Filtering Fringe Patterns

with the Extended Non Local Means Algorithm

Maciej Wielgus and Krzysztof Patorski

Institute of Micromechanics and Photonics, ul. Św. A. Boboli 8, 02-525 Warsaw, Poland

Keywords: Fringe Pattern Processing, Image Filtration, Non Local Means.

Abstract: The quality of interferometric measurements substantially benefits from the digital noise filtration.

Recently, robust non local filtration algorithms were introduced to optical metrology, the non local means

algorithm in particular. These methods allow to take advantage from the information redundancy spread in

the whole image domain for processing each pixel, constituting a powerful image denoising tool. We

evaluate how the denoising performance quality of the non local means algorithm can be further increased

by the introduction of geometrical transformations of the compared patches.

1 INTRODUCTION

Uncertainty is an intrinsic feature of every

measurement, appearing as noise in the measuring

system output. For fundamental reasons it is

impossible to fully remove its influence by hardware

setup modification. Instead of increasing the

hardware requirements most (if not all) systems for

interferometric measurements introduce some digital

noise filtration, applied to the registered pattern

before further processing. In many cases this is

a simple down-pass filtration by averaging with

binary or Gaussian mask. Median filter is a popular

choice as well. Dozens of more sophisticated

methods were proposed throughout the years.

One of the attractive novel developments in

image processing is the notion of the non local

filtration such as the non local means algorithm –

NLM (Buades et al., 2005). This group of methods

was recognized in the fringe pattern analysis just

recently. In (Wielgus and Patorski, 2012) basic

NLM algorithm was tested against several popular

filtration methods for interferometric pattern

filtration, while in (Fu and Zhang, 2012) modified

technique was proposed. The power of non local

methods lays in their ability of utilizing redundancy

in the whole image domain rather than in limited

neighbourhood of the considered pixel. Typically in

non local processing we compare patches (small

subimages containing the central pixel and its

neighbourhood) and average the intensities of their

central pixels based on established measure of patch

similarity. Unlike local averaging, the non local

method enables to avoid oversmoothing the image

and blurring its delicate features.

Robustness of non-local filtration for

photographic images, as shown in (Buades et al.,

2012), could be found as a surprising issue, as these

images do not represent any visible similarity of

distant patches. However, as noted in (Wielgus and

Patorski, 2012), situation is very different with

fringe patterns, which are quasiperiodic in nature

and therefore display similarity even between

significantly distant patches. To illustrate and

quantitatively evaluate this effect we calculate the

correlation of the fringe pattern presented in Figure

1 (a) with its chosen patch, located in the centre of

the image. This is a fragment of an experimentally

obtained interferogram of a silicone micromembrane

(Salbut et al., 2003). In Figure 1 (b) we show the

map of cross-correlation between the image and the

selected patch (brighter color = more similarity).

Note that it is a nonmonotonic function of distance

from the considered patch and that correlation

reaches high values even quite far away from the

chosen patch. This explains why non local methods

are supposed to fit particularly well for the fringe

pattern filtration. For the sake of clarity, only pixels

with normalized correlation larger than 0.3 are

shown.

In this paper we intend to exploit another

property of fringe patterns to further increase the

redundancy from which non local methods benefit,

Wielgus M. and Patorski K..

Filtering Fringe Patterns with the Extended Non Local Means Algorithm.

DOI: 10.5220/0004339500520055

In Proceedings of the International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2013), pages 52-55

ISBN: 978-989-8565-44-0

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

namely the similarity under the geometrical

operations of symmetry and rotation. In Figure 1 (c)

correlation map for the very same patch as in

Figure 1 (a-b) is shown for the case of allowing the

comparison between rotated and mirrored patches.

Clearly, there are many more patches that can be

possibly used for the filtration and therefore more

redundancy can be utilized. It can be quantitatively

expressed in terms of the norm of correlation maps,

which is 2.18 times larger for the map of Figure 1(c)

than for the map presented in Figure 1 (b).

(a)

(b)

(c)

Figure 1: Fringe pattern with indicated chosen patch (a),

correlation map of the pattern and the patch (b),

correlation map of the pattern and the patch with

geometric transformations allowed (c).

2 NLM ALGORITHM

The non local means algorithm (Buades et al., 2005)

can be considered as a basic method for the non

local patch-based filtration. It estimates the value of

image intensity I

in pixel k as a weighted average

of intensities I in other pixels

(1)

where weights are specified based on similarities of

the corresponding patches P

and P

(2)

|| . ||

denotes Gaussian-averaged Euclidean distance

between patches, h is the characteristic parameter of

the algorithm. Normalization factor N(k) is simply

(3)

The choice of the negative exponential function of

squared distance between patches is not unique, as

any smooth function monotonically decreasing for

positive arguments and reaching 0 at +∞ could be

used. It is important to emphasize that distance

between pixels have no influence on weights, which

is exactly what is meant by algorithm non-locality.

Nevertheless, in any useful implementation, patches

are compared only in certain regions limited by the

distance from considered patch. The only reason for

such limitation is the reduction of the computational

load.

3 GEOMETRIC

TRANSFORMATIONS

Proposed extension to the basic NLM algorithm lays

in consideration of not only similarities between

patches, but between mirrored and rotated patches as

well. Such an extension is very well motivated for

the fringe pattern filtration as cosine is not only

periodic, but also symmetric function as well since

orientation of fringes may strongly vary for different

pixel locations. In total, we consider 6 distinct

transformed patches I

(n)

), n = 1, 2, ..., 6:

 initial one,

 3 rotations (90, 180 and 270 degrees),

 2 symmetries (with respect to the horizontal and

vertical axis).

We calculate the weights similarly as in (2), only for

each pair of patches P

and P

we choose the best

weight from 6 transformations

(4)

In this manner we are capable of recognizing and

utilizing for denoising purposes the pixels that are

very similar but lay in the rotated (different fringe

orientation) or mirrored (different side of the fringe)

neighborhoods. Information gain was presented in

Figure 1.

FilteringFringePatternswiththeExtendedNonLocalMeansAlgorithm

4 NUMERICAL EXPERIMENTS

In the following experiment we compare filtration

results obtained by Gaussian smoothing, median

filtration, and more sophisticated anisotropic

diffusion filter based on (Tang et al., 2008), classic

NLM algorithm and finally our extended NLM

algorithm. We demonstrate the algorithm

performance on the synthetic pattern of vertical

fringes with slight phase distortion, spoiled with

additive white Gaussian noise (signal to noise ratio

smaller than 2), Figure 2 (b). Knowing the original

pattern (one without the noise, Figure 2 (a)), we are

able to quantitatively evaluate the denoising

performance in terms of the root mean square error

(the difference between fringe pattern without any

noise imposed and the filtration result), which is

summarized in the Table 1. We also include the full

processing error evaluation, which is the root mean

square error of the Fourier method based phase

calculation performed on the denoised image.

Table 1: Filtration quality evaluation for the synthetic

pattern.

Algorith

Denoising

RMSE

Noise

reduction

[%]

Phase

RMSE

filtration

64.20 0 50.42

median 34.64 46.0 26.10

Gaussian 28.84 55.1 19.82

Tang 23.20 63.9 17.95

NLM 20.16 68.6 16.23

Extended

NLM

18.57 71.1 15.48

The significant drawback of many simple filtration

algorithms is oversmoothing in the regions of

considerable fringe deformations or pattern

discontinuities, e.g., in the region of a sudden

change in the specimen height. In Fig. 4 we show

the denoising result of the extended NLM algorithm

for the case of synthetic data with fringe

discontinuity to indicate the algorithm satisfactory

behavior in such a region.

In Figure 4 the results of denoising the real

interferogram, shown in Fig. 1 (a), are presented.

Once again, high quality of the extended NLM

denoising is observable.

No fine tuning of the algorithms parameters was

done. The extended NLM method has shown

abilities to perform high quality fringe pattern

denoising.

(a)

(b)

(c)

(d)

(e) (f)

(g)

Figure 2: Synthetic interferogram (a), interferogram with

added noise (b), result of Gaussian denoising (c), median

filter denoising (d), (Tang et al., 2008) (e), the regular

NLM algorithm (f), and the extended NLM (g).

5 CONCLUSIONS

We showed how a simple extension of the non local

means algorithm, namely the consideration of

geometrically transformed patches, can increase the

fringe pattern filtration quality. The denoising

procedure quality have strong influence on the

performance of further processing, such as phase

PHOTOPTICS2013-InternationalConferenceonPhotonics,OpticsandLaserTechnology

decoding and therefore impacts on the quality and

sensitivity of the whole measurement process. One

could imagine extension of this method to any

arbitrary angle, which would vastly increase

redundancy in case of non-uniform, smoothly

varying fringe orientation angle. However, this

would as well further increase already rather high

numerical cost of computations. In fact, this seems

to be the main disadvantage and limitation of the

non-local filtration methods. The situation is

expected to change in the forthcoming years with

further growth of computer computational power as

well as with increasing popularity of computations

on parallel architecture, which is highly suitable for

the non local filtration implementations.

(a) (b)

Figure 3: Performance of the extended NLM method in

the fringe discontinuity region. The pattern (a) and the

denoising result (b).

(a) (b)

Figure 4: Denoised real interferogram (a) and difference

between the original and denoised images (b).

ACKNOWLEDGEMENTS

This research was supported by the fundings of the

Faculty of Mechatronics, Warsaw University of

Technology.

REFERENCES

Buades, A., Coll, B., Morel, J. M., 2005. A review of

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Wielgus, M., Patorski, K., 2012. Non-local fringe image

filtration: a new interferometric data filtration

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Fu, S., Zhang, C., 2012. Fringe pattern denoising using

averaging based on non-local similarity. Optics

Communications 285 2541-2544.

Salbut, L., Patorski, K., Jozwik, M., Kacperski, J.,

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micro-elements testing by interferometry using time-

average and quasi-stroboscopic techniques. Proc.

SPIE 5145, 23–32.

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FilteringFringePatternswiththeExtendedNonLocalMeansAlgorithm