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					            Sparse Aperture Imaging




    Matthew P Dierking                Nicholas J Miller
 Air Force Research Laboratory       Bradley D Duncan
       Sensors Directorate                    LOCI
Electro-Optical Combat ID Branch   University of Dayton (Ohio)
 Sparse Aperture Imaging

• Objective
• Resolution metrics (PSF and MTF)
• Designing optimized sparse arrays
• Inter-aperture phase aberrations
• Imaging experiment
• Future work
• Conclusions
             Why construct sparse
            aperture imaging arrays?
   The resolution of a conventional, diffraction-limited optical
    imaging system is determined by its aperture diameter.


   Can we synthesize a large effective aperture (yielding
    higher resolution) from multiple, strategically located sub-
    apertures? YES.


   We will construct a sparse aperture array from identical,
    unit-diameter, diffraction-limited circular apertures.
    Sparse aperture array with resolution
    equivalent to a single, large aperture
            D circ




            s


                        
D




            Dspan

                                                 
                                              D eff 
                          Sparse aperture       Aarray N
                           figure of merit,   A  D 2
                                                  eff  eff
                              fill factor
 Sparse Aperture Imaging

• Objective
• Resolution metrics (PSF and MTF)
• Designing optimized sparse arrays
• Inter-aperture phase aberrations
• Imaging experiment
• Future work
• Conclusions
I image u , v   PSF u , v   I ideal u , v 
 PSF resolution metrics
            • FWHM of central peak
            • Peak-to-Integrated-Side-
                    Lobe-Ratio, PISLR



    Deff

                          D
                                  s


                                  Dspan
raxx   MTF resolution metrics
              • Spatial frequency cutoff
rmin          • Mid-frequency MTF level

         We define, Deff = rmin(lf)


       Deff

                         D
                               s


                              Dspan
 Sparse Aperture Imaging

• Objective
• Resolution metrics (PSF and MTF)
• Designing optimized sparse arrays
• Inter-aperture phase aberrations
• Imaging experiment
• Future work
• Conclusions
                     Optimized two-dimensional arrays

                       Golay described point arrays having
                     compact nonredundant autocorrelations

                  Golay-3   Golay-6    Golay-9          Golay-12




Autocorrelation




 Point array
PSF and MTF of a Golay-3 array
PSF and MTF of a Golay-6 array
PSF and MTF of a Golay-9 array




     10.8
        PSF and MTF of a Golay-12 array



Ø14.8
       Golay-9 array
Optimization by expansion factor


                       Expansion factor = 1.4
                      Ø10.83




                               Deff

                                         D
      Expansion factor = s/D                    s


                                             Dspan
                          Golay-9 array
                             1.0 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.1 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.2 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.3 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.4 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.5 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                          Golay-9 array
                             1.6 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                           Golay-9 array
                             1.7 expansion factor

   Intensity PSF (log-scaled, 3 decades)          MTF (log-scaled, 2 decades)
-0.4                                        -50


-0.2


  0                                          0


 0.2


                                            50
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                    MTF along fx axis
                                             1

6000                                        0.8


                                            0.6
4000

                                            0.4
2000
                                            0.2


  0                                          0
  -0.4      -0.2       0     0.2              -50              0             50
              Focal plane [mm]                  Spatial frequency [cycles/mm]
                             Golay-9 array
                    Optimization by expansion factor

                                                r min  lf 
           Exp.factor                                                         Fill factor
 Array                  FWHMPSF    PISLR [dB]                   MTFmid freq
              (s)                                                                 ()
 Golay-9      1.0        do/6.98     -6.14         4.36           0.089        47.2%
 Golay-9      1.1        do/7.58     -7.09         4.71           0.076        40.6%
 Golay-9      1.2        do/8.16     -7.94         5.02           0.067        35.7%
 Golay-9      1.3        do/8.77     -8.70         5.32           0.059        31.9%
 Golay-9      1.4        do/9.38     -9.39         5.68           0.052        27.9%
                                                                                            *
 Golay-9      1.5        do/9.98     -10.0         5.92           0.046        25.7%
 Golay-9      1.6        do/10.6     -10.6         6.23           0.041        23.2%
 Golay-9      1.7        do/12.7     -11.4         2.12           voids           --
 Golay-9      1.8        do/13.4     -11.9         1.66           voids           --
 Golay-9      1.9        do/14.1     -12.4         1.20           voids           --
 Golay-9      2.0        do/14.8     -12.8         1.18           voids           --

* Expansion factor considered optimized when MTFmin ~ 0.03 in passband
               Quality measures of optimized N sub-
                       aperture Golay arrays

                Exp.factor                                                                 Fill factor
     Array
                   (s)
                             FWHMPSF       PISLR [dB]   r m in  lf    MTFmid freq
                                                                                               ()
     Single        NA           do            7.13         1.00             NA                  1.00
    Golay-3        1.6        do/2.92        -2.92         1.99            0.136                75.8%
    Golay-4        1.6        do/4.04        -5.03         2.55            0.106                61.7%
    Golay-6        1.5        do/5.83        -7.52         3.98            0.067                37.9%
    Golay-9        1.4        do/9.38        -9.39         5.68            0.052                27.9%
*
    Golay-12       1.3        do/12.7        -10.6         6.59            0.045                27.6%


                                                                                       Aarray        N
                                        Recall our figure of merit,                                   2
                                                                                       Aeff         Deff
* Selected as optimal sparse aperture array
 Sparse Aperture Imaging

• Objective
• Resolution metrics (PSF and MTF)
• Designing optimized sparse arrays
• Inter-aperture phase aberrations
• Imaging experiment
• Future work
• Conclusions
         Golay-9 array
Piston added to a single sub-aperture




                      10.83
                     ØPhase piston applied to
                     gray shaded sub-aperture
                        (up to one wave, l
                           Golay-9 array
       0l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       1l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       2l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       3l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       4l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       5l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       6l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       7l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       8l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       9l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
       10l/10 piston applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
           Golay-9 array
Tilt aberrations on a single sub-aperture




                       Ø10.83 tilt applied to gray
                        Phase
                         shaded sub-aperture
                           (up to two waves
                           Golay-9 array
           0l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
             l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           2l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           3l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           4l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           5l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           6l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           7l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           8l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
           9l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
                           Golay-9 array
          10l/5 tilt applied to single sub-aperture
   Intensity PSF (log-scaled, 3 decades)         MTF (log-scaled, 2 decades)
-0.4                                       -50


-0.2


  0                                         0


 0.2


                                           50
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]

         Intensity PSF along x-axis                   MTF along fx axis
                                            1

6000                                       0.8


                                           0.6
4000

                                           0.4
2000
                                           0.2


  0                                         0
  -0.4      -0.2       0     0.2             -50              0             50
              Focal plane [mm]                 Spatial frequency [cycles/mm]
 Sparse Aperture Imaging

• Objective
• Resolution metrics (PSF and MTF)
• Designing optimized sparse arrays
• Inter-aperture phase aberrations
• Imaging experiment
• Future work
• Conclusions
          Incoherently imaging a resolution
          target with a Golay aperture array
             Laser
                                  Resolution
                                    target
                      Diffuser                     Lens f 1                             Lens f 2    CCD
                                                                      Pupil mask



4" Spatial Filter -                   f1                       f1 + f 2                        f2
Beam expander

                       Rotating                f1 = 500mm focal length , Ø76.2mm lens
                       diffuser                f2 = 750mm focal length , Ø76.2mm lens


                                    Golay pupil masks
                                                                Ø1.278




                       Ø0.493
                                      Ø0.684

                                                                Ø1.083
ISO 12233 resolution target
  Transmissive chrome on glass chart
  Chart dimensions: 15.2 by 11.4 mm
                               Horizontal and vertical MTFs
                                    of single aperture


                 Horizontal MTF for Golay-1 array (M = 1.5)                                   Vertical MTF for Golay-1 array (M = 1.5)
       1                                                                            1
                                            Single sub-aperture                                                          Single sub-aperture
      0.9                                   Theoretical sparse array            0.9                                      Theoretical sparse array
                                            Experimental sparse array                                                    Experimental sparse array
      0.8                                                                       0.8

      0.7                                                                       0.7

      0.6                                                                       0.6
MTF




                                                                              MTF
      0.5                                                                       0.5

      0.4                                                                       0.4

      0.3                                                                       0.3

      0.2                                                                       0.2

      0.1                                                                       0.1

       0                                                                            0
        0   20          40          60          80            100       120          0   20         40          60          80           100         120
                                cycles/mm                                                                   cycles/mm
                                Horizontal and vertical MTFs
                                      of Golay-3 array


                 Horizontal MTF for Golay-3 array (M = 1.5)                                   Vertical MTF for Golay-3 array (M = 1.5)
       1                                                                            1
                                                                                                                         Single sub-aperture
                                            Single sub-aperture
      0.9                                                                       0.9                                      Theoretical sparse array
                                            Theoretical sparse array
                                                                                                                         Experimental sparse array
                                            Experimental sparse array
      0.8                                                                       0.8

      0.7                                                                       0.7

      0.6                                                                       0.6
MTF




                                                                              MTF
      0.5                                                                       0.5

      0.4                                                                       0.4

      0.3                                                                       0.3

      0.2                                                                       0.2

      0.1                                                                       0.1

       0                                                                            0
        0   20          40          60          80            100       120          0   20         40          60          80           100         120
                                cycles/mm                                                                   cycles/mm
                                Horizontal and vertical MTFs
                                      of Golay-6 array


                 Horizontal MTF for Golay-6 array (M = 1.5)                                    Vertical MTF for Golay-6 array (M = 1.5)
       1                                                                             1
                                            Single sub-aperture                                                           Single sub-aperture
      0.9                                   Theoretical sparse array                0.9                                   Theoretical sparse array
                                            Experimental sparse array                                                     Experimental sparse array
      0.8                                                                           0.8

      0.7                                                                           0.7

      0.6                                                                           0.6



                                                                              MTF
MTF




      0.5                                                                           0.5

      0.4                                                                           0.4

      0.3                                                                           0.3

      0.2                                                                           0.2

      0.1                                                                           0.1

       0                                                                             0
        0   20          40          60          80            100       120           0   20         40          60          80           100         120
                                cycles/mm                                                                    cycles/mm
                               Horizontal and vertical MTFs
                                     of Golay-9 array


                Horizontal MTF for Golay-9 array (M = 1.5)                                    Vertical MTF for Golay-9 array (M = 1.5)
      1                                                                             1
                                           Single sub-aperture                                                           Single sub-aperture
  0.9                                      Theoretical sparse array                0.9                                   Theoretical sparse array
                                           Experimental sparse array                                                     Experimental sparse array
  0.8                                                                              0.8

  0.7                                                                              0.7

  0.6                                                                              0.6



                                                                             MTF
MTF




  0.5                                                                              0.5

  0.4                                                                              0.4

  0.3                                                                              0.3

  0.2                                                                              0.2

  0.1                                                                              0.1

      0                                                                             0
       0   20          40          60          80            100       120           0   20         40          60          80           100         120
                               cycles/mm                                                                    cycles/mm
                                Horizontal and vertical MTFs
                                     of Golay-12 array


                 Horizontal MTF for Golay-12 array (M = 1.5)                                   Vertical MTF for Golay-12 array (M = 1.5)
       1                                                                             1
                                             Single sub-aperture                                                          Single sub-aperture
      0.9                                    Theoretical sparse array            0.9                                      Theoretical sparse array
                                             Experimental sparse array                                                    Experimental sparse array
      0.8                                                                        0.8

      0.7                                                                        0.7

      0.6                                                                        0.6
MTF




                                                                               MTF
      0.5                                                                        0.5

      0.4                                                                        0.4

      0.3                                                                        0.3

      0.2                                                                        0.2

      0.1                                                                        0.1

       0                                                                             0
        0   20           40          60          80            100       120          0   20          40          60          80           100        120
                                 cycles/mm                                                                    cycles/mm
      MTF attenuation means reduced
    contrast compared to filled apertures

• MTF is a measure of contrast transference. We
  observed that, MTFmid freq   6 , for a sparse
  aperture imaging system which means reduced
  contrast at middle spatial frequencies.


• Therefore, a practical sparse aperture imaging
  system should employ a post-detection restoration
  filter, such as a least squares Wiener filter, to
  improve image contrast.
               Golay-1 raw and
            Wiener restored images


Captured image of Golay-1
             Golay-9 raw and
          Wiener restored images


Captured image Golay-9
           Future work



Instead of using the large imaging lens to
combine optical beams prior to detection,
use a coherent detection method to directly
measure both the focal plane intensity and
phase at each sub-aperture.
                Synthesis of sub-aperture field by a
                      single imaging lens (a)
                Post-detection synthesis of intensity
                     images by virtual lens (b)

              Aperture mask
                                                                     Multiple small
                                                                     imaging cameras




                                                                             Virtual focal
                                Focal plane                                  plane detector
                                field, F(u,v)

Pupil field      Imaging lens                   Pupil field   Virtual
f(x,y)           (phases sub-apertures)         f(x,y)        imaging lens


               (a)                                            (b)
                  Conclusions

   Optical sparse aperture imaging was theoretically
    shown to enhance resolution over a single
    monolithic aperture of equal total area.

   The enhanced resolution was verified by calculation
    of MTF from experimental edge images.

   The resolution gain of a sparse aperture imaging
    system comes at a cost -- loss of contrast and
    reduced SNR. We successfully applied a
    restoration filter to recover lost contrast.

   Sparse arrays require accurate phasing of the
    multiple sub-apertures.

				
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