Chapter 6 Image Enhancement

1
Chapter 6Image Enhancement

• Chuan-Yu Chang (???)Ph.D.
• Dept. of Electronic Engineering
• National Yunlin University of Science
  Technology
• chuanyu_at_yuntech.edu.tw
• Office ES709
• Tel [phone] Ext. 4337

2
Image Enhancement

• The purpose of image enhancement methods is to
  process and acquired image for better contrast
  and visibility of features of interest for visual
  examination and subsequent computer-aided
  analysis and diagnosis.
• Different medical imaging modalities provide
  specific characteristic information about
  internal organs or biological tissues.
• Image contrast and visibility of the features of
  interest depend on the imaging modality and the
  anatomical regions.
• There is no unique general theory or method for
  processing all kinds of medical images for
  feature enhancement.
• Specific medical imaging applications present
  different challenges in image processing for
  feature enhancement.

3
Image Enhancement (cont.)

• Medical images from specific modalities need to
  be processed using a method that is suitable to
  enhance the features of interest.
• Chest X-ray radiographic image
• Required to improve the visibility of hard bony
  structure.
• X-ray mammogram
• Required to enhance visibility of
  microcalcification.
• A single image-enhancement method may not serve
  both of these applications.
• Image enhancement tasks and methods are very much
  application dependent.

4
Image Enhancement (cont.)

• Image enhancement tasks are usually characterized
  in two categories
• Spatial domain methods
• Manipulate image pixel values in the spatial
  domain based on the distribution statistics of
  the entire image or local regions.
• Histogram transformation, spatial filtering,
  region growing, morphological image processing
  and model-based image estimation
• Frequency domain methods
• Manipulate information in the frequency domain
  based on the frequency characteristics of the
  image.
• Frequency filtering, homomorphic filtering and
  wavelet processing methods
• Model-based techniques are also used to extract
  specific features for pattern recognition and
  classification.
• Hough transform, matched filtering, neural
  networks, knowledge-based systems

5
Spatial Domain Methods

• Spatial domain methods process an image with
  pixel-by pixel transformation based on the
  histogram statistics or neighbor.
• Faster than Fourier transform
• Frequency filtering methods may provide better
  results in some applications if a priori
  information about the characteristic frequency
  components of the noise and features of interest
  is available.
• The spike-based degradation in MRI will be remove
  by Wiener filtering method.

6
Background

• Spatial domain
• The aggregate of pixels composing an image.
• Operate directly on these pixels

Spatial domain process willbe denoted by
g(x,y)Tf(x,y) where f(x,y) input image
g(x,y) processed image T an
operator
mask filter kernel template windows
7
Background (cont.)

• Transformation Function
• sT(r)
• where T is gray-level transformation function
• Processing technologies
• Point processing
• Enhancement at any point in an image depends only
  on the gray level at that point.
• Mask processing or filtering

thresholding
Contrast stretching
8
Some Basic Gray Level Transforms

• Some basic Gray Level Transforms
• s T(r)
• r the gray level value before process
• s the gray level value after process

9
Some Basic Gray Level Transforms (cont.)

• Image Negatives
• Reversing the intensity levels of an image
• Photographic Negative
• s L-1-r
• Suited for enhancing white or gray detail
  embedded in dark regions of an image

10
Some Basic Gray Level Transforms (cont.)

• Log Transformations
• S c log (1r)
• Maps a narrow range of low gray-level values in
  the input image into a wider range of output
  levels.

11
Some Basic Gray Level Transforms (cont.)

• Power-Law Transformations
• scrr
• s c (r e )r
• Where c and r are positive constants
• Power-law curves with fractional values of r map
  a narrow range of dark input values into a wider
  range of output values, with the opposite being
  true for higher values of input levels.

12
Some Basic Gray Level Transforms (cont.)

• Gamma Correction
• The process used to correct this power-law
  response phenomena

13
Some Basic Gray Level Transforms (cont.)

• Example 3.1
• MR image of fractured human spine

c1, r0.6
c1, r0.4
c1, r0.3
14
Some Basic Gray Level Transforms (cont.)
15
Some Basic Gray Level Transforms (cont.)
Picewise-Linear Transformation Function

• Contrast Stretching
• To increase the dynamic range of the gray levels
  in the image being processed.
• Linear function
• If r1s1 and r2s2
• Thresholding
• If r1r2, s10 and s2L-1

Control points
16
Some Basic Gray Level Transforms (cont.)
Picewise-Linear Transformation Function

• Gray-level Slicing
• Highlighting a specific range of gray levels in
  an image.

17
Some Basic Gray Level Transforms (cont.)

• Bit-plane Slicing
• Highlighting the contribution made to total image
  appearance by specific bits.
• Separating a digital image into its bit planes is
  useful for analyzing the relative importance
  played each bit of the image.
• Determining the adequacy of the number of bits
  used to quantize each pixel.
• Image compression.

18
Some Basic Gray Level Transforms (cont.)

• An 8-bit fractal image

19
Some Basic Gray Level Transforms (cont.)

• The eight bit planes of the image in Fig. 3.13

20
Histogram Processing
Histogram h(rk)nk rk is the k-th gray-level nk
is the number of pixels in the image having
gray-level k
Normalized Histogram p(rk)nk/n
21
Medical Images and Histograms
T2 weighted proton density image
X-ray CT image
22
Histogram Processing (cont.)

• Histogram Equalization

• Assume that the transformation function T(r)
  satisfies the follows
• T(r) is a single-valued and monotonically
  increasing
• 0ltT(r)lt1 for 0ltr lt1

23
Histogram Processing (cont.)

• Histogram equalization automatically determines a
  transformation function that seeks to produce an
  output image that has a uniform histogram.
• The histogram equalization method forces image
  intensity levels to be redistributed with an
  equal probability of occurrence.

24
Histogram Equalization
25
Histogram Modification

• The histogram equalization method can cause
  saturation in some regions of the image resulting
  in loss of details and high frequency information
  that may be necessary for interpretation.
• If a desired distribution of gray values is known
  a priori, a histogram modification method is used
  to apply a transformation that changes the gray
  values to match the desired distribution.
• The target distribution can be obtained from a
  good contrast image that is obtained under
  similar imaging conditions.

26
Histogram Modification

• The conventional scaling method of changing gray
  values from the range a,b to c,d can be given
  by a linear transformation aswhere z and znew
  are the original and new gray values of a pixel
  in the image.

27
Histogram Modification(cont.)

• Histogram modification (Specification)
• To specify the shape of the histogram that we
  wish the processed image to have.

(6.5)
(6.6)
(6.7)
28
Histogram Processing (cont.)
2. ?????Histogram Equalization ??,??????G(z)
1. ?????Histogram Equalization
3. ???sk????zk,??zk????0L-1
29
Procedure for histogram matching

1. Obtain the histogram of the given image
2. Use E.q.(6.5) to precompute a mapped level sk for
   each level rk
3. Obtain the transformation function G(z) from the
   given pz(z) using Eq.(6.6)
4. Precompute zk for each value of sk using the
   scheme defined in Eq(6.7)
5. Use the value from step (2) and step (4), mapping
   rk to its corresponding level sk, then map level
   sk into the final level zk.

30
Histogram Processing (cont.)

• Example 3.4 Comparison between histogram
  equalization and histogram matching

31
Histogram Processing (cont.)
32
Histogram Processing (cont.)
33
Image averaging

• Signal averaging is a well-know method for
  enhancing signal-to-noise ratio.
• Sequence images can be averaged for noise
  reduction, leading to smoothing effects.
• Image averaging
• Noisy image g(x,y)
• Averaging K different noisy images
• The standard deviation at any point in the
  average image is

(6.8)
(6.9)
(6.10)
34
Enhancement using Arithmetic/Logic Operations
(cont.)

• Example 3.8 Noise reduction by image averaging

35
Enhancement using Arithmetic/Logic Operations
(cont.)
36
Image Subtraction

• Image Subtraction
• If two properly registered images of the same
  object are obtained with different imaging
  conditions, a subtraction operation on the
  acquired image can enhance the information about
  the changes in imaging conditions.
• The enhancement of difference between images

37
Image Subtraction (cont.)

• The value in a difference image can range from a
  minimum of -255 to a maximum of 255. How to solve
  this problem?
• Solution 1 g(x,y)g(x,y)255/2
• Solution 2 g(x,y)g(x,y)-min(g(x,y)) g(x,y)
  g(x,y)255/max(g(x,y))

38
Neighborhood Operations
 

• The spatial filtering methods using neighborhood
  operations involve the convolution of the input
  image with a specific mask to enhance an image.
• The gray value of each pixel is replaced by the
  new value, computed according to the mask applied
  in the neighborhood of the pixel.
• The neighborhood of a pixel may be defined in any
  appropriate manner based on a simple
  connectedness or any other adaptive criterion.

39
Basic of spatial filtering

• Basic of spatial filtering

If image size MN, mask size mn
where a(m-1)/2 b(n-1)/2
Convolving a mask with an image
40
Basic of spatial filtering (cont.)
41
Smoothing Spatial Filter

• Smoothing filters are used for blurring and for
  noise reduction.
• Smoothing Linear Filter
• Sometimes are called averaging filter , lowpass
  filter
• Box filter
• A spatial averaging filter in which all
  coefficients are equal
• Weighted average
• Pixels are multiplied at different coefficient

42
Smoothing Spatial Filter (cont.)

• Example 3.9
• Image smoothing with masks of various sizes

43
Smoothing Spatial Filter (cont.)
44
Order-Statistic Filters

• Order-Statistic Filters (Nonlinear spatial
  filters)
• Based on ordering 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.
• Median filter
• Particularly effective in the presence of impulse
  noise (salt-pepper noise)
• AlgorithmStep 1 sort the value of the pixels
  encompassed by the filter.Step 2 determine
  their median.Step 3 assign the median to the
  center pixel.

45
Order-Statistic Filters

• Max filter
• Min filter

46
Sharpening Spatial Filters

• Objectives
• To highlight fine detail in an image
• To enhance detail that has been blurred
• The derivatives of a digital function are defined
  in terms of differences
• First derivative
• Must be zero in flat segment
• Must be nonzero at the onset of a gray-level step
  or ramp
• Must be nonzero along ramps

47
Sharpening Spatial Filters

• Second derivative
• Must be zero in flat areas
• Must be nonzero at the onset and the end of
  gray-level step or ramp.
• Must be zero along ramps of constant slope

48
Sharpening Spatial Filters (cont.)
49
Sharpening Spatial Filters (cont.)

• Summary
• First-order derivatives generally produce thicker
  edges in an image.
• Second-order derivatives have a stronger response
  to fine detail
• First-order derivatives generally have a stronger
  response to a gray-level step
• Second-order derivatives produce a double
  response at step changes in gray level.

50
Use of Second Derivatives for Enhancement- The
Laplacian

• Isotropic filter (rotation invariant)
• Whose response is independent of the direction of
  the discontinuities in the image.
• Laplacian

51
Use of Second Derivatives for Enhancement- The
Laplacian
52
Laplacian Second Order Gradient for Edge
Detection
 
53
Image Sharpening with Laplacian
 
54
Use of Second Derivatives for Enhancement- The
Laplacian

• Image enhancement

55
Use of Second Derivatives for Enhancement- The
Laplacian (cont.)

• Example 3.11
• Imaging sharpening with the Laplacian.

56
Use of Second Derivatives for Enhancement- The
Laplacian (cont.)

• Example 3.12
• Image enhancement using a composite Laplacian mask

57
Use of Second Derivatives for Enhancement- The
Laplacian (cont.)

• Unsharp masking and high-boost filtering
• Used in publishing industry
• Unsharp masking To sharpen images consist of
  subtracting a blurred version of an image from
  the image itself.

58
Use of Second Derivatives for Enhancement- The
Laplacian (cont.)

• Example 3.13
• Image enhancement with a high-boost filter

59
Use of First Derivatives for Enhancement -The
Gradient

• The gradient of f at coordinates (x,y) is defined
  as the two-dimensional column vector
• The magnitude of this vector is given by

60
Use of First Derivatives for Enhancement -The
Gradient (cont.)
61
Use of First Derivatives for Enhancement -The
Gradient

• Example 3.14
• Use of the gradient for edge enhancement.

62
Image Averaging
 
63
Median Filter
64
Feature Enhancement Using Adaptive Neighborhood
Processing

• adaptive neighborhood-based image processing
  technique
• Using a low-level analysis and knowledge about
  desired features in designing a contrast
  enhancement function.
• The contrast enhancement function is then used to
  enhance mammographic features while suppressing
  the noise.
• An adaptive neighborhood structure is defined as
  a set of two neighborhoods inner and outer

65
Feature Enhancement Using Adaptive Neighborhood
Processing

• Three types of adaptive neighborhood can be
  defined
• constant ratio
• maintains the ratio of the inner to outer
  neighborhood size at 13
• constant difference
• allows the size of the outer neighborhood to be
  (cn) x (cn)
• feature adaptive
• adapts the arbitrary shape and size of the local
  features to obtain the Center and Surround
  regions is defined using the pre-defined
  similarity and distance criteria.
• Center consisting of pixels forming that
  feature
• Surround consisting of pixels forming the
  background for that feature.

66
Feature Enhancement Using Adaptive Neighborhood
Processing

• The procedure to obtain the Center and the
  Surround regions
• The inner and outer neighborhoods around a pixel
  are grow using the constant difference adaptive
  neighborhood criterion.
• To define the similarity criterion, gray-level
  and percentage thresholds are defined.
• Using these thresholds, the region around each
  pixel in the image is grown in all directions
  until the similarity criterion is violated.
• The region forming all pixels, which have been
  included in the neighborhood of the centered
  pixel satisfying the similarity criterion are
  designated as the Center region.
• The Surround region is composed of all pixels
  contiguous to the Center region.

67
Feature Enhancement Using Adaptive Neighborhood
Processing

• The local contrast C(x,y) fro the centered pixel
  is then computed as
• The Contrast Enhancement Function (CEF) is used
  as a function to modify the contrast distribution
  in the contrast domain of the image.
• The contrast histogram is analyzed and correlated
  to the requirements of feature enhancement. Using
  the CEF, a new contrast value C(x,y) is
  computed.
• The new contrast value C(x,y) is used to compute
  a new pixel value for the enhanced image g(x,y)
  as

68
Feature Adaptive Neighborhood
Region growing for a feature adaptive neighborhood
Image pixel values in a 7x7 neighborhood
Central and Surround regions for the feature
adaptive neighborhood
69
Micro-calcification Enhancement
70
Frequency-Domain Filtering

• Frequency domain filtering methods process an
  acquired image in the Fourier domain to emphasize
  or de-emphasize specified frequency components.
• The low frequency range components usually
  represent shapes and blurred structures in the
  image.
• The high frequency information belongs to sharp
  details, edges and noise.
• A low-pass filter with attenuation to
  high-frequency components would provide image
  smoothing and noise removal.
• A high-pass filter with attenuation to
  low-frequency extracts edge and sharp details for
  image enhancement and sharpening effects.

71
Filtering in the Frequency domain (cont.)

• ????
• g(x,y)h(x,y)f(x,y)
• ????
• H(u,v) is called a filter.
• The Fourier transform of the output image is
• G(u,v)H(u,v)F(u,v)
• The filtered image is obtained simply by taking
  the inverse Fourier transform of G(u,v)
• Filtered Image F-1G(u,v)

72
Filtering in the Frequency domain (cont.)

• Basics of filtering in the frequency domain
• Multiply the input image by (-1)xy to center the
  transform
• Compute F(u,v), the DFT of the image from (1)
• Multiply F(u,v) by a filter function H(u,v)
• Compute the inverse DFT of the result in (3)
• Obtain the real part of the result in (4)
• Multiply the result in (5) by (-1)xy

73
Filtering in the Frequency domain (cont.)

• Basic steps for filtering in the frequency domain

74
Frequency-Domain Methods

• ????g(x,y)??????f(x,y)??point spread function
  (PSF) h(x,y)???(convolution)??????????
• ?Fourier transform????
• ????????inverse filtering??

????????H(u,v)????????????F(u,v)
???N(u,v)??????Fourier Transfor?????,??????H(u,v)?
0?????,N(u,v)/H(u,v)?????F(u,v)??
75
Low-pass Filtering

• The ideal low-pass filter suppresses noise and
  high-frequency information providing a smoothing
  effect to the image.
• An ideal low-pass filter can be designed by
  assigning a frequency cut-off value w0. The
  frequency cut-off value can also be expressed as
  the distance D0 from the origin in the Fourier
  domain.

76
Low-Pass Filtering (cont.)

• Ideal Low-pass Filter
• 2 D ideal lowpass filter

??(u,v)????????????
(4.3-2)
(4.3-3)
77
Low-Pass Filtering (cont.)

• ????(cutoff frequency)
• H(u,v)1?H(u,v)0???????
• ????
• ?????

(4.3-4)
(4.3-5)
78
ExampleImage power as a function of distance
from the origin of the DFT
???5, 15, 30, 80, and 230
????92, 94.6, 96.4, 98, and 99.5
79
Example 4.4 Image power as a function of distance
from the origin of the DFT (cont.)
??????(ringing)
80
Low-Pass Filtering (cont.)
81
Low-Pass Filtering (cont.)

• ??????????(Butterworth low-pass filter)
• BLPF????????????
• ????????H(u,v)????????????

82
Low-Pass Filtering (cont.)

• ???BLPF??????????
• ???BLPF??????,?????
• ???BLPF??????,

83
Chapter 4 Image Enhancement in the Frequency
Domain
84
Low-Pass Filtering (cont.)

• ???????(Gaussian low-pass filter)

85
Low-Pass Filtering
The low-pass filtered MR brain image
Low-pass filter function H(u,v)
The Fourier transform of the filtered MR brain
image
The Fourier transform of the original MR brain
image
86
High Pass Filtering

• The high-pass filtering is used for image
  sharpening and extraction of high-frequency
  information such as edges.

87
High Pass Filtering (cont.)

• ???????(Ideal Highpass Filters)
• ?????????(Butterworth Highpass Filters)
• ??????? (Gaussian Highpass Filters)

(6.33)
(6.34)
(6.35)
88
High Pass Filtering (cont.)
89
High Pass Filtering (cont.)

• Spatial representations of typical (a) ideal (b)
  Butterworth, and (c) Gaussian frequency domain
  highpass filters

90
High Pass Filtering (cont.)

• Result of ideal highpass filtering (a) with
  D015, 30, and 80

91
High Pass Filtering (cont.)

• Result of BHPF order 2 highpass filtering (a)
  with D015, 30, and 80

92
High Pass Filtering (cont.)

• Result of GHPF order 2 highpass filtering (a)
  with D015, 30, and 80

93
Inverse Filtering

• Direct inverse filtering
• Compute an estimate, ,of the transform of
  the original image simply by dividing the
  transform of the degraded image, G(u,v), by the
  degradation function
• ????degradation function,?????????degraded
  image,??N(u,v)?????
• ?degradation function?0????,N(u,v)/H(u,v)???.

(5.7-1)
(5.7-2)
94
Inverse Filtering (cont.)

• One way to get around the zero or small-value
  problem is to limit the filter frequencies to
  values near the origin.
• We know that H(0,0) is equal to the average value
  of h(x,y) and that this is usually the highest
  value of H(u,v) in the frequency domain.
• Thus, by limiting the analysis to frequencies
  near the origin, we reduce the probability of
  encountering zero values.
• In general, direct inverse filtering has poor
  performance.

95
Inverse Filtering (cont.)
Cutoff H(u,v) a radius of 40
??G(u,v)/H(u,v)
Cutoff H(u,v) a radius of 85
Cutoff H(u,v) a radius of 70
96
Minimum Mean Square Error (Wiener) Filtering

• Incorporated both the degradation function and
  statistical characteristics of noise into the
  restoration process.
• The objective is to find an estimate f of the
  uncorrupted image f such that the mean square
  error between them is minimized.

(5.8-1)
?????,??? K???
(6.20)
(6.21)
97
Example 5.12

• Fig. (a) is the full inverse-filtered result
  shown in Fig. 5.27(a).
• Fig. (b) is the radially limited inverse filter
  result of Fig. 5.27(a).
• Fib. (c) shows the result obtained using
  Eq(5.8-3) with the degradation function used in
  Example 5.11.

98
Example 5.13

• From left to right,
• the blurred image of Fig. 5.26(b) heavily
  corrupted by additive Gaussian noise of zero mean
  and variance of 650.
• The result of direct inverse filtering
• The result of Wiener filtering.

99
Constrained Least Squares Filtering

• The difficulty of the Wiener filter
• The power spectra of the undegraded image and
  noise must be known
• A constant estimate of the ratio of the power
  spectra is not always a suitable solution.
• Constrained Least Squares Filtering
• Only the mean and variance of the noise are
  needed.

100
Constrained Least Squares Filtering

• We can express Eq(5.5-16) in vector-matrix form,
  as
• Suppose that g(x,y) is of size M x N, then we can
  form the first N elements of the vector g by
  using the image elements in first row of g(x,y),
  the next N elements from the second row, and so
  on.
• The resulting vector will have dimensions MN x 1.
  these are also the dimensions of f and h.
• The matrix H then has dimensions MN x MN
• Its elements are given by the elements of the
  convolution given in Eq(4.2-30).
• Central to the method is the issue of the
  sensitivity of H to noise.
• To alleviate the noise sensitivity problem is to
  base optimality of restoration on a measure of
  smoothness, such as the second derivation of an
  image.

(5.9-1)
101
Constrained Least Squares Filtering (cont.)

• To find the minimum of a criterion function, C,
  defined assubject to the constraintwhere
  is the Euclidean vector norm, and
  is the estimate of the undegraded image.
• The frequency domain solution to this
  optimization problem is given by the
  expressionwhere g is a parameter that must
  be adjusted so that the constraint inEq(5.9-3) is
  satisfied.

(5.9-2)
(5.9-3)
(5.9-4)
102
Constrained Least Squares Filtering (cont.)

• P(u,v) is the Fourier transform of the function.
• This function is the same as the Laplacian
  operator.
• Eq.(5.9-4) reduces to inverse filtering if g is
  zero.

(5.9-5)
103
Constrained Least Squares Filtering (cont.)
g were selected manually to yield the best visual
results.
104
Constrained Least Squares Filtering (cont.)

• It is possible to adjust the parameter g
  interactively until acceptable results are
  achieved.
• If we are interested in optimality, the parameter
  g must be adjusted so that the constraint in
  Eq(5.9-3) is satisfied.
• Define a residual vector r as
• Since, from the solution in Eq(5.9-4), is a
  function of g, then r also is a function of this
  parameter. It can be shown thatis a
  monotonically increasing function of g.
• What we want to do is adjust gamma so that

(5.9-6)
(5.9-7)
(5.9-8)
105
Constrained Least Squares Filtering (cont.)

• Because f(g) is monotonic, finding the desired
  value of g is not difficult.
• Step 1 specify an initial value of g.
• Step 2 Compute r2
• Step 3 Stop if Eq(5.9-8) satisfied otherwise
  return to Step 2 after increasing g ifor
  decreasing g ifUse the new value of g in
  Eq(5.9-4) to recompute the optimum estimate

106
Constrained Least Squares Filtering (cont.)

• In order to use the algorithm, we need the
  quantities and . To compute , from
  Eq(5.9-6) thatFrom which we obtain r(x,y) by
  computing the inverse transform of R(u,v).
• Consider the variance of the noise over the
  entire image, which we estimate by the
  sample-average methodwhereis the sample
  mean.

(5.9-9)
(5.9-10)
(5.9-11)
(5.9-12)
107
Constrained Least Squares Filtering (cont.)

• With reference to the form of Eq(5.9-10), the
  double summation in Eq(5.9-11) is equal to
• This gives us the expression
• We can implement an optimum restoration algorithm
  by having knowledge of only the mean and variance
  of the noise.

(5.9-13)
108
Constrained Least Squares Filtering (cont.)

• The initial value used for g was 10-5, the
  correction factor for adjusting g was 10-6, the
  value for a was 0.25.

109
Homomorphic filter

• ????f(x,y)??????????????
• ?(6.36)??????????????????,
• ?????
• ?

(6.36)
(6.38)
(6.39)
110
Homomorphic filter

• ???????H(u,v)???G(u,v) ,?
• ?????
• ?
• ?(6.41)????

(6.40)
(6.41)
111
Homomorphic filter (cont.)

• ??g(x,y)????????????,???????????????????
• ??

(6.42)
112
Homomorphic filter (cont.)

• ????????????

113
Homomorphic filter (cont.)

• The illumination component of an image generally
  is characterized by slow spatial variations.
• The reflectance component tends to vary abruptly
• The low frequencies of the Fourier transform of
  the logarithm of an image with illumination and
  the high frequencies with reflectance.

114
Homomorphic filter (cont.)

• The HF requires specification of a filter
  function H(u,v) that affects the low-and high
  frequency component of the Fourier transform in
  different ways.
• The filter tends to decrease the contribution
  made by the low frequencies (illumination) and
  amplify the contribution made by high frequencies
  (reflectance).
• The net result is simultaneous dynamic range
  compression and contrast enhancement.

??????????????????
?rHgt1, rLlt1?????????,??????????
rHgt1
rLlt1
????(??),?????(??) ,????????
115
Example 4.10

• In the original image
• The details inside the shelter are obscured by
  the glare from the outside walls.
• Fig. (b) shows the result of processing by
  homomorphic filtering, with gL0.5 and gH2.0.
• A reduction of dynamic range in the brightness,
  together with an increase in contrast, brought
  out the details of objects inside the shelter.

116
Wavelet Transform

• Fourier Transform only provides frequency
  information.
• Fourier Transform does not provide any
  information about frequency localization.
• It does not provide information about when a
  specific frequency occurred in the signal.
• Short-Term Fourier Transform
• Windowed Fourier Transform can provide
  time-frequency localization limited by the window
  size.
• The entire signal is split into small windows and
  the Fourier Transform is individually computed
  over each windowed signal.
• The STFT provide some localization depending on
  the size of the window, it does not provide
  complete time-frequency localization.
• Wavelet Transform is a method for complete
  time-frequency localization for signal analysis
  and characterization.

117
Wavelet Transform

• The wavelet transform provides a series expansion
  of a signal using a set of orthonormal basis
  function that are generated by scaling and
  translation of the mother wavelet y(t), and the
  scaling function f(t).
• The wavelet transform decomposes the signal as a
  linear combination of weighted basis functions to
  provide frequency localization with respect to
  the sampling parameter such as time or space.
• The multi-resolution approach (MRA) of the
  wavelet transform establishes a basic framework
  of the localization and representation of
  different frequencies at different scales.

118
Wavelet Transform

• In MRA
• Scaling function is used to create a series of
  approximations of a function or image, each
  differing by a factor of a from its nearest
  neighboring approximations.
• Wavelets are then used to encode the difference
  in information between adjacent approximating.

119
Wavelet Transform..

• Wavelet Transform
• Works like a microscope focusing on finer time
  resolution as the scale becomes small to see how
  the impulse gets better localized at higher
  frequency permitting a local characterization
• Provides Orthonormal bases while STFT does not.
• Provides a multi-resolution signal analysis
  approach.

120
Wavelet Transform

• Using scales and shifts of a prototype wavelet, a
  linear expansion of a signal is obtained.
• Lower frequencies, where the bandwidth is narrow
  (corresponding to a longer basis function) are
  sampled with a large time step.
• Higher frequencies corresponding to a short basis
  function are sampled with a smaller time step.

121
Wavelet Transform

• A scaling function f(t) in time t can be defined
  as
• The scaling and translation generates a family of
  functions using the following dilation equations
  (refinement equation)where hn is a set of
  filter (low-pass filter) coefficient.
• To induce a multi-resolution analysis of L2(R),
  where R is the space of all real numbers, it is
  required to have a nested chain of closed
  suspaces defined as

(6.44)
k??fj,k(t)?x????,j??fj,k(t)???(?x????)
????????????? ????????,????? ????????????????
(6.45)
(6.46)
???????????????????????????????????
122
Wavelet Transform
??????scaling function??????? ?????????scaling
function??????? ??V0???????V1????
123
Wavelet Transform

• Define a function y(t) as the mother wavelet
• The wavelet basis induces an orthogonal
  decomposition of L2(R)
• y(t) can be expressed as a weighted sum of the
  shifted y(2t) aswhere gn is a set of filter
  (high-pass filter) coefficients.

(6.47)
Wj is a subspace spanned by y(2jt-k)
(6.48)
(6.49)
124
Wavelet Transform

• The wavelet-spanned subspace is such that it
  satisfies the relation
• Since the wavelet functions span the orthogonal
  complement spaces, the orthogonality requires the
  scaling and wavelet filter coefficients to be
  related through the following
• Let xn be an arbitrary square summable sequence
  representing a signal in the time domain such that

(6.50)
(6.51)
(6.52)
125
Wavelet Transform

• The series expression of a discrete signal xn
  using a set of orthonomal basis function jknis
  given bywhere Xk ltjk (l),x(l)gtSjk
  (l)xl?????where Xk is the transform of xn
• All basis function must satisfy the
  orthonormality conditionwith

(6.53)
(6.54)
126
Wavelet Transform

• The series expansion is considered to be complete
  if every signal from l2(Z) can be expressed using
  the expression in Eq.(6.35)
• Using a set of bio-orthogonal basis function, the
  series expansion of the signal xn can be
  expressed aswhereand

??xn????bi-orthogonal basisfunctions ????
(6.55)
127
Wavelet Transform

• Using a quadrature-mirror filter theory, the
  orthonormal bases jk(n) can be expressed as
  low-pass and high-pass filters for decomposition
  and reconstruction of a signal.
• It can be shown that a discrete signal xn can
  be decomposed into Xk aswhereand

Low-pass filter
(6.56)
High-pass filter
h0?h1?????? g0?g1??????
128
Wavelet Transform

• A perfect reconstruction of the signal can be
  obtained if the orthonomal bases are used in
  decomposition and reconstruction stages as
• The scaling function provides low-pass filter
  coefficients and the wavelet function provides
  the high-pass filter coefficients.

Wavelet function (high-pass function)
(6.57)
Scaling function (low-pass filter)
129
Wavelet Transform

• A multi-resolution signal representation can be
  constructed based on the differences of
  information available at two successive
  resolutions 2j and 2j-1.
• Decomposing a signal using the wavelet transform
• The signal is filtered using the scaling function
  (low-pass filter)
• Sub-sampling the filtered signal (scale
  information)
• Filtering the signal with the wavelet (high-pass
  filter) and subsampling by a factor of two.
  (detail signal)
• The difference of information between resolution
  2j and 2j-1 is called detail signal at
  resolution 2j .

130
Wavelet Transform
Decomposition
Reconstruction

• Figure 6.19. (a) A multi-resolution signal
  decomposition using Wavelet transform and (b) the
  reconstruction of the signal from Wavelet
  transform coefficients.

131
Wavelet Transform

• The signal decomposition at the jth stage can
  thus be generalized as
• To decompose an image, the above method for 1D
  signals is applied first along the rows of the
  image, and then along the columns.
• The image at resolution 2j1, represented by
  Aj1, is first low-pass and high-pass filtered
  along the rows.
• The result of each filtering process is
  subsampled.
• Next the subsampled results are low-pass and
  high-pass filtered along each column.
• The results of these filtering processes are
  again subsampled.

(6.58)
132
Wavelet Transform

• Figure 6.20. Multiresolution decomposition of an
  image using the Wavelet transform.

133
Wavelet Transform

• This scheme can be iteratively applied to an
  image to further decompose the signal into
  narrower frequency bands.
• Each frequency band can be further decomposed
  into four narrower bands.
• Each level of decomposition reduces the
  resolution by a factor of two, the length of the
  filter limits the number of levels of
  decomposition.
• Daubechies (1992) proposed the least asymmetric
  wavelets
• Computed for different support widths as larger
  support widths provide more regular wavelets.
• See Figure 6.21 and Table 6.1

134
Wavelet and Scaling Functions
135
Wavelet Transform

• Table 6.1 Coefficients for the Corresponding
  Low-pass and High-Pass Filter for the Least
  Asymmetric Wavelet

N High-Pass Low-Pass
0 -0.107148901418 0.045570345896
1 -0.041910965125 0.017824701442
2 0.703739068656 -0.140317624179
3 1.136658243408 - 0.421234534204
4 0.421234534204 1.136658243408
5 -0.140317624179 - 0.703739068656
6 -0.017824701442 -0.041910965125
7 0.045570345896 0.107148901418
136
Wavelet Decomposition Space
137
Image Smoothing and Sharpening Using the Wavelet
Transform

• The wavelet transform provides a set of
  coefficients representing the localized
  information in a number of frequency bands.
• For denoising and smoothing is to threshold these
  coefficients in those bands that have a high
  probability of noise and then reconstruct the
  image using the reconstruction filters (Eq.6.57).
• The reconstruction process integrates information
  from specific bands with successive upscaling of
  resolution to provide the final reconstructed
  image at the same resolution as of the input
  image.
• If certain coefficients related to the noise or
  noise-like information are not included in the
  reconstruction process, the reconstructed image
  shows a reduction of noise and smoothing effects.

138
Image Decomposition
Image
139
Image Processing and Enhancemenet
MR?????????
??high-high band????MR??
??high-high band????MR??
140

• It is difficult to discriminate image features
  from the noise based on the spatial distribution
  of gray values.
• A useful distinction between the noise and image
  features may be made, if some knowledge about the
  processed image features and their behavior is
  known a prior.
• The need for some partial image analysis that
  must be performed before the image enhancement
  operations are performed.