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Chapter 8Analog Optical Information Processing

Prof. Hsuan T. Chang

June 5, 2013

Prof. Hsuan T. Chang Chapter 8 Analog Optical Information Processing

Outline

I 8.1 Historical Background

I 8.2 Incoherent Optical Information Processing Systems

I 8.3 Coherent Optical Information Processing Systems

I 8.4 The VaderLugt Filter

I 8.5 The Joint Transform Correlator

I 8.6 Applications to Character Recognition

I 8.7 Optical Approaches to Invariant Pattern Recognition

I 8.8 Image Restoration

I 8.9 Processing Synthetic-Aperture Radar (SAR) Data

I 8.10 Acousto-Optics Signal Processing Systems

I 8.11 Discrete Analog Optical Processors


Goals of this chapter:A presentation of the most important and widely used analogoptical information processing architectures and applications


8.1 Historical Background

1. Abbe and Porter’s experiments in 1873 and 1906,respectively.

2. Verification of Abbe’s theory of image formation in themicroscope and an investigation of its implications.


8.1.1 The Abbe-Porter Experiments – 1

1. Figure 8.1 shows the general nature of the experiments.

2. Object: a fine wire mesh; illuminated by collimated, coherentlight.

3. Fourier spectrum of the periodic mesh appears in the backfocal plane of the imaging lens.

4. By placing various obstructions (iris, slit, or small stop) in thefocal plane, the spectrum of the image can be directlymanipulated.


Fig. 8.1



5. Figure 8.2(a) shows the spectrum of the mesh.

6. Figure 8.2(b) is the full image of the original mesh.

7. A series of isolated spectral components, each spread awaysomewhat by the finite extent of the circular aperture.

8. Bright spots along the horizontal and vertical axescorresponding to the horizontally and vertically directedcomplex-exponential components.

9. Off-axis spots correspond to components directed atcorresponding angles in the object plane.


Fig. 8.2



10. Spatial filtering techniques – by inserting a narrow slit in thefocal plane to pass only a single row of spectral components.

11. Figure 8.3(a) shows the transmitted spectrum when ahorizontal slit is used.

12. Figure 8.3(b) shows the corresponding image, contains onlythe vertical structure of the mesh.

13. Figures 8.4(a) and 8.4(b) show the spectrum and image of themesh filter with a vertical slit in the focal plane.


Fig. 8.3


Fig. 8.4


8.1.2 The Zernike Phase-Contrast Microscope – 1

1. Many objects of interest in microscopy are largelytransparent, thus absorbing little or no light.

2. When light passes through such an object, the predominanteffect is the generation of a spatially varying phase shift.

3. This is not directly observable with a conventional microscopeand a light intensity detector.



4. A number of techniques for viewing such objects: for example,interferometric techniques.

5. Central dark ground method (see Prob. 8.2)

6. Schlieren method (see Prob. 8.3)

7. Suffer from a similar defect - observed intensity variations arenot linearly related to the phase shift and therefore Cannotdirectly indicate the thickness of the object .



8. In 1935, Frits Zernike proposed a new phase contrasttechnique which rests on spatial filtering principles.

9. The observed intensity is linearly related to the phase shiftintroduced by the object.

10. A transparent object with amplitude transmittance

tA(ξ, η) = exp[jϕ(ξ, η)]

is coherently illuminated in an image-formation system.



11. Assume a unit magnification and neglect the finite extent ofentrance and exit pupils.

12. A necessary condition to achieve linearity between phase shiftand intensity is that the variable part of the former is smallcompared with 2π, i.e., ∆ϕ < 2π.

13. In this case the crudest approximation to tA might be

tA(ξ, η) = exp[jϕ0] exp[j∆ϕ] ≈ e jϕ0[1 + j∆ϕ(ξ, η)] (8.2)

where ϕ0 represents the average phase shift, the terms in(∆ϕ)2 and higher powers are neglected.

14. So ∆ϕ(ξ, η) by definition has no zero-frequency spectralcomponents.



15. The first term in Eq. (8.2) – a strong wave component thatpasses through the sample suffering a uniform phase shift ϕ0.

16. The second term – generates weaker diffracted light that isdeflected away from the optical axis.

17. The image produced by a conventional microscope could bewritten as

Ii ≈ |1 + j∆ϕ|2 = (1 + j∆ϕ)(1− j∆ϕ) = 1 + (∆ϕ)2 ≈ 1

where the term (∆ϕ)2 has been replaced by zero.



18. Zernike realized that the diffracted light arising from the phasestructure is not observable because it is in phase quadraturewith the strong background.

19. If this phase-quadrature relation could be modified, two termsmight interfere more directly to produce observable variationof image intensity.

20. He proposed that a phase-changing plate be inserted in thefocal plane to modify the phase relation between the focusedand diffracted light.

21. The phase of the focused light is retarded by either π/2 or3π/2 radians relative to the phase retardation of diffractedlight.



22. The intensities in the image plane become

Ii ≈ | exp[j(π/2)] + j∆ϕ|2 = |j(1 + ∆ϕ)|2 ≈ 1 + 2∆ϕ (8.3)

and

Ii ≈ | exp[j(3π/2)]+j∆ϕ|2 = |−j(1−∆ϕ)|2 ≈ 1−2∆ϕ. (8.4)

23. The image intensity has become linearly related to thevariation of phase shift ∆ϕ.

24. Eqs. (8.3) and (8.4) are referred to as positive and negativephase contrast, respectively.


8.1.3 Improvement of Photographs: Marechal – 1

1. Undesired defects in photographs are arising fromcorresponding defects in the OTF of the incoherent imagingsystem.

2. If the photographic transparencies were placed in a coherentoptical system, then by insertion of appropriate attenuatingand phase-shifting plates in focal plane, a compensating filtercould be synthesized to at least partially remove the undesireddefects.

3. Small details in the image could be strongly emphasized if thelow-frequency components of the object spectrum were simplyattenuated.

4. The overall frequency response is more satisfactory.


8.1.3 Improvement of Photographs: Marechal – 2

5. To remove image blur caused by badly defocus, it will yield animpulse response which consisted of a uniform circle of light.

6. The corresponding OTF was H(ρ) ≈ 2J1(πaρ)πaρ

where a is a

constant and ρ =√

f 2X + f 2Y .

7. The compensating filter was synthesized by placing both anabsorbing plate and a phase-shifting plate in the focal plane ofthe coherent filtering system, as shown in Fig. 8.5(a).

8. Figure 8.5(b) shows that the large low-frequency peak of H isattenuated, while the first negative lobe of H is phase-shiftedby 180o .


Fig. 8.5


8.2 Incoherent Optical Information Processing Systems – 1

1. The use of spatially incoherent light in optical informationprocessing provides certain advantages and disadvantages.

2. Advantages: general freedom of incoherent systems fromcoherent artifacts such as dust speckles; allow the introductionof data into the system by means of LED arrays or CRTdisplays.

3. Generally speaking, incoherent systems are somewhat moresimple than coherent systems in their physical realization.



4. Some serious disadvantages:(1) No frequency plane in the focal plane of a coherent opticalsystem.(2) The intensity of light is fundamentally a nonnegative andreal physical quantity, which limits on the type of datamanipulations that can be carried out in purely optical form.(3) Incoherent systems often must have a heavy intrusion ofelectronics at their output in order to achieve a flexibilitycomparable with that of coherent systems.



Incoherent data processing systems can be broadly divided intothree separate categories:

1. Systems based on geometrical optics

2. Systems based on diffraction

3. Discrete systems


8.2.1 Systems based on Geometrical Optics

1. Such approaches ignore the diffraction phenomenon andsuffer from limitations on the achievable space-bandwidthproduct.


8.2.1Systems based on image casting – 1

1. Almost invariably use one form or another of what could becalled “image casting” or “shadow casting,” namely thegeometrical projection of one image onto another.

2. A system that performs a spatial integration of the product oftwo functions.

3. An intensity transmittance τ1 is imaged onto a secondtransmittance τ2, the intensity at each point immediatelybehind the second transparency is τ1τ2

4. A photodetector can be used to measure the total intensitytransmitted through the pair, yielding a photocurrent I givenby

I = κ

∫ ∫ ∞

∞τ1(x , y)τ2(x , y)dxdy .



5. Figure 8.6 shows two means of achieving the image castingoperation

6. In Fig. 8.6(a), lens L1 casts a magnified image of the uniformincoherent source onto the two directly-contactedtransparencies.

7. Lens L2 then casts a demagnified image of the lighttransmitted by τ2 onto the photodetector D.

8. In Fig. 8.6(b), sometimes the physical separation of twotransparencies is desired.

9. Lens L1 casts a magnified source image onto τ1. Lens L2images τ1 onto τ2, and lens L3 casts a demagnified image oflight transmitted by τ2 onto the detector.


Fig. 8.6



10. The transparency τ1 must be inserted in an inverted geometryto compensate for the inversion introduced by the imagingoperation performed by L2.

11. It is often desired to realize the convolution operation bymoving one of the transparencies with uniform velocity andmeasuring the photodetector response as a function of time.

12. Let τ1 be introduced without the inversion referred to earlier

I = κ

∫ ∫ ∞

∞τ1(−x ,−y)τ2(x , y)dxdy .



13. If τ1 is moved in −x direction with speed v , the detectorresponds as a function of time will be given by

I (t) = κ

∫ ∫ ∞

∞τ1(vt − x ,−y)τ2(x , y)dxdy .

14. If the scans are repeated sequentially, each for a differentdisplacement −ym, the detector response will be

Im(t) = κ

∫ ∫ ∞

∞τ1(vt − x , ym − y)τ2(x , y)dxdy .

15. The array of Im(t) represents a full 2-D convolution, albeitsampled in the y displacement.


8.2.1Convolution without motion – 1

1. To perform the convolution operation without relativemotions if the optical configuration is modified as shownin Fig. 8.7.

2. Operation of this system:I Consider the light generated by a particular point (−xs ,−ys)

on the source.I The rays from that point emerge from L1 (and τ1) parallel with

each other and illuminate τ2 with an intensity distributionproportional to τ1[−x + (d/f )xs ,−y + (d/f )ys ].

I After passing through τ2 the rays are focused onto the detectorat coordinate (xs , ys).

I The intensity distribution across the detector may be written

I (xd = xs , yd = ys) = κ

∫ ∫ ∞

∞τ1(

d

fxs−x ,

d

fys−y)τ2(x , y)dxdy ,

which is the desired convolution.


Fig. 8.7


8.2.1Impulse response synthesis with a misfocused system – 1

1. Figure 8.8: Impulse response synthesis with a misfocusedsystem.

2. The lens L1 again serves to illuminate the “input” transparency(τ1) with uniform light from the extended source S .

3. Lens L2 forms an image of τ1 across the plane P ′.

4. Assume that τ1 and P ′ are each at distance 2f from the lensL2, thus yielding magnification unity in the proper image plane.

5. The transparency τ2, having an intensity transmittance equalin form to that of the desired impulse response, is inserteddirectly against lens L2.

6. The system output is found across the plane P , locateddistance ∆ from the ideal image plane P ′.


Fig. 8.8



7. Operation of this system:7.1 Apply a unit-intensity point source at (x , y) on τ1 and find

resulting intensity distribution across P.7.2 In geometrical-optics approximation, the rays passing through

τ2 converge to an ideal point in plane P ′, and then diverge toform a demagnified projection of τ2 in plane P.

7.3 The projection is centered at coordinates

xd = −(1 +∆

2f)x , yd = −(1 +

∆

2f)y ,

and the demagnification of τ2 is ∆/2f .



8. Taking into account the reflection of τ2 when projected, theresponse to the point source becomes

|h(xd , yd ; x , y)|2

= κτ2

{−2f

∆

[xd + (1 +

∆

2f)x

],−2f

∆

[yd + (1 +

∆

2f)y

].

}9. The intensity at output coordinates (−xd ,−yd) can be written

as the convolution integral

I (−xd ,−yd) = κ′∫ ∫ ∞

∞τ1(x , y)|h(−xd ,−yd ; x , y)|2dxdy .


8.2.1Limitations

1. The system geometry must be chosen in such a way thatdiffraction effects are entirely negligible.

2. To maximize the input space-bandwidth product, wewould attempt to place as many independent data pointsas possible on the transparencies.

3. As the structure on the input transparencies gets finer andfiner, more and more of the light passing through themwill be diffracted, with less and less of the light obeyingthe laws of geometrical optics.

4. If large quantities of data are to be squeezed into anaperture of a given size, ultimately diffraction effects mustbe taken into account.

5. When a system is designed based on geometrical optics, itis used in a way that assures accuracy of the laws ofgeometrical optics.


8.2.2 Systems That Incorporate the Effects of Diffraction –1

1. Design incoherent optical information processing systems thattake full account of the laws of diffraction.

2. Two major difficulties in performing general filtering operationswith incoherent light: (1) the PSFs that can be synthesizedmust be non-negative and real; (2) There are many differentpupil-plane masks that will generate the same intensity PSF,but no known method for finding the simplest such mask.

3. One particular approach called two-pupil OTF synthesis, whichprovide a means for performing bandpass filtering usingincoherent light (Prob. 8-10).



4. Bandpass filtering is an operation that fundamentally requiressubtraction, for the large low-frequency components must beremoved by the processing operations.

5. True subtraction is not possible with purely optical operations.So, incoherent processing must be supplemented with someother form of processing (either electronic or coherent optical)to achieve the bandpass operation.



6. If we collect an incoherent image using an optical system withthe pupil shown in Fig. 8.9(a), the resulting OTF of thesystem is shown in Fig. 8.9(b).

7. We now place a phase plate over one of the two apertures inthe pupil, a phase plate that introduces a 180o phase shift.The autocorrelation function of this new pupil yields an OTFshown in Fig. 8.9(c).

8. The two image intensities collected with the two OTFs aresubtracted, perhaps by an electronic system. The effectivetransfer function is shown in Fig. 8.9(d), which indeed providesa true bandpass filter.


Fig. 8.9



9. Summary: incoherent optical information processing is oftensimpler than coherent optical processing, but in general ismuch less flexible in terms of operations that can be achieved.


8.3 Coherent Optical Information Processing Systems

1. When coherent illumination is used, filtering operation canbe synthesized by direct manipulation of the complexamplitude appearing in the back focal plane of a Fouriertransform lens.


8.3.1 Coherent System Architectures – 1

1. Linear in complex amplitude, capable of realizing operation ofthe form

I (x , y) = K

∣∣∣∣∫ ∫ ∞

−∞g(ξ, η)h(x − ξ, y − η)dξdη

∣∣∣∣2 .2. Figure 8.10 shows three of many different system

configurations that can realize this operation.


Fig. 8.10



3. Figure 8.10(a) is the most straightforward and is often referredas to a “4f ” filtering architecture.

4. Input transparency with amplitude transmittance g(x1, y1) isplaced in plane P1.

5. In the rear focal plane P2, a transparency is placed to controlthe amplitude transmittance through the plane.

6. An amplitue k1G (x2/λf , y2/λf ) is incident on P2, where G isthe Fourier transform of g and k1 is a constant.

7. A filter is inserted in plane P2 to manipulate the spectrum ofg .



8. If H represents the desired transform function, then theamplitude transmittance of the filter should be

tA(x2, y2) = k2H( x2λf,y2λf

).

9. The field behind the filter is GH .

10. The lens L3 again Fourier transforms the modified spectrum ofthe input, producing a final output in its real focal plane P3.

11. Note that the final coordinate system is inverted due to asequence of two Fourier transforms has been used.

12. Disadvantage of this architecture: vignetting can occur duringthe first Fourier transform operation.



13. Figure 8.10(b):· The system uses one fewer lens.· Lens L2 performs both the Fourier transform and the imagingoperations.· The spectrum appears in the rear focal plane P2 and thefiltered image appears in plane P3.· The magnification of the system is unity.· The spectrum of input has associated with it a quadraticphase factor of the form exp[−j k

2f(x22 + y 2

2 )].· The vignetting will be even worse than the system inFig. 8.10(a).



14. Figure 8.10(c):· Only two lenses are used.· This system has no vignetting problems.· The quadratic phase factor across the input plane iscancelled by the converging illumination.· Disadvantage: the system now is of length 6f rather than5f , which is used in previous two architectures.


8.3.2 Constraint on Filter Realization – 1

1. Figure 8.12 shows the regions of the complex plane that can bereached by the transfer functions of coherent optical systemsunder different constraints on the frequency transparency.

2. In (a), when only an absorbing transparency is used, thereachable region is limited to the positive real axis between 0and 1.

3. In (b), if binary phase control is added to this absorbingtransparency, then the reachable region is extended to theregion -1 to 1 on the real axis.

4. In (c), if a pure phase filter is used, then the values of thetransfer function would be restricted to the unit circle.

5. In (d), the region of the complex plane that one wouldgenerally desire to reach if there were no constraints, namelythe entire unit circle.


Fig. 8.12


8.3.2 Constraint on Filter Realization – 2

6. In summary, the most severe limitation to the traditionalcoherent processor arose from the difficulty of simultaneouslycontrolling the amplitude and phase transmittances in any butvery simple patterns.

7. It was until 1963, with the invention of the interferometricallyrecorded filter, this serious limitation was largely overcome.


8.4 The VanderLugt Filter

1. The frequency-plane masks generated by this techniquehave the remarkable property that they can effectivelycontrol both the amplitude and phase of a transferfunction, in spite of the fact that they consist only ofpatterns of absorption.


8.4.1 Synthesis of the Frequency-Plane Mask – 1

1. The frequency-plane mask is synthesized with the help of aninterferometric (or holographic) system, such as that shown inFig. 8.13.

2. A portion of light strikes the mask P1 with an amplitudetransmittance that is proportional to the desired impulseresponse h.

3. The lens L2 Fourier transforms h, yielding an amplitudedistribution 1

λfH(x2/λf , y2/λf ) incident on the recording

medium in P2.

4. A second portion of the light passes above the mask P1,strikes a prism P , finally is incident on the recording mediumat angle θ.

Ur (x2, y2) = r0 exp(−j2παy2)

where α = sin θλ


Fig. 8.13



5. The total intensity on the recording medium - determined bythe interference of the two mutually coherent amplitudedistributions present.

I =

∣∣∣∣r0 exp(−j2παy2) +1

λfH(

x2λf,y2λf

)

∣∣∣∣2= r 20 +

1

λ2f 2

∣∣∣H(x2λf,y2λf

)∣∣∣2 + r0

λfH(

x2λf,y2λf

) exp(j2παy2)

+r0λf

H∗(x2λf,y2λf

) exp(−j2παy2)



6. IfH(

x2λf,y2λf

) = A(x2λf,y2λf

) exp[jψ(

x2λf,y2λf

)],

then I can be expressed in the form

I(x2, y2) = r 20 +1

λ2f 2A2(

x2λf,y2λf

)

+2r0λf

A(x2λf,y2λf

) cos[2παy2 + ψ(

x2λf,y2λf

)].

7. This form illustrates the recording of a complex function H onan intensity-sensitive device: amplitude and phase informationare recorded with a high-frequency carrier introduced by theangle tilt of the reference wave.



8. There are other optical systems that will produce the sameintensity distribution.

9. Figure 8.14(a) – a modified Mach-Zehnder interferometer.

10. Figure 8.14(b) – a modified Rayleigh interferometer.


Fig. 8.14



11. The final step: the exposed film is developed to produce atransparency which has an amplitude transmittance that isproportional to the intensity distribution that was incidentduring exposure.

tA(x2, y2) ∝ r 20 +1

λ2f 2|H |2 + r0

λfH exp(j2παy2)

+r0λf

H∗ exp(−j2παy2).

12. It remains to be demonstrated how that particular term of thetransmittance can be utilized and the other terms excluded.


8.4.2 Processing the Input Data – 1

1. The generated frequency-plane mask may be inserted in any ofthe processing systems shown in Fig. 8.10.

2. If the input to be filtered is g(x1, y1), then incident on themask is a complex amplitude distribution 1

λG ( x2

λf, y2λf).

3. The field strength transmitted by the mask becomes

U2 ∝ r 20G

λf+

1

λ3f 3|H |2G +

r0λ2f 2

HG exp(j2παy2)

+r0λ2f 2

H∗G exp(−j2παy2).



4. The final lens L3 of Fig. 8.10(a) optical Fourier transforms U2.

5. The field strength in plane P3 becomes

U3(x3, y3) ∝ r 20 g(x3, y3)

+1

λ2f 2[h(x3, y3)⊗ h∗(−x3,−y3)⊗ g(x3, y3)]

+r0λf

[h(x3, y3)⊗ g(x3, y3)⊗ δ(x3, y3 + αλf )]

+r0λf

[h∗(−x3,−y3)⊗ g(x3, y3)⊗ δ(x3, y3 − αλf )] .

6. The third and fourth terms are of particular interest.



7. Note that

h(x3, y3) ⊗ g(x3, y3)⊗ δ(x3, y3 + αλf )

=

∫ ∫ ∞

−∞h(x3 − ξ, y3 + αλf − η)g(ξ, η)dξdη,

this third output term yields a convolution of h and g ,centered at coordinates (0,−αλf ) in the (x3, y3) plane.

8. Similarly, the fourth term

h(−x3,−y3) ⊗ g(x3, y3)⊗ δ(x3, y3 − αλf )

=

∫ ∫ ∞

−∞g(ξ, η)h∗(ξ − x3, η − y3 + αλf )dξdη,

which is the crosscorrelation of g and h, centered atcoordinates (0, αλf ) in the (x3, y3) plane.



9. The first and second terms are of no particular utility in theusual filtering operations and are centered at the origin.

10. If the “carrier frequency” α is chosen sufficiently high, orequivalently a sufficiently steep angle, the convolution andcrosscorrelation terms will be defected sufficiently far off-axisto be viewed indenpendently.

11. Figure 8.15 illustrates various output terms.


Fig. 8.15



12. If the maximum width of h in the y direction is Wh and thatof g is Wg , then the widths of the various output terms are asfollows:

12.1 r20 g(x3, y3) → Wg

12.2 1λ2f 2 [h(x3, y3)⊗ h∗(−x3,−y3)⊗ g(x3, y3)] → 2Wh +Wg

12.3 r0λf [h(x3, y3)⊗ g(x3, y3)⊗ δ(x3, y3 + αλf )] → Wh +Wg

12.4 r0λf [h

∗(−x3,−y3)⊗ g(x3, y3)⊗ δ(x3, y3 − αλf )] → Wh +Wg

13. Complete separation will be achieved if

α >1

λf(3Wh

2+Wg),

or equivalently, if

θ >3

2

Wh

f+

Wg

f,

where the small-angle approximation sin θ ≈ θ has been used.


8.4.3 Advantages of the VanderLugt Filter – 1

1. Remove the two most serious limitations to conventionalcoherent optical processors.(1) When a specific impulse response is desired, the taskof finding the associated transfer function is eliminated;the impulse response is Fourier transform optically by thesystem that synthesizes the frequency-plane mask.(2) The generally complicated complex-valued transferfunction is synthesized with a single absorbing mask.

2. Remain very sensitive to the exact position of thefrequency-plane mask, but less than the conventionalcoherent processor.

3. The difficulties are now transferred back to the spacedomain, but are in general much less severe.


8.5 The Joint Transform Correlator – 1

1. An alternative method for performing complex filtering using aspatial carrier for encoding amplitude and phase information isconsidered.

2. Joint transform correlator (JTC), although like theVanderLugt filter, it is equally capable of performingconvolutions and correlations.

3. Both the desired impulse response and the data to be filteredare presented simultaneously during the recording process.

4. Figures 8.16(a) and 8.16(b) show the recording and thefiltered output.


Fig. 8.16



5. The field transmitted through the front focal plane is given by

U1(x1, y1) = h(x1, y1 − Y /2) + g(x1, y1 + Y /2)

where Y is the separation of centers of two inputs.

6. In the rear focal plane of the lens,

U2(x2, y2) =1

λfH(

x2λf,y2λf

)e−jπy2Y /λf+1

λfG (

x2λf,y2λf

)e+jπy2Y /λf



7. Taking the squared magnitude, the intensity incident on therecording plane is

I(x2, y2) =1

λ2f 2

[∣∣∣H(x2λf,y2λf

)∣∣∣2 + ∣∣∣G (

x2λf,y2λf

)∣∣∣2

+ H(x2λf,y2λf

)G ∗(x2λf,y2λf

)e−jπy2Y /λf

+ H∗(x2λf,y2λf

)G (x2λf,y2λf

)e+jπy2Y /λf ] .

8. The transparency is assumed to have an amplitudetransmittance that is proportional to the intensity thatexposed it.



9. The transmitted field is Fourier transformed by lens L4 (Fig.8.16(b)).

10. Taking account of scaling factor and coordinate inversions, thefield in the rear focal plane of L4 is

U(x3, y3) =1

λf[h(x3, y3)⊗ h∗(−x3,−y3)

+ g(x3, y3)⊗ g ∗(−x3,−y3)

+ h(x3, y3)⊗ g ∗(−x3,−y3)⊗ δ(x3, y3 − Y )

+ h∗(−x3,−y3)⊗ g(x3, y3)⊗ δ(x3, y3 + Y )] .



11. Again it’s the third and fourth terms are of most interest.They can be rewritten as

h(x3, y3) ⊗ g ∗(−x3,−y3)⊗ δ(x3, y3 − Y ))

=

∫ ∫ ∞

−∞h(ξ, η)g ∗(ξ − x3, y3 + η − y3 + Y )dξdη,

and

h∗(−x3,−y3) ⊗ g(x3, y3)⊗ δ(x3, y3 + Y )

=

∫ ∫ ∞

−∞g(ξ, η)h∗(ξ − x3, η − y3 − Y )dξdη,

12. Both expressions are crosscorrelations of the functions g andh. One is centered at (0,−Y ) and the other at (0,Y ).



13. To obtain a convolution of the functions h and g , it isnecessary one of them (and only one) be introduced in theprocessor of Fig. 18.6(a) with a mirror reflection about its ownorigin.

14. Separation of the correlation (or convolution) terms from theuninteresting on-axis terms requires adequate separation of thetwo inputs at the start.

15. If Wh and Wg represent the widths of h and g , respectively,both measured in y direction, then the separation of desiredterms can be shown to occur if

Y > max{Wh,Wg}+Wg

2+

Wh

2.



16. JTC is in some cases more convenient than the VanderLugtgeometry.

17. Precise alignment of the filter transparency is required for thelater one, while no such alignment is necessary for the JTC.

18. JTC is advantageous for real-time systems that are required torapidly change the impulse response.

19. The price paid for the joint geometry is generally a reductionof the space-bandwidth production of the input transducerthat can be devoted to the data to be filtered.


8.6 Application To Character Recognition

A particular application of optical information processing:character recognition.


8.6.1 The Matched Filter – 1

1. By definition, a linear space-invariant filter is said to bematched to a particular signal s(x , y) if its impulse responseh(x , y) is given by

h(x , y) = s∗(−x ,−y).

2. If input g(x , y) is applied to a filter matched to s(x , y), thenthe output v(x , y) is

v(x , y) =

∫ ∫ ∞

−∞h(x − ξ, y − η)g(ξ, η)dξdη

=

∫ ∫ ∞

−∞g(ξ, η)s∗(x − ξ, y − η)dξdη,

which is the crosscorrelation function of g and s.



3. The concept of the matched filter first arose in the field ofsignal detection.

4. Figure 8.17 shows an optical interpretation of thematched-filtering operation.

5. Suppose that a filter, matched to the input signal s(x , y), is tobe synthesized by means of a frequency-plane mask in theusual coherent processing geometry.

6. Fourier transform of the impulse response shows that therequired transfer function is

H(fx , fy ) = S∗(fx , fy ).

7. The frequency plane filter should have an amplitudetransmittance proportional to S∗.


Fig. 8.17



8. When the signal s is present at the input, incident on the filteris a field distribution proportional to S , and transmitted by thefilter is a field distribution proportional to SS∗, which isentirely real.

9. Thus the transmitted field consists of a plane wave, which isbrought to a bright focus by the final transforming lens.

10. When an input signal other than s(x , y) is present, thetransmitted light will not be brought to a bright focus by thefinal lens.

11. If the input is not centered on the origin, the bright point inthe output plane simply shifts by a distance equal to themisregistration distance. → Space invariance of the matchedfilter.


8.6.2 A Character Recognition Problem – 1

1. Consider the input g to a processing system may be consist ofN possible alphanumeric characters, represented bys1, s2, . . . , sN .

2. The identification process can be realized by applying theinput to a bank of N filters, each matched to one of thepossible input characters.

3. Figure 8.18 shows the block diagram of the recognitionmachine.

4. The input is simultaneously applied to the N matched filterswith transfer functions S∗

1 , S∗2 , . . . , S

∗N .

5. The response of each filter is normalized by the square root ofthe total energy in the character to which it is matched.

6. Finally, the squared moduli of the outputs|v1|2, |v2|2, . . . , |vN |2 are compared.


Fig. 8.18



7. If the particular character g(x , y) = sk(x , y) is actually presentat the input, then the particular output |vk |2 will be the largestof the N responses.

8. The peak output |vk |2 is given by

|vk |2 =

[∫ ∫∞−∞ |sk |2dξdη

]2∫ ∫∞−∞ |sk |2dξdη

=

∫ ∫ ∞

−∞|sk |2dξdη.

9. The response |vn|2(n = k) of an incorrect matched filter isgiven by

|vn|2 =

[∫ ∫∞−∞ sks

∗ndξdη

]2∫ ∫∞−∞ |sn|2dξdη

.



10. From Schwarz’s inequality,∣∣∣∣∫ ∫ ∞

−∞sks

∗ndξdη

∣∣∣∣2 ≤ ∫ ∫ ∞

−∞|sk |2dξdη

∫ ∫ ∞

−∞|sn|2dξdη.

11. It follows directly that

|vn|2 ≤∫ ∫ ∞

−∞|sk |2dξdη = |vk |2,

with equality if and only if sn(x , y) = κsk(x , y).

12. This capability is not unique to the matched filter. It is oftenpossible to modify (mismatch) all the filters in such a way thatthe discrimination between characters is improved.


8.6.3 Optical Synthesis of a Character-RecognitionMachine – 1

1. Our discussion here is directed at the VanderLugt-type system.

2. Recall – one of the outputs is itself the crosscorrelation of theinput pattern with the original pattern from which the filterwas synthesized.

3. By restricting attention to the proper region of the outputspace, the matched filter output is readily observed.

4. Figure 8.19(a): the impulse response of a VanderLugt filterwhich has been synthesized for the character P.

5. Upper portion of response will generate the convolution of theinput, while the lower response will generate thecrosscorrelation of the input. The central portion is undesiredand not of interest.


Fig. 8.19



6. Figure 8.19(b): the response of the matched filter portion ofthe output to the letters Q, W, and P.

7. Note the presence of the bright point of light in the responseto P.

8. To realize the entire bank of matched filters in Fig. 8.18, itwould be possible to synthesize N separate VanderLugt filters,applying the input to each filter sequentially.

9. If N is too large, it is possible to synthesize the entire bank offilters on a single frequency-plane filter.

10. This can be done by frequency-multiplexing, or recording thevarious frequency-plane filters with different carrier frequencieson a single transparency.



11. Figure 8.20(a): one way of recording the multiplexed filter.

12. The letters Q, W, and P are at different angles w.r.t. thereference point, and then the crosscorrelations of Q, W, and Pwith the input character appear at different distances from theorigin, as illustrated in Fig. 8.20(b).


Fig. 8.20


8.6.4 Sensitivity to Scale Size and Rotation

1. Matched filters are too sensitive to scale size changes androtations of input patterns.

2. The response of the correct matched filter is reduced, anderrors arise in the pattern recognition processes.

3. One solution is to make a bank of filters, each of which ismatched to the pattern of interest of with a differentrotation and/or scale size.

4. If any of these matched filters have a large output, thenthe pattern of interest is known to have been presented tothe input.


8.7 Optical Approach To Invariant Pattern Recognition

1. A vast number of different pattern recognition approaches.

2. Note that the classical matched filter approach isinsensitive to translation, which should be preserved inany approach to reducing sensitivity to other types ofobject variation.

3. We will only briefly touch on three different approaches toinvariant pattern recognition that have receivedconsiderable attention in the literature.


8.7.1 Mellin Correlators – 1

1. Fourier-based correlators are extremely sensitive to bothmagnification and rotation of the object.

2. The Mellin transform is closely related to the Fourier transformbut exhibits a certain invariance to object magnification.

3. Mellin transform in 1-D form:

M(s) =

∫ ∞

0

g(ξ)ξs−1dξ,

where in the most general case, s is a complex variable.

4. If s is restricted to the imaginary axis, i.e. s = j2πf and usingthe substitution ξ = e−x , then

M(j2πf ) =

∫ ∞

−∞g(e−x)e−j2πfxdx

is nothing but the Fourier transform of the function g(e−x).Prof. Hsuan T. Chang Chapter 8 Analog Optical Information Processing


5. It is possible to perform a Mellin transform with an opticalFourier transforming system provided the input is introducedin a “stretched” coordinate system, in which the natural spacevariable is logarithmically stretched (x = − ln ξ).

6. Let M1 represent the Mellin transform of g(ξ) and let Ma

represent the Mellin transform of g(aξ), where 0 < a <∞.



7. The Mellin transform now is found as

Ma(j2πf ) =

∫ ∞

0

g(aξ)ξj2πf−1dξ

=

∫ ∞

0

g(ξ′)

(ξ′

a

)j2πf−1dξ′

a

= a−j2πf

∫ ∞

0

g(ξ′)ξ′j2πf−1

dξ′,

where ξ′ = aξ was made.

8. Since |a−j2πf | = 1, it proves that |Ma| is independent of scalesize a.



1. The rotation of an object by a certain angle is equivalent totranslation in 1-D if the object is presented in polarcoordinates, provided that the center chosen for the polarcoordinate system coincides with the center of rotation of theobject.

2. An approach to achieving simultaneous scale and rotationinvariance. See Ref. [50] for details.

3. A 2-D object g(ξ, η) is entered into the optical system in adistorted polar coordinate system, the distortion arising fromthe fact that the radial coordinate is stretched by a logarithmictransformation.


8.7.2 Circular Harmonic Correlation – 1

1. An approach that focuses on the problem of invariance ofobject rotation is based on a circular harmonic decompositionof the object.

2. The circular harmonic expansion: a general 2-D functiong(r , θ), expressed in polar coordinates, is periodic in thevariable θ with period 2π.

3. It is possible to express g in a Fourier series in the angularvariable

g(r , θ) =∞∑

m=−∞

gm(r)ejmθ,

where the Fourier coefficients are functions of radius,

gm(r) =1

2π

∫ 2π

0

g(r , θ)e−jmθdθ.



4. If the function g(r , θ) undergoes a rotation by an angle α toproduce g(r , θ − α), the circular harmonic expansion becomes

g(r , θ − α) =∞∑

m=−∞

gm(r)e−jmαe jmθ

and thus the mth circular harmonic is subjected to a phasechange of −mα radians.

5. Consider the crosscorrelation of the function g and h, which inrectangular coordinate is written

R(x , y) =

∫ ∫ ∞

−∞g(ξ, η)h∗(ξ − x , η − y)dξdη.



6. Of particular interest is the value of the crosscorrelation at theorigin, which in rectangular and polar coordinates is

R0 = R(0, 0) =

∫ ∫ ∞

0

g(ξ, η)h∗(ξ, η)dξdη

=

∫ ∞

0

rdr

∫ 2π

0

g(r , θ)h∗(r , θ)dθ.

7. The crosscorrelation between the function g(r , θ) and anangularly rotated version, g(r , θ − α), yields

Rα =

∫ ∞

0

rdr

∫ 2π

0

g ∗(r , θ)g(r , θ − α)dθ,

which, when the function g ∗(r , θ) is expanded in a circularharmonic expansion, is equivalent expressed as (see next page)



Rα =

∫ ∞

0

r

[∞∑

m=−∞

g ∗m(r)

∫ 2π

0

g(r , θ − α)e jmθdθ

]dr .

But1

2π

∫ 2π

0

g(r , θ − α)e−jmθdθ = gm(r)e−jmα,

and therefore

Rα = 2π∞∑

m=−∞

e−jmα

∫ ∞

0

r |gm(r)|2dr .

8. Result: each of the circular harmonic components of thecrosscorrelation undergoes a different phase shift −mα.



9. If an optical filter that is matched to the Mth circularharmonic component of a particular object is constructed,then...

10. if that same object is entered as an input to the system withany angular rotation, a correlation peak of strengthproportional to

∫∞0

r |gM(r)|2dr will be produced, independentof rotation.

11. The price paid for rotation invariance: the strength of thecorrelation peak is smaller than that of an unrotated version ofthe object.

κM =

∫∞0

r |gM(r)|2dr∑∞m=−∞

∫∞0

r |gm(r)|2dr.

12. The quality of the correlation peaks obtained depends onmaking a “good” choice of center of the circular harmonicexpansion.


8.7.3 Synthetic Discriminant Functions – 1

1. The synthetic discriminate function (SDF) is a method forconstructing a single pattern-recognition filter.

2. Its correlation properties are tailored in advance by means of acertain “training set” of images, whose desired correlationswith the reference filter are known in advance.

3. The training images may be distorted versions (scale changeand rotation) of a single object.

4. Let the training set of N images be represented by {gn(x , y)}where n = 1, 2, . . . ,N .

5. For a particular training image, we may want the correlationwith our filter’s impulse response h(x , y) to be unity, and insome cases be zero.



6. We divide the set {gn} into two subsets, {g+n } for which we

wish the correlation to be unity, and {g−n } for which we wish

the correlation to be zero.

7. Thus the constraints are∫ ∫ ∞

−∞g+n (x , y)h(x , y)dxdy = 1

and ∫ ∫ ∞

−∞g−n (x , y)h(x , y)dxdy = 0.



8. To obtain a filter impulse h(x , y) that has the desiredcorrelations with the training set, h(x , y) can be expanded asbasis functions,

h(x , y) =N∑

n=1

angn(x , y),

where an are for the moment unknown.9. Consider the correlation of any one member of the training

set, gk(x , y) with the filter function h(x , y),

ck =

∫ ∫ ∞

−∞g ∗k (x , y)h(x , y)dxdy

=N∑

n=1

an

∫ ∫ ∞

−∞g ∗k (x , y)gn(x , y)dxdy ,

and ck is known to be either zero or unity.Prof. Hsuan T. Chang Chapter 8 Analog Optical Information Processing


10. Let pkn represent the correlation between gk and gn, we seethat

ck =N∑

n=1

anpkn.

11. Considering all N members of the training set, we establish atotal set of N linear equaltions in the N unknowns an, but fora different value of k .

12. The entire collection of these equations can be expressed in asingle matrix equation

Pa = c ,

where a and c are column vectors of length N , and P is anN ×N matrix of correlations between the training images, (seenext page)



a =

a1a2...aN

, c =

c1c2...cN

,P =

p11 . . . p1Np21 . . . p2N. . .. . .. . .

pN1 . . . pNN

.

13. The vector c is a column vector of known values, the matrix Pcontains known elements, and we seek knowledge of the vectora to specify the desired impulse response of our filter. It canbe determined as

a = P−1c .


8.8 Image Restoration

Image restoration – the restoration of an image that hasbeen blurred by a known linear, invariant point-spreadfunction.


8.8.1 The Inverse Filter – 1

1. Leto(x , y): the intensity distribution associated with anincoherent object.i(x , y): the intensity distribution associated with a blurredimage of that object.s(x , y): a known space-invariant point-spread function

2. Assume the magnification of the imaging system is unity.

3. The object and image are related by

i(x , y) =

∫ ∫ ∞

−∞o(ξ, η)s(x − ξ, y − η)dξdη.

4. We seek to obtain an estimation o(x , y) of o(x , y), based oni(x , y) and the known s(x , y).

5. In other words, we wish to invert the blurring operation andrecover the original object.



6. Given the relationship between object and image in frequencydomain,

F{i(x , y)} = F{s(x , y)⊗ o(x , y)} = S(fx , fy )O(fx , fy ),

the spectrum of the original object can be obtained by simplydividing the image spectrum by the known OTF of theimaging system,

O(fx , fy ) =I (fx , fy )

S(fx , fy ).

7. A solution is to pass the detected image i(x , y) through alinear space-invariant filter with transfer function

H(fx , fy ) =1

S(fx , fy ),

commonly referred to as an “inverse filter.”Prof. Hsuan T. Chang Chapter 8 Analog Optical Information Processing


Several serious defects of such an inverse filter:

8. Diffraction limits the set of frequencies over which the transferfunction S(fx , fy ) is nonzero to a finite range. Outside thisrange, S = 0 and its inverse is ill defined.

9. Within the range shown above, it is possible that transferfunction S will have isolated zeros.

10. The inverse filter takes no account of the fact that there isinevitably noise present in the detected image, along with thedesired signal.


8.8.2 The Wiener Filter, or the Least-Mean-Square-ErrorFilter – 1

1. A new model that takes into account explicitly the presence ofnoise is now adopted,

i(x , y) = o(x , y)⊗ s(x , y) + n(x , y),

where n(x , y) is the noise associated with the detectionprocess.

2. Both the object o(x , y) and the noise term are regarded asrandom processes. Assume that the power spectral densities ofthe object and the noise are known, and are represented byΦo(fx , fy ) and Φn(fx , fy ).



3. The goal is to produce a linear restoration filter that minimizesthe mean-square difference between the true object o(x , y)and the estimate of the object o(x , y), i.e. to minimize

ϵ2 = Average[|o − o|2].

4. The derivation of the optimum filter can be referred toanother source in [119].

5. The transfer function of the optimum restoration filter is givenby

H(fx , fy ) =S∗(fx , fy )

|S(fx , fy )|2 + Φn(fx ,fy )Φo(fx ,fy )

.

This type of filter is often referred to as a Wiener filter, afterits inventor.



6. Note that at frequencies where the signal-to-noise ratio (SNR)is high (Φn/Φo ≪ 1), the optimum filter reduces to an inversefilter,

H ≈ S∗

|S |2=

1

S,

while at frequencies where the SNR is low (Φn/Φo ≫ 1), itreduces to a strongly attenuating matched filter

H ≈ Φo

ΦnS∗.



7. Figure 8.21 shows magnitudes of the transfer function of aWiener filter under the assumption of a severe focusing errorand while power spectra for the signal and noise. Severaldifferent SNRs are represented.

8. At high SNR, the Wiener filter reduces the relative strength ofthe low frequencies and boots the relative strength of the highfrequencies. At low SNR, all frequencies are reduced.


Fig. 8.21


8.8.3 Filter Realization – 1

1. Two methods for optically realizing inverse and Wienerrestoration filters, both are based on the use ofVanderLugt-type filters.

2. Assume that a transparency with amplitude transmittance tA,proportional to s(x , y) has recorded the known impulseresponse of the blurred system.


8.8.3 Filter Realization: Inverse filter – 2

3. We record two transparencies which will be sandwiched toform the frequency plane filter.

4. One component only consists of the known blur function S .The second transparency captures information only about theintensity |S |2.

tA1 ∝ S∗(fx , fy ), and tA2 ∝1

|S(fx , fy )|2.

5. When these two transparencies are placed in close contact, theamplitude transmittance of the pair is

tA = tA1tA2 =S∗(fx , fy )

|S(fx , fy )|2=

1

S(fx , fy ),

which is the transfer function of an inverse filter.


8.8.3 Filter Realization: Inverse filter – 3

6. In addition to the all the difficulties associated with an inversefilter that were mentioned earlier, this method suffers fromother problems related to the photographic medium.

7. The dynamic range of amplitude transmittance over which thisfilter can function properly is quite limited in practice.


8.8.3 Filter Realization: Wiener filter – 4

8. A superior approach: generate a Wiener filter with considerablemore dynamic range than the previous method afforded.

9. Several aspects:9.1 Diffraction, rather than absorption, is used to attenuate

frequency components.9.2 Only a single interferometrically generate filter is required,

albeit one with an unusual set of recording parameters.9.3 The filter is bleached and therefore introduces only phase shifts

in the transmitted light.



10. Certain postulates underlie this method of recording a filter.10.1 The maximum phase shift introduced by the filter is much

smaller than 2π radians,

tA = e jϕ ≈ 1 + jϕ.

10.2 ϕ ∝ D10.3 D = γ log E − D0

10.4 ∆tA ∝ ∆ϕ ∝ ∆D ∝ ∆(log E )10.5 ∆D ∝ ∆(log E ) ≈ ∆E

E, making tA ∝ ∆E

E.



11. The exposure produced by this interferometric recording is

E (x , y) = T

{A2 + a2

∣∣∣s ( x

λf,y

λf

)∣∣∣2+ 2Aa

∣∣∣s ( x

λf,y

λf

)∣∣∣ cos [2παx + ϕ( x

λf,y

λf

)]},

where11.1 A is the square root of the intensity of the reference wave at

the film plane11.2 a is the square root of the intensity of the object wave at the

origin of the film plane11.3 α is the carrier frequency introduced by the off-axis reference

wave11.4 ϕ is the phase distribution associated with the blur transfer

function S11.5 T is the exposure time



12. It is recorded with the object wave much stronger at the originof the film plane than the reference wave, i.e. A2 ≪ a2.

13. Because this condition, the average exposure E and thevarying component of exposure ∆E are expressed as

E = T

[A2 + a2

∣∣∣s( x

λf,y

λf)∣∣∣2]

∆E = 2AaT∣∣∣s ( x

λf,y

λf

)∣∣∣ cos [2παx + ϕ( x

λf,y

λf

)].

14. Choosing the term of transmittance of the processedtransparency that is proportional to S∗,

tA ∝ ∆E

E∝ S∗

K + |S |2,

where K = A2/a2 (often called beam ratio).Prof. Hsuan T. Chang Chapter 8 Analog Optical Information Processing


15. Figure 8.22 shows photographs of the blur impulse response,the magnitude of the deblur impulse response, and the impulseresponse of the cascaded blur and deblur filters, illustratingthe restoration of a blurred point source.


Fig. 8.22


8.9 Processing synthetic-aperture radar (SAR) data(Skipped!)

I


8.10 Acousto-optical signal processing systems

I


8.11 Discrete analog optical processors

I


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