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Three-dimensional optical correlator using a sub-image array

Jae-Hyeung Park, Joohwan Kim, and Byoungho Lee

Open Access Open Access

Abstract

A three-dimensional optical correlator using a lens array is proposed and demonstrated. The proposed method captures three-dimensional objects using the lens array and transforms them into sub-images. Through successive two-dimensional correlations between the sub-images, a three-dimensional optical correlation is accomplished. As a result, the proposed method is capable of detecting out-of-plane rotations of three-dimensional objects as well as three-dimensional shifts.

©2005 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Conceptual diagram of the proposed method

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Fig. 2.
Fig. 2. Sub-image (a) geometry and (b) generation

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Fig. 3.
Fig. 3. Observing-angle-invariance of the sub-image: (a) ordinary image (or elemental image) (b) sub-image

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Fig. 4.
Fig. 4. Size-invariance of the sub-image: (a) ordinary image (b) sub-image

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Fig. 5.
Fig. 5. Procedure for detecting out-of-plane rotation and 3D shift

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Fig. 6.
Fig. 6. Examples of experimentally obtained elemental images and sub-images

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Fig. 7.
Fig. 7. Example of (a) JPS captured by CCD and (b) correlation peak calculated by Fourier transforming the captured JPS digitally.

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Fig. 8.
Fig. 8. Experimental result: intensity profile of the correlation peaks between one sub-image for a reference object located at (xr, yr, zr )=(0 mm, 0 mm, 25 mm) and each sub-image of a signal object located at (xs, ys, zs )=(5 mm, 0 mm, 40 mm) with θx-z =0°, 2°, 4°, and 6° and θy-z =0°

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Fig. 9.
Fig. 9. Experimental result: detected positions of the correlation peak with various locations of the signal object when the reference object is located at (xr, yr, zr )=(0 mm, 0 mm, 25 mm) and the signal object has no out-of-plane rotation.

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Fig. 10.
Fig. 10. Experimental result: detected positions of the correlation peak with various locations of the signal object when the reference object is located at (xr, yr, zr )=(0 mm, 0 mm, 25 mm) and the signal object has θx-z =4°, and θy-z =0° rotation.

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Equations (6)

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θ sub , y − z , i = tan − 1 ( y i f ) ,
Δ u r , i , s , i = u r , i − u s , i = y r − y s + ( z r − z s ) tan θ sub , y − z , i φ .
Δ u r , i , s , j = y r − y s + z r tan θ sub , y − z , i − z s tan θ sub , y − z , j φ
= y r − y s + z r tan θ sub , y − z , i − z s tan ( θ sub , y − z , j + θ y − z ) φ .
Δ θ = tan − 1 ( y i + 1 f ) − tan − 1 ( y i f ) ≈ y i + 1 − y i f = s f ,
Ω = 2 tan − 1 ( φ 2 f ) ≈ φ f ,
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