3.7 Adaptive filtering

3.7 Adaptive filtering JoonasVanninen Antonio Palomino Alarcos

Adaptive filtering • Linear filteringdoesnottakeintoaccountthe local features of theimage • Causes forexampleblurring of theedges • Animprovement: changethefilterparametersaccordingtothe local statistics • Changetheshape and thesize of theneighborhood • Suppressthefilteringifthere are featuresthatwewantto preserve in theneighborhood

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

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

LLMMSE estimation • g is the corrupted image • μg is the local mean • σg2 is the local variance • ση2 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

MSE=360 MSE=1595 MSE=324

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 doesn’t 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 window • 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 blurring

Sub-areas

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

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

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) Ξ causal FIR filter g(m,n) Ξ 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 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

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

MSE=395 MSE=1590

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 window • Additional assumption: image processes have zero mean

Implementation • Taking into account that now we have zero mean noise, following estimate is derived: • σf2 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

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

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: κcontrolstherate at which T changes Updating the window size (II)

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 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

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 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

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 σf2 > 2ση2, it is assumed that the neighbourhood contains a significant feature • If so, the tolerance is modified by T = 2 σf2

Implementation (II) • If the foreground has only one pixel, the neighborhood is enlarged to include the backround around it (3 x 3) • Flat regions →σf2 << 2ση2 → foreground will grow to Q • Busy regions →σf2 >> 2ση2 → tolerance will be reduced

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 anassessment of the results by a specialist

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3.7 Adaptive filtering