# Sampling Density Criterion for Circular Structured Light 3D Imaging

### Deokwoo Lee, Hamid Krim

#### Abstract

3D reconstruction work has chiefly focused on the accuracy of reconstruction results in computer vision, and efficient 3D functional camera system has been of interest in the field of mobile camera as well. The optimal sampling density, referred to as the minimum sampling rate for 3D or high-dimensional signal reconstruction, is proposed in this paper. There have been many research activities to develop an adaptive sampling theorem beyond the Shannon-Nyquist Sampling Theorem in the areas of signal processing, but sampling theorem for 3D imaging or reconstruction is an open challenging topic and crucial part of our contribution in this paper. We hence propose an approach to sampling rate (lower / upper bound) determination to recover 3D objects (surfaces) represented by a set of circular light patterns, and the criterion for a sampling rate is formulated using geometric characteristics of the light patterns overlaid on the surface. The proposed method is in a sense a foundation for a sampling theorem applied to 3D image processing, by establishing a relationship between frequency components and geometric information of a surface.

#### References

- Chan, Y., Delmas, P., Gimel'farb, G., and Valkenburg, R. (2008). On fusion of active range data and passive stereo data for 3d scene modelling. In Image and Vision Computing New Zealand, 23rd International Conference. IEEE.
- de'Sperati C and P, V. (1997). The relationship between curvature and velocity in two-dimensional smooth pursuit eye movements. The Journal of Neuroscience, 17(10):3932 - 3945.
- Eldar, Y. C. and Pohl, V. (2009). Recovering signals from lowpass data. CoRR, abs/0907.3576.
- Faugeras, O., Luong, Q.-T., and Papadopoulou, T. (2001). The Geometry of Multiple Images: The Laws That Govern The Formation of Images of A Scene and Some of Their Applications. MIT Press, Cambridge, MA, USA.
- Geng, J. (2011). Structured-light 3d surface imaging: atutorial. Adv. Opt. Photon., 3(2):128-160.
- Higgins, J. (2003). A sampling theorem for irregularly spaced sample points (corresp.). IEEE Transactions on Information Theory, 22(5):621 - 622.
- Jerri, A. J. (1977). The shannon sampling theorem - its various extensions and applications: A tutorial review. Proceedings of the IEEE, 65(11):1565 - 1596.
- Kolev, K., Tanskanen, P., Speciale, P., and Pollefeys, M. (2014). Turning mobile phones into 3d scanners. 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 00(undefined):3946-3953.
- Lacquaniti, F., Terzuolo, C., and Viviani, P. (1983). The law relating the kinematic and figural aspects of drawing movements. Acta Psychologica, 54(13):115 - 130.
- Lee, D. and Krim, H. (2011). A sampling theorem for a 2d surface. In Scale Space and Variational Methods in Computer Vision - Third International Conference, SSVM 2011, Ein-Gedi, Israel, May 29 - June 2, 2011, Revised Selected Papers, pages 556-567.
- Lei, Y., Bengtson, K., Li, L., and Allebach, J. (2013). Design and decoding of an m-array pattern for low-cost structured light 3d reconstruction systems. In Image Processing (ICIP), 2013 20th IEEE International Conference on.
- Mallat, S. G. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell., 11(7):674-693.
- Oprea, J. (2007). Differential Geometry and Its Applications. Mathematical Association of America, 2nd edition.
- Papoulis, A. (1977). Signal analysis. Electrical and electronic engineering series. McGraw-Hill, New York, San Francisco, Paris.
- Song, Y., Glasbey, C. A., Polder, G., and van Heijden, G. W. (2014). Non-destructive automatic leaf area measurements by combining stereo and time-of-flight images. IET Computer Vision, 8(5):391-403.
- Unser, M. (2000). Sampling-50 years after shannon. Proceedings of the IEEE, 88(4):569-587.

#### Paper Citation

#### in Harvard Style

Lee D. and Krim H. (2017). **Sampling Density Criterion for Circular Structured Light 3D Imaging** . In *Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 6: VISAPP, (VISIGRAPP 2017)* ISBN 978-989-758-227-1, pages 478-483. DOI: 10.5220/0006147504780483

#### in Bibtex Style

@conference{visapp17,

author={Deokwoo Lee and Hamid Krim},

title={Sampling Density Criterion for Circular Structured Light 3D Imaging},

booktitle={Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 6: VISAPP, (VISIGRAPP 2017)},

year={2017},

pages={478-483},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006147504780483},

isbn={978-989-758-227-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 6: VISAPP, (VISIGRAPP 2017)

TI - Sampling Density Criterion for Circular Structured Light 3D Imaging

SN - 978-989-758-227-1

AU - Lee D.

AU - Krim H.

PY - 2017

SP - 478

EP - 483

DO - 10.5220/0006147504780483