Abstract

Holographic aperture ladar (HAL) is a variant of synthetic aperture ladar (SAL). The two processes are related in that they both seek to increase cross-range (i.e., the direction of the receiver translation) image resolution through the synthesis of a large effective aperture. This is in turn achieved via the translation of a receiver aperture and the subsequent coherent phasing and correlation of multiple received signals. However, while SAL imaging incorporates a translating point detector, HAL takes advantage of a two-dimensional translating sensor array. For the research presented in this article, a side-looking stripmap HAL geometry was used to sequentially image a set of Ronchi ruling targets. Prior to this, theoretical calculations were performed to determine the baseline, single subaperture resolution of our experimental, laboratory-based system. Theoretical calculations were also performed to determine the ideal modulation transfer function (MTF) and expected cross-range HAL image sharpening ratio corresponding to the geometry of our apparatus. To verify our expectations, we first sequentially captured an oversampled collection of pupil plane field segments for each Ronchi ruling. A HAL processing algorithm incorporating a high-precision speckle field registration process was then employed to phase-correct and reposition the field segments. Relative interframe piston phase errors were also removed prior to final synthetic image formation. By then taking the Fourier transform of the synthetic image intensity and examining the fundamental spatial frequency content, we were able to produce experimental modulation transfer function curves, which we then compared with our theoretical expectations. Our results show that we are able to achieve nearly diffraction-limited results for image sharpening ratios as high as 6.43.

© 2012 Optical Society of America

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References

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2010 (3)

2009 (1)

2008 (2)

2005 (2)

2003 (1)

2002 (1)

Beck, S. M.

Binet, R.

Buck, J. R.

Buell, W. F.

Colineau, J.

Dickinson, R. P.

Dierking, M. P.

Duncan, B. D.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005), Chaps. 3, 4, and 6.

Guizar, M.

M. Guizar, Efficient Subpixel Image Registration by Cross-Correlation (Matlab Central, MathWorks, Inc., 1994–2011). http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation .

Guizar-Sicairos, M.

Jameson, D. F.

Kozlowski, D. A.

Lehureau, J.-C.

Marechal, N. J.

Miller, J. J.

Rabb, D. J.

Richards, M. A.

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005), Chap. 8.

Stafford, J. W.

Stokes, A. J.

Stremler, F. G.

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, 1990), Chap. 3.

Thurman, S. T.

Tippie, A. E.

Wright, T. J.

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

J.-C. Lehureau and J. Colineau, “Optical synthetic aperture imagery,” Proc. SPIE 5816, 54–65 (2005).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005), Chaps. 3, 4, and 6.

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, 1990), Chap. 3.

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005), Chap. 8.

M. Guizar, Efficient Subpixel Image Registration by Cross-Correlation (Matlab Central, MathWorks, Inc., 1994–2011). http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation .

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Figures (13)

Fig. 1.
Fig. 1.

Diagram of our experimental setup.

Fig. 2.
Fig. 2.

An idealized image intensity cross section of a 50% duty cycle Ronchi ruling. The mean intensity is A, the peak-to-peak variation is 2B and the spatial period is WR.

Fig. 3.
Fig. 3.

The stripmap HAL data processing method.

Fig. 4.
Fig. 4.

An example off-axis hologram of our 0.25cyc/mm target containing a digitally cropped array of 256×256 pixels. The somewhat discernible diagonal structure of this image is due to the tilted LO beam.

Fig. 5.
Fig. 5.

Modulus of the Fourier transform of the hologram shown in Figure 4. The 256×256 array contains the image (first quadrant) and conjugate (third quadrant) terms, as well as the central LO autocorrelation peak.

Fig. 6.
Fig. 6.

(a) The cropped and zero-padded image term of Figure 5 is (b) inverse Fourier transformed back to the pupil plane. Only the moduli of the complex fields are shown in both cases.

Fig. 7.
Fig. 7.

Modulus of the composite, effective pupil plane array (a) before and (b) after amplitude equalization.

Fig. 8.
Fig. 8.

Comparing the above image to that of Fig. 6(a), we see that taking the Fourier transform of the HAL processed extended pupil plane field results in a higher resolution image of the target than is possible with a single frame of data alone.

Fig. 9.
Fig. 9.

Side-by-side comparison of (a) a single sub-aperture image and (b) a fully HAL processed image of our 0.25cyc/mm Ronchi ruling target. Both images have been cropped, up-sampled and sharpened by identical methods.

Fig. 10.
Fig. 10.

Side-by-side comparison of (a) a single sub-aperture image and (b) a fully HAL processed image of our 1cyc/mm Ronchi ruling target. Both images have been cropped, up-sampled and sharpened by identical methods.

Fig. 11.
Fig. 11.

Theoretical and experimental MTF functions for single sub-aperture imaging.

Fig. 12.
Fig. 12.

Theoretical and experimental MTF functions for synthetic images formed by using 12 partially overlapping frames of fully HAL processed data. The experimental cutoff frequency of 8.55cyc/mm is only a few percent less than the theoretically expected, diffraction limited value of 9.29cyc/mm.

Fig. 13.
Fig. 13.

Theoretical and experimental MTF functions for synthetic images formed by using 8 partially overlapping frames of fully HAL processed data. More nearly diffraction limited performance is realized in exchange for reduced image sharpening.

Equations (18)

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fo=DapλRo,
DEFF=2DSAR+Dap,
ISR=DEFFDap=2DSARDap+1=2ΔxDap(N1)+1.
V=ImaxIminImax+Imin=(A+B)(AB)(A+B)+(AB)=BA.
I(x)=A+Bn=Fnexp(jnp0x),
Fn=sinc(n2)
po=2πWR,
I(p)=2πAδ(p)+2πBn=Fnδ(pnω0),
V=π2c1c0.
MTF(fx)=VV0=1V0π2c1c0.
Fi(m,n)=Fi(m,n)exp(j2π(mropNr+ncopNc)),
gp(m,n)=gp(m,n)exp(j2π(mroiNr+ncoiNc)),
S=m=1Nsn=1Ns|I{gu(m,n)ϕdβ(m,n)}|γ
ϕd(m,n)=exp(πj((2mNs1)2+(2nNs1)2)(Ns1)2).
gs(m,n)=gu(m,n)ϕdβ(m,n).
MTFsub-aperture=0.960.72fx,
MTFHAL-12shot=0.940.11fx.
MTFHAL-8shot=0.960.16fx.

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