Mask design and fabrication in coded aperture imaging

Nuclear Instruments and Methods in Physics Research A 709 (2013) 129–142

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Mask design and fabrication in coded aperture imaging Paul M.E. Shutler n, Stuart V. Springham, Alireza Talebitaher National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Republic of Singapore

a r t i c l e i n f o

abstract

Article history: Received 29 November 2012 Received in revised form 17 January 2013 Accepted 17 January 2013 Available online 29 January 2013

We introduce the new concept of a row-spaced mask, where a number of blank rows are interposed between every pair of adjacent rows of holes of a conventional cyclic difference set based coded mask. At the cost of a small loss in signal-to-noise ratio, this can substantially reduce the number of holes required to image extended sources, at the same time increasing mask strength uniformly across the aperture, as well as making the mask automatically self-supporting. We also show that the Finger and Prince construction can be used to wrap any cyclic difference set onto a two-dimensional mask, regardless of the number of its pixels. We use this construction to validate by means of numerical simulations not only the performance of row-spaced masks, but also the pixel padding technique introduced by in ’t Zand. Finally, we provide a computer program CDSGEN.EXE which, on a fast modern computer and for any Singer set of practical size and open fraction, generates the corresponding pattern of holes in seconds. & 2013 Elsevier B.V. All rights reserved.

Keywords: Coded aperture imaging Cyclic difference set Open fraction Signal-to-noise ratio Row-spaced mask Pixel padding

1. Introduction In an earlier paper [1] we reviewed how coded masks have been widely and successfully applied not only in traditional fields such as X-ray and g-ray astronomy [2,3], but also nuclear medicine [4,5], laser-fusion imaging [6,7] and remote sensing [8,9]. In that earlier paper we extended the statistical analysis of cyclic difference set (CDS) based coded masks and showed that only a few different open fractions are required to capture almost all of the optimal signal-to-noise ratio (SNR). Nevertheless, researchers who are new to coded aperture imaging face several difficulties when trying to incorporate it into their existing research. First, save for only a very few exceptions [10–12], the low open fraction CDS required to achieve near-optimal SNR can be obtained solely as Singer sets, and while several good accounts of their construction exist [13,14], the mathematical language in which these texts are written makes them relatively inaccessible to non-specialists. Signal processing texts are much more accessible [15,16] but, because their focus is on digital communications rather than image processing, Singer sets are mentioned either only very briefly [15] or not at all [16], despite all the ingredients for their construction being present. Online depositories exist for error correcting codes [17] but not Singer sets, perhaps because signal processing is naturally one-dimensional so needs only relatively small sets (  1023 ), whereas image processing is naturally two-dimensional so requires much larger sets (  1045 ). For new

n

Corresponding author. Tel.: þ65 6790 3896; fax: þ 65 6896 9417. E-mail address: [email protected] (P.M.E. Shutler).

0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.01.032

researchers, then, obtaining suitable low open fraction CDS can represent a significant practical hurdle. Second, since cyclic difference sets are naturally one-dimensional, they generally first need to be ‘‘wrapped’’ [7] or ‘‘folded’’ [2,10] onto two-dimensional coded masks, but confusingly many apparently unrelated wrappings exist, some for rectangular masks [18–20], some for hexagonal [21,22], while some CDS seem to be already wrapped [23,24]. Moreover, the approach which most of these authors used to ensure a fully coded field of view led them to use the phrase ‘‘periodic folding’’ in the restricted sense of a mosaicking [7,10,19,20,22–24], what mathematicians would call an edge-to-edge tessellation [25], other wrappings being dismissed as ‘‘not periodic’’ [7] or ‘‘displaced periodic’’ [20]. Also, the number of pixels in the half open fraction CDS which they used, namely twin prime or pseudonoise sequences, could always be factorised as a coprime pair and hence mosaicked using the PSW wrapping [19], something which is not possible for Singer sets in general. A new researcher could therefore be forgiven for believing that most Singer sets cannot be wrapped ‘‘periodically’’ at all. Third, demands for improved resolution can be satisfied only by masks with ever more pixels, hence ever more holes, and while low open fraction masks certainly require fewer holes per mask, this may be offset by the necessity of having several such masks. Costs per hole may also be significant, for example, in fusion imaging when holes are micro-machined into heat resistant materials [26–28], or in remote sensing where individual pixels may be micro-shuttered [8,9]. There is also an increasing need for masks to be self-supporting, especially when operating in harsh environments [26–28], or if the wavelength of the radiation precludes the use of transparent backing materials

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[8,9]. Low open fraction CDS do increase overall mask strength, because the average hole density is low, but the distribution of holes is far from uniform, and where local hole density is high the mask will remain weak. Finding ways of achieving good SNR with as few holes as possible, while at the same time producing uniformly strong and self supporting masks, is therefore an important task. This paper addresses each of these issues as follows. First, in Section 2.2 we re-express the account of Singer sets given by Baumert [13] but using the signal processing language of Peterson and Weldon [16], to give a simple recipe which any researcher can use to generate their own Singer sets. In particular, we take advantage of increases in computer speeds to show that primitive polynomials, a key ingredient which traditionally could be obtained only from depositories and only for certain open fractions [29], can now be generated automatically and for any open fraction by searching lexicographically through a set of candidates until one is found. We demonstrate all of the above by uploading with the online version of this paper a computer program CDSGEN.EXE which, on a fast modern computer and for any Singer set of practical size and open fraction, generates the corresponding pattern of holes in seconds. Second, in Section 2.3 we show that the Finger and Prince construction [21] can wrap any CDS onto a 2D mask, regardless of its number of pixels, and in a way which is always periodic in the strict mathematical sense, that is, translationally invariant under a pair of linearly independent vectors. In particular, we show that regarding the Finger and Prince construction as a map p : D-C, from the set of pixels in the detector plane D to the set of pixels in the cyclic difference set C, greatly simplifies the processes of decoding and image reconstruction, since it is easier to perform the computations in C rather than in D, then pull the results back to D using p. Viewed this way, the choice of possible mask configurations can be seen to be very large. A compact choice would be of the form of a rectangle with a possibly incomplete final row. In the case where the number of pixels in the CDS factorises, so that the final row can be complete, this reduces to the wrapping of Miyamoto [18]. Third, in Section 3.1 we introduce the new concept of a rowspaced mask, where a number of blank rows are interposed between every pair of adjacent rows of holes of a conventional CDS based mask, like ‘‘no two holes touching’’ masks [6,7] except spaced along one axis only rather than both axes. This effectively splits the field of view into several interleaved sub-images, analogous to scan line interleave (SLI) in 3D graphics [30] but with many sub-images not just two. For extended as opposed to point sources, we show that the use of row-spaced masks can substantially reduce the number of holes required at the cost of only a small loss in SNR. At the same time, the presence of the blank rows between the rows of holes increases mask strength systematically and uniformly across the aperture, as well as making the mask automatically self-supporting. In Section 3.2 we verify the performance of row-spaced masks using numerical simulations of a kind made possible only by the large supply of Singer sets provided by Section 2.2 and the Finger and Prince construction described in Section 2.3. We note that row-spaced masks have already achieved good experimental results [26]. Lastly, in Section 3.3 we use the fact that any CDS can be wrapped regardless of its size to validate numerically the pixel padding technique originally introduced by in ’t Zand [3]. This is the addition or deletion of a small number of pixels so as to force a cyclic difference set which either does not factorise at all, or at least not well, into a mask of a certain shape or periodicity, which for in ’t Zand was that of a perfect square. We show that all the Singer sets of practical size and open fraction considered in this paper can be pixel padded until they decompose into

approximately equal coprime factors, again at the cost of only a small loss in SNR. Using the PSW wrapping [19], they can then all be mosaicked onto low aspect ratio rectangles, that is, ‘‘folded periodically’’ in the sense understood in the literature.

2. Mask design 2.1. Signal-to-noise ratio To make this paper relatively self-contained, in this first section we briefly summarise the important results on signal-to-noise ratio derived in our earlier work [1]. A cyclic difference set (CDS) based coded mask is generally described by three parameters ðM 0 ,M 1 ,M 2 Þ, where M0 is the number of pixels, of which M1 are holes, and where the number of holes matching those of a cyclically displaced copy of the mask is a constant equal to M 2 . Two ratios can be defined, r1 ¼ M1 =M0 and r2 ¼ M2 =M1 , but for any mask with M0 large the difference between them is negligible. They are then referred to collectively as the open fraction r. Those particles emitted from an object which pass through the holes of the mask leave tracks on a detector as in Fig. 1. Any pinhole image of the object will be pixilated by the shadow of the mask on a scale which can be varied, or zoomed, by adjusting the relative distances between object, mask and detector. To avoid complications due to the fraction r0 of the area of a single pixel occupied by a single hole, we found it convenient to define N ¼ r0 N0 to be the number of particles out of the total N0 striking the mask which would pass through if every pixel contained a hole, and n ¼ r0 n0 to be the number of such particles from any given object pixel, so for a uniformly bright object with m pixels n¼N/m. Decoding essentially consists in passing across the detector a copy of the mask scaled to match its shadow, then counting the number of particle tracks visible through the holes. This yields an image of the object superimposed on an approximately uniformly bright pedestal, whose average is then subtracted. We defined signal-to-noise ratio (SNR) to be the brightness of the object after pedestal subtraction divided by the square root of its variance, meaning either variance across any patch in the original object small enough to have approximately constant brightness, or variance of a single pixel between hypothetically many reconstructed images, with numerically identical results. We expressed SNR in terms of n, m and r, pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ SNRðn,m, rÞ ¼ n= 1=r þ m=ð1rÞ

Fig. 1. A coded mask imaging system showing (a) the pinhole image formed by a single mask hole, and (b) the shadow of the mask cast by a point in the object, where the mask is based on a ðM 0 ,M 1 ,M2 Þ ¼ ð15,7,3Þ cyclic difference set.

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where for a uniformly bright object with m pixels m ¼ m1, but for a non-uniform object, or for an object in the presence of background noise, a more general expression for m applies. We then showed that the SNR is greatest at some optimal open fraction ropt given by pffiffiffiffiffi pffiffiffiffi ropt ðmÞ ¼ 1=ð1 þ mÞ C 1= m, pffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi SNRopt ðn,mÞ ¼ ropt n C n=m, M 1 C M 0 = m ð2Þ where the equalities are always exact, but the approximations are valid in the large m limit. The ratio of the SNR of an arbitrary open fraction mask to that of an optimal mask is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SNRðrÞ=SNRðropt Þ ¼ 1= r2opt =r þð1ropt Þ2 =ð1rÞ ð3Þ which expresses the penalty incurred if a sub-optimal mask is used. For small 9rropt 9 the penalty is negligible, because this SNR ratio function has a stationary point at r ¼ ropt , but for large 9rropt 9 it can become significant. For example, if ropt ¼ 1=8 and r ¼ 1=7 this ratio SNRðrÞ=SNRðropt Þ is only 0.9987, while for ropt ¼ 1=17 and r ¼ 1=2 it is 0.7498. For any target value SNR0 , say, substituting n¼N/m into Eqs (2) yields an upper limit on the fineness with which the structure of an object of given luminosity N can be imaged, that pffiffiffiffi is, mt N =ðSNR0 Þ. Using the zooming property described above, however, m can be made arbitrarily small, so for any given N a clear albeit small image can always be obtained.

2.2. Singer sets The simplest Singer sets are pseudonoise sequences [16], being the output from a feedback shift register as shown in Fig. 2 computed over Z2 , modulo-2 arithmetic, where 0¼hole. More precisely, any state vector (Xn), n ¼ 0,1,2, . . . ,N1, is followed by ðX 0n Þ given by X 0n ¼ X n1 þ An :X N1 according to a set of feedback coefficients (An), where X 1 ¼ 0. It is possible [13] to choose (An) so that the state vectors (Xn) cycle through all 2N possible states, save for the trivial state (0,0,y,0), hence M 0 ¼ 2N 1. In this case the feedback coefficients (An) are part of what is called a primitive polynomial, which in the past could be obtained only from depositories and only for certain open fractions [29]. Arranging the state vectors in sequence as shown in Fig. 2, the CDS is then the final column, so the number of holes M 1 in the CDS equals the number of state vectors which have non-zero entries only in the other N1 columns, that is M 1 ¼ 2N1 1. It is also true, but hard to prove [13], that all the other columns are merely cyclic permutations of the last column, as is seen to be the case in Fig. 2, so the number of ‘‘double holes’’ M 2 equals the number of state vectors which have non-zero entries only in N 2 columns, that is, M 2 ¼ 2N2 1. If we replace Z2 with Zp , modulo-p a prime, the output turns out to be p1 repeat copies of the CDS differing only by overall multiplication by each of the p  1 non-zero elements of Zp in turn. In fact it is possible to replace p by any prime power q, except that the arithmetic is in general then not modulo-q, but something known as a Galois field denoted GF(q) [13]. These are also usually obtained from depositories, which can be quite troublesome as it is difficult to correlate the notational conventions for the Galois fields with those used in the other depositories for the primitive polynomials. Fortunately, for coded masks we really only need GF(4), GF(8) and GF(9), since for any larger prime power q there is always a nearby prime p which is as good, and the arithmetic tables for these three cases are given in Table 1. The parameters ðM0 ,M 1 ,M 2 Þ are easy to obtain from the pseudonoise case described above by replacing 2 with q and dividing

Fig. 2. An N ¼4 feedback shift register over modulo-2 arithmetic generating the ðM 0 ,M1 ,M 2 Þ ¼ ð15,7,3Þ cyclic difference set (vertical oval) illustrated in Fig. 1, using the feedback coefficients ðA0 ,A1 ,A2 ,A3 Þ ¼ ð1,1,0,0Þ (horizontal oval).

everything by q1 to account for the repeated copies, M0 ¼

qN 1 , q1

M1 ¼

qN1 1 , q1

M2 ¼

qN2 1 q1

ð4Þ

where again 0 ¼hole among a stream of pseudorandom numbers in the range 0,1, . . . ,q1, so r C 1=q. (Mathematical texts generally write N þ1 for N, their notation being tied to the degree of the polynomials involved, which is one less than the number of coefficients [13]). If we step lexicographically through all possible choices of feedback coefficients, and for each choice take the first M0 elements from the output stream as our candidate CDS, most will give an incorrect value for M1 , so verifying the CDS property, which is generally much slower than generating candidates in the first place, can be omitted. It is also much faster when M 1 is small to verify the CDS property by checking that cyclic differences between pairs of holes are uniformly distributed [13] than to compute the cyclic autocorrelation directly. With these precautions, valid feedback coefficients can be found for any Singer set of practical size and open fraction on a fast modern computer in seconds. Such a program CDSGEN.EXE is uploaded with the online version of this paper, and the key parameters of these

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Table 1 Arithmetic tables for Galois fields (a) GF(4), (b) GF(8) and (c) GF(9).

(a)

+ 0 1 2 3

0 0 1 2 3

1 1 0 3 2

(b)

(c)

2 2 3 0 1

× 0 1 2 3

3 3 2 1 0

0 0 0 0 0

1 0 1 2 3

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 0 3 2 5 4 7 6

2 2 3 0 1 6 7 4 5

3 3 2 1 0 7 6 5 4

4 4 5 6 7 0 1 2 3

5 5 4 7 6 1 0 3 2

6 6 7 4 5 2 3 0 1

7 7 6 5 4 3 2 1 0

× 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 3 1 7 5

3 0 3 6 5 7 4 1 2

4 0 4 3 7 6 2 5 1

5 0 5 1 4 2 7 3 6

6 0 6 7 1 5 3 2 4

7 0 7 5 2 1 6 4 3

+ 0 1 2 3 4 5 6 7 8

0 0 1 2 3 4 5 6 7 8

1 1 2 0 4 5 3 7 8 6

2 2 0 1 5 3 4 8 6 7

3 3 4 5 6 7 8 0 1 2

4 4 5 3 7 8 6 1 2 0

5 5 3 4 8 6 7 2 0 1

6 6 7 8 0 1 2 3 4 5

7 7 8 6 1 2 0 4 5 3

8 8 6 7 2 0 1 5 3 4

× 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 1 6 8 7 3 5 4

3 0 3 6 2 5 8 1 4 7

4 0 4 8 5 6 1 7 2 3

5 0 5 7 8 1 3 4 6 2

6 0 6 3 1 7 4 2 8 5

7 0 7 5 4 2 6 8 3 1

8 0 8 4 7 3 2 5 1 6

2 0 2 3 1

3 0 3 1 2

practical Singer sets are given in Table 2. Except for q¼17 and 37 for the ‘‘equally spaced’’ families described in our earlier paper [1], the range of q is confined to 2 r qr 11, for reasons which will become clear in Section 3.1. Masks smaller than 20  20 are too small to be useful, while 300  300 is a reasonable fabrication limit, so the range 400 r M 0 r100,000 was chosen.

Table 2 Prime power q, dimension N and feedback coefficients (An) which generate the ðM0 ,M 1 ,M 2 Þ Singer sets shown; p is the least number of pixels by which M 0 must be padded to achieve a coprime factorisation M 0 þ p ¼ m  n for use in the PSW wrapping, also interpretable as the number of pixels by which the final row would be incomplete were the Finger and Prince construction used. q

N

ðA0 , A1 , A2 , . . .Þ

M0

p

mn

M1

M2

2

9 10 11 12 13 14 15 16

(1,0,0,0,1,0,y) (1,0,0,1,0,y) (1,0,1,0,y) (1,1,0,0,1,0,1,0,y) (1,1,0,1,1,0,y) (1,1,0,1,0,1,0,y) (1,1,0,y) (1,0,1,1,0,1,0,y)

511 1023 2047 4095 8191 16,383 32,767 65,535

þ2 0 þ3 0 1 0 0 0

19  27 31  33 41  50 63  65 90  91 127  129 151  217 255  257

255 511 1023 2047 4095 8191 16,383 32,767

127 255 511 1023 2047 4095 8191 16,383

3

7 8 9 10 11

(2,0,1,0,y) (1,0,0,1,0,y) (1,0,1,1,0,y) (1,1,0,1,0,y) (2,0,1,0,y)

1093 3280 9841 29,524 88,573

1 4 þ3 þ5 þ5

28  39 52  63 92  107 153  193 266  333

364 1093 3280 9841 29,524

121 364 1093 3280 9841

4

6 7 8 9

(2,1,1,0,y) (3,2,1,0,y) (2,1,0,1,0,y) (2,1,1,0,y)

1365 5461 21,845 87,381

0 1 3 1

35  39 65  84 134  163 257  340

341 1365 5461 21,845

85 341 1365 5461

5

5 6 7 8

(1,1,0,y) (3,1,,0,y) (2,1,0,y) (2,1,1,0,y)

781 3906 19,531 97,656

þ2 0 1 0

27  29 62  63 126  155 312  313

156 781 3906 19,531

31 156 781 3906

7

4 5 6

(4,1,1,0,y) (3,1,0,y) (4,2,1,0,y)

400 2801 19,608

1 þ4 0

19  21 51  55 129  152

57 400 2801

8 57 400

8

4 5 6

(3,1,0,y) (3,1,1,0,y) (2,1,0,y)

585 4681 37,449

5 1 þ1

20  29 65  72 175  214

73 585 4681

9 73 585

9

4 5 6

(5,1,0,y) (3,0,1,0,y) (4,1,1,0,y)

820 7381 66,430

þ5 þ3 þ5

25  33 71  104 215  309

91 820 7381

10 91 820

11

4 5

(9,1,0,y) (2,0,1,0,y)

1464 16,105

2 þ8

34  43 123  131

133 1464

12 133

17

4 5

(6,1,0,y) (6,1,0,y)

5220 88,741

þ6 1

67  78 261  340

307 5220

18 307

37

3 4

(2,1,0,y) (6,1,0,y)

1407 52,060

1 1

37  38 201  259

38 1407

1 38

2.3. The finger and prince construction The earliest CDS masks flown on X-ray satellites [2] used 1D arrays of slits [32], reducing the 2D field of view to 1D, but a 1D array of holes, or strip mask, as shown in Fig. 3 can form 2D images if the detector is also divided into strips parallel to the mask and each strip is decoded separately. The strip mask then need be only as long as the 2D field of view is wide and, as we shall see in Section 3.1, in the near-field, when the number of object pixels is large, this is arguably the best arrangement. Such a system may suffer from poor image quality, however, if the number of object pixels is not large, since they may fall unevenly between the different 1D sub-images. Each sub-image will then have a different optimal open fraction ropt , which will be impossible to satisfy collectively using a single value for the open fraction r of the strip mask. The solution is to combine all the detector strips together, each offset from the next by a constant number of pixels b say, as in Fig. 3, like a TV line scan. Provided the number of pixels in the CDS is large enough to accommodate the whole object, there will be only a single value of m, hence a single value of ropt for the mask as a whole. This effectively yields a map pðx,yÞ ¼ x þ by (modulo M0 ) from the set of pixels in the detector plane D ¼ fðx,yÞ : x,y A Zg to the pixels of

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Fig. 3. Finger and Prince construction, where the shaded region is the pinhole image of the object cast by one hole in a ðM 0 ,M 1 ,M 2 Þ ¼ ð31,6,1Þ strip mask, the large black circles () marking the holes and small dots () the non-hole pixels.

the CDS space C ¼ f0,1, . . . ,M 0 1g. This is the Finger and Prince construction [21], which they applied to hexagonal grids but which they intended for, and has been applied to, other grids including rectangular [31]. Although it is more common in the literature to regard a wrapping as a (multi-valued) map going in the opposite direction from C to D [20,22], having it in the form p : D-C brings many advantages. For example, decoding is conceptually much simpler, since we can use p to accumulate the detector counts in C, then perform the cyclic correlations more easily there than in D. To display the reconstructed image in two dimensions we simply use p to pull back from C to D, that is, the image intensity at any pixel (x,y) in D is set to be the same as at pðx,yÞ in C. Fig. 4 shows just such a decoding using the q¼5, hence r C 1=q ¼ 1=5, ðM 0 ,M 1 ,M 2 Þ ¼ ð31,6,1Þ strip mask shown in Fig. 3 with b ¼ 7 applied to a uniformly bright m¼10 test object in the shape of a 2 by 5 rectangle, with flux N¼4000 following Poisson statistics. The one-dimensional cyclic

periodicity of the reconstructed image in the CDS space is then clearly visible as a two-dimensional periodicity in the image plane. When combining detector strips in this way, the strip mask must be as long as the total number of pixels in the 2D field of view, requiring a highly elongated detector across which the particle tracks will be thinly spread. For inexpensive but easily saturated detectors, such as the solid state track detectors used in our own experimental work [26–28], this can be an advantage, but for expensive high performance detectors something more compact is clearly required. This is easy to arrange in the Finger and Prince approach since any inverse image p1 ðCÞ will function as a mask, that is, any set of pixels in D in bijective (one-to-one) correspondence with C under p. This is because any two holes drilled in the mask and which map under p to the same hole in C, will cast pinhole images of the object which differ in D by a translation vector which also maps to zero under p. A good (compact) choice is a mask which is approximately

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Fig. 4. Reconstructed image of the uniformly bright test object shown (insert) formed by the ðM 0 ,M 1 ,M 2 Þ ¼ ð31,6,1Þ mask from Fig. 3 pulled back to the detector plane, where positive values are black (), negative values white (J), and where bubble area is proportional to the absolute value of the data represented.

rectangular, composed of sub-strips of length b stacked one above the other, possibly with an incomplete final row which, in the special case that b divides M 0 exactly, reduces to the wrapping of Miyamoto [18]. An example is shown in Fig. 5 being the strip mask of Fig. 3 cut into lengths of b ¼ 7 pixels, which tiles the plane periodically under the linearly independent vectors u and v shown, matching the periodicity in Fig. 4. This way of implementing the Finger and Prince construction is able to wrap any CDS onto a compact 2D mask regardless of its number of pixels M 0 , a fact which has not been appreciated up to now, but which will be put to good use in the rest of this paper. However, non-mosaicked tesselations of the kind shown in Fig. 5 have appeared in the literature before, although not always in this guise. For example, although Gottesman and Fenimore considered only the subset of hexagonal masks which could be mosaicked [24], those with M 0 ¼ 3n2 þ 3n þ 1, the set originally envisaged by Finger and Prince, with M 0 ¼ m2 þ mn þn2 , was far larger [21]. In general the hexagons in this larger set cannot be mosaicked, and this has been nicely illustrated in recent work by Horn [12], the special case m¼nþ1 being those which can. In fact, it is not hard to see that if such a hexagonal mask is cut in half parallel to any pair of its sides, the two halves can be reassembled to give a parallelogram with an incomplete final row, the exact counterpart on a hexagonal grid of the rectangle with incomplete final row shown on the square grid of Fig. 5. The Finger and Prince construction has also recently been generalised to the case of a square grid [12], giving a basic tile in the form of a union of two squares, which can be tessellated but not as a mosaic.

For instance, the example given in the paper [12, Fig. 1] has a basic tile with 52 þ22 ¼29 pixels which, being prime, cannot be mosaicked as any kind of rectangle, but which does have a doubly periodic tiling. This same example also illustrates how it is possible to reverse the process of mask compactification which we described earlier. Taking the bottom row of the larger of the two squares in [12, Fig. 1] and extending it to a strip 29 pixels long, it is clear by inspecting the way in which this strip intersects the other tiles that it does wrap around the basic tile, covering each pixel exactly once each, even though the sub-strips in this case are not stacked contiguously as in our Fig. 5, but instead are alternating. This approach has been further generalised by the ¨ work of Busboom and Luke [22] in that, rather than tessellating with a strip laid down so that consecutive pixels of the CDS are adjacent in the plane, larger jumps along a specified vector are allowed. Provided the coordinates of the vector are chosen carefully, the strip will then wrap around a basic tile with a hexagonal shape more general even than those considered by Finger and Prince, and which are again sometimes mosaicked, and sometimes not, depending upon the parameters chosen.

3. Mask fabrication 3.1. Row-spaced masks Decoding each 1D sub-image of a strip mask separately does work well in the near-field, when the object almost fills the 2D

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Fig. 5. A compact choice of mask for the same ðM 0 , M 1 , M 2 Þ ¼ ð31,6,1Þ CDS as for Figs. 3 and 4 wrapped using b ¼ 7, tiling the plane periodically under translation vectors u ¼ ð7,1Þ and v ¼ ð3,4Þ.

Fig. 6. (a) An s¼ 3 row-spaced mask based on a ðM 0 , M 1 , M 2 Þ ¼ ð63,31,15Þ CDS with b ¼ 21, and (b) the interleaved sub-images of the object being imaged.

field of view (m t M0 ), since each sub-image will also be almost filled, obliging the object pixels to be evenly distributed. Variations in ropt between sub-images will then be small enough not to

impact significantly the SNR produced by a single choice of r for the strip. Surprisingly, strips masks are then in a real sense better than full 2D masks. Suppose a 2D mask has an s0  s0 pixel field of view, of which the strip mask occupies one row. If the object pixels are evenly distributed between sub-images, the values of m, n and M 0 are all lower for the strip mask by a factor s0. Eqs (2), which apply both to 1D and to 2D masks, then show that, compared to the 2D mask, the strip mask has a ropt higher by a pffiffiffiffiffi factor s0 , an SNRopt about the same, and an M 1 lower by a factor pffiffiffiffiffi s0 . For example, for a 100  100 pixel field of view almost filled by object, a strip mask yieldspapproximately the same SNR as a ffiffiffiffiffiffiffiffiffi full 2D mask but with about 100 ¼ 10 times fewer holes. The same advantage can be had in the mid-field (1{m{M 0 ) using a row-spaced mask, where (s 1) blank rows are placed between each adjacent pair of rows of an unspaced mask as shown in Fig. 6a, and where the image is decoded by splitting the detector and hence the field of view into s interleaved sub-images as shown in Fig. 6b. If the object pixels are distributed evenly between each sub-image, the values of m, n and M 0 for the spaced mask will all be lower by a factor s, so again Eqs (2) give an pffiffi approximately equal SNR but with about s fewer holes than an optimal unspaced mask. Unlike the near-field case, an even distribution of object pixels cannot now be assumed, but if they are at least distributed at random, averaging k per sub-image, pffiffiffi variations between sub-images will not exceed about 7 2 k. pffiffiffi Variations in qopt ¼ 1=ropt C k between sub-images will, by the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi binomial theorem, be k 72 k C qopt 71, but the qopt ¼ 8 example in Section 2.1 implies that varying qopt by 71 has negligible impact on the SNR if qopt Z8, or k \50. Any object with m 4 100

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could therefore be imaged with an s Z 2 spaced mask with pffiffiffi only a small loss in SNR, and fewer than 1= 2 C 70% of the holes of an optimal unspaced mask. Open fractions less than ropt ð100Þ C 1=11 are therefore redundant, since a rowspaced mask with underlying open fraction in the interval 1=11 r r r 1=2, which we chose for Table 2, could be used instead. If an even distribution of object pixels between sub-images could be assured some other way, even this lower limit of k¼50 pixels per sub-image could be reduced significantly since, for low m, the approximations in Eqs (2) err in the same direction for both spaced and unspaced masks. This means that the ratio of SNRopt between spaced and unspaced masks, , pffiffi SNRopt ðn=s,m0 =sÞ 1= s 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð5Þ cðm0 ,sÞ ¼ SNRopt ðn,m0 Þ 1 þ m0 =s1 1þ m0 1 where m0 is the number of object pixels for the unspaced mask, remains close to unity for much smaller m0 and much larger s than one might expect. We can use c to define a maximal spacing smax , so that the SNR of any spacing sr smax will fall short of the optimal unspaced SNRopt by at most a factor l, by means of the equation cðm0 ,smax Þ ¼ l. Solving an easy quadratic equation yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio n pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi where smax ¼ m0  m20 ðkm0 Þ2 =2 k ¼ ð1 þ m0 1Þ=l. Row-spaced masks are automatically self-supporting, so each hole can entirely fill its pixel, whereas circular holes in square pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pixels reduce SNR by at least a factor r0 ¼ p=4 C 89%, so choosing l ¼ 0:89 gives spaced masks a performance comparable with unspaced non-self-supporting masks whose holes cannot be as large due to the presence of support structures. Although the concepts behind row-spaced masks and the details of their design are new, masks which share some of their capabilities but as an unintended consequence of their own design have been fabricated in the past, namely the ‘‘no two holes touching’’ (NTHT) masks of Fenimore and Cannon [6]. In the NTHT design, each of the r rows and s columns of a conventional CDS based mask are split into two, effectively cutting each pixel into 2  2 ¼4 smaller pixels, the holes then being shrunk to fit into one of the four. When decoded at the original r  s resolution the results are predictably poor [1,4], but Fenimore and Cannon showed that they could be decoded successfully at the higher 2r  2s resolution, even though they are no longer strictly CDS based due to the existence of zeros in the autocorrelation function [6]. They also introduced two further techniques aimed at reducing image blur: fine sampling, where the decoder is moved across the detector plane in fraction of a pixel rather than whole pixel steps, and delta decoding, where the holes of the decoder are reduced in size relative to those of the coded mask. Although both of these other techniques were very successful, because they helped retain the finer spatial information which is present in the tracks on the detector but which conventional decoding discards, nevertheless they are general techniques which can be applied to any coded mask and have no effect on the SNR. By analogy with row-spaced masks, NTHT images can be decoded at higher resolution because the mask effectively divides the field of view into four interleaved sub-images, except that the interleaving is in the form of a 2  2 checkerboard pattern rather than by scan lines. Provided the object pixels are evenly distributed between these four sub-images, all of the above formulae for row-spaced masks still apply to NTHT masks in the special case s ¼4. Since their primary concern was on smoothing the reconstructed images, however, Fenimore and Cannon compared the images produced by their NTHT mask only with finely sampled and delta decoded images,

which was easy to do as this only involved analysing the same set of data three different ways. What they failed to do was to perform any statistical analysis of the SNR of their NTHT mask, nor did they compare its performance experimentally with that of a conventional CDS based mask at the same resolution, which would have been much more difficult, as this would have required fabricating a second mask and exposing a second image not necessarily comparable with the first. As a consequence, they were unaware of the potential performance advantage of their NTHT design, but with the benefit of hindsight we can now compute what these performance benefits would have been. The approximately m0 ¼50 object pixels [6, Fig. 3] were evenly distribution between the four sub-images because the object was almost a disc, and in fact for this shape of object a checker board interleaving is probably more effective than our own scan line interleaving. Using Equation (5), the NTHT mask would have SNR reduced by a factor 91% due to the spacing, then using Equation (3) by a further factor 88% due to sub-optimality of the CDS with r ¼ 1=2 which they used compared to ropt ð50=4Þ ¼ 1=4:39, giving an overall factor of 91  88¼80%. A conventional CDS based mask with the same resolution would again by Equation (3) have SNR reduced by a factor 80% due to sub-optimality of the r ¼ 1=2 CDS compared to ropt ð50Þ ¼ 1=8, but unlike a NTHT mask, which is self-supporting, the holes could not have been expanded to fill the pixels, resulting in a further factor of about 89% as discussed above, giving an overall factor 80  89¼71%. What they would have found, therefore, was that their NTHT mask produced a significantly better SNR, but with four times fewer holes, as compared to a conventional CDS based mask with the same resolution. 3.2. Simulations For a row-spacing s ¼2, which already makes a mask selfsupporting, an even distribution of object pixels between subimages is automatic if the object is extended and not a set of point sources, as contiguous pixels will be split roughly equally between the two sub-images. An even distribution for higher spacings s 42 can also be assured if the object is elongated along a known axis and the row-spacing is oriented at right angles. This is the case in our own experimental work [26–28], where the pinch region of the plasma focus is elongated with an approximately 2:5 aspect ratio. For the total particle flux and detector geometry which is typical in our work, N ¼105 is an appropriate value so, for a target SNR0 C 5, results in Sections 2.1 and 3.1 limit the resolution to m0 t65 and the maximal spacing for l ¼ 0:89 to smax C 6. It is difficult to make a fair comparison in numerical simulations, as we cannot simply take a single CDS and make of it both spaced and unspaced masks, since the collecting areas are then quite different. Masks with comparable areas but different spacings must have different underlying CDS, but with the Singer sets provided in Section 2.2, and the Finger and Prince construction in Section 2.3, constructing such masks is now possible. For m0 ¼65, an optimal unspaced mask has ropt ð65Þ ¼ 1=9, so we chose the q ¼ 9 mask shown in Fig. 7a with M 0 ¼ 7381 ¼ ð82  90Þ þ1 and M 1 ¼ 820 holes wrapped using b ¼ 90 which, for N ¼105, should yield SNRopt ð105 =65,65Þ ¼ 4:3581. Assuming an even distribution of object pixels, an s ¼6 mask would have ropt ð65=6Þ ¼ 1=4:1358, so we chose the q ¼4 mask shown in Fig. 7c with M 0 ¼ 1365 ¼ 15  91 and M 1 ¼ 341 holes wrapped using b ¼ 91. The in principle SNRopt ð105 =65=6,65=6Þ ¼ 3:8718, but the more general Equation (1) applied to our slightly suboptimal mask yields SNRð105 =65=6,65=6,1=4Þ ¼ 3:8710, which is 89% of the unspaced SNRopt ¼ 4:3581 but with only about 40% of

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Fig. 8. SNR as a function of spacing s, for N ¼ 105 and m0 ¼ 65, where each data point (J) is the average over 100 randomly generated reconstructed images, each cross ( þ) the average over each set of 10 data points, and the theory curve is the ratio function cð65,sÞ between spaced and unspaced masks multiplied by an overall scaling of SNRopt ¼ 4:3581 to match the optimal unspaced mask.

Fig. 7. Row-spaced masks (a) M 0 ¼ 7381, q¼ 9, b ¼ 90, s ¼1 (i.e. unspaced), (b) M 0 ¼ 4681, q ¼8, b ¼ 104, s ¼2 and (c) M 0 ¼ 1365, q ¼4, b ¼ 91, s ¼6. The number of holes, respectively, are: 820, 585 and 341.

the holes. For comparison we wanted an s ¼ 2 mask with ropt ð65=2Þ ¼ 1=6:6125, but since q ¼5 and q ¼7 masks of the right size do not exist we chose the next best fit, the q ¼8 mask shown in Fig. 7b with M 0 ¼ 4681 ¼ ð45  104Þ þ 1 and M 1 ¼ 585 holes wrapped using b ¼ 104. Again, the in principle

SNRopt ð105 =65=2,65=2Þ ¼ 4:1943, but applying the more general Equation (1) to this slightly more sub-optimal mask yields an actual SNRð105 =65=2,65=2,1=8Þ ¼ 4:1812, which is 96% of the unspaced SNRopt ¼ 4:3581 for only about 70% of the holes. These predictions are confirmed by simulation results shown in Fig. 8 for N ¼105 and Poisson statistics applied to the uniformly bright m0 ¼65 test object shown, and where, as is customary in the literature, and following our earlier paper [1], the simplifying assumption was made of no overspill between adjacent pixels. For each mask, then for each of 1000 randomly generated reconstructed images, SNR was computed as the mean brightness across the set of 65 reconstructed object pixels divided by the square root of their sample variance, then SNR was averaged over sets of 100 or over all 1000 by averaging the mean brightnesses and sample variances respectively. The test object was chosen with a (realistically) uneven distribution of pixels between sub-images, which explains why the averages ( þ) for s ¼6 fall slightly below the theory curve, whereas for s ¼2 the reason is mostly the sub-optimality of the open fraction of the underlying CDS. Fig. 9 shows sample reconstructed images for the same parameters as the Fig. 8 simulations. Fig. 10 shows photographs of the masks fabricated in stainless steel [26] with damage due to the energetic ions from the plasma focus clearly visible in the first image. (The primitive polynomials for these masks came from a depository, rather than CDSGEN.EXE, so the holes are permuted relative to those in Fig. 7). 3.3. Pixel padding In Section 2.3 we saw that the Finger and Prince construction can wrap any CDS onto a 2-D mask regardless of its number of pixels M 0 . For the wrapping to be periodic in the sense used in the literature, however, (a mosaicking), we need the PSW construction [19], which requires the number of pixels to factorise as M 0 ¼ mn where m and n are coprime. As can be seen from Table 2, although many Singer sets

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Fig. 9. Sample reconstructed images for the three spaced masks in Fig. 7, where positive values are black (), negative values white (J), showing little difference in image quality despite the substantial reduction in the number of holes between the masks.

do factorise in this way, most fail to do so but only by a few pixels. For example, the q¼5 mask with M 0 ¼ 19,531 has one pixel too many to factorise as 126  155. In his paper describing the r C 1=3 triadic residue mask used in the SAX-WFC [3], in ’t Zand faced a similar problem in that his set contained M 0 ¼ 65,539 ¼ 2562 þ 3 pixels. Given the considerable convenience of having an exactly 256  256 mask, and since his triadic residues were in any case not an exact CDS, he sensibly decided simply to discard the extra three pixels and carry on. Armed with the Finger and Prince construction, however, we can now simulate in ’t Zand’s pixel padding, by which we mean the addition (p 4 0) or deletion (p o 0) of a small number p of pixels to enable a wrapping with a certain periodicity to exist, and verify whether or not its impact is indeed small. That some padding is reasonable can be judged from the flatness of the autocorrelation side lobes, as in ’t Zand did to validate his triadic residue sets [3]. This can be displayed in 2D as the reconstructed image of a point source, but where the single object pixel illuminates each mask pixel with exactly one particle, as one should do anyway to check that the decoding is correct. Fig. 11 shows such results for a q ¼5, M 0 ¼ 3906 ¼ 62  63 mask wrapped using b ¼ 63 for various amounts of padding, compared to similar results obtained from a random 1/5 open 60  60 mask. These demonstrate what is retrospectively obvious, that since a

padded autocorrelation is the sum of two essentially independent, partial autocorrelations, the departure from flatness is largely independent of the amount of padding and is always less than that for a random mask. These results are confirmed by those given in Fig. 12, which show SNR against padding for N ¼5  105 for three different m¼ 100 test objects, averaged over 1000 reconstructed images. While the average SNR of a padded mask is indeed roughly independent of padding, and always falls between that of a perfect CDS mask and of a random mask, the data are quite erratic and dependent on the form of the object. This erratic behaviour is explained by the reconstructed images in Fig. 13 for the first of the three objects, for different paddings and for the random mask. Unlike the autocorrelation images in Fig. 11 where, except very close to the object, background variations appeared to be uniformly distributed, now we see artefacts in the background almost as large as the object itself. (For the random mask this may be due to clumps in the distribution of the holes where, if the autocorrelation is high at one offset because clumps overlap, it will be high for nearby offsets since the same clumps will still mostly overlap.) A compact object tends to amplify this effect as can be see from the results in Fig. 14 for all three objects at p¼ þ 20, showing that artefacts are reduced when the object structure becomes more open. Such artefacts do not impact significantly the

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Fig. 10. Row-spaced masks fabricated in stainless steel for (a) M 0 ¼ 4681, q ¼8, s ¼2, and (b) M 0 ¼ 1365, q¼ 4, s ¼ 6, showing some damage due to the energetic ions from the plasma focus.

Fig. 11. Autocorrelation function of a q ¼5, M 0 ¼ 3906 mask with different paddings (a) p¼ þ 1 and (b) p ¼ þ 20, compared with (c) a random 1=5 open mask, where positive values are black (), negative values are white (J).

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Fig. 12. SNR as a function of padding p for the q¼ 5, M 0 ¼ 3906 mask and for the random 1/5 open mask (RAN) from Fig. 11, for the three different test objects with increasingly open structure as shown inset.

detection of point sources, and random masks have indeed flown on several space missions [2,33], but in the mid-field they are distracting. Overall, these results suggest that a padding of up to 75 is always acceptable, and up to 710 for all but the most compact objects. Fortunately, this heuristic result nicely accommodates the data in Table 2, since most Singer sets factorise up to 9p9 r3 and all do by 9p9 r 8. Of course, none of the random masks which have been flown on space mission were single randomly chosen examples like the above. Instead, all were optimised in one of two ways, either by minimising the height of the highest peak in the autocorrelation side lobe, for the MPC on the Skylark rocket [34] and for the WXM on board the HETE satellite [35], or by ensuring that the different possible patterns of holes in each small patch or sub-aperture were themselves uniformly distributed across the aperture, for the balloon borne instruments SAGE [36] and BURST [37,38]. To achieve these optimisations a search heuristic was used, either a global search, generating a large number of differently chosen hole patterns and taking the best [35,36], or a local search, closing an open pixel somewhere and opening a closed pixel somewhere else, which effectively moves a hole from one place to the other, repeated iteratively [34,38]. Random masks have since been supplanted by genuine CDS based coded masks [2], starting with the 50% open fraction random masks [36,37] displaced by the earliest CDS based masks incorporating r ¼ 1=2 twin prime or pseudonoise sequences. The lower open fraction random masks survived longer, due to the difficulty in obtaining suitable CDS with r o1=2, but eventually these too were overtaken, the 33% open random mask on the WXM [35] by the 33% open triadic residue (approximate) CDS mask on the SAX-WFC [3], and the 25% open random mask design intended for the JEM-X instrument on the INTEGRAL satellite [36,38] by the 25% open fraction CDS mask based on biquadratic residues which was actually flown [39]. Now that we can generate Singer sets in a wide range of mask sizes and open fractions, random pattern masks would appear to be entirely redundant. The optimisation criteria used to produce random masks, however, are still potentially useful for improving the performance of pixel padded masks which, as we have seen, falls below that

Fig. 13. Sample reconstructed images for the q¼ 5, M 0 ¼ 3906 mask for different paddings (a) p¼ þ 5 and (b) p ¼ þ 10, compared with (c) the 1/5 open random mask from Fig. 11, showing the emergence of background artefacts.

of the non-padded, that is, perfect masks from which they were derived. In particular, by minimising the highest side lobe peak we might hope to redress the loss in overall SNR which we saw in Fig. 12, while ensuring uniformity of the distribution of different

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as combinatorial optimisation and operations research. Although it lies beyond the scope of this current paper, we feel confident that applying these modern local search heuristics could help improve the performance of pixel padded masks to something close to the performance of perfect CDS based masks.

4. Conclusion

Fig. 14. Sample reconstructed images for the q ¼5, M 0 ¼ 3906 mask with padding p ¼ þ20, for the same three test objects as shown in Fig. 12, showing the reduction in artefacts as the structure of the objects becomes more open.

sub-aperture hole patterns could reduce the incidence of the artefacts which we saw in Fig. 13. Unlike these optimisation criteria, which are still valid, local search heuristics have changed out of all recognition in the intervening years, with a whole spectrum of techniques such as simulated annealing, tabu search and even genetic algorithms being developed in fields entirely unrelated to coded aperture imaging, such

The mathematics of Singer sets may be hard, but the CDS property itself is easy to verify. This allowed us to take advantage of increases in computer speeds to generate Singer sets with fewer mathematical prerequisites. In particular, by searching through all possible feedback shift register coefficients, we generated primitive polynomials without needing depositories, hence avoiding their notational complications and limits on open fraction. Even the use of Galois fields might have been avoided by starting with the correct number of holes, randomly distributed, then displacing the holes using a local search heuristic to find the Singer set. Similar optimised random arrays (ORA) have been fabricated in the past [2,33], but do not seem to have taken full advantage of modern local search heuristics such as simulated annealing and tabu search. Also, for the open fractions r ¼ 1=3 and 1/4 which were their target, suitable Singer sets already exist. More useful would be to relax the restriction to inverse prime powers entirely, by starting with a random mask with arbitrary open fraction, then using local search to get as close as possible to the ideal CDS property. This could help fill in the gaps between r ¼ 1=2, 1/3, and 1/4, and at r ¼ 1=6, where no Singer sets exist. Coded apertures work best in the far-field (1 t m), that is, they image small objects more clearly than large objects since, for pffiffiffi pffiffiffiffiffi given n, as m decreases both ropt C 1= m and SNRopt ¼ ropt n increase. Row-spaced masks take advantage of this by slicing a large object into several smaller objects, each clearer than the original, and this improvement cancels out almost all the reduction in SNR due to the loss of flux caused by the blank rows. NTHT masks [6,7] could achieve a similar advantage, but spacing along both axes rather than just one effectively limits the spacing factor s to the perfect squares 1, 4, 9,y, and prevents us from taking advantage of any elongation in the object. Row-spaced masks are automatically self-supporting, so their holes can be expanded to entirely fill the pixel they are in, giving almost equal performance, but with substantially fewer holes, compared to unspaced nonself-supporting masks whose holes cannot be as large due to the presence of support structures. Micro-shuttered row-spaced masks are particularly attractive, since the presence of blank rows not only reduces the number of shutters in a systematic way, but also allows more space for the shutter mechanism, making them cheaper and simpler to fabricate. Although the Finger and Prince construction produces an imperfect rectangle, in practice the deviation from a perfect rectangle is slight, as can be seen from Fig. 7 and Table 2, and, using the map p : D-C, the computations are equally simple whether the rectangle is perfect or not. Nevertheless, the Finger and Prince construction generalises the Miyamoto wrapping, but to produce a perfect mosaic requires the PSW wrapping [19], so one might wonder whether the PSW wrapping can be generalised in a similar way. In a forthcoming paper we shall show how, by extending the notion of wrapping to all linear maps of the form pðx,yÞ ¼ ax þ by, we can generalise both the PSW and Miyamoto wrappings together into a single construction. Although all the factorisations shown in Table 2 are coprime, so that the PSW wrapping can be used, even non-coprime pairs do not give rectangles with much lower aspect ratios, so one might also wonder if there are any other shapes which perform better. Our forthcoming paper will show that working in the space of all linear maps p makes clearer how to wrap Singer sets not only as hexagons,

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which are already well known [21,22], but also other more exotic shapes which are equally compact and almost as symmetrical.

Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.nima.2013.01.032.

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