Removal of Artifacts using linear and adaptive methods

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Removal of Artifacts using linear and adaptive
methods
Chapter 3 , Biomedical Image Analysis, Rangaraj
M. R. , CRC Press
Biomedical image processing
Vibhor kumar
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Convolution - 1D matrices and 2D matrices
1D convolution g(n) ? k0,n f(k)h(n-k) 2D
convolution g(m,n) ? i0,m? j0,n f(k)h(m-i ,
n-j) in fourier space the convolution can be
expressed as G(k,l) H(k,l)F(k,l) where
H(k,l) is fourier transform of matrix h
and F(k,l) is
fourier transform of matrix f . if h is a block
circulant matrix then by matrix diagonalisation
the convolved image g can be written as g
WDhW-1f where Dh is a diagonal matrix.
W-1 is a matrix of size MN
X MN, with M2 partition of size N X N.
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Optimal Filtering of images
. The Wiener Filter . Adaptive Filters . The
Local LMMSE filter . The adaptive 2D LMS
filter . The adaptive rectangular window LMS
filter . The adaptive- neighborhood filter
. Noise updating repeated Wiener filter
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Wiener filter
Taking the image as g f ? The wiener
estimation problem is to determine estimate f
Lg of f from the given noisy image g. LWiener
?f (?f ?? )-1 and filtered image is f
?f (?f ?? )-1g
1. The wiener filter is not spatially adaptive,
so it is likely to blur sharp features and
edges. 2. The gain of the wiener filter varies
from one frequency sample to another.
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The Local LMMSE filter
The LMMMSE approach computes at every spatial
location (m,n) an estimate f(m,n) of the original
image value by applying a linear operator to the
available corrupted image. as shown below f
(m,n) a(m,n) g(m,n) b(m,n)
Scalars a(m,n) and b(m,n) are found such that the
local MSE e2(m,n) E((f(m,n) - f(m,n))2) is
reduced.
estimate
original noisy image
On derivation b(m,n) f(m,n) - a(m,n) g(m,n)
and a(m,n) ? fg (m,n) / ? 2g (m,n)
Local covariance between original image (f) and
noisy image (g)
Local variance of noisy image
f(m,n) E(f(m,n)) a(m,n) g(m,n) - E(g(m,n))
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LMMSE - Analysis (pros and cons)
The final estimate estimating equation becomes
. If the variation is only due to noise then
LMMSE estimate is equal to local mean . If the
variance of the local noisy image is more than
the variance of noise , the LMMSE estimate is
closer actual noisy value g(m,n).
The filter gives good noise reduction but poor
noise filtering near edges of high frequency
region
In refined LMMSE filter, where neighborhood box
of the pixels are split around the edges, the
performance increases with additional
computational task of detecting the right
direction of edge
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Adaptive modifications of wiener filter
The noise-updating repeated Wiener filter
(NURW) NURW filter consists of an iterative
application of the LLMMSE filter, i.e. variance
of the noise is updated after every iteration,
for use in the LLMMSE estimate formula The
adaptive 2D LMS filter The 2D LMS algorithm is
derived by defining a causal FIR filter. The
estimated image f(m,n) is computed as f(m,n)
? p1,P ? q1,P w l(p,q)g(m-p, n-q)
. Noise is substantially reduced even in areas
near edges.
FIR filter
noisy image
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The adaptive 2D LMS filter
The filter coefficients for the pixel l1 or
wl1(p,q) is determined by minimizing MSE at the
present pixel location l, using the method of
steepest descent. So it can be represented by
wl1(p,q) wl(p,q) - ? ? el2, where el is
taken as el d(m,n) - f(m,n) Hadhoud and
Thoumas proposed technique to estimate d(m,n)
from the input image g(m,n) by de-correlation.
Properties 1. It suppress noise in a relatively
uniform manner i.e. Blur the edge and leave
excessive noise in the uniform region , Even
though the MSE is relatively lower. 2. The right
convergence factor ? is very important
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Adaptive rectangular window LMS filter
The assumption used A globally non-stationary
process can be considered to be locally
stationary over a small region Just like LMMSE
filter but local mean and variance used as the
mean and variance of the original non-noisy
image f(m,n) in the window whose size may be
variable . The equation is
?2f(m,n)
f(m,n) E(g(m,n))
g(m,n) - E(g(m,n))
?2f(m,n) ?2?(m,n)
?2f(m,n) is taken as ?f2(m,n)- ?2?(m,n) if
?2?(m,n) gt?2?(m,n) or 0 otherwise
Mean taken in local window (size-R,C) around the
pixel(m,n)
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The adaptive neighborhood filter
The neighborhood is variable in size and shape
and contains similar spatially connected pixels.

Adaptive-neighborhood noise subtraction
An estimate of the noise at pixel(m,n) is
obtained from the corresponding adaptive
neighborhood grown in the corrupted image g
as mean(? (m,n)) ? g(m,n) and estimate of
image as
f(m,n) ?g(m,n) (1-? ) g(m,n) - ?g(m,n)
? is taken as sqrt (?2?(m,n) / (?2f(m,n)
?2?(m,n) ) )
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Comparative Analysis of Filters for Noise Removal

1. original non-noisy image
2. image with speckle noise
3. 3x3 LLMMSE, MSE 119.15
4. NURW, MSE 116.75
5. Adaptive-neighborhood mean filter, MSE 236.09
6. Adaptive-neighborhood LLMMSE, MSE 68.01

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Comparative Analysis of Filters for Noise Removal
Area to represent clear differences

1. original non-noisy image
2. image with film-grain noise MSE275.11
3. 3x3 LLMMSE, MSE 119.81
4. NURW, MSE 166.42
5. Adaptive-neighborhood mean filter, MSE 236.27
6. Adaptive-neighborhood LLMMSE, MSE 69.98

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Comparative Analysis of Filters for Noise Removal
Benifit of adaptive neighborhood
Area to represent clear differences

1. original non-noisy image
2. image with salt and pepper noise MSE1740.86
3. 3x3 mean filter, MSE 642.20
4. 3x3 median filter MSE 206.63
5. Adaptive-neighborhood mean filter, MSE 213.02
6. Adaptive-neighborhood LLMMSE, MSE 205.72

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Review
1. convolution 2. Optimal filtering (a)
Wiener filtering (b) Adaptive filters
The local LMMSE filter The
noise updating repeated wiener filter
The Adapative 2D LMS filter The
adaptive rectangular window LMS filter
The adaptive-neighborhood filter 3.
comparisons of filters on different type of noise
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Important References

1. Wiener NE. Extrapolation, Interpolation, and
   Smooting of Stationary Time series, with
   Engineering Applications, MIT Press, Cambridge,
   MA, 1949.
2. Lee JS Digital image enhancement and noise
   filtering by use of local statistics, IEEE
   Transactions on Pattern Analysis and Machine
   Intelligence,PAMI-2.165-168, March 1980.
3. Jiang SS and Sawchuk AA. Noise updating
   repeated Wiener filter and other adaptive noise
   smoothing filters using local image statistics,
   Applied Optics, 25, 2326-2337, July 1986.
4. Hadhoud MM and Thomas DW, The two-dimensional
   adaptive LMS(TDLMS) algorithm, IEEE Transactions
   on Circuits and Systems, 35(5)485-494,1988.
5. Mahesh B, Song WJ and Pearlman WA, Adaptive
   estimators for filtering noisy images, Optical
   Engineering, 29(5), 488-494,1990.