White light optical processor for edge enhancement and

spectral filtering

Fred M. Dickey and Donald J. Moore

A white light optical processor is described using partial coherence theory. The application of the processor

to both edge enhancement and band-pass filtering of imagery is discussed. The spectral components of the

spatial frequency content of images can be separated or filtered if the various spectra do not overlap by a dis-

tance determined from the spatial spectrum of the mutual intensity function of the image illumination. The

results of an experimental and computer evaluation of the processor are presented.

Introduction

Edge enhancement and spatial spectrum band-passfiltering are important operations utilized in a numberof image processing applications. These operations arereadily accomplished in the spatial frequency domainusing coherent Fourier optical processing techniques.1

An optimum edge detection filter theory, developed forlinear space-invariant Fourier optical image processing,2

and the white light processor described in this paperresulted from a program conducted to develop opticalanalog techniques applicable to the generation of linedrawings from photographic imagery.3 The results aresuitably general and may be applied to other imageprocessing applications requiring edge enhancement.

The analysis of imaging systems employing partiallycoherent light is nonlinear and, therefore, more complexthan the case of coherent linear systems.48 A broadtreatment of partial coherence theory is available instandard texts.9' 10 Using partial coherence theory, weshow that a two-lens system can be used for edge en-hancement and band-pass filtering if the input film il-lumination is from a collimated source of sufficientlysmall size, even though the spectral width of the sourcemay be quite broad (white light). The results of acomputer analysis of the edge enhanced image of a stepfunction are presented, as well as the results of an ex-perimental demonstration of the theory.

The authors are with Boeing Wichita Company, Wichita, Kansas67210.

Received 13 November 1978.0003-6935/79/101679-05$00.50/0.© 1979 Optical Society of America.

In our optical edge enhancement studies, we haveemployed both coherent Fourier optical processingtechniques and the white light processor. The whitelight approach offers several potential advantages. Dueto multispectral averaging, the white light processor isrelatively free of speckle effects. This approach alsooffers the potential of edge enhancement of color pho-tographs or color-coded images. It should be possible,by use of color filters, to select certain edges when pro-cessing color photographs. A major disadvantage is thatavailable white light sources do not produce the opticalintensities obtainable with laser systems. However, awider range of recording films can be used with a whitelight source (image) than with laser systems.

Processor Theory

The white light processor is functionally illustratedin Fig. 1. In the figure, the incoherent source and col-limating lens provide the object film illumination. Theoptical processing of the light transmitted by the objectis achieved by the filter and telescopic lens pair L1 andL2. This configuration of elements is essentially thesame as that commonly used for a two-lens coherentFourier transform optical system.

We analyze the system by determining the mutualintensity at the object, filter, and image planes usingpartial coherence theory of a quasi-monochromaticfield. The intensity at these planes is determined byintegration of the mutual intensity over the spectralrange of the source. A 1-D treatment is presented tosimplify the analysis, but the results are readily ex-tended to the 2-D case.

The analysis proceeds from a statement of some ofthe more important relationships between the mutualcoherence function, the mutual intensity, and the mu-tual spectral density.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1679

Object Iagencoherfen tlSource A A

Collimatleg Les Filter LesLens 1t 2

Fig. 1. Schematic diagram of the white light processor.

The mutual coherence function r(x 1,x2,r) satisfiesthe wave equations9

1 2 FrcX2sT)Vjr(x1,X2,T) = , j = 1,2,

J~~~~ ~~~~ dT&r (1)

where x1 and x2 are spatial variables, and r is the tem-poral correlation interval. The corresponding spectralequations are

V2G(x1 ,x2 ,v) + k 2 G(xl,x 2 ,v) = 0, j = 1,2, (2)

where v is the temporal frequency.The mutual spectral density G(xlx 2,v) and the mu-

tual coherence function are related by the equations

INXIx2T) = f G(x 1,x 2 ,v) exp(-i27rvi)dv, (3)

G(X1,X2,V = I r(x1 ,x2 ,T) exp(i27rvT)dT, v 0,

=0 ,v<0. (4)

The intensity of the light field at position x1 is givenby

I(x1,v) = (x 1,x1,0) = J(xlx 1 ), (5)

where the mutual intensity J(xl,x 2) is

J(x1 ,x2) = (xlx 2,0). (6)

For a suitably narrow spectral width A (quasi-monochromatic approximation), we have

r(x 1,x2,T) _r(x 1,x2,0) exp(-i27rvT),= J(xI,x2 ) exp(-i2XrVT). (7)

Substituting Eq. (7) in Eq. (1) and comparing the resultwith Eq. (2) give

G(x 1 ,x 2,v) = J(XlX2), (8)

where the dependence of J(xl,x 2 ) on v is made explicit.It should be noted that, by implication, v and corre-spondingly k are mean quantities. The intensity of thefield, using Eqs. (3), (5), and (8), is given by

I(xi) = 3 J(xixi)dv. (9)

Applying the analysis to the processor, the mutual in-tensity, J(x1 ,x2), of the object illumination for meanfrequency v can be determined using the Van Cittert-Zernike theorem. The mutual intensity of the lighttransmitted by the object film is

Jo(xIx 2 ) = t(x1)t*(x 2 )J(x1,x 2 ), (10)

where t is the complex transmittance of the film, and

xl and x2 are object plane coordinates. The propaga-tion of the mutual intensity from one surface to anotheris described by the integral,

J(Q1,Q2) = JJ(P,,P2) e 8pi S 2)] AIA2*dPldP 2, (11)

where S1 is the distance between points P1 and Q1, witha corresponding relation for S2. The Al and A 2 quan-tities are the obliquity factors commonly encounteredin diffraction theory. Using this relationship and thefar-field, or Fraunhofer relationship, between the frontand back focal planes of a simple lens, and providingthat J(x 1 ,x 2 ) is of the form J(x 1- x 2), we can obtain themutual intensity at the filter plane as

J(xlf,X2f) =f 2 X2 Ji(W - W)J(c)t(W- - 2)d, (12)

where cw = (k/f)xlf and w2 = (k/f)x2 f. The tilde is usedto denote a Fourier transformation. For t real (an ap-propriate assumption in image processing) we have

(- W2) = *(CO2 - C). (13)

The intensity in the filter plane can be determined fromEqs. (12) and (13) to have the form

I(xlf) = J(xlfxlf) = f2x12 S |VJ(od -w. (14)

It is interesting to note that Eq. (14) is a convolution ofthe squared magnitude of the Fourier transform of theobject transmittance with the Fourier transform of theobject illumination mutual intensity. Further referenceto this relationship will be made in the section on spatialfiltering.

The mutual intensity of the image is obtained usingEqs. (11) and (12) and arguments similar to those above.The result is

J(zl,z 2) =32 f2X2 f( + )f(O + Z2)

X ty)t(O)J(y - O)d'ydO. (15)

The function f is the transmittance of the filter placedin the filter plane, and is the inverse Fourier transformof f. In obtaining Eq. (15), f is assumed to be a real evenfunction. The intensity of the image is then obtainedby equating z and Z2, giving

J(z1 ,zi) = 3215 f2X2 (7 + z)0 + z1)

X ty)t(0)J('y - )dydO. (16)

The intensity of the broad spectrum image is then ob-tained by substituting Eq. (16) in Eq. (9) with thespectral weighting function appropriate to the inco-herent source.

To limit the paper to a reasonable length, the inter-mediate steps in arriving at Eq. (16) have been omitted.They are accomplished by a change of variables and theuse of basic Fourier transform properties. Equation(16) is in agreement with the general result given byBorn and Wolf.1 ' Equation (16) is developed specifi-cally for the telescopic processor, Fig. 1. Also, we havean interest in the intermediate equations, (12) and (14),which describe the optical field in the filter plane.

1680 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

Edge EnhancementThe edge enhancement capability of the white light

processor system was evaluated both in the laboratoryand by computer analysis of Eq. (16). The filter func-tion f is given by

f(w) = rect[w/(2wo)] - rect[(1Ow)/(2w0)], (17)

where coo is the upper cutoff frequency, and o is definedfollowing Eq. (12). The factor of 10 between the upper

and lower cutoff frequency was determined experi-mentally to give good results. Except for a constantfactor, the inverse transform of the filter function isgiven by

1Ax) = - sinwox - sin[(wox)/1OJI.

x(18)

For an upper cutoff frequency of 25 cycles/mm at awavelength of 0.5 ,m, Eq. (18) reduces to

AX)= 1 s 507rx -sin 5rf~x) =-sm - Xo

1.0

Fig. 2. Processor step response, Eq. (16) with b = 10.0.

(19)

where Xo is the normalized wavelength, i.e., Xo = 1 cor-responds to a wavelength of 0.5 ,um. Specifically, coo isdefined following Eq. (12) as wo = (27rxf)/(XoXo.5f), andthe spatial frequency xf/(XO.5f) is assigned the value 25cycles/mm (X0 .5 = 0.5 Am).

The mutual intensity, J(y - 0), at the input trans-parency can be determined using the Van Cittert-Zer-nike theorem and neglecting constant factors, the mu-tual intensity is given by

J(,y _ 0) sin[(2rp)/(Xf)](y - 0) sin[(7rb)/Xo](y - 0) (20)

[(2p)/(,)]( - 0) [(rb)/Xo](y - 0)

where b = (4 X 10 3p)/f if X is in millimeters and p is thehalfwidth of the source, and ft is the focal length of thecollimating lens shown in Fig. 1. The transmittance ofthe object is appropriately taken to be the unit step,i.e.,

t(0) = U(f) = 11'01 O00, elsewhere.

.1.0

Fig. 3. Processor step response, Eq. (16) with b = 5.0.

(21)

Figures 2-4 are the results of a computer evaluation ofEq. (16) and / and J given by Eqs. (19) and (20) for threevalues of b with Xo = 1. It can be seen from Fig. 2 thatfor large values of b with respect to 5 in the second termin Eq. (18), poor edge enhancement is achieved. Thissituation is due to the fact that the coherence width ofthe object illumination is small compared with the widthof the inverse transform of the filter function /. Figures3 and 4 show that as b decreases (increasing coherencewidth) the edge enhancement performance of the sys-tem improves. There is little increase in performance,with decreasing b, once the coherence width is largecompared with the width of f. Figure 5 is the result of

1.0

Fig. 5. Integration of Eq. (16) over the visible spectrum for b =2.5.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1681

.10 -. 16 -. 14 -. 12 -. 10 - .08 - .06 - .04 - .02 0 .02 .04 .06 .08 .0 .1 2 .14

Fig. 4. Processor step response, Eq. (16) with b = 2.5.

(a) Input Photograph (b) Edge Enhanced Output Image

Fig. 6. Optically edge enhanced image produced with a laboratorywhite light processor.

integration of Eq. (16) according to Eq. (9) over a fre-quency range corresponding to 0.8 < X0 < 1.4 with b =2.5. The integral of Eq. (16) was computed with theassumption that the incoherent source spectral distri-bution was flat over the range of integration. In gen-eral, a weighting function would be required to includethe source spectral distribution.

An example of the results of the laboratory evaluationof the system is shown in Fig. 6. In the laboratory sys-tem, an arc lamp was used with an arc diameter of lessthan 0.25 mm, and the focal length of the collimatinglens was approximately 660 mm. For these conditionsand X = 0.5 Aim, the coherence width of the object illu-mination is determined from Eq. (19) to be 1.32 mm.The focal length of both lens L and L2 is 330 mm. Forthe filter dimensions used in the experiment, the widthof f (x) was approximately 0.20 mm.

Spatial Spectrum Filtering

The white light processor described in Fig. 1 also maybe applied to the spatial filtering or separating com-ponents of a spatially modulated (sampled) image. Oneparticular area of interest is the removal of the dotpattern in halftone images or in removal of the ziptonepattern in line drawings that has been added by theartist to shade specific areas.3

Figure 7 is representative of the spectrum of imageswe wish to filter. The image spectrum contains azero-order (central) component, Fo(x if), with lower andhigher frequency components, Fl(xlf) and F(xlf),where x f is the filter plane coordinate. The width ofthe spectral components in the figure is 2fo, and theircenter frequencies are separated by f It is desired to

F (. f)

.' /

F(. if)

\ F Ix

f -f f fC 0 0 c

-f-f -f +f0 c 0 C f f f +f

C O C O

Fig. 7. Image spectrum with low frequency and higher ordercomponents.

produce an image corresponding to the zero-order termalone. In a coherent optical system, this can be ac-complished by simple low-pass filtering, and a faithfulreproduction is obtained if the spectral components donot overlap. For the white light processor, to obtainsatisfactory results, the spectral components must beseparated by an additional distance as determined bythe spatial spectrum of the mutual intensity of theobject illumination.

To understand the band-pass filtering capability ofthe processor, it is necessary to realize that the whitelight processor is basically an imaging system. Atransparent aperture placed at the filter plane (centeredon the optical axis) will not affect the fundamentalimaging properties of the system; however, some generaldegradation of resolution may occur. The character ofthe image is determined by considering Eq. (14). FromEq. (14) it is seen that the light intensity in the filterplane is given by the convolution of the spatial spectrumof the mutual intensity and the magnitude squared ofthe object transmittance. This is in effect a broadeningof the spectrum of Fig. 7. The intensity in the filter

1682 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

/

I(0f)

xlf

Fig. 8. Intensity in the filter plane corresponding to the spectrum of Fig. 7.

(a) Ziptoned Input Line Drawing (b) Output Line Drawing

Fig. 9. An example of white light processor removal of ziptone pat-tern by low-pass spectral filtering.

plane is illustrated in Fig. 8. If the three componentsof the intensity in the filter plane do not overlap asshown in Fig. 8, they must be separated by the width ofthe spatial spectrum of the illumination mutual inten-sity scaled to the focal plane of lens L1. The spatialFourier transform of the mutual intensity, Eq. (20), isa rect function, and its width w scaled to the filter planeis given by

w = (2pf)/f,. (22)

The condition of nonoverlap is then given by

fa - 2f. > w = (2pf)/fa. (23)

When this condition is satisfied, a filter of width Wfgiven by

Wf = 2(f 0 + w) (24)

will produce an image corresponding to the low fre-quency portion of the spectrum shown in Fig. 7. Thebasis of the above argument would apply equally wellif the filter were offset to pass any of the remainingspectral components. The preceding equations areapplicable for a specific mean wavelength. In general,their application should consider the spectral range ofthe source.

Figure 9 is the result of optical removal of the ziptonedot pattern from a photographically recorded linedrawing. The system used was that depicted in Fig. 1with the focal lengths given in the preceding sectionwith an appropriately sized aperture placed at the filterplane.

Conclusion

We have described a white light optical processorcapable of image edge enhancement and spatial band-pass filtering, and the theory of the processor and theresults of computer and laboratory evaluation have beenpresented. We found that the performance of theprocessor for the two functions described comparesfavorably with that obtained with a coherent opticalprocessor.

We acknowledge the assistance provided by MaxParkhurst during the laboratory evaluation.

References1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968).2. F. M. Dickey and K. S. Shanmugam, Appl. Opt. 16, 145 (1977).

3. F. M. Dickey, J. R. White, and J. Crill, Opt. Eng. 18, No. 2(1979).

4. R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57, 1497

(1967).5. R. E. Swing and J. R. Clay, J. Opt. Soc. Am. 57, 1180 (1967).6. G. B. Parrent, Jr., J. Opt. Soc. Am. 51, 143 (1961).7. R. E. Kinzly, J. Opt. Soc. Am. 55, 1002 (1965).8. G. B. Parrent, Jr., J. Opt. Soc. Am. 49, 787 (1959).9. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

1975).10. J. Perina, Coherence of Light (Van Nostrand Reinhold, New

York, 1971).11. Ref. 9, p. 529.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1683

k , \

I