Computer generated spatial filters for large object formats

Volume 20, number 1

OPTICS COMMUNICATIONS

January 1977

COMPUTER GENERATED SPATIAL FILTERS FOR LARGE OBJECT FORMATS H. BARTELT, W.J. DALLAS and A.W. LOHMANN Physikalisches Institut der Universjtiit, 8520 Erlangen, Fed. Rep. of Germany Received 12 August 1976

A cheap computer generated hologram (CGH) contains 32 × 32 sampling cells, typically. When using a cheap CGtt as spatial filter, the input object is also restricted to 32 × 32 sampling or resolution cells, llowever, practical objects are much larger. Sufficiently large filters would be too expensive in terms of computing and plotting time. We present a new method for producing CGH spatial filters. We were able to produce a 160 X 160 cell CGH filter with computing and plotting labour for only 32 × 32 sampling ceils, We used that spatial filter as matched filter for pattern detection.

1. The problem Computer generated holograms (CGH) have several advantages and some disadvantages when compared with ordinary grey interference holograms. Advantages are: functional flexibility, more light efficiency, less noise. The functional flexibility is due to the programming flexibility of the digital computer, by which the CGH is generated. The superior signal-to-noise ratio is due to the binary transmission of the CGH [1] I: The main disadvantage of the CGH is the higher production price (for computation and plotting), to be measured in Dollars per resolution cells. In this paper we will address this economical problem. We want to show a cheap way for making CGH's, that are to be Used as spatial filters. First we will explain why a cheap CGH with only 32 × 32 cells is not very useful as spatial filter. Next, we will present a new two-step production method for CGH's with a large number of cells. This new * Presently at: Physics Dept., V.P.1. + S.U., Blacksburg, VA 24061, USA. To compare the light efficiency of a binary hologram and a grey hologram, we can consider the diffraction efficiency of regular gratings. A square wave grating sends (417r)2 times more light into the first diffraction order than a fully modulated sine wave grating does. in addition a binary hologram yields less noise from light scattered by the photographic grain structure, as shown by Strand [1].

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method is cheap but not universal, yet well suited for the production of CGH spatial filters. Finally, we will demonstrate the application of our CGH's as spatial filters for pattern detection. The economical problem, which we are addressing here, has been solved before by different means [ 2 - 4 ] . Our approach differs in that our spatial filters are binary, which is favourable in terms of signal-tonoise ratio. Yet another prior solution [5] can accommodate binary CGH's as filters, but the object is processed only in small portions, sequentially. As a consequence the image processing operation is time consuming and also mechanically complex.

2. Format requirements of spatial filters A spatial filter is a Fourier hologram, in most cases. Hence, we begin by showing in fig. la a Fourier CGH (here only 4 × 4 cells) and in fig. lb a reconstruction H(x,y) thereof, taken from the full 32 × 32 cell hologram. Besides the holographic image F(x - x O,y) we see the zeroth order and the twin image. These latter two parts of the hologram reconstruction H(x, y) cause all the trouble which we want to avoid in our new method. The hologram reconstruction H(x, y) appears in the image plane of a point source. If that point source is replaced by an object O(x,y) we will observe the con-

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OPTICS COMMUNICATIONS

January 1977

Fig. 1. (a) Section of a binary hologram here 4 × 4 cells. (b) Reconstruction from a 32 × 32 hologram. volution O * H in the reconstruction plane. This reconstruction consists of three parts, that are the convolutions of O(x,y) with the three parts of H(x,y). These three convolutions may or may not overlap, as shown in fig. 2. For the purpose of spatial filtering we cannot tolerate overlapping, since only the convolution of the object O(x,y) and the holographic image F(x -- x 0, y) is of interest. Apparently, the two different reconstructions H(x, y) in fig. 2 contain the same image structure. But they differ by the distance between image F and zero order peak. That distance is controlled by the spatial carrier frequency of the hologram, while the spatial modulation of that carrier must be the same in both cases. Hence, the digital computation of the spatial modulation should take no more time for the desirable CGH, that produces the desirable reconstruction H(x, y), shown in the lower part of fig. 2. But the OBJECT O ~ x )

RECONSTR. HIx)

f~[-xxo)~

~ ~

OUTPUT

F (X-XoI

x

x _

~-

AXI O(x]

H(x)

O(x).~ H Ix}

?'q Ax 2

Fig. 2. The overlap problem o f holographic fffltering; object O(x, y); hologram reconstruction H(x, y), with the filter response F ( x - xo,y) as part of H; convolution of O and H, as observed in the image plane.

plotting labour increases considerably, if the spatial carrier frequency is increased by a factor 16, for example. The plotting labour consists of the computation of the plotting commands and of their graphic execution. Very often the plotting labour is more costly than the basic CGH computation which, in essence, simulates the wave propagation from the hologram to the reconstruction plane. Therefore, it is the aim of our new CGH production method to simplify both parts of the plotting labour. A straightforward approach for avoiding the undesirable overlapping in the output (fig. 2, upper right) would be to design the hologram such that the holographic image F(x - Xo) is surrounded by a sufficiently large dark frame. In other words, not F(x) with width &x 1 (see fig. 2) enters into the hologram computation, but F(x) together with the dark frame, adding up to the width z3x2, as indicated in fig. 2. The number of sampling points within the quadratic image frame will increase by a factor (2t9¢2/~1) 2. As a consequence the computing time and the core space will increase by the same factor, if not more. This factor might be typically in the order of 100. Hence, this straightforward approach is not very sensible. It is not necessary either, since the addition of the dark frame did increase the input data array considerably, but only with lots of zeros. Obviously, it is wasteful to perform a Fourier transform with input data that are a priori known to be zero almost everywhere. There must be a better approach to our problem of surrounding the holographic image F with a large dark frame.

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Volume 20, number 1

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January 1977

3. The new CGH production method A binary hologram consists of N X N resolution cells. Each cell contains a rectangular, transparent aperture with variable position and size (fig. 1a). The transmittance of the whole hologram is described by:

H(g,u)= ~m ~n r e c t ( g - P m n g o - m g O ) Amngo

rect( v- Qmnvo-nVo BmnvO I "

The parameters Amn, Bran, Pmn, Qmn must be chosen so that the Fourier transformation of the hologram H(x, y) will reconstruct the desired image F(x,y), for example, around the first diffraction order (x = 1/go,y = 0):

H(x, y ) = FOU [H(g, v)] = ~ m

A mnBmn govo sine (A mn go x) sinc (Bmn roy ) r/

× exp(27riPmngoX)exp(2niQmnvoy)exp[27ri(mgox + nvov)] , F(x, y) = ~ ~ Otmnexp(i~bmn ) exp [27ri(mu0 x + nvOY)] . m

n

To achieve this, two or more parameters in each cell must be varied [1]. Two possibilities for encoding the function F(x,y) are: a) type I

Amn = l / 2 ,

Bmn = amn ,

Pmn = (Pmn/2zr ,

Qmn = O .

Bmn = 1 ,

Pmn = dPmn/2rr ,

Qmn = O .

b) type Ill

Amn = (1/zr) arcsin(zramn),

Now we are ready to discuss our method, which is a two-step process. In the first step a non-redundant basic hologram with the data of a normal synthetic hologram is plotted. It contains in each direction the M-fold number of resolution cells, but only every Mth cell is covered with a rectangular aperture, as shown in fig. 3a for M -- 2. This yields a reconstruction in the Mth order, as can be seen in fig. 3b, here for M = 5.

A mn gO

!

Bran VO

/'

H 1 (x, y ) = FO U [H(g, v) ] = ~ ~ A mnBrnn goVo sine (A mn go x) sine(Bran rOY) m

n

× exp(2niPmn go x) exp(2rriQm n roy ) exp [27ri(Mmgox + MnvoY)] . The reconstructed function F(x,y) is far removed from the zeroth order. In the region between, however, false images appear which are intolerable if this CGH would be used as spatial filter. In a second step the redundant part of the hologram is added. This can be done by means o f a step-and~repeat camera, which ~terlaces identical pattern H 1 (g, v). The result, H 2 (g, v) has been scetched in fig. 4a for M = 2. Mathematically this interlace process means a convolution of H 1 (ta, v) with an array of delta functions. The corresponding operation in the image plane is a multiplication of the reconstruction H 1 (x,y) with an array 52

factor G(x, y). Such reconstruction H 2 = H 1G is shown in fig. 4b for M = 5.

ff2(.,") = gl (., v) *

k

/

-/"o), k,j=O, . . . , M - 1 ,

H 2 (x, y) = FOU [H 2 (~, u)] = H 1 (x, y) G (x, y) with

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OPTICS COMMUNICATIONS

a

January 1977

Fig, 3, (a) Thinned hologram (1 out of 4 cells). (b) Reconstruction from thinned hologram (1 out of 25).

b

V -I- i---l-- -I-- --'--

4-

j -~

3 ,

,,,

,

,

HOLOGRAM

f't2(p,u)

k

j

exp127ri(k/~oX +B'OY)]

sin (~rM/10x) sin(nMuoY) sin (lr/a0 x)

t" £,t

RECONSTRU?TION H, Cx.y):H.(x,y)

G(x,y) = ~

f °,.;Z; 'i~+'t

sin (nu Oy) "

Apparently the array factor G suppresses fairly well the false images, except for small portions o f the immediate neighbours. The array factor G(x,y) deviates in two ways from the ideal shape, which would be a rect-function at the location of the image F(x - M/la O,y). That image can also be coiEidered as the Mth order diffraction of the hologram H. The one deviation is the non-uniform "transmission" value o f G(x, y) in the image area of F(x - M/la O,y). This non-uniformity can be precom-

~.~Z~ l

Fig. 4. (a) Filling up the thinned

hologram (1 --*4). (b) Reconstruction from filled hologram (1 ~ 25).

pensated by going with an edge-enhanced version o f F into the algorithm for computing H 1 . The second draw. back of the array factor G are the tails of the maximum, that let appear small portions of the adjacent false images of order M - 1 and M + 1, as can be seen in fig. 4b. We found a way for making these tails even smaller. We produced again a "thinned" hologram H I , that contained small rectangles only in every Mth cell. But in contrast to a normal hologram o f type I, the height of the rectangles is now one fourth of the cell height (Bran = ¼ instead ofBmn = 1). In addition we assigned different vertical positions (Qmn = +- g, +--~)" These vertical shifts were chosen such that we could do moLre than M repetition steps when producing H 2 from H 1 without ever obtaining superposed rectangles 53

Volume 20, number 1

OPTICS COMMUNICATIONS

January 1977

Fig. 5. (a) Block-interlaced hologram (thinned 1/4; repeated X 7). (b) Reconstruction from a blockinterlaced hologram.

in H2' This procedure can be seen in fig. 5a, where each small rectangle is repeated 4 × 4 times. Every sampling cell now contains four small rectangles. This corresponds to a repetition rate of 23I X 2M instead o f M × M, as earlier. Such a 2M × 2/1I repetition rate with M = 5 was applied to an original 32 × 32 hologram H 1 in order to obtain the reconstruction shown in fig. 5b. Apparently, the tails of the maximum of the array factor G are now better suppressed as compared to fig. 4b. The dot artifacts in fig. 5b did not disturb when using this particular CGH as spatial filter.

4. Application for pattern detection A well-known method for pattern detection is

Vander Lugt's matched filtering. We used one of our CGH's for such a pattern detection experiment. The size of the object O(x, y) was nine times of "target" F(x, y), which occurred three times within that object. Accordingly, the convolution of O(x, y) and F(-x, -y) in the image plane contained three detection peaks at appropriate locations (fig. 6b), Actually, this experiment was done in incoherent monochromatic light. Hence, we observed the convolution of intensities IO(x,y)l 2 and JF(-x, - y ) l 2, which is the same as the correlation of IO(x, y)l 2 and IF(x, y)l 2. In some of our experiments our actual ~ p e t i t i o n procedure, that gave us the full hologram H 2 from H 1 , was different from our earlier description. Our stepand-repeat facility was shifted only in the horizontal direction. Hence, our plotter had to paint always iden-

Fig. 6. (a) Large object, containing the target three times. (b) Output of matched filtering experiment. 54

Volume 20, number 1

OPTICS COMMUNICATIONS

tical cells vertically on top of each other. In other words, our plotted hologram was " t h i n n e d " only in one direction. The repeat process used horizontal steps. This is easy to do for " t y p e I" CGH's, where the rectangles have the full height o f the cell. Hence, all M cells in vertical column o f a M × M cell block could be plotted jointly with long vertical strokes. In conclusion, we summarize: our new method for producing computer holograms requires computing time and plotting time only for a 32 X 32 CGH. When repeating that "thinned" primary hologram 5 X 5 times, we obtained a CGH-spatial filter which could accommodate object sizes o f 160 X 160 sampling cells. We applied this CGH filter for pattern detection (incoherent matched filtering). Other fruitful applications for this new CGH filter type would be image deblur-

January 1977

ring, image differentiation and any other process where the size o f the object O(x,y) is larger than the size o f the filter response F(x, y).

References [1] B.R. Brown and A.W. Lohmann, Appl. Opt. 5 (1966) 967; A.W. kohmann and D.P. Paris, Appl. Opt. 6 (1967) 1739; A.W. Lohmann, Proc. SPIE Sem. on "Developments in Holography" 25 (1971) 43-49; T.C. Strand, Opt. Eng. 13 (1974) 219. [2] P. Chavel and S. Lowenthal, Appl. Opt. 13 (1974) 718. [3] P. Chavel and S. Lowenthal, J. Opt. Soc. Amer. 66 (1976) 14. [4] W.-HI Lee and M.O. Grear, Appl. Opt. 13 (1974) 925. [5] D.A. Ansley, Appl. Opt. 12 (1973) 2890.

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