Transcript Medical Image Analysis - National University of Kaohsiung

Medical Image Analysis
Image Enhancement
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Spatial Domain Methods

Spatial domain methods
◦
◦
◦
◦

Pixel-by-pixel transformation
Histogram statistics
Neighborhood operations
Faster than frequency filtering
Frequency filtering
◦ Better when the characteristic frequency
components of the noise and features of
interest are available
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Histogram Transformation

Histogram
h(ri )  ni
ni
p ( ri ) 
n

i  0,1,...,L  1
Histogram equalization
i
i
ni
si  T (ri )   pr (rj )  
j 0
j 0 n
i  0,1,...,L  1
Figure 6.1. An X-ray CT image (top left) and T-2 weighted proton density
image (top right) of human brain cross-sections with their respective
histograms at the bottom. The MR image shows a brain lesion.
Figure 6.2. Histogram equalized images of the brain MR images
shown in Figure 6.1 (top) and their histograms (bottom).
Histogram Modification

Scaling
znew

d c

( z  a)  c
ba
Histogram modification
i
ui  T (ri )   pr (rj )
j 0
i  0,1,...,L  1
Histogram Modification

Histogram modification
◦ Target histogram: pz ( zk )
i
vi  V ( zi )   pz ( zk )
k 0
ui  V ( zi )  T (ri )
si  V T (ri )  V (ui )
i  0,1,...,L  1
1
1
Image Averaging

Averaging
◦ Enhancing signal-to-noise ratio
g ( x, y)  f ( x, y)  n( x, y)
1
g ( x, y ) 
K
K
 g ( x, y )
i 1
i
Eg ( x, y)  f ( x, y)
 g ( x, y )
1

 n( x, y )
K
Image Subtraction

Subtraction
◦ Enhance the information about the changes in
imaging conditions
◦ Angiography: The anatomy with vascular
structure is obtained first. An appropriate dye
or tracer drug is then administered in the
body, where it flows through the vascular
structure. A second image of the same
anatomy is acquired at the peak of the tracer
flow.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Figure 6.3. An MR angiography image obtained through image
subtraction method.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Neighborhood Operations

Use a weight mask
g ( x, y ) 
1
p
p
  w( x' , y' )

x '  p y '  p
p
p
  w( x' , y' ) f ( x  x' , y  y' )
x '  p y '  p
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)
Figure 6.4: A 4-connected (left) and 8-connected neighborhood of a
pixel f(0,0).
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
1
2
1
2
4
2
1
2
1
Figure 6.5. A weighted averaging mask for image smoothing. The
mask is used with a scaling factor of 1/16 that is multiplied to the
values obtained by convolution of the mask with the image [Equation
6.11].
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Figure 6.6. Smoothed image of the MR brain image shown in Figure
6.1 as a result of the spatial filtering using the weighted averaging
mask shown in Figure 6.5.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Median Filter

Median filter
◦ Order-statistics filter
f ( x, y)  m ediang (i, j )
( i , j )N
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Figure 6.7. The smoothed MR brain image obtained by spatial filtering
using the median filter method over a fixed neighborhood of 3x3 pixels.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Adaptive Arithmetic Mean Filter

Adaptive
◦ If the noise variance of the image  is
similar to the variance of gray values in the
2
specified neighborhood of pixels,  s , the
filter provides an arithmetic mean value of the
neighborhood
2
n
ˆf ( x, y)  g ( x, y)  

2
n
2
s
g ( x, y)  gms ( x, y)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Image Sharpening and Edge
Enhancement

Sobel
◦ The first-order gradient in x and y
directions defined by  f ( x, y ) / 
x and
 f ( x, y) /  y
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
-1
-2
-1
-1
0
1
0
0
0
-2
0
2
1
2
1
-1
0
1
Figure 6.8. Weight masks for first derivative operator known as Sobel. The
mask at the left is for computing gradient in the x-direction while the mask
at the right computes the gradient in the y direction.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
-1
-1
-1
-1
0
1
0
0
0
-1
0
1
1
1
1
-1
0
1
-1
-1
0
0
1
1
-1
0
1
-1
0
1
-0
1
1
-1
-1
0
Figure 6.9. Weight masks for computing first-order gradient in (clockwise
from top left) in horizontal, 45 deg, vertical and 135 deg directions.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Image Sharpening and Edge
Enhancement

Laplacian
◦ The second-order dirivative operator
◦ Edge-based image enhancement
 f ( x, y)  f ( x, y)
 f ( x, y) 


2
2
x
 y
2
2
2
[ f ( x  1, y)  f ( x  1, y)  f ( x, y  1) 
f ( x, y  1)  4 f ( x, y)]
0
-1
0
-1
8
-1
0
-1
0
-1
-1
-1
-1
8
-1
-1
-1
-1
(a)
(b)
Figure 6.10. (a) A Laplacian weight mask using 4-connected neighborrhod
pixels only; (b) A laplacian weight mask with all neighbors in a window of 3x3
pixels; and (c) the resultant second-order gradient image obtained using the
mask in (a).
-1
-1
-1
-1
9
-1
-1
-1
-1
Figure 6.11. Laplacian based image enhancement weight mask with
diagonal neighbors and the resultant enhanced image with emphasis on
second-order gradient information.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Feature Enhancement Using
Adaptive Neighborhood Processing

Three types of adaptive neighborhoods
◦ Constant ratio: an inner neighborhood of
size c c and an outer neighborhood of
size 3c  3c
◦ Constant difference: the outer neighborhood
of size (c  n)  (c  n)
◦ Feature adaptive
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Feature Enhancement Using
Adaptive Neighborhood Processing

Feature adaptive
◦ Center region: consisting of pixels forming the
feature
◦ Surround region: consisting of pixels forming
the background
◦ 1. The local contrast. Pc ( x, y) : the average
of the Center region. Ps ( x, y) : the average
of the Surround region
C ( x, y) 
Pc ( x, y)  Ps ( x, y)
max{Pc ( x, y), Ps ( x, y)}
Feature Enhancement Using
Adaptive Neighborhood Processing

Feature adaptive
◦ 2. The Contrast Enhancement Function (CEF)
: modify the contrast distribution by the
contrast histogram
C ' ( x, y)
◦ 3. The enhanced image
C ( x, y) 
Pc ( x, y)  Ps ( x, y)
max{Pc ( x, y), Ps ( x, y)}
Feature Enhancement Using
Adaptive Neighborhood Processing

Feature adaptive
◦ 3. The enhanced image
Ps ( x, y )
g ( x, y ) 
if Pc ( x, y )  Ps ( x, y )
1  C ' ( x, y )
g ( x, y )  Ps ( x, y )(1  C ' ( x, y )) if Pc ( x, y )  Ps ( x, y )
Xc
Xc
Center Region
Surround Region
Figure 6.12. Region growing for a feature adaptive neighborhood:
image pixel values in a 7x7 neighborhood (left) and Central and
Surround regions for the feature adaptive neighborhood.
(a)
(b)
Figure 6.13. (a) A part of a digitized breast film-mammogram with
microcalcification areas. (b): Enhanced image through feature adaptive
contrast enhancement algorithm. (c): Enhanced image through histogram
equalization method.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
(c)
Figure 6.13. (a) A part of a digitized breast film-mammogram with
microcalcification areas. (b): Enhanced image through feature adaptive
contrast enhancement algorithm. (c): Enhanced image through histogram
equalization method.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Frequency Domain Filtering
 g ( x, y )
 f ( x, y)
 h ( x, y )
 n ( x, y )
: an acquired image
: the object
: a Point Spread Function (PSF)
: additive noise
g ( x, y)  h( x, y)  f ( x, y)  n( x, y)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Frequency Domain Filtering

The Fourier transform
G(u, v)  H (u, v) F (u, v)  N (u, v)

Inverse filtering
G(u, v) N (u, v)
ˆ
F (u, v) 

H (u, v) H (u, v)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Wiener Filtering
 S f (u, v)
 Sn (u, v)
: the power spectrum of the signal
: the power spectrum of the noise





2

H
(
u
,
v
)


1


ˆ

G (u, v)


F (u, v)  
S n (u, v) 
2
 H (u, v) 
 H (u, v) 


S
(
u
,
v
)
f



Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Wiener Filtering
 Sn (u, v)
: if it is white noise
 1  H (u, v) 2 
G(u, v)

Fˆ (u, v)  
2

H
(
u
,
v
)

 H (u, v)  K 
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Constrained Least Square Filtering

Acquired image
g  Hf  n

Optimization
e2  TraceE{(f  fˆ )(f  fˆ )t }

Subject to the smoothness constraint
min{fˆ t [C]t [C]fˆ}
Constrained Least Square Filtering
1
 2 1

 1 2 1

1 2

[C] 

1














1
 2

1 
.
.
.
.
The estimated image
1
ˆf  [H]t [H]  1 [C]t [C] [H]t g



Low-Pass Filtering

Ideal
◦ D0 : the frequency cut-off value
◦ D(u, v) : the distance of a point in the
Fourier domain from the origin representing
the dc value
Fˆ (u, v)  H (u, v)G(u, v)
1 if D(u, v)  D0
H (u, v)  
otherwise
0
Low-Pass Filtering

Reduce ringing artifacts
◦ Butterworth or Gaussian

Butterworth
1
H (u, v) 
2n
1  [ D(u, v) / D0 ]

Gaussian
H (u, v)  e
 D2 (u ,v ) / 2 2
Figure 6.14: From top left clockwise: A low-pass filter function H(u,v) in the
Fourier domain, the low-pass filtered MR brain image, the Fourier transform of
the original MR brain image shown in Figure 6.1, and the Fourier transform of
the low-pass filtered MR brain image
High-Pass Filtering

High-pass filtering
◦ Image sharpening and extraction of highfrequency information
◦ Edges

Ideal
1 if D(u, v)  D0
H (u, v)  
otherwise
0
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
High-Pass Filtering

Reduce ringing artifacts
◦ Butterworth or Gaussian

Butterworth
1
H (u, v) 
2n
1  [ D0 / D(u, v)]

Gaussian
H (u, v)  1 e
 D2 (u ,v ) / 2 2
Figure 6.15: From top left clockwise: A high-pass filter function H(u,v) in the
Fourier domain, the high-pass filtered MR brain image, and the Fourier
transform of the high-pass filtered MR brain image.
Homomorphic Filtering
 i( x, y)
 r ( x, y )
: illumination
: reflectance
f ( x, y)  i( x, y)r ( x, y)

In general
f ( x, y)  f1 ( x, y) f 2 ( x, y)
g ( x, y)  ln f ( x, y)  ln f1 ( x, y)  ln f 2 ( x, y)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Homomorphic Filtering

Frequency filtering in the logarithmic
transform domain
{g ( x, y)}  {ln f1 ( x, y)  ln f 2 ( x, y)}
G(u, v)  F1 (u, v)  F2 (u, v)
S (u, v)  H (u, v)G(u, v) 
H (u, v) F1 (u, v)  H (u, v) F2 (u, v)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Homomorphic Filtering
s( x, y)
 F 1{H (u, v) F1 (u, v)}  F 1{H (u, v) F2 (u, v)}
 f '1 ( x, y)  f '2 ( x, y)
fˆ ( x, y)  es ( x, y )  fˆ1 ( x, y) fˆ2 ( x, y)
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
f ( x, y)
ln
FT
H(u,v)
IFT
exp

f (x, y)
Figure 6.16. A schematic block diagram of homomorphic
filtering.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Homomorphic Filtering

An example
◦ f1 ( x, y) and f 2 ( x, y) components can
represent, respectively, low- and highfrequency components
◦ A circularly symmetric homomorphic filter
function

H (u, v)  ( H   L ) 1  e
c ( D 2 ( u ,v ) / D 2 )
 
L
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
H(u,v)
H
L
D(u,v)
Figure 6.17: A circularly symmetric filter function for Homomorphic
filtering.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Figure 6.18 The enhanced MR image obtained by Homomorphic
filtering using the circularly symmetric function in Equation 3.43.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Wavelet Transform for Image
Processing
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
x[n]
H1
X(1)[2k+1]
2
X(1)[2k]
H0
H1
2
2
X(2)[2k+1]
X(2)[2k]
H0
H1
2
2
X(3)[2k+1]
X(3)[2k]
H0
2
(a)
X(1)[2k+1]
X(2)[2k+1]
X(3)[2k+1]
X(3)[2k]
2
G1
2
G0
+
2
G1
2
G0
+
2
G1
2
G0
+
(b)
Figure 6.19. (a) A multi-resolution signal decomposition using Wavelet transform
and (b) the reconstruction of the signal from Wavelet transform coefficients.
H
2
High-High Dj3
1
H
H
2
High-Low Dj2
1
H
H
2
0
H
H
2
Low-High Dj1
1
2
0
H
2
Low-Low Aj
0
Horizontal
Subsampling
Vertical
Subsampling
Figure 6.20. Multiresolution decomposition of an image using the
Wavelet transform.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.
Figure 6.21. The least asymmetric wavelet with eight coefficients.
Figure 6.22. Three-level wavelet decomposition of the MR brain image
shown in Figure 6.1.
Figure 6.23. The MR brain image of Figure 6.1 reconstructed from the
low-low frequency band using the wavelet decomposition shown in
Figure 6.21.
Figure 6.24. The MR brain image of Figure 6.1 reconstructed from the
low-high, high-low and high-high frequency bands using the wavelet
decomposition shown in Figure 6.21.
Figures come from the textbook: Medical Image Analysis, by Atam P.
Dhawan, IEEE Press, 2003.