulse noise: random occurrences of white
intensity values

Gaussian noise: impulse noise but its intensity
values are drawn from a Gaussian distribution

noise intensity value:

k: random value in [a,b]

σ
: width of Gaussian

models sensor noise (due to camera electronics)
2
2
2
)
,
(
σ
k
e
y
x
g

=
E.G.M. PetrakisFiltering21
Examples of Noisy Images
a.
Original image
b.
Original image
c.
Salt and pepper noise
d.
Impulse noise
e.
Gaussian noise
E.G.M. PetrakisFiltering22
Noise Filtering

Basic Idea: replace each pixel intensity value with
an new value taken over a neighborhood of fixed
size

Mean filter

Median filter

The size of the filter controls degree of smoothing

large filter 
large neighborhood 
intensive
smoothing
E.G.M. PetrakisFiltering23
Mean Filter

Take the average of intensity values in a m x
n
region of each pixel (usually m = n)

take the average as the new pixel value

the normalization factor mn
preserves the range
of values of the original image
()
()
∑∑
∈∈
=
m
kn
l
l
k
f
mn
j
i
h
,
1
,
E.G.M. PetrakisFiltering24
Mean Filtering as Convolution

Compute the convolution of the original
image with

simple filter, the same for all types of noise

disadvantage: blurs image, detail is lost
()

=
1
1
1
1
1
1
1
1
1
3
3
1
,
j
i
g
E.G.M. PetrakisFiltering25
Size of Filter
•T
h
e

size
of the filter controls the amount of
filtering (and blurring).

5 x 5, 7 x 7 etc.

different weights might also be used

normalize by sum of weights in filter

=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
5
1
)
,
(
j
i
g
E.G.M. PetrakisFiltering26
Examples of Smoothing

From left to right: results of 3 x 3, 5 x 5 and 7 x 7 mean filters
E.G.M. PetrakisFiltering27
Median Filter

Replace each pixel value with the median
of the gray values in the region of the
pixel:
1.
take a 3 x 3 (or 5 x 5 etc.) region centered
around pixel (i,j)
2.
sort the intensity values of the pixels in the
region into ascending order
3.
select the middle value as the new value of
pixel (i,j)
E.G.M. PetrakisFiltering28
Computation of Median Values

Very effective
in removing salt and pepper or impulsive
noise while preserving image detail

Disadvantages: computational complexity, non linear filter
E.G.M. PetrakisFiltering29
Examples of Median Filtering

From left to right: the results of a 3 x 3, 5 x 5 and 7 x 7 median filter
E.G.M. PetrakisFiltering30
Gaussian Filter

Filtering with a m
x
m

the weights are computed according to a
Gaussian function:

σ
is user defined
2
2
2
2
)
,
(
σ
j
i
e
c
j
i
g
+

=

1
4
7
10
7
4
1
4
12
26
33
26
12
4
7
26
55
71
55
26
7
10
33
71
91
71
33
10
7
26
55
71
55
26
7
4
12
26
33
26
12
4
1
4
7
10
7
4
1
Example:
m = n = 7
σ2
= 2
E.G.M. PetrakisFiltering31
Properties of Gaussian Filtering

Gaussian smoothing
is very effective for removing
Gaussian noise

The weights
give higher significance to pixels near the
edge (reduces edge blurring)

They are linear
low pass filters

Computationally efficient
(large filters are implemented
using small 1D filters)

Rotationally symmetric
(perform the same in all directions)

The degree of smoothing
is controlled by σ
(larger σ
for
more intensive smoothing)
E.G.M. PetrakisFiltering32

A 3-D plot of a 7 x & Gaussian mask: filter symmetric and isotropic
E.G.M. PetrakisFiltering33
Gaussian Smoothing

The results of smoothing an image corrupted with
Gaussian noise with a 7 x 7 Gaussian mask
E.G.M. PetrakisFiltering34
Computational Efficiency

Filtering twice
with g(x) is equivalent to
filtering with a larger filter
with

Assumptions
()
(
)
(
)
()
()
()
()
y
x
g
y
x
g
y
x
f
y
x
h
y
x
g
y
x
f
y
x
h
,
,
,
,
,
,
,

=

=
()
2
2
2
,
σ
x
e
y
x
g

=
σ
σ
2
=

E.G.M. PetrakisFiltering35
()
()
()
()
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
σ
σ
ξ
σ
ξ
σ
ξ
σ
ξ
σ
ξ
σ
π
ξ
ξ
ξ
x
x
x
x
x
e
d
e
d
e
e
d
e
e
x
g
x
g

+

+

+

+

+∞

=
=
=
=
=
=

E.G.M. PetrakisFiltering36
Observations

Filter an image with a large Gaussian

equivalently, filter the image twice with a
Gaussian with small
σ

filtering twice with a m x
n
Gaussian is
equivalent to filtering with a (n + m -
1) x
(n +
m -
1)
filter

this implies a significant reduction in
computations
σ
2
E.G.M. PetrakisFiltering37
Gaussian Separability
()
(
)
(
)
()
(
)
()
()
=

=
=

=
=

=
∑∑
∑∑
==
+

==
m
k
n
l
l
k
m
k
n
l
l
j
k
i
f
e
l
j
k
i
f
l
k
g
j
i
g
j
i
f
j
i
h
11
2
11
,
,
,
,
,
,
22
2
σ
E.G.M. PetrakisFiltering38
()
()

∑∑
=

==

=
=

=
m
k
k
m
k
h
n
l
l
k
j
k
i
h
e
l
j
k
i
f
e
e
1
2
1
'
1
2
2
,
,
2
2
2
2
2
2
σ
σ
σ
4
4
43
4
4
42
1
1-D Gaussian
horizontally
1-D Gaussian
vertically

The order of convolutions can be reversed
E.G.M. PetrakisFiltering39

An example of the separability
of Gaussian
convolution

E.G.M. PetrakisFiltering40
Gaussian Separability

Filtering with a 2D Gaussian
can be
implemented using two 1D Gaussian
horizontal filters as follows:

first filter with an 1D Gaussian

take the transpose of the result

convolve again with the same filter

transpose the result

Filtering with two 1D Gausians
is faster
!!
E.G.M. PetrakisFiltering41
a.
Noisy image
b.
Convolution with
1D horizontal