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Robust Computation of the Polarisation ImageGule Saman and Edwin R. Hancock Department of Computer Science, University of York, York, YO10 5DD, UK saman,erh@cs.york.ac.uk

AbstractIn this paper we show how to render the computation of polarisation information from multiple polariser angle images robust. We make two contributions. First, we show how to use M-estimators to make robust moments estimates of the mean intensity, polarisation and phase. Second, we show how directional statistics can be used to smooth the phase-angle, and to improve its estimation when the polarisation is small. We apply the resulting techniques to polariser images and perform surface quality inspection. Compared to polarisation information delivered by the three-point method, our estimates reveal ner surface detail.

Using a polaroid lter as analyser, the degree of both diffuse and specular polarisation may be measured together with the phase angle. Physically, the degree of polarisation is determined by the zenith angle between the remitted light and surface, together with the refractive index of the surface. The phase angle is determined by the azimuth angle of the remitted light to the surface. Depending on the geometry of the scattering process and whether the origin of the polarisation is specular or diffuse, the Fresnel theory may be used to infer surface orientation given known refractive index, or refractive index given known surface orientation. Hence, polarisation information can be used to infer information concerning surface quality via refractive index determination or surface shape for surfaces of constant refractive index. Unfortunately, the recovery of the polarisation image pose an obstacle to the reliable realisation of these two tasks. A polarisation camera, can determine the three components of the polarisation image (i.e. mean intensity, polarisation and phase) from just three images obtained with the polariser set to three different angles [4]. However, when the polarisation is small then accurate determination of the phase angle becomes unstable. Moreover, impulse noise at a pixel or slight misalignments of the analyser images can make the estimates unreliable. The aim in this paper is to explore different strategies for improving the robustness of the estimated polarisation image, and to investigate the knock-on benets for surface inspection tasks. We make two contributions. First we explore how to make moments estimates from multiple polarisation images, and use Mestimators to weight against outlying pixel values. In this way we improve the robustness of both the polarisation and phase measurements. Second, we make use of techniques from directional statistics to improve the reliability of the phase estimates. When the polarisation is small the phase-angle can prove difcult to locate due to small variations in the pixel values with polariser angle. To overcome this problem, we make use of recently

1. IntroductionIn computer vision polarisation information has been used to develop a diverse range of algorithms for problems ranging from surface inspection to surface reconstruction. The approach adopted is to use the Fresnel theory to determine the parallel and perpendicular components of the electric eld for incident light, and to model the transmission and reection of these components[2]. Generally, speaking the modelling of dielectrics is more straightforward than the modelling of metals, since in the latter case the incident electromagnetic eld induces surface currents and the Fresnel theory alone is insufcient. For dielectrics, it is convenient to distinguish between the processes of specular and diffuse polarisation. In the case of specular polarisation, initially polarised light is reected from the surface and the plane of polarisation is determined by the orientation of the reecting surface. In the case of diffuse polarisation, initially unpolarised light spontaneously acquires polarisation as it is subjected to subsurface scattering and is remitted [7]. Edwin Hancock is supported by a Royal Society Wolfson Research Merit Award.

developed methods for smoothing vector elds from directional statistics [8]. We experiment with the methods that have been used to develop techniques for computer vision for both surface shape geometry recovery [4], [3] and surface quality inspection [6], [5]. In the former case, we use images of objects with known geometry and refractive index, where we test the consistency of the estimated zenith and azimuth angles with ground truth. In the latter case, we use samples of damaged fruits to explore how well the method reveals local variations in refractive index.

can compute a weight-function based on the difference between predicted and measured pixel brightness. Here we use the Gaussian, and the weight associated with the measured brightness at a pixel with analyser setting i is wi = exp[k1 {Ii I(i )}2 ] where k1 is a constant. The weighted moments estimators for the polarisation image are computed as follows. The estimated mean intensity is simply the weighted mean of the intensity values at a pixel over the different analyser angles, i.e. I0 =N i=1 wi Ii N i=1 wi

2. Robust Estimation of the Polarisation ImageWe follow the method that was proposed in [1] for diffuse reection, where subsurface reections occur before the light is re-emitted. As already mentioned the Fresnel theory provides the relationship between the state of polarisation of the reected light and the angle of reection. The approach that has been adopted is to take a succession of images of the object at different orientations of the analyser polaroid angle for measuring the polarisation state. The images were obtained using a Nikon D200 digital SLR camera, with xed exposure and aperture settings. Both polarised and unpolarised light sources have been used. In both cases sequences of images are captured as the analyser angle is increased by increments of 10 to give 19 images per object. The reections from the environment are minimal as the experiments are conducted in a dark room with walls painted matte-black and the equipment isolated by black screens.

To compute the polarisation, we commence by computing the weighted variance of the pixel brightness values over the analyser angles, i.e. 2 =N i=1 ((Ii

I0 )/(I0 ))2N i=1

wi

and the estimated degree of polarisation is 2 =

(2)

Finally, to compute the phase angle we commence by computing Y =N i=1 I wi IiI0 0 cos 2i N i=1

wi

then the moment estimator of the phase angle is = arccos 2Y (3)

2.1

Robust Moments Estimators

The conventional way to estimate the components of the polarisation image (mean-intensity I0 , degree of polarisation and phase ) is to use the least-squares tting to 3 images [1]. Here we explore how use robust moments estimators to recover the components of the polarisation image from large sets of analyser images. Suppose that the angle of the analyser is i , where i is the index of the analyser setting. The predicted pixel brightness as a function of analyser angle is given by I(i ) = I0 1 + cos(2i 2) (1)

Our estimation of the polarisation image is an iterative process. We commence with the weights set to unity. With each iteration we estimate, I0 , and . Using the parameters we compute the predicted pixel brightness I(i ) and the t residual (Ii I(i ))2 which are then used to rene the weight wi . The revised weights are then used to make new moments estimators of the polarisation parameters, and this process is iterated. Here we use 10 iterations. Finally, we note that from the Fresnel theory for unpolarised incident light the relationship between the degree of diffuse polarisation, the angle between the surface normal and the remitted light and the refractive index n is =1 2+2n2 (n+ n )2 1 (n n )2 sin2 (4) sin2 +4 cos n2 sin2

To develop our M-estimators for the polarisation image we require a weighting function. Let Ii be the measured pixel brightness with analyser angle i , then we 2

(a)

where k2 is a scale-factor. We treat the polarisation weight and phase at the pixel indexed j as a 2-vector with magnitude w(j ) and direction j . We can resolve this vector into components xj = w(j ) cos j and yj = w(j ) sin j . Our idea is to smooth the resulting vector eld over the pixel sites of a neighbourhood N of j, weighting against pixels with small values of . To this end, we compute weighted averages of the components of the weight vector eld, xj = jNj w(j )cos j jNj w(j )

(b)

andj = y

jNj w(j )sin j jNj w(j )

(6) and the smoothed value of the phase angle is(c)

yj j = arctan xj

(7)

Figure 1. Shows the (a) mean intensity, (b) Polarisation and (c) Phase Image for iteration 1, 5 and 10

When k 2 is small then the smoothing becomes equivalent to smoothing a vector eld in which the length of the vectors is .

3

Results

(a)

(b)

Figure 2. Shows the Mean Intensity , Polarisation, Phase and Smoothed Phase Image for the (a) proposed method and (b) three images

2.2

Improving estimates of

One of the problems the hinders the estimation of the polarisation image, is that when the degree of polarisation is small then the variation in the measured pixel brightness are small. As a result it becomes difcult to reliably estimate the phase angle, . As a second step in the robust estimation of the polarisation image we use ideas from directional statistics to smooth the eld of phase angles. To this end we introduce the weight function for the pixel indexed j w(j ) = 1 e k22

We have tested our new method for computing the polarisation image on various objects. These include fruits and vegetables, Figure 4 where our aim is to use the polarisation image