Depth Sensing Using Geometrically Constrained Polarization Normals

Abstract

Analyzing the polarimetric properties of reflected light is a potential source of shape information. However, it is well-known that polarimetric information contains fundamental shape ambiguities, leading to an underconstrained problem of recovering 3D geometry. To address this problem, we use additional geometric information, from coarse depth maps, to constrain the shape information from polarization cues. Our main contribution is a framework that combines surface normals from polarization (hereafter polarization normals) with an aligned depth map. The additional geometric constraints are used to mitigate physics-based artifacts, such as azimuthal ambiguity, refractive distortion and fronto-parallel signal degradation. We believe our work may have practical implications for optical engineering, demonstrating a new option for state-of-the-art 3D reconstruction.

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Notes

  1. 1.

    For example, (Yu et al. 2013; Han et al. 2013; Wu et al. 2014) using SfS, and (Nehab et al. 2005; Haque et al. 2014) using PS.

  2. 2.

    Formalized by Augustin-Jean Fresnel (1788–1827) as the famous “Fresnel Equations”.

  3. 3.

    For this paper, we rotate the polarizer to the desired angle. However, the mechanical process of rotation could lead to small errors, for which Schechner (2015) has devised a self-calibrating solution.

  4. 4.

    In practice, the degree of polarization is generally an order of magnitude larger for specular dominant reflections.

  5. 5.

    Refer to Sect. 5 for a detailed analysis of key assumptions.

  6. 6.

    For all experiments, this distance was set to 20 millimeters. To find the neighborhood, we use the kd-tree search algorithm, which can be implemented by the rangesearch command in MATLAB.

  7. 7.

    \({\mathbf {N}^{\text {polar}}}\) is obtained through shape from polarization and this normal map will suffer from the physics-based artifacts described previously. This can be seen visually in Fig. 1c.

  8. 8.

    The 2D-TV implementation parallels the optimization program from prior work (Kadambi and Boufounos 2015).

  9. 9.

    Calculation of \(2\times 2^K - V\): each facet can have 2 possible normal orientations due to the azimuthal ambiguity, leading to \(2^K\) possible surface configurations due only to the degrees of freedom of the facets. By constraining the rest of the surface using the anchor point as a boundary condition, this leads to \(2\times 2^K\) surface configurations. Since we assume that the facet is continuous with the anchor point, the overall dimensionality is reduced to \(2\times 2^K - V\).

  10. 10.

    Implementing high-frequency ambiguity correction. First, a difference image is formed of the depth normals and polarization normals. The difference image only contains detailed features, as it would not show up in the former (otherwise the method from Sect. 4.1.1 would have been sufficient). From the difference image, the pixel at an edge from 0 to a non-zero value represents an anchor point. An edge corresponding to a change in sign (e.g., at the V-groove of a corner) is a pivot point. A greedy approach is used to flip surface facets, which enforces a closed surface constraint. Details about the closed surface constraint can be found in Miyazaki et al. (2003).

  11. 11.

    Note: this notion of a patch operates on the pixel grid and is thus different from our convention of defining a point cloud neighborhood (cf. Sect. 4.1). For our paper, we use a 7x7 patch.

  12. 12.

    Estimation of the sinusoidal parameters from Eq. 1 becomes unstable when there is little contrast between \(I_{\min }\) and \(I_{\max }\).

  13. 13.

    A value of \(\lambda = 0.02\) is recommended.

  14. 14.

    In this paper, we use the term “reflection components” following an optical imaging convention. This is identical to the term “reflectance components”, which may be more familiar to some readers.

  15. 15.

    Laser Scanner: http://nextengine.com/assets/pdf/scanner-techspecs.pdf.

  16. 16.

    While there are variants of generalized photometric stereo for non-Lambertian objects and natural environment lighting, they usually require about 50-100 images according to a recent survey in Shi et al. (2016).

  17. 17.

    Polarization mosaic: http://moxtek.com/optics-product/pixelated-polarizer.

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Acknowledgements

The authors thank G. Atkinson, T. Boult, S. Izadi, D. Miyazaki, G. Satat, N. Naik, I.K. Park, H. Zhao for valuable feedback. Achuta Kadambi is supported by the Charles Draper Doctoral Fellowship and the Qualcomm Innovation Fellowship. Boxin Shi is supported by a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO). The work of the MIT-affiliated coauthors was supported by the Media Lab Consortium members.

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Correspondence to Achuta Kadambi or Boxin Shi.

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Communicated by Rene Vidal, Katsushi Ikeuchi, Josef Sivic and Christoph Schnoerr.

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Kadambi, A., Taamazyan, V., Shi, B. et al. Depth Sensing Using Geometrically Constrained Polarization Normals. Int J Comput Vis 125, 34–51 (2017). https://doi.org/10.1007/s11263-017-1025-7

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Keywords

  • Computational photography
  • Light transport
  • Depth sensing
  • Shape from polarization