4. Lecture Image enhancement: Filtering 1 Image preprocessing - - PowerPoint PPT Presentation

• 4. Lecture

Image enhancement: Filtering

1

• Aims:

– Improvement of image data – Suppress unwanted distortions – Highlight interesting features

• Image preprocessing does not increase the

information content

• Best preprocessing is no preprocessing

– better: high quality image acquisition

Image preprocessing

2

Classification of preprocessing

• Point operations – Grayvalue transformations
• Local operations – Spatial filters, look at local

neighbourhoods

• Global operations – Inverse filtering
• Geometric Transformations

3

Grayvalue transformations

• Alteration of the brightness of the pixels independent of the

position in the image

• Function T to transform original grayvalue p to grayvalue q

q = T(p)

• Fast implementation possible by creation of a look-up table
• Used for interactive image viewers

4

Examples

p q p1 p2 c Image negative Contrast stretching Thresholding q p q p

5 [Matlab]

• Image values can be to large for the display or for the image

data format

• E.g. storing a Fourier transformed image in a 8Bit grayvalue

image

T : q = c ¢ log(1 + p)

Dynamic range compression

6

Image can be made brighter or darker

• < 1 image gets brighter
• > 1 image gets darker

Gamma correction

T : q = p° ° °

7 [Matlab]

Histogram equalization

• Histogram counts the instances of each grayvalues in the image
• It is the grayvalue distribution of the image
• Can be computed with the Matlab command imhist

8

Plotting histograms in Matlab

9

Histogram equalization

• Transform the initial distribution to match a uniform

distribution

• All the grayvalues appear similarly often
• Transformation leads to enhanced contrast

H(p) p G(p) p

10

Example

11

500 1000 1500 2000 2500 3000 50 100 150 200 250 200 400 600 800 1000 1200 1400 1600 1800 2000 50 100 150 200 250

[Matlab]

Pseudo color transformation

• Each grayvalue gets a color value
• Grayvalues that are very similar can get totally different color

values and are therefore distinguishable

13

Illumination correction

• When is this necessary?
• Ideally the sensitivity of an image sensor is

independent of the pixel location. This is however not always true.

• Artificial lighting might lead to non uniform

illumination

• If the deviation is systematic it can be corrected

14

Example: Illumination

15 [Matlab]

Illumination correction

• f(i,j) = e(i,j)g(i,j)

– g(i,j)...original image (not distorted) – e(i,j)... distortion – f(i,j)... distorted images

• Distortion can be computed by use of reference image of

constant color/grayvalue:

c j i f j i e c j i e j i f c j i g

c c

) , ( ) , ( ) , ( ) , ( ) , (   

) , ( ) , ( ) , ( ) , ( ) , ( j i f j i cf j i e j i f j i g

c

 

correction:

16

Local image preprocessing

• The local neighbourhood of a pixel will be used to compute the

new pixel value (spatial filtering)

• Smoothing

– to suppress noise – image gets blurred – high frequencies vanish

• Gradient operators

– using local derivatives – Sharpening/Edge detection

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noisy image

Spatial filtering

• Definition: A filter kernel (window) will be moved from pixel to

pixel and for every pixel a new value is computed as a linear combination of the filter kernel and the grayvalues of the local neighborhood.

• This process is mathematically a convolution
• Convolution of the kernel w and the image f:

18

Illustration

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) 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)

x y

• rigin

mask

Mask coefficients Image section under mask

) , ( ) , ( ) , ( t y s x f t s w y x g

a a s b b t

  



   

19

Example1: Averaging

• One simple example is smoothing using a 3x3 mask.

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) 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

 

) 1 , 1 ( ... ) 1 , 1 ( ) , 1 ( ) 1 , 1 ( 9 1 ) , (             y x f y x f y x f y x f y x g

20

What happens at the borders?

• Problem: What happens at the borders?

– Ignore the edges

• The resulting image is

smaller than the original – Pad with zeros

• Introducing unwanted artifacts

– Mirror image at edges

21

Values Outside the Range

• Linear filtering might bring the intensity outside the display

range.

• Solutions :

– Clip values – Scaling transformation

• Transform values in [gL,gH] to [0,255]

255 if 255 if if 255           x x x x y

L H L

g g g x y    255

New min New max 22

Separable Filters

• A filter matrix with rank(w)=1 can be implemented by successive

application of two simpler filters.

• A rank=1 matrix can be written as the outer product of 2

vectors.

• Separating filters results in great time saving.
• By how much?

– For a mask of size nxn, for each image pixel

• Originally, n2 multiplications and n(n-1) additions
• After separation, 2n multiplications and 2(n-1) additions

 

1 1 1 3 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 9 1                     

23

Some terminology on Frequency…

• Frequencies: a measure of the amount by which gray values

change with distance.

• High/low-frequency components: large/small changes in gray

values over small distances.

• High/low-pass filter: passing high/low components, and reducing
• r eliminating low/high-frequency filter.

24

Low-Pass Filter

• Mostly for noise reduction/removal and smoothing

– 3x3 averaging filter to blur edges – Gaussian filter,

• based on Gaussian probability distribution function
• a popular filter for smoothing
• more later when we discuss image restoration

 

2 2 2

2 2

2 1 ,





y x

e y x G

 



25

Gauss filtering

• A Gaussian filter is a weighted average filter. Pixel in the center
• f the filter window are weighted stronger than pixel on the

border (according to the Gaussian distribution).

• The only parameter is (sigma), which is the standard
• deviation. This value controls the degree of smoothing.

– as this parameter goes up, more pixels in the neighborhood are involved in “averaging”, – the image gets more blurred, and – noise is more effectively suppressed



26

Coefficients for filtering mask

• In theory, the Gaussian distribution is non-zero everywhere,

which would require an infinitely large convolution mask.

• but in practice it is effectively zero more than about three from

the mean, and so we can truncate the mask at this point. Discrete approximation to Gaussian function with

4 . 1  



0.1299 0.2793 0.3604 0.2793 0.1299 0.2793 0.6004 0.7748 0.6004 0.2793 0.3604 0.7748 1.0000 0.7748 0.3604 0.2793 0.6004 0.7748 0.6004 0.2793 0.1299 0.2793 0.3604 0.2793 0.1299

1299 0.2793 0.

27

Separating a 2D Gauss filter

• 2-D isotropic Gaussian is separable into x and y components.
• Thus the 2-D convolution can be performed by first convolving

with a 1-D Gaussian in the x direction, and then convolving with another 1-D Gaussian in the y direction.

• 1-D x component for the 2D kernel.
• The y component is exactly the same but is oriented vertically.

0.3604 0.7748 1.0000 0.7748 0.3604

1299 0.2793 0.

28

Example: Gauss filtering

sigma =1 sigma =2 sigma =3

• riginal

29

High-Pass Filter

• The Laplacian

is a high-pass filter, based on second derivatives.

• Sum of coefficients is zero
• Insight: in areas where the gray values are similar (low-

frequency), application of the filter makes the gray values close to zero.

 

y x f ,

2



               1 1 4 1 1

2

h             1 1 4 1 1

1

h

• r

30

Laplace Filter: Demo

Original Laplacian-filtered

31

Edge sharpening: Laplace Filter

• Laplacian highlights discontinuities, and turn

featureless regions into dark background.

• How can we make good use of the property?

– One example is edge enhancement

         

       

2 2 1 2

using , , , using , , , , h y x f y x f h y x f y x f y x g

• riginal

Laplacian enhanced

32

Edge sharpening:Unsharp Masking

• To make edges slightly sharper and crisper.
• This operation is referred to as edge enhancement, edge

crispening or unsharp masking.

• A very popular practice in industry.
• Subtracting a blurred version of an image from the image itself.

     

, 2 , ,

s

f x y f x y f x y  

Sharpened image

       

 

, , , ,

s

f x y f x y f x y f x y   

Blurred image Original image Edge image

     

2

1 , , , 2

s

f x y f x y f x y  

33

Unsharp Masking: Details

f(x,y) blurred image Sharpened

34

Unsharp Masking: Filter mask

• Combining filtering and subtracting in one filter.
• Schematically,

1 9 1 9 1 9 1 9 1 9 1 9 1 1 1 9 1 9 1 9 1 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 h A A                                            

Original Blur with low-pass filter Scale with 1/A (A>1) Subtract Scale for display

35

High-Boost Filtering

• Generalization of unsharp masking
• Here A is called boost coefficient, and
• We rewrite the equation as
• Applicable to any sharpening operation

– e.g. sharpening with Laplace operator – The filter becomes

     

y x f y x Af y x fhb , , ,   1  A

         

y x f y x f y x f A y x fhb , , , 1 ,    

     

y x f y x f A

s

, , 1   

2 s

f f  

                1 1 4 1 1 A h

36

High-Boost Filtering: Demo

• By varying A, better overall brightness can be improved.
• riginal

A=0 A=1 A=1.7, much brighter

37

Image Smoothing

• To suppress noise
• Computing an average of the grayvalues in a local

neighborhood

• Edges get blurred by averaging (high frequencies get

suppressed)

• Filter can be edge-preserving
• Averaging works well when single pixels are wrong
• Large noise regions (blobs) cannot really get removed

38

Averaging over multiple images

Original

image with noise averaging over 10 images

   







n k k

j i g n j i f

1

, 1 ,

39

Spatial filtering

           1 1 1 1 1 1 1 1 1 9 1 h

                       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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49 1 h

40

Spatial filtering

           1 1 1 1 1 1 1 1 1 9 1 h            1 1 1 1 2 1 1 1 1 10 1 h            1 2 1 2 4 2 1 2 1 16 1 h

Approximation of Gauss filtering

41

Averaging with rotating filter mask

• Avoids blurring of edges
• Analyses the homogeneity of the local neighborhood
• Only pixels from the homogeneous part of the filter mask are

used for averaging

• Variance is used as criteria for homogeneity

   

   

                 

 

  R j i R j i

j i g n j i g n

, 2 , 2

, 1 , 1 

42

Rotating filter masks

1 2 7 8 ...

Algorithm: for each pixel:

• compute variance for each of the masks
• select mask with smallest variance
• use pixels indexed by mask to compute the average value

43

Nonlinear Filters

• Maximum filter: the output the maximum value under

the mask

• Minimum filter: the output the minimum value under

the mask

• Rank-order filter:

– Elements under the mask are ordered, and a particular value is returned. – Both maximum and minimum filter are instances

• f rank-order

– Another popular instance is median filter

44

More Nonlinear Filters

• Alpha-trimmed mean filter:

– Order the values under the mask – Trim off elements at either end of the ordered list – Take the mean of the remainder – E.g. assuming a 3x3 mask and the ordered list – Trimming of two elements at either end, the result of the filter is

9 3 2 1

... x x x x    

  5

7 6 5 4 3

x x x x x    

45