Near-field artifact reduction in planar coded aperture imaging Roberto Accorsi and Richard C. Lanza
Coded apertures for imaging problems are typically based on arrays having perfect cross-correlation properties. These arrays, however, guarantee a perfect point-spread function in far-field applications only. When these arrays are used in the near-field, artifacts arise. We present a mathematical derivation capable of predicting the shape of such artifacts. The theory shows that methods used in the past to compensate for the effects of background nonuniformities in far-field problems are also effective in reducing near-field artifacts. The case study of a nuclear medicine problem is presented to show good agreement of simulation and experimental results with mathematical predictions. © 2001 Optical Society of America OCIS codes: 080.0080, 110.0110, 110.1220.
1. Introduction
The most common application of coded aperture imaging 共for review articles, see Refs. 1 and 2兲 is in far-field applications, the best example of which is the imaging of high-energy x- and gamma-ray sources in astronomy.3,4 The technique has also been applied to problems in fusion5,6 and nuclear medicine.7–9 The mathematics of coded aperture imaging shows that apertures with perfect correlation properties provide ideal point-spread functions 共e.g., Ref. 10兲, but this is true only in far-field cases. In near-field geometry 共i.e., when rays coming from the same point source in the object cannot be considered parallel兲 many factors potentially cause deviation from ideality. The most serious factors are the finite thickness of the object and the modulation of the intensity of the projection of the aperture on the detector because of the varying incidence angle of gamma-rays. Thick objects cause artifacts because a one-view coded aperture camera can focus on only one plane of the object at a time: All other planes are out of focus and contribute artifacts to the image.10 In this paper we concentrate on artifacts that are caused by nearfield geometry in planar imaging in which artifacts from object thickness are not a concern. Planar stud-
ies, however, are not limited to somewhat academic examples such as point or line sources and thin test objects, but also include the important case of threedimensional objects of thickness of the order of the depth of focus of the instrument. In studies such as in molecular imaging, this is the case for small animals 共typically mice兲, which are not thicker than the depth of focus of a near-field coded aperture camera that we built 共approximately 1 cm兲— even when magnification is high enough to achieve high resolution—and can be considered two-dimensional 共2-D兲 objects. The variation of the incidence angle adds an intensity modulation to the projection of the aperture. The result is that classic deconvolution techniques11 produce near-field images with significant artifacts. To overcome this problem, more advanced deconvolution algorithms have been suggested.12–14 In this paper we present our results of applying to near-field problems a technique originally developed to mitigate the effects of a nonuniform background in far-field applications.15–17 Unlike other artifact reduction techniques, the method is implemented in hardware, does not rely on heavy computation, and does not require any significant extension of the time needed to produce an image.18 2. Mathematical Formulation of Coded Aperture Imaging
The authors are with the Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. The e-mail address for R. C. Lanza is
[email protected]. Received 1 December 2000; revised manuscript received 16 May 2001. 0003-6935兾01兾264697-09$15.00兾0 © 2001 Optical Society of America
From geometrical optics, in a coded aperture experiment the number of counts recorded at the detector position ri is given by P共ri兲 ⬀
兰兰
ro
O共ro兲 A
冉
冊
b a ri ⫹ ro cos3共兲d2ro, z z
10 September 2001 兾 Vol. 40, No. 26 兾 APPLIED OPTICS
(1)
4697
a convolution because of the near-field term cos3共兲. The effect of the near field is not a modulation of the object by a constant factor cos2共0兲, as is the case for off-axis sources in far-field problems.19 From coded aperture theory we know that the reˆ is obtained from P by periodic constructed image O correlation with the decoding pattern G⬘ associated with the mask A 共Ref. 10兲: ˆ ⫽ P 丢 G⬘, O
(4)
where the prime indicates that G was scaled accordingly to A⬘. If no assumptions are made, this form is not analytically tractable. A.
If the object is far from the detector so that 兩ri ⫺ ro兩 ⬍⬍ z, @共ri, ro兲, then
Fig. 1. Coded aperture geometry.
cos3共兲 ⬵ 1.
where ⫽ arctan共兩ri ⫺ ro兩兾z兲, O is the activity distribution on the object plane, A is the transmission of the aperture 共at the mask plane兲, and all other symbols are explained in Fig. 1. characterizes the trajectories of skew rays through the system: It is the origin of the difference between the near-field and the far-field cases. If we define ⫽⫺
冉 冊 冉 冊
A⬘共r兲 ⫽ A
P ⫽ O⬘ ⴱ A⬘,
ˆ ⫽ 共O⬘ ⴱ A⬘兲 丢 G⬘ ⫽ ᑬ关O⬘ ⴱ 共G⬘ 丢 A⬘兲兴 O
(2)
⫽ ᑬ共O⬘ ⴱ ␦兲 ⫽ ᑬ共O⬘兲,
兰兰
O⬘共兲 A⬘共ri ⫺ 兲
冤
冏冢
⫻ cos arctan 3
ri ⫹
冏 冣冥
a b
z
d2 .
B.
(3)
冤
cos3 arctan
4698
ri ⫹ z
a b
冏 冣冥
冦
冋 冉 冊册 冉
⬵ cos3 arctan
兩ri兩 z
1⫺
Near-Field Case
In some applications it is imperative to collect data as close to the source as possible. For example, in nuclear medicine, sensitivity must be maximized to keep the dose to the patient as low as reasonably achievable. Hence, in such applications, 兩ri ⫺ ro兩 is often comparable to z and relation 共5兲 does not hold. To make the problem still mathematically treatable we expanded the near-field term of relation 共3兲 in a Taylor series to the second order with center ri. The result is
Here O⬘ and A⬘ are, respectively, a rescaled and reflected form of the object and a rescaled version of the mask pattern. The scaling coefficient for O⬘ is the magnification of a pinhole camera with the same object-to-mask and mask-to-detector distances 共the negative sign indicates that the object is inverted兲, whereas the scaling coefficient for A⬘ is the ratio of the size of the mask to the size of its projection on the detector. Note that relation 共3兲 is not in the form of
冢冏
(7)
where ᑬ is the reflection operator. The reconstructed object is the object itself apart from a rescaling constant. However, if the condition in relation 共5兲 does not hold, relation 共3兲 does not assume the simplified form of Eq. 共6兲, and we do not reach this result; the image is corrupted in some way.
we obtain the form P共ri兲 ⬀
(6)
which is the same as the correlation of the rescaled 共but not reflected兲 object with the rescaled mask. This is the starting point of all literature concerned with finding the optimal decoding pattern G such that the reconstruction of the object is perfect. For ideal pairs 共 A, G兲, i.e., pairs such that A R G ⫽ ␦, Eq. 共4兲 becomes
a r , b
a r , z
(5)
This is the far-field approximation. Under this hypothesis, relation 共3兲 is reduced to the convolution
b ro, a
O⬘共r兲 ⫽ O ⫺
Far-Field Approximation
3兾z 2 兩ri兩 2 1⫹ 2 z
APPLIED OPTICS 兾 Vol. 40, No. 26 兾 10 September 2001
冊冤
ri ⴰ
a 1a 2 ⫹ 兩兩 ⫺ b 2b
冉
5兾2z 2 兩ri兩 2 1⫹ 2 z
冊冉
ri ⴰ
冊
a b
2
冥冧
,
(8)
where ⴰ indicates a scalar product. This expansion is more accurate the larger the margin by which the condition
冏 冏
a 兩ro兩 b ⫽ ⬍1 兩ri兩 兩ri兩
(9)
is true. When high resolution is sought, magnification tends to be high, so a兾b is generally small. Another way of looking at the same condition is to recognize that ro is a variable spanning the object. Because of high magnification, if the object fits in the field of view 共FOV兲, it is typically much smaller than the detector and ro ⬍ ri, which makes the Taylor approximation a good one over most of the detector. Indeed, in our applications, stopping at second order was sufficient to explain the artifacts we were seeing. Note that high magnification is a sufficient condition only for Eq. 共9兲, which can be satisfied in many other cases. Relation 共8兲 breaks the near-field term cos3共兲 into the sum of a zero, with a first- and second-order contribution; Eq. 共4兲 is decomposed to a sum of parts that can be examined one at a time. 3. Artifact Prediction
In the following subsections we attempt to predict the shape of near-field artifacts in the reconstructed images. Examples are taken from a high-resolution study of a thyroid phantom 共see Section 5兲. Images were taken for an object-to-detector distance of 40 cm and a magnification factor z兾a ⬵ 4.3. The mask used was a 62 ⫻ 62 pixel no-two-holes-touching 共NTHT兲 array11 based on a 31 ⫻ 31 pixel modified uniformly redundant array 共MURA兲20 and was made of 1.5-mm-thick tungsten. Its pinholes were 1.1 mm wide. A.
Zero-Order Correction
冤 冢冏 冏冣冥 冋 冉 冊册
cos3 arctan
⬵ cos3 arctan
兩ri兩 z
,
which does not depend on ; substituted into relation 共3兲, we obtain
冋 冉 冊册 兰兰
兩ri兩 P共ri兲 ⫽ cos arctan z
B. First Order: Object
Centering the Mask Pattern and the
From relation 共8兲 the expression for the first-order term is
冤
冏冢
cos arctan
ri ⫹ z
O⬘共兲 A⬘共ri ⫺ 兲d . 2
(11) Because P is not the convolution of O⬘ and A⬘, correlation with G⬘ will not produce the object. However, the near-field effect is reduced to a prefactor depending on only the detector coordinate ri. In other words, in this approximation the artifacts are
冏 冣冥冨
a b
冋 冉 冊册 冉
⬵ ⫺cos3 arctan
(10)
3
the same as if the object were concentrated at the center of the FOV. Thus it is easy to correct zeroorder artifacts exactly when we divide the projection data P by this prefactor. The projection is now a convolution, and we are reduced to the far-field case of Section 2.A. All our decoding programs include this correction. The effects can be seen in Fig. 2. Even if some improvement is achieved, the artifacts, the bright bows that can be seen at the top and bottom of Fig. 2, are hardly eliminated. Higher-order terms are not negligible and must be analyzed. Also note that the noise level of the images in Fig. 2 is somewhat increased. By dividing by a constant we are artificially increasing the number of counts of side pixels. The average value is restored, but the variance remains that of a lower number of counts.
3
If the expansion is stopped at zero order, from relation 共8兲 one obtains a ri ⫹ b z
Fig. 2. Effect of zero-order correction. The white rims surrounding the cold spots and the right lobe of the thyroid are not artifacts but part of the simulated object.
兩ri兩 z
I
3兾z 2 兩ri兩 2 1⫹ 2 z
冉冊
ri ⴰ
冊
a , b
(12)
where the bar at the left-hand side indicates that this expression includes the first-order term only. Substitution into relation 共3兲 leads, after zero-order correction, to P共ri兲 ⬀
ri ⴰ z ⫹ 兩ri兩 2 2
兰兰
O⬘共兲 A⬘共ri ⫺ 兲d2.
(13)
To reach a useful interpretation, it is important to bear in mind that A⬘ is a function describing the aperture. Typical coded apertures are binary, i.e., they can assume only two values, 0 and 1, indicating, respectively, the closed and open positions of the mask. In light of this, one can recognize that the integrand is the center of mass of the object “cut” by 10 September 2001 兾 Vol. 40, No. 26 兾 APPLIED OPTICS
4699
A⬘. Convolution makes the result a function of the shift of A⬘. Now, if A⬘ covers the FOV uniformly and the object is also reasonably uniform, the result is not a strong function of shift and gives an approximately constant contribution. Numerically we verified that, in our case study, the integral in relation 共13兲 gives P only a small modulation of the main structure coming from the first factor. Consequently, we replace the integral with a constant vector Oⴕ. The result can then be substituted into Eq. 共4兲. Because Oⴕ is constant, it can be taken out of the correlation integral. The result is ˆ ⬀ Oⴕ ⴰ O
兰兰
ri
ri G⬘共ri ⫹ 兲d2ri. z ⫹ 兩ri兩 2 2
(14)
The important consequence here is that, under the above-mentioned hypotheses, the shape of first-order artifacts depends on G only. This noted, we proceed to make a further approximation with the only scope of discussing this same result in more intuitive terms, but the following observations could have been referred to this more involved form as well. In our case study, ri spans at most a 38.6 cm ⫻ 38.6 cm area and z is 40 cm. For these values, substitution of the fraction in the integral with ri兾z2 is accurate within 28% at the worst points. We obtain ˆ ⬀ Oⴕ ⴰ O
兰兰
ri G⬘共ri ⫹ 兲d2ri.
(15)
ri
Both factors are important in the discussion of first-order artifacts. The integral, just like relation 共13兲 for O⬘, gives the position of the center of mass of the open positions of the decoding pattern G as a function of the decoding position 共which is the same as the shift of G兲. Therefore the object reconstrucˆ involves an additive term depending on the tion O particular form of G. The case of a MURA pattern20 is shown in Fig. 3. The image at the bottom left shows that this function has a sudden drop at its center, resulting in a vertical and a horizontal line at the center of the reconstruction. If G 共and thus the mask pattern A兲 is not taken as generated by the rules found in the literature but is shifted by half a period in both directions, these drops can be moved to the borders of the image, which removes the most unpleasant part of the artifact. This is what we mean by pattern centering. The case of other patterns 共a 31 ⫻ 33 pixel M sequence,19 a 33 ⫻ 33 pixel product array,21,22 and a NTHT MURA11兲 is shown in Fig. 4. Centering of the pattern was first indicated by Fenimore, with specific reference to the case of the M sequences, as an improvement for effects that are due to “distortions in the mask or nonuniformities in the detector” in 2-D far-field imaging.23 The first factor in relation 共15兲, the center of mass of the object, can be used to remove this artifact completely. In fact, if the object were centered on the FOV, Oⴕ would be zero, canceling the term. From a practical point of view one can take a first, raw image 4700
APPLIED OPTICS 兾 Vol. 40, No. 26 兾 10 September 2001
Fig. 3. Center of mass of the decoding pattern G as a function of a shift for a MURA. The expected form of the artifact is at the bottom left. Cross sections of this function are shown on its top and right. If a shift different from that in which the arrays are given by generation rules had been used, the result would have been that at the top right. This shift corresponds to solid black row and the solid white column of a MURA pattern being put at the center of the mask.
to estimate Oⴕ and then make the necessary adjustments before taking a second picture. However, we show that this is not necessary. C.
Second Order:
Mask and Antimask
If the object is centered on the FOV, first-order artifacts disappear, but second-order artifacts, normally hidden by the stronger lower-order ones, become vis-
Fig. 4. First-order artifacts for three different array families. Top, 31 ⫻ 33 pixel M sequence; middle, 33 ⫻ 33 pixel pseudonoise product; bottom, 34 ⫻ 34 pixel NTHT MURA. Masks and decoding arrays are shown in the shift that mitigates the artifact. Noncorrected artifacts are shown for the pattern as obtained from the generation rule found in the literature. The correction is substantial for M sequences and NTHT patterns, whereas it is of dubious effectiveness for product arrays.
ible. This must be the case in Fig. 2 because the terms so far analyzed still do not explain the bows corrupting the image. The starting point is the substitution into relation 共3兲 of the second-order terms of relation 共8兲:
冢冏
冤
ri ⫹
cos3 arctan
冏 冣冥冨
a b
z
冋 冉 冊册 冉 冊 冤 冉 冊冉 冊冥
兩ri兩 ⬵ cos3 arctan z
⫻
II
3兾z 2 兩ri兩2 1⫹ 2 z
5兾2z 2 兩ri兩2 1⫹ 2 z
1a 2 兩兩 ⫺ 2b
ri ⴰ
a b
2
.
(16)
This time we have to consider two terms. The first is P共ri兲 ⬀
1 z ⫹ 兩ri兩 2 2
兰兰
O⬘共兲 A⬘共ri ⫺ 兲兩兩 2d2.
(17)
The integrand is the second moment of inertia of O⬘, cut by A⬘, with respect to an axis perpendicular to the object plane and passing through its center. If the open positions of A⬘ are uniformly distributed 共which is the case for all patterns discussed in this paper兲, the shift of A⬘ does not greatly modify the result. The integral is then approximately constant. If we note, as we did for first-order artifacts, that 兩ri兩2 ⬍ z2, the factor outside the integral does not depend strongly on ri. Therefore, when we decode, G is convolved with a constant, giving a constant term that can be neglected. The second term of relation 共16兲 yields 兩ri兩 2 P共ri兲 ⬀ 2 共 z ⫹ 兩ri兩 2兲 2
兰兰
O⬘共兲 A⬘共ri ⫺ 兲共rˆ i ⴰ 兲 2d2,
(18) where rˆi is the unit vector having the same direction as ri. The integrand is the moment of inertia of O⬘, cut by A⬘, with respect to the axis in the plane of the object perpendicular to rˆi as a function of the shift of A⬘. In the usual assumption that the open positions of A⬘ are uniformly distributed, it is a function with little dependence on 兩ri兩. The integrand can be approximated with I共rˆi 兲, where is the open fraction of A and I is the moment of O⬘, which we just discussed above. The contribution to the decoded image is obtained by substitution into Eq. 共4兲: ˆ ⬀ O
兰兰
ri
兩ri兩 I共rˆ i 兲G⬘共ri ⫹ 兲d2ri. 共 z ⫹ 兩ri兩 2兲 2 2
2
Fig. 5. Second-order artifacts for the arrays of Figs. 3 and 4 共after pattern centering兲 calculated with relation 共20兲. Note the similarity of MURAs and NTHT MURAs. This is also true at first order. Dimensions are in pixels.
(19)
Dependence on the object enters relation 共19兲 only through its moment of inertia I. Therefore it is stronger the less isotropic the object, as, for example,
a line source. For isotropic objects, like a circle, I is a constant and comes out of the integral. In this case, artifacts depend again on G only, i.e., the mask family we are using:
ˆ ⬀ I O
兰兰
ri
兩ri兩 2 G⬘共ri ⫹ 兲d2ri. 共 z 2 ⫹ 兩ri兩 2兲 2
(20)
As a first approximation, one can ignore the denominator and conclude that second-order artifacts have the shape of the moment of inertia of G 共with respect to an axis perpendicular to its plane and passing through its center兲 as a function of shift. In the calculations, however, one can implement as easily all of relation 共20兲. The application to different array families is shown in Fig. 5. Not surprisingly, at second order the second moments of O and G appear. For isotropic objects, artifacts depend only on a quantity we know 共G兲 and can be predicted independently of the object. However, as we show in Subsection 3.D, we found empirically that the prediction is accurate even for objects that, as with our test object, are not particularly isotropic. D.
Mask Transparency
The derivation above can be expanded to take into account transmission through the opaque portion of the mask. Let t be the average fraction of incident photons passing through an opaque part of the mask instead of being blocked. In this case, the pattern recorded at the detector is the contribution of a fraction 共1 ⫺ t兲 of photons that “see” an ideal mask 关and which are then described by relation 共1兲兴, plus a frac10 September 2001 兾 Vol. 40, No. 26 兾 APPLIED OPTICS
4701
tion t of photons that do not see the mask at all. The starting point is to rewrite relation 共1兲 as
兰兰
P共ri兲 ⬀ 共1 ⫺ t兲
O共ro兲 A
ro
⫹t
兰兰
冉
冊
a b ri ⫹ ro cos3共兲d2 ro z z
O共ro兲cos3共兲d2ro.
(21)
ro
After applying the same arguments presented above, we found that transparency does not change the shape of the artifacts but intensifies them. This should be expected, because in the previous derivation the effect of our approximations was to ultimately consider A as a constant, leading to terms in the form of the second term of relation 共21兲. This does not mean that the shape of the artifacts is independent of A. In fact, this dependence on the array comes back through the deconding array G, which is intimately and unequivocally related to A. E.
Background Nonuniformity
The theory can also be extended to the case of background nonuniformity, which is also relevant to farfield applications. In these problems the recorded pattern is P共ri兲 ⬀
兰兰
O⬘共兲 A⬘共ri ⫺ 兲d2 ⫹ B共ri兲 ⫽ O⬘ⴱA⬘
⫹ B共ri兲,
(22)
which, with use of Eq. 共7兲, upon decoding becomes ˆ ⬀ 共O⬘ⴱA⬘兲 丢 G ⫹ B共ri兲 丢 G ⫽ ᑬ共O⬘兲 ⫹ B共ri兲 丢 G. O (23) The artifacts are then given by B共ri兲 R G, which is a constant in the common assumption of uniform background. Otherwise, B can be expanded in a Taylor series, and first- and second-order contributions are found again. For example, in the case of a linearly varying background, B共ri兲 ⫽ s ⴰ ri ⫹ q,
(24)
Fig. 6. Simulation of first-order artifacts. The mask was a 62 ⫻ 62 pixel NTHT MURA. 共a兲 Centering artifact for a non-patterncentered mask. For a pattern-centered mask: 共b兲 first-order artifact for an off-center object, and 共c兲 if the object is centered the artifact vanishes.
4. Verifying the Theory A.
B.
where s and q are constants, and the artifact has the shape ˆ ⫽ O
兰兰
B共ri兲G共ri ⫹ 兲d2ri
ri
⫽sⴰ
兰兰
ri G共ri ⫹ 兲d2ri ⫹ const,
(25)
ri
which has the form of relation 共15兲 despite originating from a completely different term. 4702
APPLIED OPTICS 兾 Vol. 40, No. 26 兾 10 September 2001
Simulation Results
Our first batch of simulations were aimed at reproducing first-order artifacts. Results for a NTHT MURA mask are shown in Fig. 6. The centering artifact is evident in the simulation of the thyroid test object. To show the plane that can be seen at the top right of Fig. 4 we needed to center the mask pattern and then use an off-centered object. We simulated a square source close to the top left corner of the FOV. As expected, a linear modulation of the intensity appears in the image: The bottom right corner of Fig. 6共b兲 is brighter, with the brightness decreasing linearly as the top left corner is approached. The slope of this plane depends on the position of the object as well as on the rotation of the mask and can be predicted from relation 共15兲. Some other structures can be seen to the right and below the object. These could be due to the approximations made, in particular that about the uniformity of the object, which in this case is probably not valid. Finally, if the object is moved to the center of the FOV, the artifact, as expected, disappears 关Fig. 6共c兲兴. To reproduce second-order artifacts, projections of the thyroid test object were simulated for different patterns 共after pattern centering兲. The results are in the last two columns of Fig. 7 共for an explanation of the other columns, see Section 5兲. Artifacts match the shape predicted by theory and are shown in Fig. 5. Published Results
Results of earlier research on coded apertures appear in the literature. In Fig. 8 we show two coded aperture images of a hand and of an emission-computed tomography cold-rod phantom from Ref. 12. We know that the authors used a 1-mm-thick lead mask with 1.5-mm holes arranged in an uniformly redundant array pattern of dimensions unknown to us. The artifacts for this family are almost identical to those expected from NTHT MURAs 共see Figs. 4 and 5兲. We believe that the large background of the hand image to be the plane predicted by theory and seen in
Fig. 7. Simulation results. Four arrays were considered: a 79 ⫻ 79 pixel MURA, a 77 ⫻ 77 pixel new system array 关a product array similar to the 33 ⫻ 33 pixel pseudonoise product of Figs. 4 and 5 共Ref. 22兲兴, a 62 ⫻ 62 pixel NTHT array based on a 31 ⫻ 31 pixel MURA, and a 63 ⫻ 65 pixel M sequence. The five columns show, for each mask, the two views, the sum and the difference picture 共see Section 5兲, and the prediction of the artifact according to relation 共20兲. Note the poor signal-to-noise ratio for the new system mask and especially the antimask.
our simulations. Such an artifact is replaced by the second-order artifact in the phantom images, consistently with our observations on the centering of the object. In this image, a horizontal line is also evident. Unfortunately we do not know the shift of the pattern used in the study of Ref. 12 or whether it was different from that used for the hand image. Here we can only note that it would be consistent with a first-order centering artifact because of the mask centering along one dimension only. The presence at the same time of a first- and second-order artifact could be an indication that the center of mass of the
Fig. 8. Hand and an emission-computed tomography cold-rod phantom. In this phantom the smallest rod diameter is 6.4 mm. From Ref. 12.
object was close to, but not exactly on, the center of the FOV. In fact, in this situation first-order artifacts would be reduced to the point that second-order artifacts become visible. In conclusion, the artifacts that can be seen in the images of Fig. 8 are consistent with the predictions of our theory. 5. Artifact Reduction
Given an array and decoding array pair 共 A, G兲 with ideal correlation properties 共i.e., A R G ⫽ ␦兲, the pair 共1 ⫺ A, ⫺G兲 also has ideal correlation properties 共except for a flat pedestal that can be subtracted and does not distort the image兲. This array is the negative of the previous array because in A closed elements are substituted with open elements and vice versa. This is often called the antimask 共e.g., see Ref. 17兲. Note that when the image is reconstructed, the two sign changes cancel in Eq. 共7兲, so that the reconstructed object does not change sign; i.e., we are not taking a negative picture. However, first- and second-order artifacts are seen to depend only on G 关relations 共15兲 and 共20兲兴 and must change sign. When an image of the same object is taken with an antimask, artifacts change sign whereas the reconstructed object does not. When the two images are 10 September 2001 兾 Vol. 40, No. 26 兾 APPLIED OPTICS
4703
Fig. 9. Simulation of two exposures of a thyroid: mask 共top left兲 and antimask 共top right兲. Note that the artifacts change sign. When the two images are added 共bottom left兲, they cancel out and the signal is reinforced. If the images are subtracted, the opposite is true and the artifact is reinforced whereas the object cancels out 共bottom right兲.
added, they should cancel the artifacts and reinforce the reconstruction, whereas when they are subtracted, they should cancel the object and reinforce the artifacts. We simulated this experiment with the test object: The remarkably successful outcome is shown in Fig. 9. In Fig. 7 the same experiment was repeated with equal success for different array families. From these simulations we learn that, when applying this artifact reduction technique, one should be aware of the noise properties of both the mask and the antimask in the choice of the mask pattern. For example, even if low-throughput arrays such as the new system arrays of Ref. 22 have reasonable noise properties, the associated antimask is noisy, causing a considerable signal-to-noise loss in the reconstructed image. In our experience, URA and MURA patterns 共or patterns based on URA or MURA patterns兲 provide the most balanced performance. To verify our simulations, we carried out a series of experiments using a thyroid phantom in conjunction with a Siemens E-Cam. We were particularly interested in the potential for high-resolution, highsensitivity imaging with a coded aperture and an existing Anger camera for nuclear medicine applications. The results from the exposure of a 2-D phantom injected with ⬃200 Ci of 99mTc are reported in Fig. 10. With the phantom at 40 cm from the detector and approximately 9 cm from the mask, distances suggested as optimal by our simulations, the two exposures took approximately 8 min each. Agreement with the computer simulation of Fig. 9 is good. The techniques of pattern centering and mask and antimask have long been used in far-field applica4704
APPLIED OPTICS 兾 Vol. 40, No. 26 兾 10 September 2001
Fig. 10. Experimental results for a thyroid phantom. Figure 9 provides a comparison with the simulation. To obtain a truly 2-D image, only the bottom of the phantom was filled. The spike coming out of the bottom left lobe is the injection channel. The measured resolution of this image is approximately 1.5 mm.
tions where they were used to reduce the effects of mask distortion, nonuniformities in the detector, and artifacts arising from systematic differences in detector background.15–17,23 The formalism developed above shows that these techniques are also effective in correcting a different kind of artifact, caused by near-field geometry, whose origin is different in nature and that is indeed present even for perfect masks, ideal detectors, and in total absence of background. 6. Conclusions
The arrays that provide an ideal point-spread function in far-field coded aperture imaging do not operate under ideal conditions in near-field imaging. The images produced are affected by artifacts that have already been encountered by other researchers and are found in published results. A theory capable of predicting the shape of such artifacts in the case of planar imaging has been developed, providing valuable insights in image analysis. A few image improvement strategies are suggested. Centering the object and the mask pattern was shown to eliminate first-order artifacts, the strongest, but was not effective in eliminating residual second-order artifacts. A technique previously used for nonuniform background reduction15–17 proved successful: Two images are taken, one with a mask and one with its antimask, and then added. The exposure time, divided into two halves, is not increased by more than the time necessary to physically change the mask pattern and start a new acquisition. Furthermore, if the mask is antisymmetric 共a pattern in which some rotation or reflection results in the replacement of open with closed mask
positions and vice versa兲, the antimask is simply a rotation or reflection of the mask and there is no need to fabricate two masks. Also, the mechanisms to change the mask, when needed, are greatly simplified.17 The price paid is a reduced range of available patterns: In our case we were forced to choose a 62 ⫻ 62 pixel pattern in place of a 74 ⫻ 74 pixel pattern, with a 17% loss in resolution at constant FOV. Moving the object closer to the detector may possibly increase the sensitivity, but artifacts are also reinforced, especially because the assumption of a thin object may break down and artifacts from out-of-focus planes may dominate the image. All simulations and experiments were performed with an object-todetector distance of 40 cm. We are planning to explore the practicality of shorter distances in the near future. A second factor expected to strengthen artifacts is the FOV. In this application we were concerned mainly with high resolution. For a detector of given dimensions, this results in a relatively small FOV, in our case 9 cm ⫻ 9 cm. In applications with a wider FOV, expansion to the second order may not be sufficient; but one can also argue that the method can be extended to higher orders, and the ensuing artifacts may turn out to be eliminated with the same mask and antimask technique. This method has indeed already given a proof of robustness. In fact, collimation effects from a finite mask thickness, even if included in our simulations, were not taken into account in the theory. However, both simulations and experiments show that when two pictures are taken, it seems to overcome this additional difficulty. Finally, the mask and antimask technique can be coupled with other artifact reduction techniques,12,13 perhaps as a preprocessing stage to provide a better first iteration and thus accelerating convergence. Francesca Gasparini of the Politecnico di Milano has been involved from the inception of this study. We also thank Berthold Horn for his careful review of the manuscript and Robert Zimmerman of the Harvard Medical School and Brigham and Women’s Hospital for his kind support and assistance with the experiments. This material is based on research supported by the Office of National Drug Control Policy under contract DAAD07-98-C-0117 and by the Federal Aviation Administration under grant 93-G053. References 1. E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349 – 403 共1987兲. 2. G. K. Skinner, “Imaging with coded-aperture masks,” Nucl. Instrum. Methods Phys. Res. A 221, 33– 40 共1984兲. 3. R. H. Dicke, “Scatter-hole cameras for X-rays and gamma rays,” Astrophys. J. 153, L101–L106 共1968兲.
4. J. G. Ables, “Fourier transform photography: a new method for X-ray astronomy,” Proc. Astron. Soc. Aust. 1, 172–173 共1968兲. 5. E. E. Fenimore, T. M. Cannon, D. B. Van Hulsteyn, and P. Lee, “Uniformly redundant array imaging of laser driven compressions: preliminary results,” Appl. Opt. 18, 945–947 共1979兲. 6. Y. W. Chen, M. Yamanaka, N. Miyanaga, T. Yamanaka, S. Nakai, C. Yamanaka, and S. Tamura, “Three-dimensional reconstruction of laser-irradiated targets using URA coded aperture cameras,” Opt. Commun. 71, 249 –255 共1989兲. 7. K. F. Koral, J. E. Freitas, W. L. Rogers, and J. W. Keyes, “Thyroid scintigraphy with time-coded aperture,” J. Nucl. Med. 20, 345–349 共1979兲. 8. W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, J. H. Thrall, T. J. Brady, and J. W. Keyes, Jr., “Coded-aperture imaging of the heart,” J. Nucl. Med. 21, 371–378 共1980兲. 9. H. H. Barrett, “Fresnel zone plate imaging in nuclear medicine,” J. Nucl. Med. 13, 382–385 共1972兲. 10. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 共1978兲. 11. E. E. Fenimore and T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods,” Appl. Opt. 20, 1858 – 1864 共1981兲. 12. Y. H. Liu, A. Rangarajan, D. Gagnon, M. Therrien, A. J. Sinusas, F. J. Th. Wackers, and I. G. Zubal, “A novel geometry for SPECT imaging associated with the EM-type blind deconvolution method,” IEEE Trans. Nucl. Sci. 45, 2095–2101 共1998兲. 13. K. F. Koral and W. L. Rogers, “Application of ART to timecoded emission tomography,” Phys. Med. Biol. 24, 879 – 894 共1979兲. 14. J. S. Fleming and B. A. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instrum. Methods Phys. Res. A 221, 242–246 共1984兲. 15. M. L. McConnell, D. J. Forrest, E. L. Chupp, and P. P. Dunphy, “A coded aperture gamma ray telescope,” IEEE Trans. Nucl. Sci. NS-29, 155–159 共1982兲. 16. P. P. Dunphy, M. L. McConnell, A. Owens, E. L. Chupp, and D. J. Forrest, “A balloon-borne coded aperture telescope for low-energy gamma-ray astronomy,” Nucl. Instrum. Methods Phys. Res. A 274, 362–379 共1989兲. 17. U. B. Jayanthi and J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instrum. Methods Phys. Res. A 310, 685– 689 共1991兲. 18. R. Accorsi, F. Gasparini, and R. C. Lanza, “An improved method for radiation imaging using coded apertures,” U.S. patent application 60/236, 878 共29 September 2000兲. 19. T. M. Cannon and E. E. Fenimore, “Tomographical imaging using uniformly redundant arrays,” Appl. Opt. 18, 1052–1057 共1979兲. 20. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344 – 4352 共1989兲. 21. S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 共1986兲. 22. K. Byard, “Synthesis of binary arrays with perfect correlation properties— coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 336, 262–268 共1993兲. 23. E. E. Fenimore, “Large symmetric transformations for Hadamard transforms,” Appl. Opt. 22, 826 – 829 共1983兲.
10 September 2001 兾 Vol. 40, No. 26 兾 APPLIED OPTICS
4705
