A Differential Approach to Shape from Polarisation: A LevelSet Characterisation
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Abstract
Despite the longtime research aimed at retrieving geometrical information of an object from polarimetric imaging, physical limitations in the polarisation phenomena constrain current approaches to provide ambiguous depth estimation. As an additional constraint, polarimetric imaging formulation differs when light is reflected off the object specularly or diffusively. This introduces another source of ambiguity that current formulations cannot overcome. With the aim of deriving a formulation capable of dealing with as many heterogeneous effects as possible, we propose a differential formulation of the Shape from Polarisation problem that depends only on polarimetric images. This allows the direct geometrical characterisation of the levelset of the object keeping consistent mathematical formulation for diffuse and specular reflection. We show via synthetic and realworld experiments that diffuse and specular reflection can be easily distinguished in order to extract meaningful geometrical features from just polarimetric imaging. The inherent ambiguity of the Shape from Polarization problem becomes evident through the impossibility of reconstructing the whole surface with this differential approach. To overcome this limitation, we consider shading information elegantly embedding this new formulation into a twolight calibrated photometric stereo approach..
Keywords
Shape from polarisation Linear PDEs Photometric stereo Quasilinear PDEs Specular reflection Diffuse reflection Variational method1 Introduction
Retrieving 3D shape features from polarisation cues acquired from a single point of view is a concept introduced by Koshikawa (1979) to constrain the surface normals of objects made out of dielectric materials. Together with the increasing impact of the Computer Vision for 3D scanning techniques, the socalled Shape from Polarisation problem (SfP) became one of the most physically based approaches in the Shape from X family. It is based on the physical property that unpolarised light becomes partially polarized once it reflects off an object. The acquisition process consists of taking images of a static object under the same illumination conditions, using a linear polariser in front of a camera. Each image differs from any other by the fact that the polariser has been rotated by a known angle from its initial position. What makes the SfP approach very interesting, at least in theory, is that polarised image formation is both albedo and lighting independent. However, despite the theoretical and practical progress achieved over almost 40 years of research, the SfP still provides limited constraints of the surface preventing the shape recovery (due to the periodicity of the polarisation information). Indeed, most of the existing SfP approaches usually combine other cues coming from multiview (Atkinson and Hancock 2006a, 2007a; Rahmann and Canterakis 2001), shading (Ngo et al. 2015; Drbohlav and Sara 2001; Morel et al. 2006; Huynh et al. 2013; Atkinson and Hancock 2007b; Smith et al. 2016) and recently RGBD data (Kadambi et al. 2015) in order to increase the information of the 3D shape and fully reconstruct the surface under observation.
Furthermore, another important limitation of the SfP is its mathematical formulation. Instead of being expressed within a single framework, it is fragmented into several steps derived from physical models that describe the behaviour of light when it propagates between media of differing refractive indices (MartinezHerrero et al. 2009). This makes the SfP less straightforward to understand and less practical to solve than other Shape from X approaches.

deriving a more general formulation of the PDE presented in Mecca et al. (2017) for general polarisation angles

characterizing the novel PDEs for diffuse and specular reflection

computing the levelset of an object having mixture of diffuse and specular reflections

merging the new formulation with twolight Photometric Stereo data for 3D shape reconstruction.
2 Previous Works
Polarimetric cues have been used in Computer Vision for a number of different tasks mostly related to the difficulty of dealing with specular highlights. In addition to using polarised image formation to separate diffuse from specular reflection (Nayar et al. 1993; Umeyama and Godin 2004; Wang et al. 2016; Wolff and Boult 1991), the potential of polarimetric approaches has been demonstrated by its use in determining the normal orientation of glossy surfaces (Koshikawa 1979; Wolff and Boult 1991; Wolff 1994a, b; Taamazyan et al. 2016). For the same reason, particular attention has been given to the SfP problem attempting to recover the shape of transparent objects (Miyazaki and Ikeuchi 2005; Saito et al. 1999; Miyazaki et al. 2004, 2003; Miyazaki and Ikeuchi 2007; Chen et al. 2007). However, polarimetric analysis provides geometrical information for dielectric surfaces, which has allowed the SfP to be adopted to reconstruct surfaces reflecting light diffusively too (Atkinson and Hancock 2006).
In any case, the features of the surface provided by the polarization reflectance model are limited and this results to ambiguous shape recovery. To disambiguate polarisation normals, most of the approaches enhance the SfP with supplementary cues to make the overall methodology wellposed. Several works use multiple views of the object together with polarisation imaging (Atkinson and Hancock 2005; Miyazaki et al. 2012). Other approaches are more related to what we propose here, where shading cues are merged with polarisation imaging. For example, Drbohlav and Sara (2001) employed polarisation imaging to recover the zenith angle of the surface normal adding integrability constraints to reduce the ambiguity of the uncalibrated photometric stereo to concave and convex case. Morel et al. (2006) used a lighting system in a diffuse dome composed of a ring with numerous LEDs. This provides a uniform and unpolarized light onto the object to be digitalized. The object is placed inside the dome and the light reflected by its surface is analysed by the camera and the liquid crystal polariser. The ring of LEDs is split into four parts that can be independently electrically controlled. Huynh et al. (2013) proposed an iterative method where diffuse polarisation modelling is considered together with two additional constraints of the problem, including the surface integrability and the material dispersion equation using a hyperspectral imaging system. They adopted a preliminary disambiguation proposed by Zhu and Shi (2006) extended by the use of fast marching and patch stitching. Atkinson and Hancock (2007b) disambiguated the polarisation normal using the shading information from three distant light sources, which are placed in a strategic position, such that the angles subtended by the camera and the light sources from the object are equal and also that the distances between the object and the light sources are equal too. Ngo et al. (2015) proposed a very interesting approach based on the ratio of both lambertian irradiance equations (with uniform light direction) and polarisation image formation equations to compute the surface orientation and the refractive index using at least 3 light sources. By increasing the number of input images (i.e. light sources), they eventually extended the approach to the uncalibrated case, also computing the light directions.
Lastly, Smith et al. (2016) proposed a SfP approach aided by shading information which is provided by a distant light source with orthographic viewing geometry. Besides estimating the shape up to a global concave/convex ambiguity, an important limitation for real applications is the assumption of uniform albedo.
The differential formulation for the Shape from Polarisation problem presented in Mecca et al. (2017) has been the first to exploit the monocular feature of this technique where polarimetric images of a static object are taken from a stationary point of view. The nonlinear redundant spatial information has been simplified by considering image ratio leading to homogeneous linear PDEs that describe the geometry of the object via its levelset. After that, other approaches based on image ratio have been presented (Tozza et al. 2017; Yu et al. 2017) assuming diffuse reflection only and considering twolight uncalibrated photometric stereo data and smoothness/convexity priors respectively, for computing the 3D shape.
In this work, we extend (Mecca et al. 2017) by describing the SfP through a differential formulation, where a simple mathematical formulation is capable of describing the geometry of the object through its levelset in the same fashion for both diffuse and specular reflections. This brings a twofold advantage: to conceive the expected ambiguity of the SfP problem by extracting the levelset of the surface, and to embed such PDE into a differential formulation, modelling the Photometric Stereo problem using image ratios (Mecca et al. 2016; Mecca and Falcone 2013; Smith and Fang 2016; Chandraker et al. 2013).
Indeed, our approach considers shading information coming from a minimum of two point light sources of known position. The ratio of the respective irradiance equations leads to an albedo invariant PDE that ambiguously describes the surface (Mecca et al. 2014). The combination of those two into a differential system makes the problem solvable.
Experimental validation is provided for synthetic and realworld tests. In particular, real tests have been performed using very general lighting setup (e.g. light desk, light bulb) and, most importantly, taking into account challenging materials (such as glass) to prove the robustness of our differential approach.
3 Shape from Polarization: A Differential Approach
In this section, we describe the mathematical derivation of a novel framework for the SfP problem resulting to a homogeneous linear partial differential equation. To do so, before recalling the theoretical principles for polarised imaging, we firstly consider an important aspect of the surface normal parameterisation as a function of the depth.
3.1 Camera Parameterisation
The pinhole camera modeling is an important aspect to consider while carefully taking into account image based shape reconstruction algorithms. When perspective viewing geometry comes into play, there have been several works proposing different parameterisations (Bruckstein 1988; Tankus et al. 2003; Prados and Faugeras 2005; Papadhimitri and Favaro 2013). However, to mathematically describe the SfP, we do not use any specific parameterisation of the camera since we only consider a general aspect occurring when surface normal deforms due to perspective viewing geometry. For this purpose, let us call as \({{\varvec{\chi }}}(\mathbf {x})\in \varSigma \), \(z(\mathbf {x})\) and \(\nabla z(\mathbf {x})=(z_x(\mathbf {x}),z_y(\mathbf {x}))\) the point belonging to the surface, the depth and the gradient of the surface at pixel \(\mathbf {x}=(x,y)\). Then, the first two components of the nonunit normal vector to the surface \({\overline{\mathbf{n}}}(\mathbf {x})=(\overline{n}^1(\mathbf {x}),\overline{n}^2(\mathbf {x}), \overline{n}^3(\mathbf {x}))\) are proportional to \(\nabla z(\mathbf {x})\) up to a factor depending on the focal length f.
3.2 Polarization Imaging
To give an overview of our method, it is important to mention that the idea of considering mixed polarisation (i.e both diffuse and specular polarisation) comes from the fact that our formulation based on levelset is particularly suitable for identifying regions on the objects having different type of polarisation. In fact, the \(\frac{\pi }{2}\)phase shift between the diffuse and specular polarisation transfers to the levelsets characterisation. We will show how the levelsets of diffuse and specular polarised pixels are orthogonal allowing to visual recognition of the different polarisation effects (see Fig. 7).
In order to derive the mathematical model based on levelset, without loss of generality we firstly consider the purely diffuse polarisation image formation model (i.e \(\alpha _{spec}=0\)). After that, the specular case will be easily extended and experimental tests will be showed to prove the concept.
3.3 The Differential Model for the Shape from Polarisation
In the next section we show how the new differential approach (13) extends also to specular reflection with minimal changes due to shift of the phase angle.
3.4 Specular polarisation
In the next section we describe how (17) elegantly fits into a wellposed differential system of hyperbolic PDEs when additional shading information is provided to reconstruct the 3D shape.
4 Enhancing SfP with TwoLights Photometric Stereo
In this section we describe how to exploit the new differential formulation for the SfP (17) into a single framework where shading information is provided.
In order to take advantage of the fact that SfP is albedo independent, we consider a fully calibrated Photometric Stereo (SfPS) approach having the same specific feature by basing the derivation on the irradiance equations ratio (Davis and Soderblom 1984). Unlike polarisation theory, in order to parametrize the SfPS approach, we require some additional information to be known: the camera parameters, the lighting and the type of reflection (i.e. diffuse or specular).
4.1 Numerical Approach to the Shape from Polarisation and Photometric Stereo
Note that the term \( \lambda \left\ z  z_0\right\ _{L^2}\) is a zeroorder Tikhonov regularizer that constraints the mean depth and ensures that the differential problem has a unique solution.
Equation (31) is discretized with finite differences and solved with simple least squares. Furthermore, since \(\mathbf {b}_{ps}\) and \(s_{ps}\) (but not \(\mathbf {b}_{pol}\)) implicitly depend on z, the solution of Eq. (31) is embedded in an iterative process that calculates all relevant quantities (a, \(\mathbf {h}\), \(\mathbf {b}_{ps}\) and \(s_{ps}\)) by using the current estimates of the depth values in a similar manner with Mecca et al. (2016). The optimisation is initialised with a flat plane at the mean distance.
5 Experimental Results
The proposed approach was evaluated with a range of synthetic and real data sets. In addition, we calculated the level set of the head dataset from Kadambi et al. (2015). The algorithm was implemented in matlab with a total running time of about two minutes (for the 7x3MPixel images) on a laptop with a quad, corei7, 2.6GHz CPU.
5.1 Synthetic Data
First of all, we generated \(800 \times 600\) pixel synthetic data using the “bimba” from the AIM@Shape Repository. The data were rendered with realistic effects including nonuniform albedo, perspective viewing geometry, and near point light sources. We generated a Lambertian and a specular dataset, the latter rendered with the Cook & Torrance BRDF (see Fig. 2).
We rendered the minimum required polarisation images \(I_0\), \(I_{\frac{\pi }{4}}\) and \(I_{\frac{\pi }{2}}\) generated from (3) and assuming index of refraction \(\mu =1.6\). Finally, to make the experiments realistic, we limited the precision of the data to 3 decimal digits^{1} and added 0.5% Gaussian noise.
The level sets obtained are evaluated quantitatively by calculating the angle at each pixel with the level set computed from the ground truth. The mean angular error (MAE) is used as an overall metric for comparing datasets. For reconstructing the specular dataset, we used \(c=0.25\) (see (21)). The results are shown in Fig. 3. We note that the specular dataset outperforms the diffuse one, due to the much higher polarisation effects for specular materials.
It has to be noted that these kind of synthetic data do not model a lot of interesting effects arising in real data such as selfreflections and semitransparency (e.g. Fig. 13) and so their usefulness lies on understanding robustness to noise for diffuse versus specular reflection as well as overall geometry. Thus, as diffuse regions with low elevation angle (i.e. almost flat) are quite inaccurate even on synthetic data, these kind of regions are not expected to perform well in real data either (see Fig. 12 where the diffuse, almost flat region has a very minimal polarisation sinewave).
5.2 Real Data
The setup we used for acquiring data suitable for SfP and SfPS consists of a FLIR camera FL3U332S2CCS having maximum resolution 2080 \(\times \) 1552 mounting a TECHSPEC 8mm UC series fixed focal length lens, OSRAM Platinum Dragon high power LEDs white and a linear polariser mounted on a rotary mount with post. We captured images at polariser angles \([90\), \(60\), \(30\), 0, 30, 60, \(90^\circ ]^\circ \) which is well above the minimum requirement of three images. To test the robustness of our approach, experiments with 3 and 5 images are discussed in Sect. 5.2.1. The obtained isocontorus and reconstructions are shown in Figs. 4 and 5 respectively. We note that real data experiments confirm the fact that the level sets are more accurately calculated on specular than diffuse materials which is a clear advantage of the SfP approach compared to SfPS.
In addition, we present quantitative evaluation of the levelset accuracy by using two objects of known geometry; namely a soup plate and a billiard ball (Fig. 6).
We segment diffuse from specular regions using the readily available structured Forests edge detector (Dollár and Zitnick 2013) and hierarchical segmentation of Arbelaez et al. (2009) with default parameters^{2} on the normalised polarisation orientation image (see. Fig. 8). The \(180^\circ \) phase difference between diffuse and specular ensures that the regions are easily automatically segmented as even for nearby points on the 2 sides of the boundary of the two regions, the polarisation sine waves are significantly different (see Fig. 12). Finally, one single region manual labelling is required and the rest are propagated using the segmentation boundaries.
The complete pipeline is then presented from “raw” contours assuming uniform diffuse reflection (Eq. (17)) in Fig. 7, to phase map segmentation in Fig. 8 to final levelset correction for specular regions and quantitative evaluation using the mean angular error (MAE) in Fig. 9.
Finally, another two datasets containing a marble statue, a glass bottle and a glass bowl are presented in Fig. 13. Most of the inaccuracies are at the badly illuminated regions like the top of the bottle. Notice that the strong reflection in the middle of the bowl does not reduce the quality of the levelsets. That is because the levelset equation is invariant to type of illumination; any kind of specular reflection exhibits the same polarisation phase.
5.2.1 Polarization Angle Uncertainty
In this section we consider the effect of error of the polarisation angles to the orientation of the level sets. In addition, we also examine the effect of using different number of polarisation images. The experiments are performed using the real datasets of the plate and billiard ball (see Fig. 6) so as to be able to get a quantitative evaluation. We note two distinct cases namely systematic shift of the angles (Fig. 10) and random “noise” (Fig. 11). The first case corresponds to uncertainty of the zero point of the polariser (something that has to be calibrated) and leads to a systematic shift of the phases and thus rotations of the contours. The second case models inaccurate rotation of the polariser by simply adding random Gaussian noise to the polarisation angles. The levelset orientation is pretty robust to low levels of angular uncertainty and performs poorly only under unreasonable levels of noise (as it is unreasonable to expect \(20^\circ \) error on the polarisation angles). Another more surprising conclusion is the fact that more polarisation images can lead to reduced quality under high angular uncertainty; this means that the additional images act as outliers and cause a systematic error to the level set calculation (Figs.12, 13).
6 Conclusion and Perspective
In this work we present a new differential approach to Shape from Polarisation leading to a linear PDE that describes the levelset of the object under observation. By combining surface depth related parameters into the polarisation image formation, we derived an homogeneous linear PDE that describes the geometry of the surface through its isocontours.
This approach allows to have equivalent formulations for diffuse and specular light reflection since the shift of the phase corresponds to the same shift of the vector field describing the levelset. From the experimental point of view, this new model permits to distinguish diffuse regions from specular region very easily and then characterise the shape through its levelset entirely. The fully automatic procedure for separating diffuse from the specular reflection is an interesting future work. This could be accomplished with some sort of global consistency optimisation (such as graphcut) which can also take into account the fact that pixels with high degree of polarisation are necessarily specular; for diffuse polarisation, \(\rho \) peaks around 0.4 (Atkinson and Hancock 2005).
With the aim of providing full 3D shape recovery, we added shading cues using accurate Photometric Stereo model describing sharing information from a point light source with perspective deformation and diffuse/ specular reflection. We showed that the new Shape from Polarisation differential formulation merges very elegantly into a system of hyperbolic PDEs which is albedo independent and wellposed by considering two lights sources at least.
By introducing this levelset characterisation for the Shape from Polarisation problem, there could be several extension for merging this problem with others like multiview or Shape from Defocusing.
Footnotes
 1.
Standard cameras offer 10bit precision raw data.
 2.
Using code from https://github.com/pdollar/edges.
Notes
References
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