Image Enhancement and Filtering Techniques

1
Image Enhancement and Filtering Techniques

• EE4H, M.Sc 0407191
• Computer Vision
• Dr. Mike Spann
• m.spann_at_bham.ac.uk
• http//www.eee.bham.ac.uk/spannm

2
Introduction

• Images may suffer from the following
  degradations 
• Poor contrast due to poor illumination or finite
  sensitivity of the imaging device
• Electronic sensor noise or atmospheric
  disturbances leading to broad band noise
• Aliasing effects due to inadequate sampling
• Finite aperture effects or motion leading to
  spatial

3
Introduction

• We will consider simple algorithms for image
  enhancement based on lookup tables
• Contrast enhancement
• We will also consider simple linear filtering
  algorithms
• Noise removal

4
Histogram equalisation

• In an image of low contrast, the image has grey
  levels concentrated in a narrow band
• Define the grey level histogram of an image h(i)
  where
• h(i)number of pixels with grey level i
• For a low contrast image, the histogram will be
  concentrated in a narrow band
• The full greylevel dynamic range is not used

5
Histogram equalisation
6
Histogram equalisation

• Can use a sigmoid lookup to map input to output
  grey levels
• A sigmoid function g(i) controls the mapping from
  input to output pixel
• Can easily be implemented in hardware for maximum
  efficiency

7
Histogram equalisation
8
Histogram equalisation

• ? controls the position of maximum slope
• ? controls the slope
• Problem - we need to determine the optimum
  sigmoid parameters and for each image
• A better method would be to determine the best
  mapping function from the image data

9
Histogram equalisation

• A general histogram stretching algorithm is
  defined in terms of a transormation g(i)
• We require a transformation g(i) such that from
  any histogram h(i)

10
Histogram equalisation

• Constraints (N x N x 8 bit image)
• No crossover in grey levels after
  transformation

11
Histogram equalisation

• An adaptive histogram equalisation algorithm can
  be defined in terms of the cumulative histogram
  H(i)

12
Histogram equalisation

• Since the required h(i) is flat, the required
  H(i) is a ramp

h(i)
H(i)
13
Histogram equalisation

• Let the actual histogram and cumulative histogram
  be h(i) and H(i)
• Let the desired histogram and desired cumulative
  histogram be h(i) and H(i)
• Let the transformation be g(i)

14
Histogram equalisation

• Since g(i) is an ordered transformation

15
Histogram equalisation

• Worked example, 32 x 32 bit image with grey
  levels quantised to 3 bits

16
Histogram equalisation
17
Histogram equalisation
18
Histogram equalisation
19
Histogram equalisation
20
Histogram equalisation

• ImageJ demonstration
• http//rsb.info.nih.gov/ij/signed-applet

21
Image Filtering

• Simple image operators can be classified as
  'pointwise' or 'neighbourhood' (filtering)
  operators
• Histogram equalisation is a pointwise operation
• More general filtering operations use
  neighbourhoods of pixels

22
Image Filtering
23
Image Filtering

• The output g(x,y) can be a linear or non-linear
  function of the set of input pixel grey levels
  f(x-M,y-M)f(xM,yM.

24
Image Filtering

• Examples of filters

25
Linear filtering and convolution

• Example
• 3x3 arithmetic mean of an input image (ignoring
  floating point byte rounding)

26
Linear filtering and convolution

• Convolution involves overlap multiply add
  with convolution mask

27
Linear filtering and convolution
28
Linear filtering and convolution

• We can define the convolution operator
  mathematically
• Defines a 2D convolution of an image f(x,y) with
  a filter h(x,y)

29
Linear filtering and convolution

• Example convolution with a Gaussian filter
  kernel
• s determines the width of the filter and hence
  the amount of smoothing

30
Linear filtering and convolution
s
31
Linear filtering and convolution
Noisy
Original
Filtered s3.0
Filtered s1.5
32
Linear filtering and convolution

• ImageJ demonstration
• http//rsb.info.nih.gov/ij/signed-applet

33
Linear filtering and convolution

• We can also convolution to be a frequency domain
  operation
• Based on the discrete Fourier transform F(u,v) of
  the image f(x,y)

34
Linear filtering and convolution

• The inverse DFT is defined by

35
Linear filtering and convolution
36
Linear filtering and convolution
37
Linear filtering and convolution

• F(u,v) is the frequency content of the image at
  spatial frequency position (u,v)
• Smooth regions of the image contribute low
  frequency components to F(u,v)
• Abrupt transitions in grey level (lines and
  edges) contribute high frequency components to
  F(u,v)

38
Linear filtering and convolution

• We can compute the DFT directly using the formula
• An N point DFT would require N2 floating point
  multiplications per output point
• Since there are N2 output points , the
  computational complexity of the DFT is N4
• N44x109 for N256
• Bad news! Many hours on a workstation

39
Linear filtering and convolution

• The FFT algorithm was developed in the 60s for
  seismic exploration
• Reduced the DFT complexity to 2N2log2N
• 2N2log2N106 for N256
• A few seconds on a workstation

40
Linear filtering and convolution

• The filtering interpretation of convolution can
  be understood in terms of the convolution theorem
• The convolution of an image f(x,y) with a filter
  h(x,y) is defined as

41
Linear filtering and convolution
42
Linear filtering and convolution

• Note that the filter mask is shifted and inverted
  prior to the overlap multiply and add stage of
  the convolution
• Define the DFTs of f(x,y),h(x,y), and g(x,y) as
  F(u,v),H(u,v) and G(u,v)
• The convolution theorem states simply that

43
Linear filtering and convolution

• As an example, suppose h(x,y) corresponds to a
  linear filter with frequency response defined as
  follows
• Removes low frequency components of the image

44
Linear filtering and convolution
DFT
IDFT
45
Linear filtering and convolution

• Frequency domain implementation of convolution
• Image f(x,y) N x N pixels
• Filter h(x,y) M x M filter mask points
• Usually MltltN
• In this case the filter mask is 'zero-padded'
  out to N x N
• The output image g(x,y) is of size NM-1 x NM-1
  pixels. The filter mask wraps around truncating
  g(x,y) to an N x N image

46
Linear filtering and convolution
47
Linear filtering and convolution
48
Linear filtering and convolution

• We can evaluate the computational complexity of
  implementing convolution in the spatial and
  spatial frequency domains
• N x N image is to be convolved with an M x M
  filter
• Spatial domain convolution requires M 2 floating
  point multiplications per output point or N 2 M 2
  in total
• Frequency domain implementation requires 3x(2N 2
  log 2 N) N 2 floating point multiplications (
  2 DFTs 1 IDFT N 2 multiplications of the
  DFTs)

49
Linear filtering and convolution

• Example 1, N512, M7
• Spatial domain implementation requires 1.3 x 107
  floating point multiplications
• Frequency domain implementation requires 1.4 x
  107 floating point multiplications
• Example 2, N512, M32
• Spatial domain implementation requires 2.7 x 108
  floating point multiplications
• Frequency domain implementation requires 1.4 x
  107 floating point multiplications

50
Linear filtering and convolution

• For smaller mask sizes, spatial and frequency
  domain implementations have about the same
  computational complexity
• However, we can speed up frequency domain
  interpretations by tessellating the image into
  sub-blocks and filtering these independently
• Not quite that simple we need to overlap the
  filtered sub-blocks to remove blocking artefacts
• Overlap and add algorithm

51
Linear filtering and convolution

• We can look at some examples of linear filters
  commonly used in image processing and their
  frequency responses
• In particular we will look at a smoothing filter
  and a filter to perform edge detection

52
Linear filtering and convolution

• Smoothing (low pass) filter
• Simple arithmetic averaging
• Useful for smoothing images corrupted by additive
  broad band noise

53
Linear filtering and convolution
Spatial domain
Spatial frequency domain
54
Linear filtering and convolution

• Edge detection filter
• Simple differencing filter used for enhancing
  edged
• Has a bandpass frequency response

55
Linear filtering and convolution

• ImageJ demonstration
• http//rsb.info.nih.gov/ij/signed-applet

56
Linear filtering and convolution
57
Linear filtering and convolution

• We can evaluate the (1D) frequency response of
  the filter h(x)1,0,-1 from the DFT definition

58
Linear filtering and convolution

• The magnitude of the response is therefore
• This has a bandpass characteristic

59
Linear filtering and convolution
60
Conclusion

• We have looked at basic (low level) image
  processing operations
• Enhancement
• Filtering
• These are usually important pre-processing steps
  carried out in computer vision systems (often in
  hardware)