Digital Image Processing: Introduction -...
Transcript of Digital Image Processing: Introduction -...
University of Ioannina - Department of Computer Science
Spatial Filtering
Christophoros [email protected]
Digital Image Processing
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Contents
In this lecture we will look at spatial filtering techniques:
– Neighbourhood operations– What is spatial filtering?– Smoothing operations– What happens at the edges?– Correlation and convolution– Sharpening filters– Combining filtering techniques
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Neighbourhood Operations
Neighbourhood operations simply operate on a larger neighbourhood of pixels than point operationsNeighbourhoods are mostly a rectangle around a central pixelAny size rectangle and any shape filter are possible
Origin x
y Image f (x, y)
(x, y)Neighbourhood
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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
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Simple Neighbourhood Operations Example
123 127 128 119 115 130
140 145 148 153 167 172
133 154 183 192 194 191
194 199 207 210 198 195
164 170 175 162 173 151
Original Image x
y
Enhanced Image x
y
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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
FilterSimple 3*3
Neighbourhood e 3*3 Filter
a b c
d e f
g h iOriginal Image
Pixels
*
The above is repeated for every pixel in the original image to generate the filtered image
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Spatial Filtering: Equation Form
∑∑−= −=
++=a
as
b
bttysxftswyxg ),(),(),(
Filtering can be given in equation form as shown aboveNotations are based on the image shown to the left
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Smoothing Spatial Filters
One of the simplest spatial filtering operations we can perform is a smoothing operation
– Simply average all of the pixels in a neighbourhood around a central value
– Especially useful in removing noise from images
– Also useful for highlighting gross detail
1/9 1/9 1/9
1/9 1/9 1/9
1/9 1/9 1/9
Simple averaging filter
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Smoothing Spatial Filtering
1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9
Origin x
y Image f (x, y)
e = 1/9*106 + 1/9*104 + 1/9*100 + 1/9*108 + 1/9*99 + 1/9*98 + 1/9*95 + 1/9*90 + 1/9*85
= 98.3333
FilterSimple 3*3
Neighbourhood106
104
99
95
100 108
98
90 85
1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9
3*3 SmoothingFilter
104 100 108
99 106 98
95 90 85
Original Image Pixels
*
The above is repeated for every pixel in the original image to generate the smoothed image.
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Image Smoothing Example
The image at the top left is an original image of size 500*500 pixelsThe subsequent images show the image after filtering with an averaging filter of increasing sizes
– 3, 5, 9, 15 and 35Notice how detail begins to disappear
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Weighted Smoothing Filters
More effective smoothing filters can be generated by allowing different pixels in the neighbourhood different weights in the averaging function
– Pixels closer to the central pixel are more important
– Often referred to as a weighted averaging
1/162/16
1/16
2/164/16
2/16
1/162/16
1/16
Weighted averaging filter
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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
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Original Image Smoothed Image Thresholded Image
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Averaging Filter Vs. Median Filter Example
Filtering is often used to remove noise from imagesSometimes a median filter works better than an averaging filter
Original ImageWith Noise
Image AfterAveraging Filter
Image AfterMedian Filter
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Spatial smoothing and image approximation
Spatial smoothing may be viewed as a process for estimating the value of a pixel from its neighbours.
What is the value that “best” approximates the intensity of a given pixel given the intensities of its neighbours?
We have to define “best” by establishing a criterion.Im
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Spatial smoothing and image approximation (cont...)
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[ ]2
1
( )N
i
E x i m=
= −∑ [ ]2
1arg min ( )
N
m im x i m
=
⎧ ⎫⇔ = −⎨ ⎬⎩ ⎭∑
0Em∂
=∂
( )1
2 ( ) 0N
ix i m
=
⇔ − − =∑1 1
( )N N
i ix i m
= =
⇔ =∑ ∑
1( )
N
ix i Nm
=
⇔ =∑1
1 ( )N
im x i
N =
⇔ = ∑
A standard criterion is the the sum of squares differences.
The average value
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Spatial smoothing and image approximation (cont...)
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1
( )N
i
E x i m=
= −∑1
arg min ( )N
m im x i m
=
⎧ ⎫⇔ = −⎨ ⎬⎩ ⎭∑
0Em∂
=∂
( )1
( ) 0,N
isgn x i m
=
⇔ − − =∑1 0
( ) 0 01 0
xsign x x
x
>⎧⎪= =⎨⎪− <⎩
Another criterion is the the sum of absolute differences.
There must be equal in quantity positive and negative values.
median{ ( )}m x i=
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Spatial smoothing and image approximation (cont...)
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– The median filter is non linear:
– It works well for impulse noise (e.g. salt and pepper).
– It requires sorting of the image values.– It preserves the edges better than an average
filter in the case of impulse noise.– It is robust to impulse noise at 50%.
median{ } median{ } median{ }x y x y+ ≠ +
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Spatial smoothing and image approximation (cont...)
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Example x[n] 1 1 1 1 1 2 2 2 2 2
Impulse noise x[n] 1 3 1 1 1 2 3 2 2 3
Median(N=3) x[n] - 1 1 1 1 2 2 2 2 -
Average (N=3)
x[n] - 1.7 1.7 1 1.3 2 2.3 2.3 2.2 -
edge
The edge is smoothed
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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
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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 artefacts
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Strange Things Happen At The Edges! (cont…)
OriginalImage
Filtered Image: Zero Padding
Filtered Image: Replicate Edge Pixels
Filtered Image: Wrap Around Edge Pixels
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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 kernelConvolution 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 zFilter
a b c
d e e
f g hOriginal Image
Pixels
*
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Effect of Low Pass Filtering on White Noise
Let f be an observed instance of the image f0corrupted by noise w:
with noise samples having mean value E[w(n)]=0 and being uncorrelated with respect to location:
0f f w= +
2 ,[ ( ) ( )]
0,m n
E w m w nm n
σ⎧ == ⎨
≠⎩
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Effect of Low Pass Filtering on White Noise (cont...)
Applying a low pass filter h (e.g. an average filter) by convolution to the degraded image:
The expected value of the output is:
The noise is removed in average.
*g h f= 0*( )h f w= + 0* *h f h w= +
0[ ] [ * ] [ * ]E g E h f E h w= + 0* * [ ]h f h E w= +
0* *0h f h= + 0*h f=
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Effect of Low Pass Filtering on White Noise (cont...)
What happens to the standard deviation of g?Let where the bar represents filtered versions of the signals, then
( )22 2[ ] [ ]g E g E gσ = −
0 0* *g h f h w f w= + = +
2 20 0[( ) ] ( )f w f= Ε + −
2 2 20 0 0[( ) ( ) 2 ] ( )f w f w f= Ε + + −
20[( ) ] 2 [ ] [ ]E w E f E w= + 2[( ) ]E w=
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Effect of Low Pass Filtering on White Noise (cont...)
Considering that h is an average filter, we have at pixel n:
Therefore,
2[( ( )) ]E w n
( ) ( * )( )w n h w n=( )
1 ( )k n
w kN ∈Γ
= ∑
2
( )
1 ( )k n
E w kN ∈Γ
⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑
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Effect of Low Pass Filtering on White Noise (cont...)
2
( )
1 ( )k n
E w kN ∈Γ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑
{ }22
( )
1 ( )k n
E w kN ∈Γ
⎡ ⎤= ⎣ ⎦∑
[ ]2( ) ( )
2 ( ) ( )l n m n
m l
E w n l w n mN ∈Γ ∈Γ
≠
+ − −∑ ∑
Sum of squares
Cross products
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Effect of Low Pass Filtering on White Noise (cont...)
{ }2 22 2
( ) ( )
1 1( )k n k n
E w kN N
σ∈Γ ∈Γ
⎡ ⎤ =⎣ ⎦∑ ∑
[ ]2( ) ( )
2 ( ) ( ) 0l n m n
m l
E w n l w n mN ∈Γ ∈Γ
≠
+ − − =∑ ∑
Sum of squares
Cross products (uncorrelated as m∫l)
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Effect of Low Pass Filtering on White Noise (cont...)
Finally, substituting the partial results:
The effect of the noise is reduced.This processing is not optimal as it also smoothes image edges.
2
2
( )
1 ( )gk n
E w kN
σ∈Γ
⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑ 22
( )
1k nN
σ∈Γ
= ∑
22
1 NN
σ=2
Nσ
=
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Sharpening Spatial Filters
Previously we have looked at smoothing filters which remove fine detailSharpening spatial filters seek to highlight fine detail
– Remove blurring from images– Highlight edges
Sharpening filters are based on spatial differentiation
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Spatial Differentiation
Differentiation measures the rate of change of a functionLet’s consider a simple 1 dimensional example
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Spatial DifferentiationIm
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A B
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Derivative Filters Requirements
First derivative filter output– Zero at constant intensities– Non zero at the onset of a step or ramp– Non zero along ramps
•Second derivative filter output– Zero at constant intensities– Non zero at the onset and end of a step or ramp– Zero along ramps of constant slope
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1st Derivative
The formula for the 1st derivative of a function is as follows:
It’s just the difference between subsequent values and measures the rate of change of the function
)()1( xfxfxf
−+=∂∂
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1st Derivative (cont.)
The gradient points in the direction of most rapid increase in intensity.
• The gradient of an image:
•
Gradient direction
The edge strength is given by the gradient magnitude
Source: Steve Seitz
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1st Derivative (cont…)
0
1
2
3
4
5
6
7
8
-8
-6
-4
-2
0
2
4
6
8
5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
0 -1 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0
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2nd Derivative
The formula for the 2nd derivative of a function is as follows:
Simply takes into account the values both before and after the current value
)(2)1()1(2
2
xfxfxfxf
−−++=∂∂
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2nd Derivative (cont…)
0
1
2
3
4
5
6
7
8
5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
-15
-10
-5
0
5
10
-1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0
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Using Second Derivatives For Image Enhancement
Edges in images are often ramp-like transitions– 1st derivative is constant and produces thick
edges – 2nd derivative zero crosses the edge (double
response at the onset and end with opposite signs)
A common sharpening filter is the Laplacian– Isotropic– One of the simplest sharpening filters– We will look at a digital implementation
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The Laplacian
The Laplacian is defined as follows:
where the partial 1st order derivative in the xdirection is defined as follows:
and in the y direction as follows:
yf
xff 2
2
2
22
∂∂
+∂∂
=∇
),(2),1(),1(2
2
yxfyxfyxfxf
−−++=∂∂
),(2)1,()1,(2
2
yxfyxfyxfyf
−−++=∂∂
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The Laplacian (cont…)
So, the Laplacian can be given as follows:
We can easily build a filter based on this
),1(),1([2 yxfyxff −++=∇)]1,()1,( −+++ yxfyxf
),(4 yxf−
0 1 0
1 -4 1
0 1 0
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The Laplacian (cont…)
Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities
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OriginalImage
LaplacianFiltered Image
LaplacianFiltered Image
Scaled for Display
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But That Is Not Very Enhanced!
The result of a Laplacian filtering is not an enhanced imageWe have to do more work in order to get our final imageSubtract the Laplacian result from the original image to generate our final sharpened enhanced image
LaplacianFiltered Image
Scaled for Display
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fyxfyxg 2),(),( ∇−=
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Laplacian Image Enhancement
In the final sharpened image edges and fine detail are much more obvious
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- =
OriginalImage
LaplacianFiltered Image
SharpenedImage
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Laplacian Image EnhancementIm
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Simplified Image Enhancement
The entire enhancement can be combined into a single filtering operation
),1(),1([),( yxfyxfyxf −++−=)1,()1,( −+++ yxfyxf
)],(4 yxf−
fyxfyxg 2),(),( ∇−=
),1(),1(),(5 yxfyxfyxf −−+−=)1,()1,( −−+− yxfyxf
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Simplified Image Enhancement (cont…)
This gives us a new filter which does the whole job for us in one step
0 -1 0
-1 5 -1
0 -1 0
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Simplified Image Enhancement (cont…)Im
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Variants On The Simple Laplacian
There are lots of slightly different versions of the Laplacian that can be used:
0 1 0
1 -4 1
0 1 0
1 1 1
1 -8 1
1 1 1
-1 -1 -1
-1 9 -1
-1 -1 -1
SimpleLaplacian
Variant ofLaplacian
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Unsharp masking
Used by the printing industrySubtracts an unsharped (smooth) image from the original image f(x,y).
–Blur the image b(x,y)=Blur{f(x,y)}
–Subtract the blurred image from the original (the result is called the mask)
gmask(x,y)=f(x,y)-b(x,y)–Add the mask to the original
g(x,y)=f(x,y)+k gmask(x,y) with k non negative
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Unsharp masking (cont...)Im
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When k>1 the process is referred to as highboost filtering
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Unsharp masking (cont...)Im
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Blurred image
Mask
Unsharp masking
Highboost filtering (k=4.5)
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1st Derivative Filtering
Implementing 1st derivative filters is difficult in practiceFor a function f(x, y) the gradient of f at coordinates (x, y) is given as the column vector:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂∂∂∂
=⎥⎦
⎤⎢⎣
⎡=∇
yfxf
GG
y
xf
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1st Derivative Filtering (cont…)
The magnitude of this vector is given by:
For practical reasons this can be simplified as:
)f(∇=∇ magf
[ ] 2122
yx GG +=
21
22
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
=yf
xf
yx GGf +≈∇
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1st Derivative Filtering (cont…)
There is some debate as to how best to calculate these gradients but we will use:
which is based on these coordinates
( ) ( )321987 22 zzzzzzf ++−++≈∇
( ) ( )741963 22 zzzzzz ++−+++
z1 z2 z3
z4 z5 z6
z7 z8 z9
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Sobel Operators
Based on the previous equations we can derive the Sobel Operators
To filter an image it is filtered using both operators the results of which are added together
-1 -2 -1
0 0 0
1 2 1
-1 0 1
-2 0 2
-1 0 1
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Sobel Example
Sobel filters are typically used for edge detection
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1st & 2nd Derivatives
Comparing the 1st and 2nd derivatives we can conclude the following:
– 1st order derivatives generally produce thicker edges (if thresholded at ramp edges)
– 2nd order derivatives have a stronger response to fine detail e.g. thin lines
– 1st order derivatives have stronger response to grey level step
– 2nd order derivatives produce a double response at step changes in grey level (which helps in detecting zero crossings)
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Combining Spatial Enhancement Methods
Successful image enhancement is typically not achieved using a single operationRather we combine a range of techniques in order to achieve a final resultThis example will focus on enhancing the bone scan to the right
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Combining Spatial Enhancement Methods (cont…)
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Laplacian filter ofbone scan (a)
Sharpened version ofbone scan achievedby subtracting (a)and (b) Sobel filter of bone
scan (a)
(a)
(b)
(c)
(d)
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Combining Spatial Enhancement Methods (cont…)
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z &
Woo
ds, D
igita
l Im
age
Pro
cess
ing
(200
2)
The product of (c)and (e) which will beused as a mask
Sharpened imagewhich is sum of (a)and (f)
Result of applying apower-law trans. to(g)
(e)
(f)
(g)
(h)
Image (d) smoothed witha 5*5 averaging filter
63
C. Nikou – Digital Image Processing (E12)
Combining Spatial Enhancement Methods (cont…)
Compare the original and final images
Imag
es ta
ken
from
Gon
zale
z &
Woo
ds, D
igita
l Im
age
Pro
cess
ing
(200
2)
64
C. Nikou – Digital Image Processing (E12)
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– Sharpening filters– Combining filtering techniques