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3.7 Adaptive filtering


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

3.7 Adaptive filtering
  • Joonas Vanninen
  • Antonio Palomino Alarcos

Adaptive filtering
  • 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

  • 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

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
  • Minimizes the local mean squared error by
    applying a linear operator to each pixel in the
  • The local mean and variance are estimated from a
    rectangular window around the pixel

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

Interpretation of the equation
  • If the area processed is uniform, the second term
    is small
  • Estimate is near the local mean of the noisy
  • 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

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
  • The direction of the edge is calculated
  • Gradient operator
  • 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

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

The Adaptative 2D LMS Filter

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

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

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

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
  • 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

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


The Adaptative Rectangular Window LMS Filter

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

  • Taking into account that now we have zero mean
    noise, following estimate is derived
  • sf2 is the variance of the original image
  • 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

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

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

The adaptive-neighborhood filter
  • 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

Region growing
  • The absolute difference between each of the
    8-neighbor pixels of the region and the seed is
  • If the value is under a threshold T, pixel is
    included in the region
  • The process is iterated until there are no new
  • 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

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

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

Adaptive-neighborhood noise subtraction (ANNS)
  • Used for removing additive signal-independent
  • 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

  • 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

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

  • 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

  • You can only cure retail but you can prevent
  • Brock Chisholm

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