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## 3.7 Adaptive filtering

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### Title: The Adaptative 2D LMS Filter Author: apalomin Last modified by: Yo Created Date: 2/11/2007 12:12:30 PM Document presentation format: Presentaci n en pantalla ... – PowerPoint PPT presentation

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Title: 3.7 Adaptive filtering

1
• Joonas Vanninen
• Antonio Palomino Alarcos

2
• Linear filtering does not take into account the
local features of the image
• Causes for example blurring of the edges
• An improvement change the filter parameters
according to the local statistics
• Change the shape and the size of the neighborhood
• Suppress the filtering if there are features that
we want to preserve in the neighborhood

3
Filters
• The local LMMSE filter
• The noise-updating repeated Wiener filter
• The adaptive 2D LMS filter
• The adaptive rectangular window LMS filter
• The adaptive-neighborhood filter

4
The local LMMSE filter
• Local Linear Minimum Mean Squared Error filter,
also known as the adaptive Wiener filter
• Assumes that the image is corrupted by additive
noise
• Minimizes the local mean squared error by
applying a linear operator to each pixel in the
image
• The local mean and variance are estimated from a
rectangular window around the pixel

5
LLMMSE estimation
• g is the corrupted image
• µg is the local mean
• sg2 is the local variance
• s?2 is the variance
• Constant if noise is signal-independent
• Varies if noise is signal-dependent
• In the latter case it should be estimated locally
with the knowledge of the type of the noise

6
MSE360
MSE1595
MSE324
7
Interpretation of the equation
• If the area processed is uniform, the second term
is small
• Estimate is near the local mean of the noisy
image
• Noise is reduced
• If there is a edge in the processing window, the
variance of the image is larger than the variance
of the noise
• Estimate is closer to the actual noisy value
• Edge doesnt get blurred

8
A refined version
• If the local variance is high, it is assumed that
there is an edge in the processing window
• The edge is assumed to be straight in a small
window
• The direction of the edge is calculated
• 8 different directions
• The processing window is divided to two uniform
sub-areas over the edge
• The statistics in the sub-area that holds
processed pixel are used
• Noise is reduced near the edges without causing
blurring

9
Sub-areas
10
The noise-updating repeated Wiener Filter
• An iterative application of the LLMMSE filter
• The variance of noise used in the LLMMSE estimate
formula is updated in each iteration
• The noise is reduced even near the edges
• To avoid blurring, a smaller window size may be
choosen for each iteration

11
The Adaptative 2D LMS Filter

12
Why?
• Wiener filter has to be used with stationary and
statistically independent signal and noise models
• Adaptative 2D LMS algorithm circunvent this
problem
• It takes into account the nonstationarity of the
given image

13
2D LMS Algorithm
• Same concept fixed-window Wiener filter
• New idea filter coefficients vary depending upon
the image characteristics
• Algorithm based on the steepest descent method
• It tracks the variations in the local statistics
adapting to the changes in the images

14
• Estimate of the original pixel value f(m,n)
• Convolution
• wl(p,q) ? causal FIR filter
• g(m,n) ? noise-corrupted input image

15
Updating for the filter coefficients
• New coefficients determined by minimazing MSE
between f(m,n) and the estimation
• Steepest descent method
• µ controls the rate of convergence and filter
stability.
• Error is estimated using an approximation to the
original signal d(m,n)
• d(m,n) obtained by decorrelation from the input
image g(m,n)
• 2D delay operator of (1,1) samples, use the
previous pixel as an estimate

16
Advantages of the Algorithm
• It does not require a priori information about
• Image
• Noise statistics
• Correlation properties
• It does not require averaging, differentiation
and matrix operations

17

MSE395
MSE1590
18
The Adaptative Rectangular Window LMS Filter

19
ARW LMS Algorithm
• Same concept 2D LMS filter based on standard
Wiener filter
• New Idea use of an adaptative-sized rectangular
window
• Additional assumption image processes have zero
mean

20
Implementation
• Taking into account that now we have zero mean
noise, following estimate is derived
• sf2 is the variance of the original image
(estimated)
• Idea globally nonstationary process can be
considered locally stationary and ergodic over
small regions
• Target to identify the size of a stationary
square region for each pixel in the image
• Sample statistics can approximate the a
posteriori pareameters needed

21
Updating the window size
• Large window It may include pixels form other
ensembles
• Small window The statistics needed are poorly
estimated
• ARW lengths (Lr and Lc) are varied using a signal
activity parameter
• A similar signal activity parameter is defined in
the colum direction

22
Updating the window size (II)
• If S is large N is decremented
• If S is small N is incremented
• In order to make this decision, S is compared to
a threshold T as follows
• Threshold is defined as
• ? controls the rate at which T changes

23
• Uses a variable-size, variable-shape neighborhood
determined individually for every pixel
• Neighborhood contains only spatially connected
pixels that are similar to the seed
• This way the estimated statistics are likely to
be closer to the true statistics of the signal
• Adaptive neighbourhoods should not mix the pixels
of an object with the pixels of the background ?
no blurring of the edges

24
Region growing
• The absolute difference between each of the
8-neighbor pixels of the region and the seed is
calculated
• If the value is under a threshold T, pixel is
included in the region
• The process is iterated until there are no new
pixels
• Additionally a background region is grown by
expanding the foreground boundary by a
prespecified number of pixels
• The regions are grown for each pixel in the image

25
• An example of region growing
• The same neighborhood can be used for every pixel
in the area with the same gray-value

26
Adaptive-neighborhood mean and median filters
• The mean and median values of adaptive
neighborhoods can be used to filter noise
• They provide a larger population to compute local
statistics than fixed 3 x 3 or 5 x 5 windows
• Edge distortion is pervented because the
neighbourhood does not cross object boundaries

27
Adaptive-neighborhood noise subtraction (ANNS)
• Used for removing additive signal-independent
noise
• Estimates the noise value in the seed pixel with
an adaptive neighborhood and subtracts it to
obtain an estimate of the original
• The local statistics are estimated as in the
ARW-LMS filter

28
Implementation
• A maximum foreground limit of Q pixels is set.
• At first, the tolerance is set to the full
dynamic range
• An estimate of the variance of the uncorrupted
signal is calculated from the neighborhood
• It is compared to the noise variance, and if sf2
gt 2s?2, it is assumed that the neighbourhood
contains a significant feature
• If so, the tolerance is modified by T 2 sf2

29
Implementation (II)
• If the foreground has only one pixel, the
neighborhood is enlarged to include the backround
around it (3 x 3)
• Flat regions ? sf2 ltlt 2s?2 ? foreground will grow
to Q
• Busy regions ? sf2 gtgt 2s?2 ? tolerance will be
reduced

30
Conclusion
• Use of local statistics in an adaptive filter is
a powerful approach to remove noise
• while retaining the edges in the images with
minimal distortion
• Some of the implicit assumptions may not apply
well to the image or noise processes
• It is common to try several previously
established techniques
• It is quite difficult to compare the results
provided by different filters
• Generally MSE or RMS error is used
• In real applications, it is important to obtain
an assessment of the results by a specialist

31
Quote
• You can only cure retail but you can prevent
wholesale
• Brock Chisholm

32
any question?