Chapter 6
Image Enhancement
Chuan-Yu Chang (張傳育)Ph.D.
Dept. of Electronic Engineering
National Yunlin University of Science & Technology
chuanyu@yuntech.edu.tw
Office: ES709
Tel: 05-5342601 Ext. 4337
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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


X-ray mammogram



Required to improve the visibility of hard bony structure.
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Image Enhancement (cont.)

Image enhancement tasks are usually characterized
in two categories:

Spatial domain methods



Frequency 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…
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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Background

Spatial domain


The aggregate of pixels composing an image.
Operate directly on these pixels
Spatial domain process will
be denoted by
g(x,y)=T[f(x,y)]
where f(x,y): input image
g(x,y): processed image
T: an operator
mask
filter
kernel
template
windows
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Background (cont.)

Transformation Function



s=T(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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

Log Transformations


S = c log (1+r)
Maps a narrow range of low gray-level values in the input
image into a wider range of output levels.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

Power-Law Transformations




s=crr
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

Gamma Correction

The process used to correct this power-law response phenomena
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

Example 3.1

MR image of fractured human spine
c=1,
r=0.6
c=1,
r=0.4
c=1,
r=0.3
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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 r1=s1 and r2=s2
Thresholding

If r1=r2, s1=0 and s2=L-1
Control
points
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)
Picewise-Linear Transformation Function
Gray-level
Slicing
Highlighting a
specific range of
gray levels in an
image.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

An 8-bit fractal image
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Some Basic Gray Level Transforms (cont.)

The eight bit planes of the image in Fig. 3.13
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Medical Images and Histograms
X-ray CT image
T2 weighted proton density image
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)

Histogram Equalization
s  T (r )
0  r 1
Assume that the transformation function T(r) satisfies the follows
(a) T(r) is a single-valued and monotonically increasing
(b) 0<=T(r)<=1 for 0<=r <=1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)
nk
pr (rk ) 
n
k  0,1,2,...,L  1
k
sk  T (rk )   pr (rj )
j 0
k
nj
j 0
n



k  0,1,2,...,L  1
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Equalization
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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 as
z new
d c
z  a   c

ba
where z and znew are the original and new
gray values of a pixel in the image.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Modification(cont.)

Histogram modification (Specification)

To specify the shape of the histogram that we wish the
processed image to have.
k
k
nj
j 0
j 0
n
sk  T (rk )   pr (rj )  
k
vk  G ( z k )   p z ( z i )  s k
i 0
k  0,1,2,...,L  1
k  0,1,2,...,L  1
z k  G 1 T (rk ) G 1 sk  k  0,1,2,...,L  1
(6.5)
(6.6)
(6.7)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)
1. 計算原圖的Histogram
Equalization
2. 給予特定的Histogram
Equalization 形狀，求出轉換函數
G(z)
3. 對每個sk求對應的zk，其
中zk的灰階由0~L-1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Procedure for histogram matching
1.
2.
3.
4.
5.
Obtain the histogram of the given image
Use E.q.(6.5) to precompute a mapped level sk for
each level rk
Obtain the transformation function G(z) from the
given pz(z) using Eq.(6.6)
Precompute zk for each value of sk using the
scheme defined in Eq(6.7)
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)

Example 3.4 Comparison between histogram
equalization and histogram matching
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Histogram Processing (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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)
g ( x, y)  f ( x, y)   ( x, y)

Averaging K different noisy images
1
g ( x, y ) 
K

(6.8)
K
 g ( x, y )
i 1
i
(6.9)
The standard deviation at any point in the average image is
 g ( x, y )
1

  ( x, y)
K
(6.10)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Enhancement using Arithmetic/Logic
Operations (cont.)

Example 3.8 Noise
reduction by image
averaging
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Enhancement using Arithmetic/Logic
Operations (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
g ( x, y)  f ( x, y)  h( x, y)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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))
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
f(-1,0)
f(0,-1)
f(0,0)
f(1,0)
f(0,1)
f(-1,-1)
f(-1,0)
f(-1,0)
f(0,-1)
f(0,0)
f(0,1)
f(0,-1)
f(1,0)
f(1,1)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Basic of spatial filtering

Basic of spatial filtering
R  w(1,1) f ( x  1, y  1)  w(1,0) f ( x  1, y)  ...
 w(0,0) f ( x, y)  ...  w(1,0) f ( x  1, y)  w(1,1) f ( x  1, y  1)
If image size M*N, mask size m*n
g ( x, y) 
a
b
  w(s, t ) f ( x  s, y  t )
s   at   b
where a=(m-1)/2
b=(n-1)/2
Convolving a mask with an image
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Basic of spatial filtering (cont.)
R  w1 z1  w2 z2  ...  wmn zmn
mn
  wi zi
i 1
R  w1 z1  w2 z2  ...  w9 z9
9
  wi zi
i 1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
1 9
R   zi
9 i 1

Weighted average

Pixels are multiplied at different coefficient g ( x, y ) 
a
b
  w(s, t ) f ( x  s, y  t )
s   at   b
a
b
  w(s, t )
s   at   b
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Smoothing Spatial Filter (cont.)

Example 3.9

Image smoothing with
masks of various sizes
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Smoothing Spatial Filter (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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 (saltpepper noise)
Algorithm:
Step 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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Order-Statistic Filters


Max filter
Min filter
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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
f
 f ( x  1)  f ( x)
x
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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
2 f
  f ( x  1)  f ( x)   f ( x)  f ( x  1)
x 2
 f ( x  1)  f ( x  1)  2 f ( x)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Sharpening Spatial Filters (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for EnhancementThe Laplacian

Isotropic filter (rotation invariant)


Whose response is independent of the direction of the
discontinuities in the image.
Laplacian
2 f 2 f
 f  2  2
x
y
2
2 f
 f ( x  1, y )  f ( x  1, y )  2 f ( x, y )
x 2
2 f
 f ( x, y  1)  f ( x, y  1)  2 f ( x, y )
2
y
 2 f   f ( x  1, y )  f ( x  1, y )  f ( x, y  1)  f ( x, y  1)  4 f ( x, y )
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for
Enhancement- The Laplacian
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Laplacian: Second Order Gradient for
Edge Detection
2
2

f
(
x
,
y
)

f ( x, y)
 2 f ( x, y) 

 x2
 y2
 [ f ( x  1, y)  f ( x  1, y)  f ( x, y  1)  f ( x, y  1)  4 f ( x, y)]
-1
-1
-1
-1
8
-1
-1
-1
-1
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Image Sharpening with Laplacian
-1
-1
-1
-1
9
-1
-1
-1
-1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for
Enhancement- The Laplacian

Image enhancement
 f ( x, y)   2 f ( x, y)
g ( x, y)  
2
f
(
x
,
y
)


f ( x, y )

if the center coefficient is negative
if the center coefficient is positive
g ( x, y)  f ( x, y)   f ( x  1, y)  f ( x  1, y)  f ( x, y  1)  f ( x, y  1)  4 f ( x, y)
 5 f ( x, y)   f ( x  1, y)  f ( x  1, y)  f ( x, y  1)  f ( x, y  1)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for
Enhancement- The Laplacian (cont.)

Example 3.11

Imaging sharpening
with the Laplacian.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for
Enhancement- The Laplacian (cont.)

Example 3.12

Image
enhancement using
a composite
Laplacian mask
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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.
f s ( x, y)  f ( x, y)  f ( x, y)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Use of Second Derivatives for
Enhancement- The Laplacian (cont.)

Example 3.13

Image enhancement
with a high-boost filter
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Use of First Derivatives for Enhancement
-The Gradient


The gradient of f at coordinates (x,y) is defined
as the two-dimensional column vector:
 f 
G x   
f      fx 
G y   
 y 
The magnitude of this vector is given by
f  m ag(f )

 G x2  G 2y

1/ 2
 f  2  f  2 
      
 x   y  
1/ 2
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
59
Use of First Derivatives for Enhancement
-The Gradient (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
60
Use of First Derivatives for Enhancement
-The Gradient

Example 3.14

Use of the gradient for edge enhancement.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
61
Image Averaging
g ( x, y) 
1
p

p
p
p
  w( x, y) f ( x  x, y  y)
x   p y    p


w
(
x
,
y
)

x   p y    p
1
2
1
2
4
2
1
2
1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
62
Median Filter

m edian
g (i, j )
f ( x, y) 
(i, j )  N
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
63
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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
64
Feature Enhancement Using Adaptive
Neighborhood Processing

Three types of adaptive neighborhood can be
defined:

constant ratio:


constant difference:


maintains the ratio of the inner to outer neighborhood size at
1:3
allows the size of the outer neighborhood to be (c+n) x (c+n)
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
65
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
66
Feature Enhancement Using Adaptive
Neighborhood Processing

The local contrast C(x,y) fro the centered pixel is then
computed as
Pc ( x, y)  Ps ( x, y)
C ( x, y) 
maxPc ( x, y), Ps ( x, y)

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


Ps ( x, y)
if Pc ( x, y )  Ps ( x, y)

1

C
(
x
,
y
)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
g ( x, y)  Ps ( x, y)(1  C ( x, y )) if Pc ( x, y)  Ps ( x, y)
g ( x, y) 
67
Feature Adaptive Neighborhood
Region growing for a feature adaptive neighborhood
Image pixel values in a 7x7 neighborhood
Xc
f ( x, y)  ( xc  3)
( xc  3)  f ( x, y)  ( xc  3)
Xc
Center Region
Surround Region
f ( x, y)  ( xc  3)
Central and Surround regions for the feature adaptive neighborhood
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Micro-calcification Enhancement
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
69
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Filtering in the Frequency domain (cont.)

在空間域


g(x,y)=h(x,y)*f(x,y)
在頻率域


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-1[G(u,v)]
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Filtering in the Frequency domain (cont.)
Basics of filtering in the frequency domain
1.
2.
3.
4.
5.
6.
Multiply the input image by (-1)x+y 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)x+y
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Filtering in the Frequency domain (cont.)

Basic steps for filtering in the frequency domain
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
73
Frequency-Domain Methods

一張影像g(x,y)可視為是物體f(x,y)經由point spread
function (PSF) h(x,y)的摺積(convolution)與額外的雜
訊所組成。
g ( x, y)  h( x, y)  f ( x, y)  n( x, y)

其Fourier transform可表示成
G(u, v)  H (u, v) F (u, v)  N (u, v)

物體的資訊可有由inverse filtering得到
G (u, v) N (u, v)
ˆ
F (u, v) 

H (u, v) H (u, v)
即使知道退化函數
H(u,v)也無法完全復
原無退化影像F(u,v) 。
因為N(u,v)為一個不
知其Fourier Transfor
的隨機函數，或當退
化函數H(u,v)為0或
極小值時，
N(u,v)/H(u,v)將嚴重
影響F(u,v)的值
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
74
Low-pass Filtering

The ideal low-pass filter suppresses noise and highfrequency information providing a smoothing effect to the

image.
F u, v  H u, v Gu, v

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.
1 if Du, v   D0
H u, v   
otherwise
0
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
75
Low-Pass Filtering (cont.)

Ideal Low-pass Filter

2 D ideal lowpass filter
從點(u,v)到傅立葉轉換
中心點的距離
1
H (u, v)  
0
if D(u, v)  D0
(4.3-2)
if D(u, v)  D0
D(u, v)  (u  M / 2)  (v  N / 2)
2
2
(4.3-3)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
76
Low-Pass Filtering (cont.)

截止頻率(cutoff frequency)


H(u,v)=1和H(u,v)=0之間的過渡點。
整體功率
M 1 N 1
PT   P(u, v)
(4.3-4)
u 0 v 0

百分比功率


  100 P(u, v) / PT 

u
v

(4.3-5)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
77
Example：Image power as a function of
distance from the origin of the DFT
半徑為5, 15, 30,
80, and 230
功率比為92,
94.6, 96.4, 98,
and 99.5
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
78
Example 4.4 Image power as a function of
distance from the origin of the DFT
(cont.)
存在振鈴現象
(ringing)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
79
Low-Pass Filtering (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
80
Low-Pass Filtering (cont.)

巴特沃斯特低通濾波器（Butterworth low-pass filter）
1
H (u, v) 
2n
1  D(u, v) / D0 


BLPF沒有銳利不連續的截止頻率
將截止頻率定義在H(u,v)降到最大值的某個比例時。
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
81
Low-Pass Filtering (cont.)



一階的BLPF沒有振鈴也
沒有負值。
二階的BLPF有輕微的振
鈴，有小負值。
高階的BLPF有明顯的振
鈴，
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
82
Chapter 4
Image Enhancement in the
Frequency Domain
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
83
Low-Pass Filtering (cont.)

高斯低通濾波器(Gaussian low-pass filter)
H (u, v)  e
 D 2 u ,v  / 2 2
H (u, v)  e
 D 2 u ,v  / 2 D0 2
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
84
Low-Pass Filtering
Low-pass filter function H(u,v)
The low-pass filtered MR brain image
The Fourier transform of the filtered MR brain image
The Fourier transform of the original MR brain image
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
85
High Pass Filtering

The high-pass filtering is used for image
sharpening and extraction of high-frequency
information such as edges.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
86
High Pass Filtering (cont.)

理想高通濾波器(Ideal Highpass Filters)
0
H (u, v)  
1

if D(u , v)  D0
(6.33)
巴特沃斯高通濾波器(Butterworth Highpass Filters)
H (u, v) 

if D(u, v)  D0
1
2n
1  D0 / D(u, v)
(6.34)
高斯高通濾波器 (Gaussian Highpass Filters)
H (u, v)  1  e
 D2 u ,v / 2 D02
(6.35)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
87
High Pass Filtering (cont.)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
88
High Pass Filtering (cont.)

Spatial representations of typical (a) ideal (b)
Butterworth, and (c) Gaussian frequency domain
highpass filters
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
89
High Pass Filtering (cont.)

Result of ideal highpass filtering (a) with D0=15,
30, and 80
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
90
High Pass Filtering (cont.)

Result of BHPF order 2 highpass filtering (a) with
D0=15, 30, and 80
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
91
High Pass Filtering (cont.)

Result of GHPF order 2 highpass filtering (a) with
D0=15, 30, and 80
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
92
Inverse Filtering

Direct inverse filtering

Compute an estimate, Fˆ (u, v),of the transform of the original image
simply by dividing the transform of the degraded image, G(u,v), by
the degradation function:
G (u , v)
ˆ
F (u , v) 
H (u , v)
 G (u , v)  H (u , v) F (u , v)  N (u , v)

G (u, v) N (u, v)
Fˆ (u, v) 

H (u, v) H (u, v)


(5.7-1)
(5.7-2)
即使已知degradation function，仍然無法完全回復未degraded
image，因為N(u,v)是亂數函數
若degradation function為0或很小時，N(u,v)/H(u,v)會支配.Fˆ (u, v)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
93
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
94
Inverse Filtering (cont.)
Cutoff H(u,v) a
radius of 40
直接
G(u,v)/H(u,v)
Cutoff H(u,v) a
radius of 70
Cutoff
H(u,v) a
radius of
85
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
95
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
2
minimized.
(5.8-1)
e 2  E  f  fˆ 


*


H
(u , v) S f (u, v)
ˆ
G (u , v)
F (u, v)  
2
 S f (u, v) H (u , v)  S (u , v) 


H * (u, v)
G (u , v)

2
 H (u , v)  S (u, v) / S f (u, v) 
2

H (u , v)
通常為未知，因此以 K來估計  1
G (u , v)

(6.20)
2
H
(
u
,
v
)
H
(
u
,
v
)

S
(
u
,
v
)
/
S
(
u
,
v
)



f
2
 1

H (u , v)
(6.21)

G (u, v)
2
 H (u, v) H (u , v)  K 
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
96
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
97
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
98
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
99
Constrained Least Squares Filtering

We can express Eq(5.5-16) in vector-matrix form, as
g  Hf  η



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 .
The matrix H then has dimensions MN x MN


(5.9-1)
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
100
Constrained Least Squares Filtering (cont.)

To find the minimum of a criterion function, C, defined as
M 1 N 1

C    2 f ( x, y)

2
(5.9-2)
x 0 y 0
subject to the constraint
2
g  Hfˆ  
(5.9-3)
where w  w w is the Euclidean vector norm, and fˆ is the
estimate of the undegraded image.
The frequency domain solution to this optimization problem is given
by the expression
2

2
T
*


H
(u, v)
ˆ
F (u, v)  
G(u, v)
2
2
 H (u, v)  g P(u, v) 
(5.9-4)
where g is a parameter that must be adjusted so that the constraint
inEq(5.9-3)
is satisfied.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
101
Constrained Least Squares Filtering (cont.)

P(u,v) is the Fourier transform of the function.
 0 1 0 
p ( x, y )   1 4  1
 0  1 0 


(5.9-5)
This function is the same as the Laplacian operator.
Eq.(5.9-4) reduces to inverse filtering if g is zero.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
102
Constrained Least Squares Filtering (cont.)
g were selected manually to yield the best visual results.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
103
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
r  g  Hfˆ

Since, from the solution in Eq(5.9-4), fˆ is a function of g,
then r also is a function of this parameter. It can be shown
that
 g   r r  r
T

(5.9-6)
2
(5.9-7)
is a monotonically increasing function of g.
What we want to do is adjust gamma so that
r  η a
2
2
(5.9-8)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
104
Constrained Least Squares Filtering (cont.)

Because (g) is monotonic, finding the desired value
of g is not difficult.



Step 1: specify an initial value of g.
Step 2: Compute ||r||2
Step 3: Stop if Eq(5.9-8) satisfied; otherwise return to Step
2 after
2
2
increasing g if r  η  a
2
2
r

η
a
or decreasing g if
Use the new value of g in Eq(5.9-4) to recompute the
optimum estimate Fˆ u, v 
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Constrained Least Squares Filtering (cont.)

In order to use the algorithm, we need the quantities
and  2. To compute r ,2 from Eq(5.9-6) that
r
Ru, v  Gu, v  H u, vFˆ u, v
2
(5.9-9)
From which we obtain r(x,y) by computing the inverse
transform of R(u,v).
M 1 N 1
r   r 2 x, y 
2
x 0 y 0

(5.9-10)
Consider the variance of the noise over the entire image,
which we estimate by the sample-average method:
2
M 1 N 1
1

2 
 x, y   m 

(5.9-11)
MN x 0 y 0
where
1 M 1 N 1
m 
 x, y 

(5.9-12)
MN x 0 y 0
is the
sample mean.
醫學影像處理實驗室(Medical
Image Processing Lab.) Chuan-Yu Chang Ph.D.
106
Constrained Least Squares Filtering (cont.)


With reference to the form of Eq(5.9-10), the double
2
summation in Eq(5.9-11) is equal to 
This gives us the expression
  MN  2  m 
2

(5.9-13)
We can implement an optimum restoration algorithm by
having knowledge of only the mean and variance of the
noise.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Homomorphic filter



一幅影像f(x,y)可表示成照度與反射成分的乘積
f ( x, y)  i( x, y)r ( x, y)
但(6.36)無法直接作用在照度和反射的頻率成分上，
 f ( x, y)  ix, y r x, y 
若重新定義
g x, y   ln f x, y 
 ln ix, y   ln r x, y 

(6.36)
則
(6.38)
g ( x, y)  ln f x, y 
 ln ix, y  ln r x, y 
Gu, v  Fi u, v  Fr u, v
(6.39)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Homomorphic filter


假設以濾波函數H(u,v)來處理G(u,v) ，則
S u, v   H u, v Gu, v 
 H u, v Fi u, v   H u, v Fr u, v 
在空間域上
sx, y   1S u, v 

(6.40)
令
 1H u, v Fi u, v  1H u, v Fr u, v 
(6.41)
i ' x, y   1H u, v Fi u, v
r ' x, y   1H u, v Fr u, v 

則(6.41)可表示成
sx, y   i ' x, y   r ' x, y 
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Homomorphic filter (cont.)

因為g(x,y)是由原始影像取對數所形成，所以需藉由反對數
運算產生所要的增強影像 fˆ x, y 
fˆ x, y   e s  x , y 
 e i '( x , y ) e r ' ( x , y )
 i0 ( x, y )r0 ( x, y )

(6.42)
其中
i0 ( x, y)  e i '( x, y )
r0 ( x, y)  e r '( x, y )
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Homomorphic filter (cont.)

用於影像增強的同態濾波法
fˆ ( x, y)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
若rH>1, rL<1
則濾波器將
rH>1
減少照度，
並放大反射
所做的貢獻
rL<1
2
2
H u, v   g H  g L 1  e c D u ,v / D0    g L


下圖可用修改過的高
斯高通濾波器來近似
抑制低頻(照明)，並放大高頻(反射)
增加影像的對比度
醫學影像處理實驗室(Medical Image Processing，
Lab.)
Chuan-Yu Chang Ph.D.
114
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 gL=0.5 and gH=2.0.
 A reduction of dynamic range in the brightness, together with an
increase in contrast, brought out the details of objects inside the
shelter.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
116
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 (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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
119
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Wavelet Transform…

scaling parameter
A scaling function (t) in time t can be defined as
 j , k (t )  2 j / 2 (2 j t  k )
(6.44)
translation parameter
k決定j,k(t)沿x軸的位置，j決定j,k(t)的寬度(沿x軸的寬度)

The scaling and translation generates a family of
functions using the following dilation equations
(refinement equation)
透過此式可產生函數家族；任何
 (t )  2  hn (2t  n) 子空間的展開函式，可由它們自
(6.45)
nZ

己解析度加倍的複製版本建構出來。
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
2
(6.46)
  V1  V0  V1  V2    L
以較低尺度函數所延展之子空間被逐層包含於以較高尺度函數所延展之子空間。
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
121
Wavelet Transform…
以較低尺度之scaling function所延展之子空間
被逐層包含於以較高scaling function所延展之子空間
所有V0的展開函數都是V1的一部分
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Wavelet Transform…

Define a function y(t) as the “mother wavelet”
y j , k (t )  2 j / 2y (2 j t  k )

The wavelet basis induces an orthogonal decomposition
of L2(R)
Wj is a subspace spanned by y(2jt-k)
  W1  W0  W1  W2    L2

(6.47)
(6.48)
y(t) can be expressed as a weighted sum of the shifted
y(2t) as
y (t )  2  gny (2t  n)
(6.49)
n
where gn is a set of filter (high-pass filter) coefficients.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
123
Wavelet Transform…

The wavelet-spanned subspace is such that it satisfies
the relation
Vm1  Vm  Wm

Since the wavelet functions span the orthogonal
complement spaces, the orthogonality requires the
scaling and wavelet filter coefficients to be related
through the following
gn  (1) h1 n
n

(6.50)
(6.51)
Let x[n] be an arbitrary square summable sequence
representing a signal in the time domain such that
x[n]  l2 (Z )
(6.52)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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Wavelet Transform…

The series expression of a discrete signal x[n] using a set of
orthonomal basis function jk[n]is given by
x[n]   jk (l ), x(l ) jk [n]   X [k ]jk [n]
k Z

(6.53)
k Z
where X[k] = <jk (l),x(l)>=Sj*k (l)x[l]為展開函數
where X[k] is the transform of x[n]
All basis function must satisfy the orthonormality condition
0 k  l
1 k  l
j k (n), j l (n)   [k  1]  
with
(6.54)
|| x ||2 || X ||2
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
125
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 x[n] can be expressed as
~
x[n]   j k (l ), x(l ) j~k [n]   X [k ]j~k [n]
where
and
k Z
k Z
k Z
k Z
  j~k (l ), x(l ) j k [n]   X [k ]j k [n]
訊號x[n]可由一組
bi-orthogonal basis
functions 所組成。
~
X [k ]  jk (l ), x(l ) and X [k ]  j~k (l ), x(l )
0 k  l
~
j k [n], j l [n]   [k  1]  
1 k  l
(6.55)
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
126
Wavelet Transform…


is a filter most commonly used to
implement a filter bank that splits
an input signal into two bands.
Using a quadrature-mirror filter theory, the orthonormal
bases jk(n) can be expressed as low-pass and highpass filters for decomposition and reconstruction of a
signal.
It can be shown that a discrete signal x[n] can be
decomposed into X[k] as
x[n]   jk (l ), x(l ) jk [n]   X [k ]jk [n]
k Z
where
and
k Z
j2k [n]  h0[2k  n]  g0[n  2k ]
j2k 1[n]  h1[2k  n]  g1[n  2k ]
X [2k ]  h0 [2k  l ], x[l ]
X [2k  1]  h1[2k  l ], x[l ]
Lowpass
filter
(6.56)
Highpass
filter
h0和h1用來分解訊號
g0和g1用來重建訊號
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
127
Wavelet Transform…

A perfect reconstruction of the signal can be obtained if
the orthonomal bases are used in decomposition and
reconstruction stages as
Wavelet function
x[n]   X [2k ]j2 k [n]   X [2k  1]j2 k 1[n]
k Z

k Z
(high-pass function)
(6.57)

X [2k ]g0 [n  2k ]  X [2k  1]gt [n  2k ]
Scaling function 
k Z
k Z
(low-pass filter)

The scaling function provides low-pass filter coefficients
and the wavelet function provides the high-pass filter
coefficients.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
128
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 .
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
129
Wavelet Transform…
x[n]
H1
X(1)[2k+1]
2
Detail signal
X(1)[2k]
H0
H1
2
Decomposition
2
Scale information
X(2)[2k+1]
X(2)[2k]
H0
H1
2
2
H0
2
X(3)[2k+1]
X(3)[2k]
(a)
X(1)[2k+1]
X(2)[2k+1]
X(3)[2k+1]
2
G1
X(3)[2k]
2
G0
+
2
G1
2
G0
+
2
G1
2
G0
+
Reconstruction
(b)

Figure 6.19. (a) A multi-resolution signal decomposition using Wavelet transform
and (b) the reconstruction of the signal from Wavelet transform coefficients.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
130
Wavelet Transform…

The signal decomposition at the jth stage can thus be
generalized as
J
x[n]  
j 1
( j)
j
( j)
j
X
[
2
k

1
]
g
[
n

2
k
]

X
[
2
k
]
g
[
n

2
k]


1
0
( j)
k Z
( j)
k Z
X ( j ) [2k ]  h0 [2 j k  l ], x[l ]
( j)
(6.58)
X ( j ) [2k  1]  h1 [2 j k  l ], x[l ]
( j)

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 2j+1, represented by Aj+1, 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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
131
Wavelet Transform…
H
H
H
1
2
H
H
H
H
0
2
H
Horizontal
1
2
0
1
0
2
2
2
High-Low Dj2
Low-High Dj1
Low-Low Aj
Subsampling
Vertical

High-High Dj3
Subsampling
Figure 6.20. Multiresolution decomposition of an
image using the Wavelet transform.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
132
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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
133
Wavelet and Scaling Functions
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
134
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
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
135
Wavelet Decomposition Space
V0 data
W1
V1
V2
V3
W2
W3
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
136
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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
137
Image Decomposition
sub-sa mple
sub- sample
H-L
H
Image
H- H
H
L
X
H
L-H
L
L-L
L
horizontally
Level 0
ve rtically
Level 1
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
138
Image Processing and Enhancemenet
MR影像的三
階小波分解
排除high-high
band所重建的
MR影像
僅由high-high
band所重建的
MR影像
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
It is difficult to discriminate image features from
the noise based on the spatial distribution of gray
values.

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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.
醫學影像處理實驗室(Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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