Removal of Artifacts using linear and adaptive

methods

Vibhor kumar

Biomedical image processing

Chapter 3 , Biomedical Image Analysis, Rangaraj M. R. , CRC Press

Convolution - 1D matrices and 2D matrices

1D convolution

g(n) = k=0,n f(k)h(n-k)

2D convolution

g(m,n) = i=0,m j=0,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.

’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

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.

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)

original noisy imageestimate

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.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))]

LMMSE - Analysis (pros and cons)

f´(m,n) = E(g(m,n)) + [ ] [g(m,n) - E(g(m,n))] 2

g(m,n) - 2(m,n)

2g(m,n)

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

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) = p=1,P q=1,P w l(p,q)g(m-p, n-q)

FIR filter noisy image

. Noise is substantially reduced even in areas near edges.

The adaptive 2D LMS filter

The filter coefficients for the pixel l+1 or wl+1(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 :

wl+1(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

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

Mean taken in local window (size-R,C) around the pixel(m,n)

f´(m,n) = E(g(m,n)) + [ ] [g(m,n) - E(g(m,n))] 2

f(m,n) 2

f(m,n)+ 2η(m,n)

2f(m,n) is taken as

f2(m,n)- 2

η(m,n) if 2η(m,n) >2

η(m,n)

or 0 otherwise

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) / (2

f(m,n) +2(m,n) ) )

Comparative Analysis of Filters for Noise Removal

(a) original non-noisy image

(b) image with speckle noise

(c) 3x3 LLMMSE, MSE = 119.15

(d) NURW, MSE = 116.75

(e) Adaptive-neighborhood mean filter, MSE = 236.09

(f) Adaptive-neighborhood LLMMSE, MSE = 68.01

(a) original non-noisy image

(b) image with film-grain noise MSE=275.11

(c) 3x3 LLMMSE, MSE = 119.81

(d) NURW, MSE = 166.42

(e) Adaptive-neighborhood mean filter, MSE = 236.27

(f) Adaptive-neighborhood LLMMSE, MSE = 69.98

Comparative Analysis of Filters for Noise Removal

Area to represent clear differences

Comparative Analysis of Filters for Noise Removal – Benifit of adaptive neighborhood

(a) original non-noisy image

(b) image with salt and pepper noise MSE=1740.86

(c) 3x3 mean filter, MSE = 642.20

(d) 3x3 median filter MSE = 206.63

(e) Adaptive-neighborhood mean filter, MSE = 213.02

(f) Adaptive-neighborhood LLMMSE, MSE = 205.72

Area to represent clear differences

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

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.