Abstract

An application of varifocal mirror autostereoscopic imaging to 3-D computer-generated movies is described. A high speed movie projector and oscillating varifocal mirror project moving autostereoscopic images at 15 volumetric images per second. In order lo distribute evenly the component images along the depth axis, linear time scans of the image volume are required. However, the image position is a nonlinear function of the mirror displacement, which, in turn, has a nonlinear frequency response to the mirror driving voltage. Analytical and experimental investigations are reported in which approximately linear scans for duty cycles approaching 90% were attained.

© 1968 Optical Society of America

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References

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  1. A. C. Traub, Appl. Opt. 6, 1085 (1967).
    [CrossRef] [PubMed]
  2. J. C. Muirhead, Rev. Sci. Instrum. 32, 210 (1961).
    [CrossRef]

1967

1961

J. C. Muirhead, Rev. Sci. Instrum. 32, 210 (1961).
[CrossRef]

Muirhead, J. C.

J. C. Muirhead, Rev. Sci. Instrum. 32, 210 (1961).
[CrossRef]

Traub, A. C.

Appl. Opt.

Rev. Sci. Instrum.

J. C. Muirhead, Rev. Sci. Instrum. 32, 210 (1961).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Autostereoscopic display using a vibrating varifocal mirror

Fig. 2
Fig. 2

Schematic diagram of the system used to project autostereoscopic computer-generated movies.

Fig. 3
Fig. 3

The two top photographs show oblique views of the 3-D display from two different directions. The blurring is due both to figure movement and image jitter during the exposure, and to the shallow depth of focus of the camera. The bottom photograph shows a single movie frame that is included to assist in visually interpreting the top photographs.

Fig. 4
Fig. 4

Optical diagram showing the terms used in the analysis.

Fig. 5
Fig. 5

Diagrams of (a) the sawtooth waveform D(t), and (b) the second order correction term D2(t) used in deriving an improved mirror displacement waveform.

Fig. 6
Fig. 6

Comparison of image motion ∑(t) and mirror motion Φ(t) over one oscillation period for three different driving waveforms. Ideally, what is desired is a straight; line (sawtooth) waveform in the image position graph (the top graph). It is apparent; that, the second order image motion (B, top) is more linear than the first, order image motion (broken line, top). The sine wave case (A) is shown for comparison.

Fig. 7
Fig. 7

Experimental apparatus used to record (a) mirror displacement waveforms, and (b) swept line images of a pinhole point light source.

Fig. 8
Fig. 8

This figure illustrates the distorting effects of higher order modes that appear when a sinusoidal waveform (top row) is replaced by a sawtooth waveform (second row). Some improvement is obtained by heat treating the Mylar mirror (third row). The voltages indicated are peak to peak. The mirror diameter is 17 cm.

Fig. 9
Fig. 9

This figure shows the results of removing high frequency Fourier components from the driving waveform by low pass filtering. Tie filter cutoff frequencies are indicated at the extreme left. At a cutoff frequency of 200 Hz, this mirror displays a smooth ring-free, nearly linear mirror displacement waveform, which lasts almost 90% of the oscillation period.

Equations (10)

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s f = ( - s f + Δ ) ( 1 + Δ 2 / R 2 ) 1 + ( 4 s f / R 2 ) Δ - ( 3 / R 2 ) Δ 2 + Δ .
σ = ( - σ + δ ) ( 1 + δ 2 ) ( 1 + 4 σ δ - 3 δ 2 ) + δ .
σ = - σ / ( 1 + 4 σ δ ) .
Σ = 1 / ( 1 - Φ ) .
m - 1 = ( 1 + 4 σ δ - 3 δ 2 ) / ( 1 + δ 2 ) .
Σ ( t ) = 1 / [ 1 - Φ ( t ) ] = 1 + D ( t ) ,
Φ ( t ) = D ( t ) / [ 1 + D ( t ) ] ,
Φ ( t ) = D ( t ) - D 2 ( t ) + D 3 ( t ) - .
Φ 2 ( t ) D ( t ) - D 2 ( t ) = A 0 + 1 ( A m cos m ω t + B m sin m ω t ) ,
A 0 = - α 2 / 3 , B m = ( - 1 ) m - 1 ( 2 α / m π ) , A m = ( - 1 ) m - 1 ( 4 α 2 / m 2 π 2 ) = ( - 1 ) m - 1 B m 2 .

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