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Algorithms for Reducing Noisein Synthetic Aperture Radar Images

Troy Peterson Kling & Jeffrey Kidwell

Abstract—Images of Earth’s surface gathered by UninhabitedAerial Vehicle Synthetic Aperture Radar (UAVSAR) contain aplethora of information about the nature of the scattering mediacovering the ground. Depending on the type of instrument used toscan the landscape, a combination of radiometric, polarimetric,and interferometric data can be gathered. This data can be usedfor a wide range of purposes, from conducting biomass studiesto building digital elevation maps.

Despite its usefulness, synthetic aperture radar confers aspeckled appearance to the images it takes of Earth’s surface.This speckle noise hinders efforts to analyze image contents, andtherefore it is necessary to develop new, efficient techniques forremoving this noise. Non-local approaches are particularly well-suited for this task, since they are able to reduce speckle noisewhile preserving important structures in the image.

In this paper several different speckle filters are described andanalyzed. A brief overview of local filters is conducted to showwhy more advanced de-noising techniques are necessary. Non-local filters are then introduced and applied to optical images.Finally, each of the filters is applied to a synthetic aperture radarimage of Earth’s surface to see how it performs on a problemthat has significant real-world applications.

I. INTRODUCTION

Remote sensing images of Earth’s surface gathered byNASA JPL’s Uninhabited Aerial Vehicle Synthetic ApertureRadar (UAVSAR) provide valuable information about thenature of the scanned landscape. The type of scattering mediaon the surface (e.g. forests, vegetation, buildings, etc.) affectsthe polarization of the radio waves received by the SARinstrument. In general, three different channels are used forpolarization information - HH, VV, and HV - and the intensityand phase of each channel varies based on the type ofscattering media [1]. This polarimetric SAR (PolSAR) dataallows researchers to characterize images of the Earth’s surfaceby different types of land cover. In this paper, polarimetric ismostly ignored; only the magnitude of the HH channel is used.

Interferometric SAR (InSAR), on the other hand, performsmultiple parallel passes over a single area [2]. These passesare separated by a spatial baseline and are acquired eithersimultaneously or in a repeat-pass configuration. By detectingsmall changes in the phase of the received signals it is possibleto determine the height of the scattering media. This makesit possible to construct digital elevation models (DEMs) ofthe Earth’s surface [3]. InSAR can also be used to detecttemporal changes in a landscape by comparing data fromtwo passes separated by time. In this paper, interferometryis mostly ignored; only the image taken on the first pass isused.

Despite its usefulness, SAR imagery is inherently affectedby speckle noise. This random fluctuation in the received

signal degrades the quality of the image and prevents accurateanalysis of its contents. As a result, many filters have beendeveloped to reduce speckle noise in SAR imagery. A specklefilter is a mathematical technique for reducing noise in imagesand potentially recovering information that was lost due tonoise. By reducing the granularity of the image, one hopesto get a more accurate representation of the actual landscape.Once this is accomplished, existing algorithms can be em-ployed to gather qualitative and quantitative information aboutthe contents of the image.

A. Problem Description

1) Natural Language Description: Given a noisy syntheticaperture radar image and a selection of image filters (Boxcar,Gaussian convolution, Lee’s filter, non-local means, Fouriertransform filter, and modified ART-2 neural networks), the goalis to select the filter that best reduces noise in the image whilealso retaining image clarity by preserving edge details.

2) Formal Description: Given an n-by-m synthetic aper-ture radar image, X , with speckle noise modeled as a standardcomplex Gaussian distribution [4]

p(z) =1

πexp(−|z|2) (1)

where z = Aeiθ is a vector with amplitude A and phase θ,and given a set of noise reduction filters F = {f1, f2, ..., fk},find fp ∈ F such that

RMSE(fp(X)) ≤ RMSE(fq(X)) ∀ fq ∈ F (2)

where

RMSE(fq(X)) =

√√√√ 1

nm

n∑i=0

m∑j=0

(fq(Xij)− Tij)2 (3)

and where T is the ideal, non-noisy image.Equivalently, one could also attempt to find fp ∈ F such

that the objective function defined by the peak signal-to-noiseratio (PSNR) is maximized.

PSNR(fp(X)) = 20 · log10

(MAXI

RMSE(fp(X))

)(4)

where MAXI = 2B − 1 and B is the number of bytes usedto represent pixel values in the image.

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II. IMAGE FILTERING ALGORITHMS

Image filtering algorithms first came onto the scene in thelate 1970s and early 1980s, and they are still being developedand improved upon to this day. Statistical image filters can bebroken down into two major categories: local and non-local.Local filters attempt to estimate each pixel’s true intensityby looking at the pixels immediately surrounding it (hereafterknown as that pixel’s neighborhood). This assumes that pixelsnearby to one another are relatively similar. Non-local filtersreject this idea and are instead built upon the principle thattwo pixels can belong to the same statistical distribution evenif they are located quite far from one another in the image.In general, this allows non-local filters to perform better noisereduction than local filters.

Filters from both of these paradigms - and some from otherparadigms entirely - will be introduced and analyzed in thissection. The standard image of Lena is used throughout thispaper to show how well different filters perform. Figure 1shows what the image of Lena looks like before and after noisewas programmatically added to it. Each of the image filtersused in this section will be taking the noisy image as inputand outputting a denoised image that hopefully looks similar tothe original, non-noisy image of Lena. The RMSE and PSNRstatistics will be used throughout this paper to gauge how wellthe filters perform relative to one another. As a benchmark,the RMSE between the original image of Lena and the noisyversion is 14.628, and the PSNR is 24.827.

A. Boxcar Filter

The Boxcar filter is one of the simplest local filters inexistence. It reduces the overall variation present in an imageby setting each pixel’s intensity equal to the average of itsneighborhood. In this way, it acts as a moving average thatuniformly blurs the image.

Let Xi be a pixel from the unfiltered image, and let Yi bethe corresponding pixel in the filtered image. Then the Boxcarfilter can be written as follows:

Yi = Box(Xi) =1

|Wi|∑j∈Wi

Xj (5)

where Wi denotes the set of all pixels in pixel i’s neighbor-hood.

Figure 2 shows how well the 5x5 Boxcar filter works. Asstated previously, the noisy image is uniformly blurred. Thisdoes reduce the granularity of the image, but it also blursimportant edges in the image, causing crucial information tobe lost. In this example, the RMSE between the original imageand the denoised image is 10.444, and the PSNR is 27.753.

This RMSE value is slightly lower than the RMSE for thenoisy image, which should indicate to the reader that theBoxcar filter produced a denoised image that more closelyapproximated the true, non-noisy image. Similarly, the higherPSNR value indicates that the level of noise has been reduced,allowing the original signal to more accurately cut through theremaining noise.

B. Gaussian Convolution

Before discussing how the Gaussian convolution filterworks, it is essential to first define what convolution is. Forone-dimensional signals f and g, convolution is defined as

(f ∗ g)(t) =

∫ ∞−∞

f(τ)g(t− τ)dτ (6)

This procedure can be visualized by sliding the signal fover the length of g and recording the area under the curveat each infinitesimal step. This is similar to calculating thecross-correlation between two signals.

Gaussian convolution is simply the process of convolvingthe original signal with a Gaussian kernel. This terminologylends itself to a one-dimensional description, but Gaussianconvolution can easily be extended to work with images intwo dimensions.

The multivariate normal distribution is defined as

f(x) =1√

(2π)k|Σ|exp−1

2(x− µ)TΣ−1(x− µ) (7)

where x = [x1, x2, ..., xk]T , µ is a vector of means, and Σ isthe covariance matrix.

For a symmetric bivariate normal distribution, one cancreate a discrete version of a Gaussian kernel by simplysampling the density at certain points and assuming all valuesbelow some threshold equal zero. For example, the following5x5 truncated Gaussian kernel could be used.

KG =1

159

2 4 5 4 24 9 12 9 45 12 15 12 54 9 12 9 42 4 5 4 2

(8)

Note that KG has the nice property that its values sum to1, as is true of all probability density functions.

Gaussian convolution can then be achieved by convolvingan image with this kernel. Similar to the Boxcar filter, thisamounts to a moving average achieved using a sliding windowthat has the same dimensions as KG.

Yi =∑j∈Wi

KG(j) ·Wi(j) (9)

The key difference between this method and the Boxcarfilter is that Gaussian convolution weights pixels closer to thecenter of the window more heavily than those near the border,whereas the Boxcar filter weights all pixels equally.

Figure 3 shows the result of applying a Gaussian convo-lution filter to the noisy image of Lena. Unsurprisingly, thedenoised image looks very similar to the one obtained by usingthe Boxcar filter. However, the Gaussian convolution filter isslightly more powerful. The RMSE between the original imageand the denoised image is 9.506, and the PSNR is 28.571.This indicates that Gaussian convolution is capable of reducingnoise better than the Boxcar filter, assuming a good value ischosen for σ.

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Fig. 1. Left: The non-noisy image of Lena used throughout this paper. Right: Image with Gaussian white noise (σ = 15) added programmatically.

Fig. 2. Left: Original image. Center: Noise added programmatically. Right: Image after applying a 5x5 Boxcar filter.

C. J. S. Lee’s Filter

The Lee filter [5], [6], developed by J. S. Lee, is anexample of an adaptive local filter. Compared to the Boxcarand Gaussian convolution filters, the Lee filter performs betterat preserving sharp edges and point scatterers. It does thisby using an adaptive filtering coefficient and attempting tominimize the mean squared error between the actual denoisedimage and our estimate of it.

The Lee filter can be written as follows.

Yi = Lee(Xi) = Box(Xi) + ki(Xi −Box(Xi)) (10)

where Box(Xi) is defined in Equation (5) and ki is an adaptivefiltering coefficient defined by

ki =V ar(Y )

V ar(Wi)=V ar(Wi)− E2(Wi)σ

2

V ar(Wi)[1 + σ2](11)

where V ar(·) is the variance operator, E(·) is the expectation,and σ2 is the a priori variance of the speckle noise.

The adaptive filtering coefficient, ki, is affected by the levelof heterogeneity in a given region. If the neighborhood of agiven pixel is mostly homogeneous, then ki ≈ 0, and the Leefilter acts as a simple Boxcar filter. However, over regions

of heterogeneity, ki ≈ 1 makes it so that Yi = Xi, whichsets the filtered pixel intensity equal to the unfiltered pixelintensity, effectively doing no noise reduction whatsoever.In regions of moderate heterogeneity, ki taken on valuesbetween 0 and 1, which allows the Lee filter to find a middle-ground between the two extremes of perfect homogeneityand perfect heterogeneity. This malleability allows Lee’s filterto effectively reduce noise over homogeneous regions whileleaving fine structures and details intact.

Lee’s filter can be improved by implementing directionalmasks. While this addition does help the filter better pre-serve sharp edges and point scatterers, it can also introduceunwanted artifacts into the filtered image.

Figure 4 shows the result of applying a Lee filter to thenoisy image of Lena. The denoised image possesses a levelof clarity that could not be achieved using the previouslydiscussed filters. Regions of the image that contain highvariability (e.g. Lena’s eyes and the brim of her hat) havebeen preserved, whereas regions of low variability have beensmoothed over. This behavior is made possible by the adaptivefiltering coefficient. The RMSE between the original imageand the denoised image is 8.740, and the PSNR is 29.300.This indicates that the Lee filter is a better candidate for noisereduction than the Boxcar and Gaussian convolution filters.

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Fig. 3. Left: Original image. Center: Noise added programmatically. Right: Image after applying a 5x5 Gaussian convolution filter, σ = 1.25.

Fig. 4. Left: Original image. Center: Noise added programmatically. Right: Image after applying a 5x5 Lee filter.

D. Non-local MeansLocal filters assume that, for any given neighborhood in

an image, all nearby pixels come from the same statisticaldistribution as the center pixel. In other words, pixels thatare close to one another are assumed to be relatively similar.Non-local filters reject this idea. From a non-local perspective,there is no reason to suspect that proximity between pixels isa good measure of similarity [7]. Instead, these filters workbased on the principle that two pixels can belong to the samestatistical distribution even if they are located quite far fromone another.

One approach in particular, the non-local means algorithm[3], [8], [7], attempts to filter an image by defining each pixel’sneighborhood in a new way. Instead of restricting a pixel’sneighborhood to the area immediately surrounding it, the non-local means algorithm considers all pixels in the entire image.By comparing a small window around one pixel to a smallwindow around all others, the filter is able to gauge theirsimilarity. Using this knowledge about similarity within animage, the filter is capable of more accurately estimating eachpixel’s true intensity.

In the non-local paradigm, each pixel is assigned a newvalue equal to a weighted sum of all pixels in the image (orall pixels in a large search window around the center pixel).

Therefore, the non-local means filter can be written as follows

Ys = NL(Xs) =∑t∈Ws

w(s, t)Xt (12)

where w(s, t) is a weight associated with each pair of pixels,calculated as follows.

w(s, t) =1

Z(s)e−

d2(s,t)

h2 (13)

where h is a tuning parameter that controls the rate ofexponential decay, d2(s, t) is the Euclidean distance betweenthe two similarity windows, and Z(s) is the normalizingconstant

Z(s) =∑

je−

d2(s,t)

h2 (14)

An illustration of the non-local means algorithm is shownin Figure 5. In this diagram, the search window Ws has size7x7 and the similarity windows ∆s and ∆t have size 3x3,but any size can be chosen for these windows. In general,the larger the search window the better the filter performs.However, very large window sizes may cause the algorithm torun quite slowly.

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Fig. 5. An illustration of how the non-local means algorithm works, for a single pixel, s.

Pseudocode for the non-local means algorithm can bewritten as follows:

Input a noisy image, X .For each pixel s ∈ X:

Generate a large, square search window, Ws, around s.Generate a smaller similarity window, ∆s, around s.For each pixel t ∈Ws:

Generate a similarity window, ∆t, around t.Calculate w(s, t) using Equation (13).

End for loop.Estimate the filtered pixel intensity using Equation (12).

End for loop.Output the denoised image, Y .

This pseudocode hides some of the complexity of the algo-rithm, but it gives a good overview of how the filter works.Combined with the diagram in Figure 5, this should be enoughinformation to assist readers in implementation.

Figure 6 shows the result of applying a non-local meansfilter to the noisy image of Lena. 5x5 similarity windowsand 35x35 search windows were used to produce this result.Larger windows can be used for slightly better results, but thetime trade-off is likely not worth it. The value of the filteringparameter, h, was chosen by hand.

By visually inspecting the images in Figure 6, it is plainto see that the non-local means filter does an extraordinaryjob of reducing noise. It does everything the Lee filter does(smoothing over homogeneous regions, preserving edges andother fine structures), but it also accurately reduces noise inregions of high variance.

The RMSE between the original image and the denoisedimage is 6.158, the lowest value we have achieved so far.Similarly, the PSNR is 32.342, which is the highest value wehave achieved so far. These statistics reflect the fact that thenon-local means filter performs much better than any of thefilters previously discussed.

E. Fourier Transform

A Fourier transform takes a signal in the time domain andtransforms it into the complex-valued frequency domain. TheFourier transform of a function f is defined as

f(x) =

∫ ∞−∞

F (k)e2πikxdk (15)

This transformation can be reversed with the inverse Fouriertransform

F (x) =

∫ ∞−∞

f(k)e−2πikxdk (16)

The transformations defined above are in terms of one-dimensional signals, but a similar process can be used tocompute the Fourier transform of an image.

Noise reduction can be achieved by taking the Fast FourierTransform (FFT) of an image, filtering certain frequencies,and then performing the inverse FFT. The filtering process cantake any number of forms (e.g. low-pass filter, high-pass filter,band-pass filter). We chose to keep only those frequencies thatwere in the range (−12.75, 12.75), setting the rest to zero.

Figure 7 shows the result of applying a Fourier transformthat utilizes a bandpass filter to the noisy image of Lena. Theresults are not ideal. There is still a fair amount of noise inthe image, and many of the sharp edges have been blurred.In this way, the simplistic Fourier transform filter is similar inperformance to the Boxcar and Gaussian convolution filters.The RMSE between the original image and the denoised imageis 10.197, and the PSNR is 27.961. While these values areslightly better than the results for the Boxcar filter, they arenot anywhere close to the results produced by non-local means.

It should be noted that this is not the only way in whichFourier transforms can be applied to the problem of noisereduction. More advanced algorithms exist that filter the imagein a more adaptive manner, producing better results. Onecan also investigate wavelet transforms for even greater noisereduction.

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Fig. 6. Left: Original image. Center: Noise added programmatically. Right: Image after applying a non-local means filters, using 5x5 similarity windows,35x35 search windows, and h = 13.5.

Fig. 7. Left: Original image. Center: Noise added programmatically. Right: Image after applying a band-pass Fourier transform filter.

F. Modified Adaptive Resonance Theory

Developed by Stephen Grossberg and Gail Carpenter, theadaptive resonance theory model (ART-2) is a neural networkthat was originally designed for classifying input patterns intocategories based on their similarity [9]. This neural networkcan be trained to recognize patterns in signals and imagesso that it can correctly classify new input patterns that itreceives. ART-2 can be applied to many different classificationproblems, but it can also be used for noise reduction.

Figure 8 shows a diagram of the ART-2 neural network.There are six nodes in the F1 layer (which can be thought ofas one big node itself), labeled wi, xi, vi, ui, pi, qi. This layerfunctions as the neural network’s short-term memory. There isalso an F2 layer, which functions as long-term memory. Thislayer contains a number of nodes equal to the desired numberof categories. The bottom-up and top-down weights zji andzij are used to communicate between the F1 and F2 layers.

The short-term memory is governed by the following equa-tions. Note that e can be set to zero, which effectively turnsxi into the norm of wi, ui into the norm of vi, and qi intothe norm of pi. This normalization step is indicated by filledblack circles in Figure 8.

pi = ui +∑j

g(yj)zji qi =pi

e+ ||pi||

ui =vi

e+ ||vi||vi = f(xi) + bf(qi)

wi = Ii + aui xi =wi

e+ ||wi||

One should also note that there exists a filter function,f , that is used for noise suppression. This function cantake different forms, but for our purposes we define it as apiecewise linear function.

f(x) =

{0 if 0 ≤ x < θx if x ≥ θ (17)

where θ is a threshold variable.This filter function is where the built-in noise reduction step

takes place in ART-2. However, we would like to extend ART-2’s noise reduction capability by implementing some ideasfrom the non-local means filter.

Given a noisy image, X , we can divide it up into a series ofoverlapping windows. If we were to take these windows and

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Fig. 8. A diagram of the ART-2 neural network.

treat them as signal patterns, then we could feed them into theART-2 neural network. ART-2 would then place them into apre-determined number of categories based on their similarityto one another. After this process completes, we would knowwhich windows in the noisy image are most similar to oneanother.

Given this knowledge about window similarity, we couldattempt to filter the noisy image in several different ways. Oneoption is to take a page out of the non-local means book andcompute a weighted average within each category. Each intra-category average could then be substituted for the originalwindows belonging to that category, thereby reducing noise inthe image. However, it is unclear whether or not this wouldwork properly.

We could also take a more simplistic approach. Instead ofperforming a weighted average within each category basedon Euclidean distances, we could compute the unweightedaverage within each category and average that with the originalwindow in order to get our new, denoised window. This wouldallow for less uniformity in the resulting image, which is likelya good thing.

At this point in time, all of these ideas are theoretical. It iseven possible that the standard ART-2 neural network performsenough noise reduction on its own, and no additional steps arenecessary. We will not know which scheme is the best at noisereduction until we implement them all and test them out. Sincewe have not yet implemented ART-2, we have no results forthis section.

III. CONCLUSION

A. Results for Optical Images

From the figures shown previously, it is clear to see that thenon-local means filter performs the best out of all the filters weresearched. Although it remains to be seen whether or not themodified ART-2 neural network will outperform the standardnon-local means algorithm, we can safely conclude that non-local means is at least a front-runner for the best image filter.

Filter RMSE PSNR Big-ONon-local means 6.16 32.34 O(nmw2k2)J. S. Lee’s filter 8.74 29.3 O(nmk2)Gaussian convolution 9.51 28.57 O(nmk2)Fourier transform filter 10.20 27.96 O(nm log(nm))Boxcar filter 10.44 27.75 O(nmk2)Modified ART-2 N/A N/A O(?)

Fig. 9. RMSE and PSNR statistics for the various image filters discussed inthis paper, along with their Big-O complexity.

Figure 9 summarizes the results obtained in the previoussection, and it also includes the algorithmic complexity (usingBig-O notation) for each of the filters. The filters are orderedby ascending RMSE; those higher in the list produced lowerRMSE values, which indicates that they were better at reduc-ing noise. There are no entries for the modified ART-2 filter,because we have not yet finished implementing it. ConsideringSAR images are many thousands of pixels wide, non-localmeans would take a very long time filtering the entire image.

Figure 10 shows how long each filtering algorithm tookto execute for images of varying sizes, from 16x32 pixels to512x1024 pixels. The Boxcar, Gaussian convolution, and FFTfilters all run extremely quickly. The Lee filter takes somewhatlonger to execute, because it spends a lot of time computinglocal statistics. The non-local means filter takes longest of allto execute, with average runtime approaching one minute whenthe image is quite large.

B. Results for SAR Images

So far, we have only seen image filters applied to standardphotographs. Images taken by a synthetic aperture radar in-strument are an entirely different beast, so how can we besure our algorithms will work equally well on them? SinceSAR images are inherently noisy, there is no surefire way todetermine what the true non-noisy image should look like.

However, since we understand how noise is modeled inSAR imagery, we should be fairly confident that the imagefilters we discussed previously will work equally well onthem. And since we’re only working with single-channelmagnitudes, we don’t have to worry about coherence betweenpolarimetric or interferometric channels. Denoising the single-channel magnitude of a SAR image is very similar to denoisingan optical image, so our algorithms should work just fine.

Indeed, Figures 11 and 12 show just how well the differentfilters perform on a SAR image. The landscape being im-aged in these figures is a track of land in Harvard Forest,Massachusetts. For the most part, the forested areas shouldappear to be homogeneous. However, these regions are heavily

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Fig. 10. Average run times for images of varying sizes.

affected by speckle noise. Applying the different filters to thisnoisy SAR images produces varying results.

The Boxcar filter uniformly blurs the image, just as itdid for the test image of Lena. Gaussian convolution hasa similar effect, although the result appears to be slightlybetter. Much the same can be said about the Fourier transformfilter. The Lee filter manages to significantly reduce noise,while preserving edges and point scatterers. However, theseareas of high variance are not smoothed over at all. The non-local means filter improves upon this by performing additionalsmoothing over the forested regions and also reducing noisearound edges and point scatterers, without excessively blurringthem. This agrees with results obtained for optical images.

In conclusion, we have shown that the non-local means filteris by far the best image filter we have investigated. However, itis also the least tractable of the filters if large images are beingused. SAR images are usually much larger than the exampleswe have included here, so running them through the non-localmeans filter could take an appreciable amount of time. Wetherefore leave the choice of image filter up to the researcher.The Lee filter is great if you need an image filtered quickly,and the non-local means filter is excellent if you need an imagefiltered optimally.

C. Future Work

This research was conducted during the Spring semesterof 2015 for the course CSC 380 - Design and Analysis ofAlgorithms. Over the course of the semester, we only hadenough time to research and implement a handful of noisereduction algorithms. We also restricted our tests to opticalimages and single-channel HH magnitude SAR images.

In the future, we plan to:• Finish implementing ART-2.• Test out Daubechies wavelet transforms as a replacement

for the Fourier transform filter.• Extend the existing filters to work with polarimetric

and/or interferometric SAR images.Currently, the most promising area of research seems to be

combining ART-2 with the non-local means filter. We expectgood results from this hybrid model.

D. Acknowledgements

Thanks to Dr. Maxim Neumann of NASA’s Jet PropulsionLaboratory for supplying the synthetic aperture radar imagesof Harvard Forest, Massachusetts, and to Dr. Gene Tagliarinifor introducing us to ART-2.

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Fig. 11. Top row: Original SAR image; Boxcar; Gaussian convolution. Bottom row: Lee’s filter; non-local means; FFT filter.

E. Questions

This sections contains questions that readers can beexpected to answer after familiarizing themselves with thecontent of this paper.

Q1. What is the difference between local and non-localfilters, and why do non-locals filters generally performbetter at noise reduction than local filters?

A1. Local filters estimate each pixel’s intensity by lookingat the pixels immediately surrounding it. Non-localfilters estimate each pixel’s intensity by looking atevery pixel in the entire image (or at least a very largesearch window). Assuming the image has some degreeof self-similarity, this will result in the non-local meansfilter performing better than a local filter.

Q2. The non-local means filter gave the lowest RMSE out ofall the filters we discussed, but it’s not the best in everyrespect. What is one drawback of non-local means?

A2. The non-local means algorithm has complexityO(n · m · w2 · k2), where n ∗ m is the size of theimage, w is the width of the search window, and k isthe width of the similarity window. For even modestlylarge values of n, w, and k, non-local means takes along time to finish running.

Q3. For the Lee filter, how does the adaptive filtering coef-ficient, ki, affect the filtering procedure?

A3. Homogeneous regions of the image produce ki valuesclose to zero, which causes the Lee filter to act morelike a Boxcar filter. Heterogeneous regions of the imagethat contain a lot of variation produce ki values closeto one, which causes the filtered pixel value to be setequal to the unfiltered pixel value (thereby preservingfine structures in the image). For regions of the imagein between the extremes of perfect homogeneity andperfect heterogeneity, ki adapts accordingly.

Q4. What was the ART-2 system originally designed for, andhow can it be applied to noise reduction?

A4. The ART-2 neural network was originally designed forcategorization of signal patterns. ART-2 can be appliedto noise reduction in several ways - one of which isby categorizing overlapping windows in the image andthen applying a filter function for each category, ina similar manner to how the non-local means filterworks. However, ART-2 can also perform noise reduc-tion/suppression on its own - it is built into the system.

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Fig. 12. Top row: Original SAR image; Boxcar; Gaussian convolution. Bottom row: Lee’s filter; non-local means; FFT filter.

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