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Topic 6 - Image Filtering - I
DIGITAL IMAGE PROCESSING
Course 3624
Department of Physics and Astronomy
Professor Bob Warwick
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6. Image Filtering I: Spatial Domain Filtering
The goal of spatial filtering is typically either to (a) reduce theimpact of noise in the image via image smoothing or (b) to
sharpen the detail within the image. Both processes involve thesuppression and/or enhancement of particular spatial frequencycomponents in the image (see Topic 8).
Point process (eg. Mapping) Neighbourhood process (eg. Spatial filtering)
gxy=hij
j=0
j=n-1
i=0
i=n-1
fx-m+i, y-m-j
hijj=0
j=n-1
i=0
i=n-1
For a mask of dimension n x n (n odd),
where m = (n-1)/2 is the central element.
Calculate for all x & y
Note that the denominator is a normalisationfactor sometimes not applicable!
(ii) Compute a new image gxy from the old image fxy via:
Procedure
(i) Define an array of values known variously as the mask, filter,
operator etc.. (usually with a 3 x 3 or 5 x 5 or 7 x 7 etc.. format)
h01 h02h00
h11 h12h10
h21 h22h20i
j
h ij
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3
Computing the 2-d Result
original 3x3 average
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6.1 Smoothing FiltersSmoothing filters involve calculating the average value (withsome defined weighting) within the masked region.
Assuming a 3 x 3 format, some possibilities are:
1 1 1 Replaces the original value with
1 1 1 the average of 9 values (the central
1 1 1 pixel plus its 8 nearest neighbours)
1 1 1 Double weighting of the central pixel
1 2 1
1 1 1
0 1 0 Replaces the original value with
1 1 1 the average of 5 values (the central
0 1 0 pixel plus its 4 nearest neighbours)
1 1 1
1 2 1
1 1 1
1 1 1
1 1 11 1 1
0 1 0
1 1 1
0 1 0
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Author: Richard Alan Peters II
Smoothing Examples: Original Images
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Smoothing Examples: 33 Blur
111
111
111
9
1
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Smoothing Examples: 55 Blur
11111
11111
11111
11111
11111
25
1
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Smoothing Examples: 99 Blur
111111111
111111111
111111111
111111111
111111111
111111111
111111111
111111111
111111111
81
1
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Smoothing Examples: 1717 Blur
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
11111111111111111
289
1
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Image Smoothing Examples
Original Smoothed
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Noise Suppression Example
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Noise Suppression Example
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Calculating the effective noise reduction
Example
Applying a 3 x 3smoothing filter withunit coefficientsresults in a factor 3reduction in thenoise.
gxy=
fxy9
9
m g = m fs g = s f9
P(f) = 1s 2p e- (f- m )2
2s 2
mean and standard deviation,s
Gaussian Distribution
f
P(f)
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The Median FilterSmoothing filters can be used to reduce the impact of the noise in animage but also cause unwanted blurring. The larger the dimension
of the smoothing mask the greater the noise reduction factor butalso the greater the blurring.
Non-linear filters, such as the Median Filter, can to some degreedecouple these two effects but in a way which is hard to quantify.
133
133
140
147
152154
163
164
171
The median filter wouldreplace the central value withthe median value containedwithin the mask (or window)
region.
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Suppression of Impulsive Noise
Smoothing Filter Median Filter
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Suppression of Impulsive Noise
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6.2 Sharpening Filters
Smoothing Averaging Sharpening Differencing
In practice with discretevariables differencing isequivalent to differentiationand both the 1st and 2nd
differentials can be used tomark the position in animage where there arerapid changes in gray level(ie edges).
f(x) + f(x)
f(x) - f (x)
f(x)
f(x)
f (x)
Continuous variables
The objective is to enhance the detail in the image by accentuatingedges and boundaries.
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Types of Filters Used in Image Sharpening - I
Pairs of masks of various formats which give the vertical (axy)and horizontal (bxy) gradients:
(i) The Roberts Operators
(ii)
(iii) The Prewitt Operators
- 1 1
bxy =fx,y+1-fxy
- 1
1
a xy =fx+1,y -fxy
- 1 0 1
bxy =fx,y+1-fx,y-1
- 10
1
axy =fx+1,y-fx-1,y
- 1 0 1
- 1 0 1
- 1 0 1
gxy = axy2 +bxy
2
or
gxy
= axy
+ bxy
Where the finalgradient image iscalculated as:
- 1 - 1 - 1
0 0 0
1 1 1
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Image Sharpening Examples
Consider an image consisting of:
Applying the following filters gives:
(i)
(ii)
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 0 0 0 0 ..
0 0 0 0 0 0 0 ..
0 0 0 0 0 0 0 ..
0 0 0 0 0 0 0 ..
- 1 0 1
- 1 0 1
- 1 0 1
0 0 0 3 3 0 0 ..
0 0 0 3 3 0 0 ..
0 0 0 3 3 0 0 ..
0 0 0 3 3 0 0 ..
- 1 - 1 - 1
0 0 01 1 1
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Calculating the Gradient Image
Lena
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The Sobel Operators
Calculating the Gradient Image
- 1 - 2 - 1
0 0 0
1 2 1
- 1 0 1
- 2 0 2
- 1 0 1
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Types of Filters Used in Image Sharpening - II
(i)
(ii)
(iii)- 1 - 1 - 1
- 1 8 - 1
- 1 - 1 - 1
0 - 1 0
- 1 4 - 1
0 - 1 0
- 1 2 - 1
- 1
2
- 1
For discrete data :
fx'= fx-fx-1
fx''= fx
' -fx-1'
= (fx-fx-1) - (fx-1-fx-2)
= fx - 2fx-1 + fx-2
- fx-1 + 2fx - fx+1
Masks which give the 2nd differential or Laplacian image
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2fxy-fx,y-1-fx,y+1
4fxy-
fx-1,y-fx+1,y-
fx,y-1-fx,y+1
fxy
0
255
-255
510
2 -1-1 2
-1
-1
4 -1-1
-1
-1
The 1-d and 2-d Laplacian Mask in Action
2fxy-fx-1,y-fx+1,y
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The Image Sharpening Process
In image sharpening the usual approach is to add thederived gradient or Laplacian image to the originalimage in a process known as high frequency emphasis.Thus:
Final image = original image + gradient or Laplacian image
0 - 1 0
- 1 5 - 1
0 - 1 0
- 1 - 1 - 1
- 1 9 - 1
- 1 - 1 - 1
For the Laplacian case this can be achieved in onestep by modifying the 2-d mask to:
or
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Image Sharpening Examples
Consider an image consisting of:
Applying the following filters gives:
(i)
(ii)
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 0 1 1 1 ..
0 0 0 -3 3 0 0 ..
0 0 0 -3 3 0 0 ..
0 0 0 -3 3 0 0 ..
0 0 0 -3 3 0 0 ..
0 0 0 -3 4 1 1 ..
0 0 0 -3 4 1 1 ..
0 0 0 -3 4 1 1 ..
0 0 0 -3 4 1 1 ..
- 1 - 1 - 1
- 1 9 - 1
- 1 - 1 - 1
- 1 - 1 - 1
- 1 8 - 1
- 1 - 1 - 1
I Sh i E l
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Image Sharpening Examples
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The Mathematical Process of Spatial Filtering
Consider the 1-d case of spatial filtering with an n element mask(n odd), where m= (n-1)/2 and ignoring the normalization factor.
gx = h i'h
fx+i' (2)
gx = h i''h
fx-i'' (3)
gx = hx- ah
fa (4)
gx= hii=0
i=n-1 fx-m+i (1) f0 f1 fx fN-1
h0 hm hn-1
x
ii'
i''
a
i'=i-m
i''=-i'a = x-i''Equation 4 is the discrete version ofthe CONVOLUTION INTEGRAL.
g(x)= f(a )h(x - a )da-
Shorthand versions :
g(x) = f(x) * h(x)
gx = fx * hx
SPATIAL FILTERING IS A
CONVOLUTION PROCESS
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The Mathematical Process of Convolution
P ti l C id ti
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Practical Considerations
(i) Edge Errors
For an n x n mask there is a (n-1)/2 border to the new image where exact valuescannot be calculated. Solutions are:
(a) Reduce the dimensions of the new image (usually impractical)(b) Calculate approximate values for the border pixels
(c) Compute the cyclic convolution (beyond our scope).
(ii) Separability
A 2-d mask is separable if the successive application of two 1-d masks gives theequivalent result to the application of the 2-d mask.
For example, considered as matrices:
Therefore this smoothing mask is separable.
There are computational advantages if the mask is separable.
1 1 1
1 1 1
1 1 1
1 1 1( ) 1
1
1
Note: Both gradient and Laplacian filters can also be used in
EDGE DETECTION see topic 10
