medical image analysis image enhancement figures come from the textbook: medical image analysis, by...
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Medical Image AnalysisMedical Image AnalysisImage Enhancement
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Spatial Domain MethodsSpatial Domain Methods
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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
Histogram TransformationHistogram TransformationHistogram
Histogram equalization
ii nrh )( 1,...,1,0 Li
n
nrp ii )(
i
j
i
j
ijrii n
nrprTs
0 0
)()(
1,...,1,0 Li
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 ModificationHistogram ModificationScaling
Histogram modification
cazab
cdznew
)(
i
jjrii rprTu
0
)()(
1,...,1,0 Li
Histogram ModificationHistogram ModificationHistogram modification
◦Target histogram:
i
kkzii zpzVv
0
)()(
1,...,1,0 Li
)( kz zp
)()( iii rTzVu
)()( 11iii uVrTVs
Image AveragingImage AveragingAveraging
◦Enhancing signal-to-noise ratio
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K
ii yxg
Kyxg
1
),(1
),(
),(),( yxfyxgE
),(),(
1yxnyxg
K
Image SubtractionImage Subtraction
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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.
Neighborhood OperationsNeighborhood OperationsUse a weight mask
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px
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p
px
p
py
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' '
' '
)','()','(
)','(
1),(
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
f(-1,0)
f(0,-1) f(0,0) f(0,1)
f(1,0)
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.
Median FilterMedian Filter
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Median filter◦Order-statistics filter
),(),(),(
jigmedianyxfNji
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.
Adaptive Arithmetic Mean Adaptive Arithmetic Mean FilterFilter
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Adaptive◦If the noise variance of the image
is similar to the variance of gray values in the specified neighborhood of pixels, , the filter provides an arithmetic mean value of the neighborhood
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2s
),(),(),(),(ˆ2
2
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n
Image Sharpening and Edge Image Sharpening and Edge EnhancementEnhancement
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Sobel◦The first-order gradient in and
directions defined by and
x yxyxf /),(
yyxf /),(
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
-1 -2 -1
0 0 0
1 2 1
-1 0 1
-2 0 2
-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
0 0 0
1 1 1
-1 0 1
-1 0 1
-1 0 1
-1 -1 0
-1 0 1
-0 1 1
0 1 1
-1 0 1
-1 -1 0Figure 6.9. Weight masks for computing first-order gradient in (clockwise from top left) in horizontal, 45 deg, vertical and 135 deg directions.
Image Sharpening and Edge Image Sharpening and Edge EnhancementEnhancementLaplacian
◦The second-order dirivative operator◦Edge-based image enhancement
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),(
),(),(
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22
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x
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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).
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
-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.
Feature Enhancement Using Feature Enhancement Using Adaptive Neighborhood Adaptive Neighborhood ProcessingProcessing
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Three types of adaptive neighborhoods◦Constant ratio: an inner
neighborhood of size and an outer neighborhood of size
◦Constant difference: the outer neighborhood of size
◦Feature adaptive
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)()( ncnc
Feature Enhancement Using Feature Enhancement Using Adaptive Neighborhood Adaptive Neighborhood ProcessingProcessingFeature adaptive
◦Center region: consisting of pixels forming the feature
◦Surround region: consisting of pixels forming the background
◦1. The local contrast. : the average of the Center region. : the average of the Surround region
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Feature Enhancement Using Feature Enhancement Using Adaptive Neighborhood Adaptive Neighborhood ProcessingProcessingFeature adaptive
◦2. The Contrast Enhancement Function (CEF) : modify the contrast distribution by the contrast histogram
◦3. The enhanced image
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Feature Enhancement Using Feature Enhancement Using Adaptive Neighborhood Adaptive Neighborhood ProcessingProcessingFeature adaptive
◦3. The enhanced image
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),(),( if),('1
),(),(
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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.
Xc Xc
Center Region
Surround Region
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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.
(a) (b)
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.
Frequency Domain Frequency Domain FilteringFiltering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
: an acquired image : the object : a Point Spread Function
(PSF) : additive noise
),( yxg
),( yxf
),( yxh
),( yxn
),(),(),(),( yxnyxfyxhyxg
Frequency Domain Frequency Domain FilteringFiltering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
The Fourier transform
Inverse filtering
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),(
),(
),(
),(),(ˆ
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vuN
vuH
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Wiener FilteringWiener Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
: the power spectrum of the signal
: the power spectrum of the noise
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Wiener FilteringWiener Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
: if it is white noise),( vuSn
),(),(
),(
),(
1),(ˆ
2
2
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vuH
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Constrained Least Square Constrained Least Square FilteringFilteringAcquired image
Optimization
Subject to the smoothness constraint
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}ˆ][][ˆmin{ ff CC tt
Constrained Least Square Constrained Least Square FilteringFiltering
The estimated image
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Low-Pass FilteringLow-Pass FilteringIdeal
◦ : the frequency cut-off value◦ : the distance of a point in
the Fourier domain from the origin representing the dc value
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),( vuD
otherwise0
),( if1),( 0DvuDvuH
Low-Pass FilteringLow-Pass FilteringReduce ringing artifacts
◦Butterworth or GaussianButterworth
Gaussian
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1),(
22 2/),(),( vuDevuH
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 FilteringHigh-Pass Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
High-pass filtering◦Image sharpening and extraction of
high-frequency information◦Edges
Ideal
otherwise0
),( if1),( 0DvuDvuH
High-Pass FilteringHigh-Pass FilteringReduce ringing artifacts
◦Butterworth or GaussianButterworth
Gaussian
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20 )],(/[1
1),(
22 2/),(1),( vuDevuH
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 FilteringHomomorphic Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
: illumination : reflectance
In general
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),( yxr
),(),(),( yxryxiyxf
),(),(),( 21 yxfyxfyxf
),(ln),(ln),(ln),( 21 yxfyxfyxfyxg
Homomorphic FilteringHomomorphic Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Frequency filtering in the logarithmic transform domain
)},(ln),({ln)},({ 21 yxfyxfyxg
),(),(),( 21 vuFvuFvuG
),(),(),(),(
),(),(),(
21 vuFvuHvuFvuH
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Homomorphic FilteringHomomorphic Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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)},(),({)},(),({
),(
21
21
11
yxfyxf
vuFvuHFvuFvuHF
yxs
),(ˆ),(ˆ),(ˆ21
),( yxfyxfeyxf yxs
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
),( yxf ),( yxfln FT H(u,v) IFT exp
Figure 6.16. A schematic block diagram of homomorphic filtering.
Homomorphic FilteringHomomorphic Filtering
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
An example◦ and components
can represent, respectively, low- and high-frequency components
◦A circularly symmetric homomorphic filter function
),(1 yxf ),(2 yxf
LDvuDc
LH evuH )/),(( 22
1)(),(
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
H
L
H(u,v)
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.
Wavelet Transform for Image Wavelet Transform for Image ProcessingProcessing
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Figure 6.19. (a) A multi-resolution signal decomposition using Wavelet transform and (b) the reconstruction of the signal from Wavelet transform coefficients.
2H1
2H0 2H1
2H0 2H1
2H0
22H1
22H0 2H1
2H0
22H1
2H0
H1
22H0 2H1
2H0
22H1
2H0
H1
22H0
x[n] X(1)[2k+1]
2 G1 +
G022 G1 +
G022 G1 +
G02
2 G122 G1 +
G0222 G1 +
G02
22 G1 +
G0222 G1 +
G02
22 G1 +
G022
(a)
(b)
X(1)[2k] X(2)[2k+1]
X(2)[2k]X(3)[2k+1]
X(3)[2k]
X(3)[2k+1]
X(3)[2k]
X(2)[2k+1]
X(1)[2k+1]
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
2H1
2H0
2H1
2H0
2H1
2H0
Horizontal SubsamplingVertical Subsampling
2H1
2H0
2H1
2H0
22H
1
22H0
2H1
2H0
2H1
2H0
22H1
22H0
2H1
2H0
2H1
2H0
22H1
22H
0
Horizontal SubsamplingVertical Subsampling
Low-Low Aj
High-High Dj3
High-Low Dj2
Low-High Dj1
Figure 6.20. Multiresolution decomposition of an image using the Wavelet transform.
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.
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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.