Medical Image Analysis Image Enhancement Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003

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


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

),(),(),( yxnyxfyxg

K

ii yxg

Kyxg

1

),(1

),(

),(),( yxfyxgE

),(),(

1yxnyxg

K

Image SubtractionImage 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.


Figure 6.3. An MR angiography image obtained through image subtraction method.

Neighborhood OperationsNeighborhood OperationsUse a weight mask

p

px

p

py

p

px

p

py

yyxxfyxw

yxwyxg

' '

' '

)','()','(

)','(

1),(


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).


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].


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


Median filter◦Order-statistics filter

),(),(),(

jigmedianyxfNji


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


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

2n

2s

),(),(),(),(ˆ2

2

yxgyxgyxgyxf mss

n

Image Sharpening and Edge Image Sharpening and Edge EnhancementEnhancement


Sobel◦The first-order gradient in and

directions defined by and

x yxyxf /),(

yyxf /),(


-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.


-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

)],(4)1,(

)1,(),1(),1([

),(

),(),(

2

2

2

22

yxfyxf

yxfyxfyxf

y

yxf

x

yxfyxf

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.

Feature Enhancement Using Feature Enhancement Using Adaptive Neighborhood Adaptive Neighborhood ProcessingProcessing


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

cccc 33

)()( 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

),( yxPs

),( yxPc

)},(),,(max{

),(),(),(

yxPyxP

yxPyxPyxC

sc

sc


◦2. The Contrast Enhancement Function (CEF) : modify the contrast distribution by the contrast histogram

◦3. The enhanced image

),(' yxC

)},(),,(max{

),(),(),(

yxPyxP

yxPyxPyxC

sc

sc


◦3. The enhanced image

),(),( if)),('1)(,(),(

),(),( if),('1

),(),(

yxPyxPyxCyxPyxg

yxPyxPyxC

yxPyxg

scs

scs

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


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)


(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


: an acquired image : the object : a Point Spread Function

(PSF) : additive noise

),( yxg

),( yxf

),( yxh

),( yxn

),(),(),(),( yxnyxfyxhyxg

Frequency Domain Frequency Domain FilteringFiltering


The Fourier transform

Inverse filtering

),(),(),(),( vuNvuFvuHvuG

),(

),(

),(

),(),(ˆ

vuH

vuN

vuH

vuGvuF

Wiener FilteringWiener Filtering


: the power spectrum of the signal

: the power spectrum of the noise

),( vuS f

),( vuSn

),(

),(),(

),(

),(

),(

1),(ˆ

2

2

vuG

vuSvuS

vuH

vuH

vuHvuF

f

n

Wiener FilteringWiener Filtering


: if it is white noise),( vuSn

),(),(

),(

),(

1),(ˆ

2

2

vuGKvuH

vuH

vuHvuF

Constrained Least Square Constrained Least Square FilteringFilteringAcquired image

Optimization

Subject to the smoothness constraint

nHfg

})ˆ)(ˆ{(Trace2 tEe ffff

}ˆ][][ˆmin{ ff CC tt

Constrained Least Square Constrained Least Square FilteringFiltering

The estimated image

1.

2.

1.

.1

21

121

12

1

][C

gHCCHHf ttt ][][][1

][][ˆ1

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

),(),(),(ˆ vuGvuHvuF

0D

),( vuD

otherwise0

),( if1),( 0DvuDvuH

Low-Pass FilteringLow-Pass FilteringReduce ringing artifacts

◦Butterworth or GaussianButterworth

Gaussian

nDvuDvuH

20 ]/),([1

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


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

nvuDDvuH

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


: illumination : reflectance

In general

),( yxi

),( yxr

),(),(),( yxryxiyxf

),(),(),( 21 yxfyxfyxf

),(ln),(ln),(ln),( 21 yxfyxfyxfyxg



Frequency filtering in the logarithmic transform domain

)},(ln),({ln)},({ 21 yxfyxfyxg

),(),(),( 21 vuFvuFvuG

),(),(),(),(

),(),(),(

21 vuFvuHvuFvuH

vuGvuHvuS



),('),('

)},(),({)},(),({

),(

21

21

11

yxfyxf

vuFvuHFvuFvuHF

yxs

),(ˆ),(ˆ),(ˆ21

),( yxfyxfeyxf yxs


),( yxf ),( yxfln FT H(u,v) IFT exp

Figure 6.16. A schematic block diagram of homomorphic filtering.



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)(),(


H

L

H(u,v)

D(u,v)

Figure 6.17: A circularly symmetric filter function for Homomorphic filtering.


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


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]


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.


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.

Documents

Medical Image Analysis Image Enhancement Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003