Abstract

Recently, an axially distributed sensing system was proposed for three-dimensional (3D) imaging where the sensors are distributed along the optical axis. In this previously reported system, a priori knowledge of exact sensor positions was required for 3D image reconstruction. In this Letter, we present an axially distributed sensing with unknown sensor positions along the optical axis. In this system, only the relative positions of two sensors are needed, whereas all other sensor positions are assumed unknown. Experiments are presented to illustrate the feasibility of the proposed system and illustrate the visual quality of reconstructed 3D images by using the proposed calibrated sensor positions. To the best of our knowledge, this is the first report on axially distributed sensing with unknown sensor positions.

© 2011 Optical Society of America

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References

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  1. R. Schulein, M. DaneshPanah, and B. Javidi, Opt. Lett. 34, 2012 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
    [CrossRef]
  5. A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
    [CrossRef]

2010

2009

2006

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

2004

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Cho, M.

Cornelis, K.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

DaneshPanah, M.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

Javidi, B.

Koch, R.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Panah, M. Danesh

Pollefeys, M.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Schulein, R.

Stern, A.

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

Tops, J.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Van Gool, L.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Verbiest, F.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Vergauwen, M.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

Xiao, X.

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

Int. J. Comput. Vis.

M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Cornelis, J. Tops, and R. Koch, Int. J. Comput. Vis. 59, 207 (2004).
[CrossRef]

J. Display Technol.

Opt. Lett.

Proc. IEEE

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

Other

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

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Figures (6)

Fig. 1
Fig. 1

3D imaging setup in the axially distributed sensing system.

Fig. 2
Fig. 2

Calibration of sensor positions.

Fig. 3
Fig. 3

Elemental images obtained longitudinally: (a) sensor farthest from the scene k = 1 and (b) sensor closest to the scene. Eighteen recordings were made ( k = 18 ).

Fig. 4
Fig. 4

Distribution of all the sensor (pickup) positions. Circular points and stars represent the measured and the calibrated pickup (sensor) positions, respectively.

Fig. 5
Fig. 5

Comparison of the pickup positions. Figures in the left column show the coordinate differences between the measured sensor positions and the calibrated (computed) ones on the X, Y, Z axes, respectively. Figures in the right column illustrate the distances (error) between the calibrated positions and the measured positions for each sensor position.

Fig. 6
Fig. 6

In-focus reconstructed images and their local details at distance z = 610 mm (a) using the calibrated sensor positions and (b) using the measured sensor positions.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

m 1 i K [ R 1 t 1 ] M i ,
m 2 i K [ R 2 t 2 ] M i ,
K = [ f x 0 a x ; 0 f y a y ; 0 0 1 ] ,
[ R 1 t 1 ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 ] , [ R 2 t 2 ] = [ 1 0 0 x 2 0 1 0 y 2 0 0 1 z 2 ] ,
c k i m k i = K k [ R k t k ] M i = [ f x 0 a x 0 f y a y 0 0 1 ] [ 1 0 0 x k 0 1 0 y k 0 0 1 z k ] [ X i Y i Z i 1 ] ,
c k i u k i = f x X i + a x Z i + f x x k + z k a x , c k i v k i = f y X i + a y Z i + f y y k + z k a y , c k i = Z i + z k .
[ f x 0 a x u k i 0 f y a y v k i ] [ x k y k z k ] = [ Z i ( u k i a x ) f x X i Z i ( v k i a y ) f y Y i ] .
R z r ( x , y ) = 1 K k = 1 K I k ( x A k , y A k ) ,

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