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Coded Aperture Imaging in High-Energy Astronomy banner

Coded aperture camera imaging concept

(c) Jean in 't Zand, 1992, 1996

This is a short review on coded aperture imaging. This paper appeared in a longer version in In 't Zand (1992). A postscript version of the complete paper (original version, without figures) is available (220 kB).

The text is regularly updated for new developments in the field. Last update: March 7, 1996. NEW!

Introduction: multiplexing techniques as alternative to focusing techniques

Focusing of high-energy radiation is sofar technically feasible only for photon energies up to about 10 keV through grazing incidence reflection (Aschenbach 1985). This method can provide a very good angular resolution, i.e. down to 0.5 arcsec which is the value proposed for AXAF (van Speybroeck 1987). The collecting area is optimized through the use of nested grazing incidence mirrors. The field of view (FOV) is limited by the grazing incidence to about 1 degree, but can be enlarged by using a special configuration of the mirrors ('Lobster-eye' telescopes, Aschenbach 1985). At higher energies than 10 keV, focusing is possible in a limited way (very narrow field of views and narrow passbands) through a Laue diffraction lens (Von Ballmoos & Smither 1994).

An alternative class of imaging techniques employs straight-line ray optics that offer the opportunity to image at higher photon energies and over larger FOV's. These techniques have one common signature: the direction of the incoming rays is, before detection, encoded; the image of the sky has to be reconstructed by decoding the observation afterwards. It is apparent that this method of producing sky images is a two-step procedure, in contrast to the direct or one-step imaging procedure of focusing techniques. These alternative techniques are referred to as multiplexing techniques. Another important difference in an astrophysical application between both types of imaging concepts is that in multiplexing techniques an imaged point source experiences the noise of all photons detected over the whole detector while in focusing techniques it are only the photons in a small part of the detector. Thus, for equal collecting areas the sensitivity of a focusing instrument is always better than of a multiplex instrument. Multiplexing techniques can be divided in two classes: those based on temporal and those on spatial multiplexing (Caroli et al. 1987). A straightforward example of temporal multiplexing is the scanning collimator: when the direction of a collimator is moved across a part of the sky which contains an X-ray point source, the number of counts per second that is detected as a function of time has a triangular shape. The position of the maximum of the triangle provides the position of the source along the scanning direction and the height of the triangle provides the flux of the source. A second scan along another direction completes the two-dimensional position determination of the source. More scans may be necessary if the source is extended or when there are more sources in the FOV of the collimator. The Large Area Counter (LAC) of the Japanese X-ray satellite Ginga (Makino et al. 1987 and Turner et al. 1989) is a recent example of an instrument employing a collimator. A more sophisticated device that is based on time multiplexing was introduced by Mertz (1968) and further developed by Schnopper et al. (1968): the rotation modulation collimator (RMC). RMCs are often used as all-sky monitor. Several RMCs have flown, for instance in Ariel-V (Sanford 1975), SAS-3 (Mayer 1972) and Hakucho (Kondo et al. 1981) and in several balloon experiments (see e.g. Theinhardt et al. 1984). The most recent example is the Granat observatory which carries 4 RMCs (Brandt et al. 1990). In its basic form an RMC has the disadvantage of being insensitive to short term fluctuations of X-ray intensity (with respect to the rotation period of the aperture), because the very same temporal information must be used for reconstructing the position of sources. However, techniques to circumvent this problem have been proposed (Lund 1985). Temporal multiplexing techniques in principle do not need a position-sensitive detector, contrary to spatial multiplexing techniques. Spatial multiplexing techniques can be divided in two subclasses: in the first subclass two or more collimator grids, widely separated, are placed in front of a detector, and in the second subclass one or more arrays of opaque and transparent elements are placed there. Instruments of the former class are called 'Fourier transform imagers' (Makishima et al. 1978 and Palmer & Prince 1987). These instruments record a number of components of the Fourier transform of the observed sky, and the observed sky can be reconstructed by an inverse Fourier transform in a way that is common to the 'CLEAN' algorithm in radio astronomy. Instruments of the second subclass are called 'coded-mask systems'. In the remainder of this text, a short review is given on the imaging concept of coded-mask systems, dealing separately with each important component of such a system. Requirements to arrive at an optimum imaging capability of the whole system are discussed.

Basic concept of coded-mask imaging

A coded-mask camera, to be used to image the sky for photon energies between E1 and E2, basically consists of:

a coded mask, i.e. a plate with areas that are transparent or opaque to photons with energies between E1 and E2. The transparent and opaque areas, or shortly 'mask elements', all have an equal size and are distributed in a pre-determined pattern, which is placed on a regular grid. The form of a mask element may be arbitrary.

a position-sensitive detector, whose spatial resolution is sufficiently matched to the mask-pattern grid size and which is sensitive to photons with energies between E1 and E2.

Figure: basic concept of coded-mask imaging. Two point sources illuminate a position-sensitive detector through a mask. The detector thus records two projections of the mask pattern. The shift of each projection encodes the position of the corresponding point source in the sky; the 'strength' of each projection encodes the intensity of the point source

The principle of the camera is straightforward: photons from a certain direction in the sky project the mask on the detector; this projection has the same coding as the mask pattern, but is shifted relative to the central position over a distance uniquely correspondent to the direction of the photons. The detector accumulates the sum of a number of shifted mask patterns. Each shift encodes the position and its strength encodes the intensity of the sky at that position. It is clear that each part of the detector may detect photons incident from any position within the observed sky. After a certain illumination period, the accumulated detector image may be decoded to a sky image by determining the strength of every possible shifted mask pattern. Proper performance of a coded-mask camera requires that every sky position is encoded on the detector in a unique way. This can be stated in terms of the autocorrelation function of the mask pattern: this should consist of a single peak and flat side-lobes (a delta function). This puts constraints on the type of mask pattern and on the way its (shifted) projections are detected. An important difference to direct-imaging systems is the fact that Poisson noise from any source in the observed sky is, in principle, induced at any other position in the reconstructed sky. The imaging quality of the camera is determined by the type of mask pattern, the optical design of the camera, the spatial response of the detector and the decoding (or reconstruction) method.

Mask patterns

In view of the imaging performance, one would want the mask pattern to satisfy the following conditions:

the autocorrelation function of the mask pattern should be a delta function;

the signal-to-noise ratio of a coded sky source should be optimum.

In the early days of multiplex imaging, two types of mask patterns were proposed: a pattern of Fresnel zones (Mertz & Young 1961) and the random pinhole pattern (Dicke 1968 and Ables 1968). While a camera with a Fresnel zone plate was never applied in extra-solar X-ray astronomy, concepts based on the random pinhole pattern have. The random pinhole pattern was proposed as an extension of a pinhole camera. The pinhole camera has ideal imaging properties with respect to the first condition (in fact it is a direct-imaging system), but delivers a poor signal-to-noise ratio, since the sensitive area is severely restricted by the size of the pinhole. The sensitivity may be increased by enlarging the pinhole, but at the expense of angular resolution. The idea behind the random pinhole camera is to increase the open area of the plate, while preserving the angular resolution, by placing many duplicate pinholes at random in the plate. The random character is necessary to meet the first condition as closely as possible.

Both Fresnel zone and random pinhole mask patterns are not ideal with respect to the first condition, the patterns possess autocorrelation functions whose sidelobes are not perfectly flat. Later work concentrated on finding patterns, based on the idea of the random pinhole pattern, that do have flat side-lobes. Ideal patterns were found that are based on cyclic difference sets (Gunson & Polychronopulos 1976, Fenimore & Cannon 1978).

A cyclic difference set D, characterized by the parameters n, k and z, is a collection of k integer numbers {I1, I2,...,Ik} with values Ii between 0 and n such that for any J=/0 (mod n) the congruence Ii-Ij=J (mod n) has exactly z solution pairs (Ii,Ij) within D (Baumert 1971). An example of a cyclic difference set D with n=7, k=4 and z=2 is the collection {0,1,2,4}. Cyclic difference sets can be represented by a binary sequence a_i (i=0,...,n-1) with a_i=1 if i is a member of D and a_i=0 otherwise. In the above example a_i is given by 1110100. a_i in turn can stand for the discretized mask pattern, assigning a transparent element to a_i=1 and an opaque one to a_i=0. The cyclic autocorrelation c_l of a_i is (Baumert 1971):

i.e. a single peak on a flat background. A mask pattern based on a_i consequently satisfies condition 1. a_i has the characteristic that every difference i-j between a pair of a_i,a_j=1 is equally sampled and therefore these arrays are also called Uniformly Redundant Arrays (URA, Fenimore & Cannon 1978).

From the autocorrelation it can be anticipated that it is advantageous with respect to condition 2 to have a difference between k and z that is as large as possible, for k determines the signal and z the background level (and its noise) (note: the argument followed here to meet condition 2 is simplified. In fact, the optimum open fraction of the mask pattern is also dependent on specific conditions concerning the observed sky. See e.g. Skinner 1984 and In 't Zand, Heise & Jager 1994). The maximum difference is reached if n=4t-1, k=2t-1 and z=t-1 if t is integer. These cyclic difference sets are called Hadamard difference sets (Hall 1967 and Baumert 1971) and can be classified in at least three types, according to the value of n:

Quadratic residue sets: n is prime (the members of this set are given by the squares, modulo n, of the first (n+1)/2 integers);

Twin prime sets: n=p(p+2) for integer p, p and p+2 being prime;

Pseudo-noise sets: n=2^m-1 (m>1 is integer).

Some Hadamard difference sets may belong to more than one class, the existence of a set with a value for n given by a class is guaranteed. The above example is a quadratic residue set as well as a pseudo-noise set. Characteristic for Hadamard sets is that k=(n-1)/2, i.e. for large n the mask pattern is about half open. The cyclic autocorrelation then is:

Another collection of cyclic difference sets are the so-called Singer sets, that are characterized by n=(t^{m+1}-1)/(t-1), k=(t^m-1)/(t-1) and z=(t^{m-1}-1)/(t-1), where t is a prime power. The equivalent mask pattern will have smaller open fractions than those based on Hadamard sets; for t>>1 the open fraction approximates 1/t.

A way to construct a pseudo-noise Hadamard set is the following (Peterson 1961): if p(0),...,p(m-1) are the factors of an irreducible polynomial of order m (p(i) is 0 or 1) then a_i is defined by a shift register algorithm:
a_{i+m} = sum over j from 0 to m-1 p(j) a_{i+j} (i = 0,..., 2^m-2-m) (mod 2)
The first m values of this recursive relation, a_0,...,a_{m-1}, can be chosen arbitrarily: a different choice merely results in a cyclic shift of a_i.

If n can be factorized in a product of two integers (n=p X q), it is possible to construct a two-dimensional array a_{i,j} (i=0,...,p-1; j=0,...,q-1) from the URA a_i (i=0,...n-1). The mask pattern thus arranged is called the 'basic pattern'. The ordering of a_i in two dimensions should be such, that the autocorrelation characteristic is preserved. This means that in a suitable extension of the basic p X q pattern, any p X q section should be orthogonal to any other p X q section. A characteristic of a URA a_i is that any array a_i^s, formed from a_i by applying a cyclic shift to its elements (a_i^s = a_{mod(i+s,n)}), is again a URA which is orthogonal to a_i. Therefore, the autocorrelation characteristic of the expanded a_{i,j} is fulfilled if every p X q section is a cyclic shift of the basic pattern. Two examples of valid ordering methods are shown in the following figure:

ordering by rows (Miyamoto 1977) and ordering along extended diagonals (Proctor et al. 1979). In the latter case p and q should not have a common divisor (they should be mutually prime), otherwise one is not able to fill the basic pattern completely. Other valid orderings are possible (Heise 1992, personal communication).

The pseudo-noise arrays have the convenient property that they can easily be wrapped in almost a square of n>>1: if m is even, n can be written as n=2^m-1=(2^{m/2}}-1)(2^{m/2+1), so that p and q only differ by 2.

Several practical problems arise in the manufacturing of a two-dimensional mask plate. One in the X-ray regime is that an opaque mask element may be completely surrounded by transparent elements. In the X-ray regime it is necessary to keep transparent elements completely open, because the use of any support material at open mask elements soon results in too much attenuation of flux. Thus, an isolated opaque mask element will not have any support. Two methods may be applied to solve this problem:

Choose a mask pattern where no isolated opaque elements occur. Such patterns exist and are called 'self-supporting'.

Include a support grid in the mask plate.

Generally the second option is chosen, because every opaque element is then supported completely along its sides while in the first option opaque elements might still be only weakly supported at some of its corners. This does not affect the ideal autocorrelation characteristic, because the pattern is not altered and remains two-valued. However, the open area will be decreased somewhat, resulting in less sensitive area.

At photon energies above 10 keV this issue of support is less constraining because at these energies transparant materials can easily be found that support opaque mask elements from above or below instead of from the sides.

Another practical problem of masks occurs in applications beyond a few hundred keVs: the opaque elements generally need to be very thick, in the order of centimeters instead of 100s of microns. This means the mask element sizes cannot be smaller than that because otherwise the mask itself would act as a narrow-field collimator.

The autocorrelation characteristic remains valid only if the coding is performed by the use of a complete cycle of a basic pattern. As soon as the coding is partial, systematic noise will emerge in the side-lobes of the autocorrelation function. This noise can be interpreted as false peaks and thus deteriorates the imaging quality. In order to be able to record for every position in the observed sky a full basic pattern, one needs a special optical configuration of mask and detector (see next section). Sometimes also a mask is needed that consists of more than 1 basic pattern. How such a mosaic mask is constructed has been discussed above.

Recent developments in mask design seem to concentrate on the introduction of two-scaled mask patterns (Skinner & Grindlay 1992) and masks with open fractions of less than 50% (In 't Zand, Heise & Jager 1994). A two-scaled mask has 2 potential advantages: such a mask might increase the passband where it can be applied and it might enable a two-stepped CPU-efficient search for transient events (first, searching at a rough resolution to limit the field-of-view to search in and, second, locate the event accurately). A low open-fraction mask employs early suggestions to optimize the signal-to-noise ratio of point sources, and at the same time limit the telemetry rate. Both two-scaled and low-open-fraction masks bring along one problem which has not been satisfactorily solved yet: no such patterns have been found with ideal autocorrelation functions (with a few exceptions at particular open fractions). In certain applications this is less of a problem, and validates even the return of the random patterns (In 't Zand et al. 1995).

Optical design

The optical design of a coded-mask camera is defined by the sizes of the mask, the mask elements and the detector, the number of basic patterns used in the mask, the distance between mask and detector and the size and place of an optional collimator. Apart from the imaging quality, the design determines the angular resolution and the FOV (the latter is usually expressed in the full-width at half maximum, FWHM, of the collecting area across the observed sky).

Figure: schematic drawings of the two types of 'optimum' configurations discussed in the text. The left configuration is called 'cyclic'. Note the collimator, placed on top of the detector, necessary to confine the FOV to that part of the sky in which every position will be coded by one full basic pattern. From Hammersley (1986)

Figure: Schematic drawing of the 'simple' configuration. The sizes of the mask and detector are equal. Note that instead of a collimator, as in the optimum configurations, a shielding is used. The shielding prevents photons not modulated by the mask pattern to reach the detector. From Hammersley et al. (1992)

Optimum and simple configurations

As concluded above, for ideal imaging properties it is necessary to record for every position in the observed sky a complete cycle of the basic pattern. This can be accomplished by configuring the mask and detector in one of the following two ways (note: other configurations can be thought of (Proctor et al. 1979) that are extensions of the two mentioned here).

The mask consists of one p X q basic pattern, while the detector has a size of (2p-1) X (2q-1) mask elements. By the implementation of a collimator in front of the detector, the observed sky is restricted to those positions in the sky from where the mask is completely projected on the detector.

The mask consists of a mosaic of almost 2 X 2 cycles of the basic pattern ((2p-1) X (2q-1) mask elements), while the detector is as large as one basic pattern. A collimator is implemented, to restrict the observed sky, as seen from any position on the detector, to that part of the sky seen through one complete cycle of the basic pattern. This type of configuration often is called 'cyclic' because of the nature of the mask.

In both types the collimator gives a pyramidical-to-zero response function to the collecting detector area over the observed sky and its use may be coupled to the need for a support structure for the entrance window of the detector (Gunson & Polychronopulos 1976). In practice the choice between both types may depend on which of the components is restricted most in size by exterior conditions: the mask or the detector. However, in astronomical applications, the argument of preservation of the collecting area prevails and the choice is usually for the cyclic type.

The above types of mask/detector configurations are called 'optimum systems' (Proctor et al. 1979) in the sense that the imaging property is optimum. An alternative configuration is the 'simple' or 'box-type' system. In this system the need for full coding is relaxed. The detector has the same size as the mask, which consists of one basic pattern. No collimator is then needed on the detector; instead a shielding is used to prevent photons that do not pass the mask from entering the detector. In a simple system only the on-axis position is coded with the full basic pattern, the remainder of the FOV is partially coded. Obviously, the off-axis sources will cause false peaks in the reconstruction. However, as will be discussed later on, this coding noise can be eliminated to a large extent in the data-processing, provided not too many sources are contained in the observed part of the sky.

If one assumes for the moment that coding noise is not relevant, the question arises how the simple system compares to the cyclic system. In order to do this comparison, it seems fair to impose on both systems the same FOV and sensitivity. This means that both have a detector of equal size, but in the cyclic system the 2 X 2 mosaic mask is two times closer to the detector than in the simple system, with an appropriate adjustment of the collimator's dimensions. Therefore, the angular resolution in the cyclic system is two times worse in each dimension than in case of the simple system. Most important in the comparison is the following difference between the cyclic and the simple system, concerning the reconstruction of the flux from an arbitrary direction within the observed sky: in the cyclic system all detected photons on the complete detector may potentially come from that direction, while in the simple system only photons from the section of the detector not obscured by the shielding are relevant. Therefore, Poisson noise will affect the reconstruction in the cyclic system stronger than in the simple system. Thus, regarding the Poisson noise, the simple system is superior in sensitivity to the cyclic system (except for the on-axis position where both systems have equal properties). This conclusion is in agreement with the findings of Sims et al. (1980), who have studied the performance of both systems via computer simulations.

Reconstruction methods

Coded-mask imaging is basically a two-step procedure. After the accumulation of spatially coded detector data, the second step involves the decoding of this data, in other words the reconstruction of the observed part of the sky. Since a powerful computer is needed for the reconstruction process, this is usually done off-line (particularly if the number of mask elements is large). Several reconstruction algorithms are in use. The choice for a certain algorithm depends on the specific aim (e.g. search for detections of unexpected events or timing analysis of a restricted part of the sky), the available computer resources and the type of instrument configuration. Several types of algorithms may in fact be subsequently used on the same set of detector data. This section gives a short resume of various algorithms. For clarity the discussion is illustrated with one-dimensional examples and space is discretized in steps of a size equal to that of a mask element; the conclusions do not basically differ for two dimensions and smaller steps. The basic problem to be solved concerns the following. Let the one-dimensional vector d describe the detector (in units of counts per detector-element area), s the sky (in counts per mask element area) and b the detector background (this includes all flux which is not modulated by the mask pattern, in counts per detector element area). Suppose the detector elements are as large as a mask element. The detection process can then be described by:
d = C s + b
where C is a matrix whose rows contain cyclic shifts of the basic mask pattern. An element of C is 1 if it corresponds to an open mask element and 0 otherwise. In the case of an optimum system with a basic mask pattern of n elements, d, s and b contain n elements and C n X n elements. In the case of a simple system with a mask of n elements, d and b contain n elements, the sky vector s contains 2n-1 elements and the matrix C contains (2n-1) X n elements. The problem to be solved is to reconstruct s out of this set of linear equations. Although b is unknown also, one is in principle not interested in it. An approximation for b is: b is homogeneous over the detector plane, i.e.
b = b i
where i is the unity vector (consisting of only ones), b is now a single unknown scalar.

Inversion

This is a straightforward method to find s: multiply d with the inverse of C, C^{-1}, yielding:
C^{-1} d = s + b C^{-1} i
Obviously C should be non-singular. This is not the case for a simple system, since C is not square. However, even if C is non-singular, the possibility exists that entries of C^{-1} are so large that the term bC^{-1}i dominates the reconstruction. This will happen if rows of C are almost dependent on other rows, meaning that change of a few elements in a row might make it linearly dependent on other rows. C is then said to be 'ill-conditioned'. Fenimore and Cannon (1978) have shown that this is quite a common feature, especially for random mask patterns. This makes inversion an unfavorable reconstruction method.

Cross correlation

This is another obvious method for the reconstruction, cross correlating the detector image d with the mask pattern via a multiplication with a matrix. The mask pattern may be given by C, but in practice a modified matrix is used: the so-called reconstruction matrix M. M is constructed in such a way, that Md evaluates directly s and cancels contributions from b. Fenimore and Cannon (1978) introduced this method and called it 'balanced cross correlation'. Specifically, M is defined as:

M=(1+P) C^T-P U

where C^T is the transposed of C and U is the unity matrix (consisting of only 1's and having the same dimensions as C^T). P is a constant and is determined from an analysis of the predicted cross correlation value: a prediction of the reconstruction can be easily evaluated if one assumes that the mask pattern is based on a cyclic difference set and the camera configuration is optimum. The expected value of the cross correlation is then (using the autocorrelation value for cyclic difference sets):

Md = M (Cs + bi) = (1+P )(k-z) s + [ { (1+P )z - P k} sum s_i + { (1+P )k - P n} b ]i

Apart from a scaled value of s, which is the desired answer, the result also includes a bias term (the i-term). It is not possible to eliminate this bias by a single value of P. Rather, the sum s_i factor or the b factor can be canceled separately. Canceling the sum s_i factor involves a value P of

P_1 = z/(k-z) = (k-1)/(n-k),

canceling the b-factor involves P to be (note: An interesting characteristic of the reconstructed sky is the sum of the reconstructed values. In case P =P_1 this is sum Md = k sum s_i + nb, i.e. the sum of all detected counts. If P =P_2 this sum is equal to 0.):

P_2 = k/(n-k)

Since k/n is the open fraction, t, of the mask pattern, P_2 can be written as:

P_2 = t/(1-t).

P_1 approximates P_2 if k>>1. The reconstruction value then reduces to:

Md = ks.

Normalizing M results in:

M/k=n/k(n-k)X(C^T-k/nXU.

In the case of a simple system, this would also apply if the 'hard' zeros in C (i.e. zeros that do not arise from zero a_i-values, were replaced by cyclically shifted a_i's. The result would then imply: (Md)_i=(Md)_(mod(i+n,2n-1)); this is a consequence of the fact that 2n-1 unknowns are to be determined from a set of only n linear equations, which is under-determined in general. In this formulation a source at position i causes a false peak of the same strength at position mod(i+n,2n-1). If the 'hard' zeros in C are not replaced, this does not apply directly, but an interdependence in the solution for the reconstruction remains. This interdependence is not so strong: one real peak will cause many small ghost peaks rather than a single false peak which is just as strong as the real peak. It is then possible to find a unique solution for s as long as it does not have more non-zero values than n. This search is accomplished by testing reconstructed peaks on their authenticity, the reconstruction process is then necessarily iterative.

Photon tagging (or URA-tagging)

This reconstruction method is very similar to cross correlation and has been introduced by Fenimore (1987). It involves back-projecting every detected photon through the mask, towards a particular (possibly all) position in the sky field from which it could originate. If a closed mask element is 'encountered' in the back-projection, the photon is accumulated in the background contribution. If it 'encounters' an open mask element, it is accumulated in the source contribution for that position in the sky. Once all photons have been processed, the subtraction of the background from the source contributions (after proper normalization) completes the reconstruction. It is clear that this method is advantageous, in terms of computation time relative to a cross correlation, if the number of detected photons is very small with respect to the number of mask elements. The advantage may also hold if only a restricted part of the observed sky needs to be reconstructed. This latter situation may be applicable if all point sources in the sky have already been found in an analysis of the same data (e.g. by a total cross correlation) and one intends to analyze each point source in more detail, i.e. extract spectra and/or lightcurves. Skinner & Nottingham (1993) improved the method by extending it, taking into account imperfections of the detector (limited spatial resolution, 'dead spots' etc.), the support grid of the mask plate and a telescope motion.

Wiener filtering

Sims et al. (1980), Sims (1981) and Willingale et al. (1984) introduced Wiener filtering for use as a reconstruction method of coded-mask-system images. This filtering can be regarded as a weighted cross correlation (Sims 1981), weighing the Fourier transform components of the detector image with the inverse power density of the mask pattern. Thus, fluctuations in the modulation transfer function (note: The modulation transfer function is defined as the square root of the Fourier power spectrum) of the mask pattern are smoothed. If S/N(omega) is the ratio of the signal power density (due to spatial fluctuations by sky sources) to the noise power density at spatial frequency omega, C(omega) the Fourier transform of the mask pattern, the Wiener filter W(omega) is defined as:

W(omega) = C(omega) / ( |C(omega)|^2 + S/N(omega)^{-1} )

Because S/N is not known before the reconstruction is completed, a frequency-independent expression is used for it.

It is clear that Wiener filtering is especially helpful if the mask pattern is not ideal, which is the case for random and Fresnel zone patterns. However, ideal patterns such as those based on cyclic difference sets are characterized by flat modulation transfer functions (all spatial frequencies are equally present for URA-patterns, which is apparent from the definition of URAs. Sims et al. (1980) confirmed this via computer simulations and found this also to be the case if an ideal pattern is used in partial coding, such as in a simple system.

Iterative methods, Maximum Entropy Method

A separate class of reconstruction methods are formed by the iterative methods. These try to solve the sky vector s by an iterative search for the solution that is most consistent with the detector data. Three of such methods have been investigated for use in coded-mask imaging.

One iterative method is the maximum entropy method (MEM). MEM has gained widespread favor in different areas as a tool to restore degraded data. Introductions to the theory behind MEM as applied to image restoration can be found in Frieden (1972) and Daniell (1984), while a review is given by Narayan & Nityananda (1986). Examples of applications of MEM are given by Gull and Daniell (1978), Bryan and Skilling (1980) and Willingale (1981), while the application specifically to images from coded-mask systems are described by Sims et al. (1980) and Willingale et al. (1984). MEM was introduced in the field of coded-mask imaging by Willingale (1979). Despite the good results that can be obtained with this method, a major drawback is the large amount of computer effort required, as compared to linear methods such as cross correlation.

Another iterative method is iterative removal of sources (IROS). IROS is in fact an extension of the cross correlation method and was introduced by Hammersley (1986) as a procedure to eliminate problems due to incomplete coding (also called 'missing data') in simple systems. The advantage of IROS is that it is much more CPU efficient than MEM. The principle of IROS is as follows: 1) do a cross correlation; 2) find the strongest point source; 3) subtract the expected detector exposure by this point source from the observed detector; 4) goto 1 or, if there is no point source left, put point sources back into last cross correlation image. This procedure will ensure that any coding noise due to point sources is suppressed to a level below the statistical noise and thus ensures an unbiased determination of point source intensities and positions. One can enhance the CPU efficiency by dealing with more than 1 point source in each iteration (in a smart way which is not discussed here).

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These pages have been compiled by Jean in 't Zand. They are intended to provide general information for those interested in coded aperture imaging. Any citations should reference original papers as noted in the bibliography, and requests for further information about any of the papers should be directed to the authors thereof.

March 7, 1996

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