2. Background
Filter term in “Digital image processing”
is referred to the subimage
There are others term to call subimage
such as mask, kernel, template, or
window
The value in a filter subimage are
referred as coefficients, rather than
pixels.
3. Basics of Spatial Filtering
The concept of filtering has its roots in
the use of the Fourier transform for
signal processing in the so-called
frequency domain.
Spatial filtering term is the filtering
operations that are performed directly
on the pixels of an image
4. Mechanics of spatial filtering
The process consists simply of moving
the filter mask from point to point in an
image.
At each point (x,y) the response of the
filter at that point is calculated using a
predefined relationship
5. Neighbourhood Operations
Neighbourhood operations simply
operate on a larger neighbourhood of
pixels than point operations
Neighbourhoods are
mostly a rectangle
around a central pixel
Any size rectangle
and any shape filter
are possible
Origin x
y Image f (x, y)
(x, y)
Neighbourhood
6. Simple Neighbourhood
Operations
Some simple neighbourhood operations
include:
Min: Set the pixel value to the minimum in
the neighbourhood
Max: Set the pixel value to the maximum
in the neighbourhood
Median: The median value of a set of
numbers is the midpoint value in that set
(e.g. from the set [1, 7, 15, 18, 24] 15 is
the median). Sometimes the median works
better than the average
7. The Spatial Filtering Process
r s t
u v w
x y z
Origin x
y Image f (x, y)
eprocessed = v*e +
r*a + s*b + t*c +
u*d + w*f +
x*g + y*h + z*i
Filter
Simple 3*3
Neighbourhood
e 3*3 Filter
a b c
d e f
g h i
Original
Image
Pixels
*
The above is repeated for every pixel in the original
image to generate the filtered image
8. Spatial Filtering: Equation Form
a
a
s
b
b
t
t
y
s
x
f
t
s
w
y
x
g )
,
(
)
,
(
)
,
(
Filtering can be given
in equation form as
shown above
Notations are based on
the image shown to the
left
9. Spatial Filtering: Equation Form
a
a
s
b
b
t
t
y
s
x
f
t
s
w
y
x
g )
,
(
)
,
(
)
,
(
Filtering can be given
in equation form as
shown above
Notations are based on
the image shown to the
left
10. Linear spatial filtering
f(x-1,y-1) f(x-1,y) f(x-1,y+1)
f(x,y-1) f(x,y) f(x,y+1)
f(x+1,y-1) f(x+1,y) f(x+1,y+1)
w(-1,-1) w(-1,0) w(-1,1)
w(0,-1) w(0,0) w(0,1)
w(1,-1) w(1,0) w(1,1)
The result is the sum of
products of the mask
coefficients with the
corresponding pixels directly
under the mask
Pixels of image
Mask coefficients
w(-1,-1) w(-1,0) w(-1,1)
w(0,-1) w(0,0) w(0,1)
w(1,-1) w(1,0) w(1,1)
)
1
,
1
(
)
1
,
1
(
)
,
1
(
)
0
,
1
(
)
1
,
1
(
)
1
,
1
(
)
1
,
(
)
1
,
0
(
)
,
(
)
0
,
0
(
)
1
,
(
)
1
,
0
(
)
1
,
1
(
)
1
,
1
(
)
,
1
(
)
0
,
1
(
)
1
,
1
(
)
1
,
1
(
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
)
,
( y
x
f
11. Linear filtering
The coefficient w(0,0) coincides with image
value f(x,y), indicating that the mask is
centered at (x,y) when the computation of
sum of products takes place.
For a mask of size mxn, we assume that
m=2a+1 and n=2b+1, where a and b are
nonnegative integer. Then m and n are odd.
12. Linear filtering
In general, linear filtering of an image f
of size MxN with a filter mask of size
mxn is given by the expression:
a
a
s
b
b
t
t
y
s
x
f
t
s
w
y
x
g )
,
(
)
,
(
)
,
(
13. Discussion
The process of linear filtering similar to
a frequency domain concept called
“convolution”
mn
i
i
i
mn
mn z
w
z
w
z
w
z
w
R
1
2
2
1
1 ...
9
1
9
9
2
2
1
1 ...
i
i
i z
w
z
w
z
w
z
w
R
Simplify expression
w1 w2 w3
w4 w5 w6
w7 w8 w9
Where the w’s are mask coefficients, the z’s are the value of
the image gray levels corresponding to those coefficients
14. Nonlinear spatial filtering
Nonlinear spatial filters also operate on
neighborhoods, and the mechanics of
sliding a mask past an image are the
same as was just outlined.
The filtering operation is based
conditionally on the values of the pixels
in the neighborhood under
consideration
15. Smoothing Spatial Filters
Smoothing filters are used for blurring
and for noise reduction.
– Blurring is used in preprocessing steps,
such as removal of small details from an
image prior to object extraction, and
bridging of small gaps in lines or curves
– Noise reduction can be accomplished by
blurring
16. Type of smoothing filtering
There are 2 way of smoothing spatial
filters
Smoothing Linear Filters
Order-Statistics Filters
17. Smoothing Linear Filters
Linear spatial filter is simply the
average of the pixels contained in the
neighborhood of the filter mask.
Sometimes called “averaging filters”.
The idea is replacing the value of every
pixel in an image by the average of the
gray levels in the neighborhood defined
by the filter mask.
18. Two 3x3 Smoothing Linear Filters
1 1 1
1 1 1
1 1 1
1 2 1
2 4 2
1 2 1
9
1
16
1
Standard average Weighted average
20. Smoothing Linear Filters
The general implementation for filtering
an MxN image with a weighted
averaging filter of size mxn is given by
the expression
a
a
s
b
b
t
a
a
s
b
b
t
t
s
w
t
y
s
x
f
t
s
w
y
x
g
)
,
(
)
,
(
)
,
(
)
,
(
22. Image Smoothing Example
The image at the top left
is an original image of
size 500*500 pixels
The subsequent images
show the image after
filtering with an averaging
filter of increasing sizes
3, 5, 9, 15 and 35
Notice how detail begins
to disappear
29. Order-Statistics Filters
Order-statistics filters are nonlinear spatial
filters whose response is based on ordering
(ranking) the pixels contained in the image
area encompassed by the filter, and then
replacing the value of the center pixel with
the value determined by the ranking result.
Best-known “median filter”
30. Process of Median filter
Crop region of
neighborhood
Sort the values of
the pixel in our
region
In the MxN mask
the median is MxN
div 2 +1
10 15 20
20 100 20
20 20 25
10, 15, 20, 20, 20, 20, 20, 25, 100
5th
31. Result of median filter
Noise from Glass effect Remove noise by median filter
32. No reduction in contrast across steps, since
output values available consist only of those
present in the neighborhood (no averages).
– Median filtering does not shift boundaries, as
can happen with conventional smoothing filters
(a contrast dependent problem).
– Since the median is less sensitive than the
mean to extreme values (outliers), those
extreme values are more effectively removed.
33. The median is, in a sense, a more robust
“average” than the mean, as it is not
affected by outliers (extreme values).
• Since the output pixel value is one of the
neighboring values, new “unrealistic”
values are not created near edges.
• Since edges are minimally degraded,
median filters can be applied repeatedly, if
necessary.
34. The median filter is more expensive to
compute than a smoothing filter. Clever
algorithms can save time by making use of
repeating values as the neighborhood
window is slid across the image.
– Median filters are nonlinear:
This must be taken into account if you
plan on summing filtered images.
35. Another Smoothing Example
By smoothing the original image we get
rid of lots of the finer detail which leaves
only the gross features for thresholding
Original Image Smoothed ImageThresholded Image
36. Averaging Filter Vs. Median Filter Example
Filtering is often used to remove noise
from images
Sometimes a median filter works better
than an averaging filter
Original Image
With Noise
Image After
Averaging Filter
Image After
Median Filter
41. Strange Things Happen At The Edges!
Origin x
y Image f (x, y)
e
e
e
e
At the edges of an image we are missing pixels
to form a neighbourhood
e e
e
42. Strange Things Happen At The Edges!
(cont…)There are a few approaches to dealing
with missing edge pixels:
Omit missing pixels
• Only works with some filters
• Can add extra code and slow down processing
Pad the image
• Typically with either all white or all black pixels
Replicate border pixels
Truncate the image
Allow pixels wrap around the image
• Can cause some strange image artifacts
44. Strange Things Happen At The Edges!
(cont…)
Original
Image
Filtered
Image: Zero
Padding
Filtered Image:
Replicate Edge
Pixels
Filtered Image:
Wrap Around Edge
Pixels
Images
taken
from
Gonzalez
&
Woods,
Digital
Image
Processing
(2002)
48. Correlation & Convolution
The filtering we have been talking about so
far is referred to as correlation with the filter
itself referred to as the correlation kernel
Convolution is a similar operation, with just
one subtle difference
For symmetric filters it makes no difference
eprocessed = v*e +
z*a + y*b + x*c +
w*d + u*e +
t*f + s*g + r*h
r s t
u v w
x y z
Filter
a b c
d e e
f g h
Original
Image
*
49. Summary
In this lecture we have looked at the
idea of spatial filtering and in particular:
Neighbourhood operations
The filtering process
Smoothing filters
Dealing with problems at image edges
when using filtering
Correlation and convolution
50. Sharpening Spatial Filters
The principal objective of sharpening is
to highlight transitions in intensity or to
enhance detail that has been blurred,
either in error or as an natural effect of
a particular method of image
acquisition.
Applications: electronic printing,
medical imaging to industrial inspection
& autonomous guidance in military
systems.
51. Introduction
The image blurring is accomplished in
the spatial domain by pixel averaging in
a neighborhood.
Since averaging is analogous to
integration.
Sharpening could be accomplished by
spatial differentiation.
52. Foundation
We are interested in the behavior of
these derivatives in areas of constant
gray level(flat segments), at the onset
and end of discontinuities(step and
ramp discontinuities), and along gray-
level ramps.
These types of discontinuities can be
noise points, lines, and edges.
53. Definition for a first derivative
Must be zero in flat segments(constant
intensity/areas)
Must be nonzero at the onset of a gray-
level step or ramp; and
Must be nonzero along ramps of
constant slope.
54. Definition for a second derivative
Must be zero in flat areas;
Must be nonzero at the onset and end
of a gray-level step or ramp;
Must be zero along ramps of constant
slope
55. Definition of the 1st-order derivative
A basic definition of the first-order derivative
of a one-dimensional function f(x) is
)
(
)
1
( x
f
x
f
x
f
56. Definition of the 2nd-order derivative
We define a second-order derivative as the
difference
).
(
2
)
1
(
)
1
(
2
2
x
f
x
f
x
f
x
f
60. Analyze
The 1st-order derivative is nonzero
along the entire ramp, while the 2nd-
order derivative is nonzero only at the
onset and end of the ramp.
The response at and around the point is
much stronger for the 2nd- than for the
1st-order derivative
1st make thick edge and 2nd make thin edge
61. The Laplacian (2nd order derivative)
Discrete formulation
Constructing filter based on formulation
Shown by Rosenfeld and Kak[1982] that the
simplest isotropic derivative operator is the
Laplacian is defined as
2
2
2
2
2
y
f
x
f
f
62. Discrete form of derivative
)
,
(
2
)
,
1
(
)
,
1
(
2
2
y
x
f
y
x
f
y
x
f
x
f
f(x+1,y)
f(x,y)
f(x-1,y)
f(x,y+1)
f(x,y)
f(x,y-1)
)
,
(
2
)
1
,
(
)
1
,
(
2
2
y
x
f
y
x
f
y
x
f
y
f
63. 2-Dimentional Laplacian
The digital implementation of the 2-Dimensional
Laplacian is obtained by summing 2 components
2
2
2
2
2
x
f
x
f
f
)
,
(
4
)
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
f
1
1
-4 1
1
66. Effect of Laplacian Operator
Laplacian-derivative operator-highlights
intensity discontinuities.
deemphasizes regions with slowly
varying intensity levels.
Produce images that have grayish edge
lines & other discontinuities all
superimposed on a dark featureless
background.
67. Background features recovered while
preserving sharpening by simply adding
Laplacian image to original image.
Centre coefficient –ve subtract
otherwise add
74. Simplification
We will apply two step to be one mask
)
,
(
4
)
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
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,
(
)
,
( y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
g
)
1
,
(
)
1
,
(
)
,
1
(
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,
1
(
)
,
(
5
)
,
(
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
g
-1
-1
5 -1
-1
0 0
0 0
-1
-1
9 -1
-1
-1 -1
-1 -1
75. Unsharp masking
A process used by printing & publishing
industry to sharpen images consists of
subtracting a blurred version of an image from
the image itself. This process, called unsharp
masking, is expressed as
)
,
(
)
,
(
)
,
( y
x
f
y
x
f
y
x
fs
)
,
( y
x
fs
)
,
( y
x
f
)
,
( y
x
f
Where denotes the sharpened image obtained by
unsharp masking, and is a blurred version of
76. High-boost filtering
A high-boost filtered image, fhb is defined at
any point (x,y) as
1
)
,
(
)
,
(
)
,
(
A
where
y
x
f
y
x
Af
y
x
fhb
)
,
(
)
,
(
)
,
(
)
1
(
)
,
( y
x
f
y
x
f
y
x
f
A
y
x
fhb
)
,
(
)
,
(
)
1
(
)
,
( y
x
f
y
x
f
A
y
x
f s
hb
This equation is applicable general and does not state
explicity how the sharp image is obtained
77. High-boost filtering and Laplacian
If we choose to use the Laplacian, then we
know fs(x,y)
)
,
(
)
,
(
)
,
(
)
,
(
2
2
y
x
f
y
x
Af
y
x
f
y
x
Af
fhb
If the center coefficient is negative
If the center coefficient is positive
-1
-1
A+4 -1
-1
0 0
0 0
-1
-1
A+8 -1
-1
-1 -1
-1 -1
80. The Gradient (1st order derivative)
First Derivatives in image processing are
implemented using the magnitude of the
gradient.
The gradient of function f(x,y) is
y
f
x
f
G
G
f
y
x
81. Gradient
The magnitude of this vector is given by
y
x
y
x G
G
G
G
f
mag
2
2
)
(
-1 1
1
-1
Gx
Gy
This mask is simple, and no isotropic.
Its result only horizontal and vertical.
82. Robert’s Method
The simplest approximations to a first-order
derivative that satisfy the conditions stated in
that section are
2
6
8
2
5
9 )
(
)
( z
z
z
z
f
z1 z2 z3
z4 z5 z6
z7 z8 z9
Gx = (z9-z5) and Gy = (z8-z6)
6
8
5
9 z
z
z
z
f
84. Sobel’s Method
Mask of even size are awkward to apply.
The smallest filter mask should be 3x3.
The difference between the third and
first rows of the 3x3 mage region
approximate derivative in x-direction,
and the difference between the third and
first column approximate derivative in y-
direction.
85. Sobel’s Method
Using this equation
)
2
(
)
2
(
)
2
(
)
2
( 7
4
1
9
6
3
3
2
1
9
8
7 z
z
z
z
z
z
z
z
z
z
z
z
f
-1 -2 -1
0 0 0
1 2 1 1
-2
1
0
0
0
-1
2
-1
