Analysis of computed tomographic imaging
spectrometers. I. Spatial and spectral resolution
Nathan Hagen1,* and Eustace L. Dereniak2
1

Fitzpatrick Institute for Photonics, Duke University, Durham, North Carolina 27708, USA
2

College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA*Corresponding author: nhagen@optics.arizona.edu
Received 7 March 2008; revised 22 April 2008; accepted 30 May 2008;
posted 9 June 2008 (Doc. ID 93420); published 25 July 2008

Computed tomographic imaging spectrometers measure the spectrally resolved image of an object scene
in an entirely different manner from traditional whisk-broom or push-broom systems, and thus theirnoise behavior and data artifacts are unfamiliar. We review computed tomographic imaging spectrometry(CTIS) measurement systems and analyze their performance, with the aim of providing a vocabulary fordiscussing resolution in CTIS instruments, by illustrating the artifacts present in their reconstructeddata and contributing a rule-of-thumb measure of their spectral resolution. We also show how the data
reconstruction speed can be improved, at no cost in reconstruction quality, by ignoring redundant projections within the measured raw images. © 2008 Optical Society of America
OCIS codes:
110.4234, 300.6190, 110.3000, 120.6200.

1. Introduction

Computed tomographic imaging spectrometry(CTIS) is a technique by which a scene is viewed directly through a 2D grating (as in Fig. 1), producing a
number of prismatically dispersed images of the object on the focal plane. This is much like placing apair of crossed gratings in front of one’s eyes (asin Fig. 2), except that the CTIS imaging sensor is typically not spectrally selective like the human eye.
Inserting a field stop in the system limits the instrument’s field of view and allows the prismatic images
to be spatially separated (producing data like thoseshown in Fig. 3).
Looking at any one of these prismatic images, we
see that the data are what we would get with a slit
spectrometer instrument if the slit were opened very
wide (and the grating tilted to the appropriate angle), allowing the object’s spatial and spectral information to overlap. If only one diffraction order ismeasured, the result is a disastrous loss of spectral
0003-6935/08/280F85-11$15.00/0© 2008 Optical Society of America

resolution, but if we design the system to measure
multiple diffraction orders, then we can use the nonredundant information present within them to simultaneously recover the spectral resolution andobtain the spatial distribution as well. Thus, the complete spectrally resolved image (the data cube) can beobtained in a snapshot. This process of estimating a
3D distribution from a set of 2D images is a familiar
problem from computed tomography, from which we
recognize that each of the prismatically dispersed
images is a projection of the data cube, as illustrated
in Fig. 3.
The historical development of the CTIS disperser
element has been to find ways of steadily increasing
the number of diffraction orders (going from left toright in Fig. 4) . Descour’s original design involved
using sets of cosine gratings, such as the 3 × 3 pattern produced by a pair of cosine gratings crossed
at 90° [1], or the hexagonal pattern produced bycrossing three cosine gratings at 60° to one another(Fig. 4, first and second panels) [2]. Descour et al. expanded the range of dispersion patterns available bydesigning efficient computer-generated holographic
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Fig. 1. (Color online) CTIS instrument views the target scenethrough a 2D grating. The field stop limits the field of view, suchthat the dispersed diffraction orders are spatially separated.

dispersers capable of producing diffraction orders inboth the x and y directions, creating the 5 × 5 and 7 ×
7 patterns shown in Fig. 4, third and fourth panels[3]. This substantially increased the number of projections measurable by CTIS, packing them all within the same physical dimensions of the focal planearray. CTIS instruments in current use typicallywork with the 5 × 5 diffraction pattern on the focal
plane array (Fig. 4, third panel), so that they sample
the data cube with 25 projections. Additionally, computer-generated holographic design techniques are
flexible enough to channel light away from the zerothorder (where the detector array is prone to saturate)and into the dispersed orders, which are otherwise
light starved by comparison. This prevents a lossof dynamic range in the measured data.
In a given CTIS design, there is a simple trade-off
between the spatial resolution and the spectral resolution. In terms of sampling rate, the former is givenby the size, in pixels, of the field stop image on the
focal plane array (FPA)—the white square at the center of Fig. 3 (left). The latter is given by the length ofthe longest projection, minus the size of the field
stop. (The actual spectral resolution of CTIS instrument is a complicated issue, which we discuss at
length in Section 6 below.) For a 2048 × 2048 FPA,
a typical CTIS design will image the field stop to a
100 × 100 region, giving a projection along an outerdiagonal that is 626 pixels long. The above trade-Fig. 2. (Color online) Nondescript scene taken with an RGB camera and a pair of crossed gratings inserted in front of the lens.
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Fig. 3. (Color online) The 5 × 5 CTIS design produces a pattern of
25 projections onto the detector array, each of which is a prismatic
version of the zero-order image. Each of these prismatic images
produces a footprint on the detector array that is a tomographic
projection through the object scene’s 3D data cube.

off between spatial and spectral resolution can be expressed approximately by the relation (see Fig. 5)
pﬃﬃﬃ
3 2
L≈
ðZ − DÞ;14

ð1Þ

where L is the maximum possible spectral sampling,
Z is the length in pixels of the FPA (e.g., 2048), and D
is the spatial sampling (e.g., 100).
2.

Fourier Slice Representations

An important means of visualizing the characteristics of a CTIS instrument and of comparing various
dispersion patterns is the central slice theorem ([4],p. 205). This theorem states that for a 2D projectionof a 3D volume, the Fourier transform of the projection is equivalent to evaluating the Fourier transform of the 3D volume along a plane passing
through the origin in Fourier space. (The orientationof the evaluation plane is determined by the angle ofprojection.) This result allows us to draw a planewithin the “Fourier cube” (i.e., the Fourier transform
representation of the data cube) corresponding to thefrequencies sampled by a given projection. Doing thisfor all of the projections of a dispersion pattern gives
a Fourier slice diagram representation of that particular CTIS design. While the Fourier slice theorem
is strictly true only for continuous mappings, itremains a useful approximation for sufficiently
sampled discrete systems.
Note that the spectral dimension of the data cube
is of course not a physical dimension, and so its physical size can be defined arbitrarily. The choice madehere is to define the data cube as being a perfect cube

Fig. 4. (Color online) Past CTIS dispersion patterns: from left toright, 3 × 3 pattern produced by a pair of crossed cosine gratings,
3XCos pattern produced by three cosine gratings oriented at 60° to
one another, 5 × 5 pattern, 7 × 7 pattern.

Fig. 5. (Color online) Illustration of the spatial–spectral sampling trade-off in a given CTIS design. For a focal plane arrayof dimension Z and a data cube dimension of D (this latter giving
the spatial sampling), the longest wavelength is diffracted to theedge of the array: a distance of ðZ=2Þ − ðD=2Þ in pixel units. If weassume a spectral range of 400–700 nm, then the shortest wavelength image is diffracted by 4=7 of the same distance. The spectral
dispersion distance is therefore ð3=7Þ½ðZ − DÞ=2.
pﬃﬃﬃ Along an outerdiagonal order, this distance is increased by 2.

in size (the physical dimensions of the cube along x, y,and λ axes are identical). Typically, CTIS samples the
spectrum at a greater rate than for the spatial information of the scene, so that an individual voxel takes
on a flattened pillbox appearance by this definition.
The Fourier slice diagrams for the various CTIS
dispersion patterns are shown in Fig. 6. Evident ineach of them is a large polygonal double-cone region,
called the “missing cone,” which is not sampled by
the CTIS system. (For the corner designs, this cone
is extended such that two octants in Fourier spaceare entirely unsampled.) Naturally, the shape ofthese regions affects the reconstruction quality by altering the shape and size of the system blurring
kernel.
3. CTIS Voxel Spread Function

The presence of a missing cone in the Fourier cube
indicates that the voxel spread function (VSF) of a
CTIS system will have peculiar characteristics. Herethe VSF is described as the image of a point source(that is, one pointlike both spatially and spectrally)
not on the CTIS FPA but in the reconstructed datacube—the VSF is thus a 3D analog of the 2D point
spread function.
An instrument that has characteristics largely
similar to CTIS instruments, but which does not operate in snapshot mode, is a chromotomographic hyperspectral imaging spectrometer (CTHIS) [5,6]. The
CTHIS is capable of measuring as many projections
as desired, a quality that can be used to advantage inanalysis here, since a well-sampled CTHIS measurement has a missing cone that is a true right-circular
double-cone [Fig. 7(a)]. In the CTHIS Fourier cube,
any planar slice containing the h axis (the reciprocalof the λ axis) will contain four triangular regions, containing sampled and unsampled regions [Fig. 7(b)].
Taking the Fourier transform of the slice gives theFig. 6. (Color online) Left, various CTIS dispersion patterns and,right, their corresponding Fourier slice diagrams.

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Fig. 7. (Color online) (a) For a CTHIS instrument taking a large
number of projections, the missing cone becomes a true right-circular double cone. (b) In the corresponding Fourier cube, any planar slice containing the h axis will have the form shown. (Black
indicates the missing cone region; white indicates the sampled regions.) (c) Corresponding VSF. Here (ρ; h) and (x; λ) are Fourier conjugate coordinates.

VSF in terms of cylindrical radial coordinates r ¼
ðx; yÞ and the λ coordinate [Fig. 7(c)].
Figure 7(c) indicates that any pointlike object creates a signal within the data cube that is a maximum
at the appropriate voxel, but which also produces
weak but prolonged tails in the form of a double-cone(the shape shown in Fig. 7(c) rotated about the λaxis). The theoretical treatment of the missing cone
provides a useful perspective, but the small numberof projections available in a snapshot CTIS instrument means that the continuous approach gives only
a rough approximation to the true VSF. Examining
data cubes reconstructed from simulated and experimentally fabricated point sources gives an empiricalmeasure of the true 3D VSF. Figure 8 shows several
spectra in the VSF, determined by imaging a point
source in the center of the field of view (here the fieldof view is mapped to a 25 × 25 pixel region on the
CTIS FPA). The displayed spectra are taken fromvarious spatial positions within the data cube, progressively farther from the point-source image.
These show a close correspondence to the VSF illustrated in Fig. 7(c).
Figure 9 illustrates the same data set from anotherperspective, showing the horizontal slices throughthe VSF at various wavelength locations. It is herethat we see the discrete number of projections of a
CTIS instrument (relative to that of an ideal CTHIS
measurement) significantly alter the shape of the
VSF, and we begin to see a 5 × 5 pattern reminiscentof the dispersion pattern used. This becomes evenmore apparent when we look at the backprojected
point-source image, shown in Fig. 10. The gray-scale
map of the backprojected images indicates that thetails of the backprojection VSF are much stronger
than in the final reconstruction. The reconstruction
algorithm has thus filtered the backprojection,
greatly improving the fidelity of the estimated data
cube.
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Fig. 8. (Color online) Spectra from different spatial positions in
the reconstructed 25 × 25 × 145 data cube for a point source withλ ¼ 557 nm. The locations sampled in the data cube are indicated.
Note that each of the spectra shares the same abscissa, ranging
from 400 to 700 nm, and that the vertical axes (in arbitrary irradiance units) have a different scale in each plot: as we move further
out from the center of the data cube, the peaks show a drop in amplitude by about a factor of 10 from one plot to the next.

Fig. 9. Several planes shown from the reconstructed data cube
for a point source at λ ¼ 557 nm. The gray-scale mapping on thesefigures is logarithmic to accentuate the low-irradiance pixels. Each
image is also scaled to have its brightest pixel map to white.

3. Number of projections, P: increasing the number of projections improves reconstruction quality.
(Medical imaging systems typically use hundredsof projection angles for performing reconstructions.)
4. Fill factor: The ratio of illuminated to unused
pixels on the FPA. To maximize its ability to measure
the object data cube, a good CTIS design will maximize the use of space on the FPA (and minimize thenumber of unused pixels).
5. Blur factor: the average number of voxels mapping to a given illuminated pixel on the CTIS FPA.
This is a rough measure of the amount of blurring
incurred when performing a backprojection operation (an essential step in the reconstruction algorithms). Increased blurring of spatial and spectral
components of the data cube reduces the theoretically achievable spatial and spectral resolution.
6. Missing cone angle, ϕ: the half-angle of themissing cone inside the Fourier cube. While this metric is dependent on both the spectral sampling L andthe spatial sampling D, the missing cone angle provides another convenient way of comparing CTIS
designs.
Fig. 10. Several planes shown from the backprojected data cube
for a point source at λ ¼ 557 nm. The gray-scale mapping on thesefigures is linear. Each image is also scaled to have its brightest
pixel map to white.

Table 1 presents a set of CTIS dispersion patterns,
together with their respective design parametersand metrics.
5.

Figures 8–10 all show simulated data; corresponding plots from experimental data are quite similar,
but the presence of noise hides some of the salientfeatures.
4. First-Order Metrics for Comparing the
Various Patterns

Having a set of CTIS dispersion patterns, we would
like to have a way to characterize each CTIS designin terms of what we want: spectral and spatial resolution. Resolution by itself is a vague quantity until
we define how we plan to measure it. In the case of
CTIS, it is also an object-dependent quantity, as we
will see below. The first step toward defining what weare looking for is to create a set of metrics that weknow correlate to desirable characteristics in a CTIS
instrument. These provide a rapid and simple meansof comparing designs.
There are at least six important metrics commonlyused for characterizing the various CTIS designs.
1. Spatial dimension, D: the number of pixels in
x, and also in y, in the data cube if we assume that thedata cube is square. [This definition coincides with
the D used in Eq. (1).]2. Spectral dimension, L: the number of spectral
samples as expressed by Eq. (1). If spectral resolution
is particularly important for a given CTIS system,
then the number of spectral samples is a metric closely related to the realizable spectral resolution of
the system.

Simulation Results

A CTIS simulator was developed to provide a moreobjective means of comparing and assessing CTIS
Table 1.

Comparison of Linear CTIS Patternsa

Design Parameters
Design TypeD

Full patterns
3Xcos
75
5×5
75
7×7
75
3Xcos
100
5×5
100
7×7
100
3Xcos
200
5×5
200
7×7
200
Corner patterns
5 × 5 Corner75
7 × 7 Corner75
5 × 5 Corner100
7 × 7 Corner100
5 × 5 Corner200
7 × 7 Corner
200
Outer patterns
5 × 5 Outer75
7 × 7 Outer75
5 × 5 Outer100
7 × 7 Outer100
5 × 5 Outer200
7 × 7 Outer
200

Design Metrics
P

L

Fill

Blur

ϕ

19
25
49
19
25
49
19
25
49

558
570
570
550
526
526
522
349
349

0.18
0.24
0.42
0.25
0.31
0.54
0.57
0.57
0.88

111
81
70
141
102
112
236
147
157

7.66
7.50
7.50
10.28
10.76
10.76
20.95
29.80
29.80

9
16
9
16
9
16

1261
1261
1208
1208
996
996

0.15
0.25
0.19
0.33
0.37
0.61

103
107
133
138
231
247

17
25
17
25
17
25

570
570
526
526
349
349

0.16
0.29
0.22
0.37
0.50
0.61

80
66
107
83
180
135

7.50
7.50
10.76
10.76
29.80
29.80

a

D, dimension of the zero order; P, number of projections; L, pixels of spectral dispersion. The patterns are assumed to be dispersed onto a 2048 × 2048 FPA.
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designs. While the six metrics D, P, L, fill, blur, and ϕprovide a simple means of comparing the various
CTIS patterns, they can be ambiguous. For example,
if one design has a larger P but a smaller fill thananother design, Table 1 provides no way of knowing
which of the two metrics is more important. A moreobjective measure is necessary. The simulation isable to create data cubes, images, and system matrices of a variety of CTIS systems and to perform
complete reconstructions. All of the simulations below are performed for a 512 × 512 FPA and a data
cube with spatial dimensions 25 × 25 (the data cube’s
λ dimension is determined by the specific dispersion
pattern used). The reconstruction algorithm used isthat of Eq. (3) of [3].
The first step toward providing an objective measure is to realize that any comparison is bound to be
object dependent. Although this complicates comparisons, we can use object data cube models that closely
correspond to useful objects for a CTIS to measure,and thus we can gain a rough idea of how well we
can expect the instrument to perform in practice.
Table 2.

Allowing the simulator to generate an image andthen perform a reconstruction on this data allows
us to compare the result against the input. The normalized root mean square error (RMSE) ϵ of the reconstructed data cube is defined by

ϵ¼

PM

1

M
X

m¼1 f m m¼1

ðf m − ^f m Þ2

1=2

;

where M is the number of voxels in the true data
cube f (whose elements are f m ), the reconstructed estimate of which is ^f. The RMSE, though not a truly
objective measure of performance, provides a goodstarting point for assessing the capability of each
CTIS design.
To use the RMSE measure, we first need to construct model data cubes for the simulator. CTIS objects can take on a variety of forms, so no one single
data cube can be taken as representative, and thus avariety of models were used (see Table 2). For the simulations, each of the data cube models was taken toData Cube Models Used in the CTIS SimulationaaFigures show projections of the data cube in two ways: (left) a perspective or orthographic projection and (right) a color-mapped panchromatic projection.

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have fixed spatial dimensions of 25 × 25 pixels,
whereas the spectral dimension is taken to dependon the CTIS dispersion
pattern being used. The norPmalization by f m in the RMSE should prevent any
given pattern from being penalized by this approach.
The detector was taken to be a noiseless 512 × 512
pixel array, and the PSF for each projection to be a
Gaussian of FWHM ¼ 1 pixel.
Figure 11 shows the performance for various CTIS
dispersion patterns on the 16 data cube models. The
reconstruction algorithm uses only 20 iterations forthe simulation results, even though the data cube estimate for the more complex objects is generally not
yet near convergence there. (This is done to keep thesimulation close to typical operating conditions for aphysical CTIS system.) The RMSE values shown are
for the final iteration. The feature to note in the plotsis the spread in RMSE among the various CTIS designs for a given data cube model. For example, for
the uniform model (no. 1) all designs perform well,
but for the point-source model (no. 2) the two corner
designs perform much worse than do the three full
designs. For the more complicated data cube models,such as UA Logo (no. 14), Starfield (no. 15), andCheckerboard (no. 16), it appears that even for anoiseless CTIS system we cannot expect to achieve
better than 25% mean error for data cube voxels, regardless of the dispersion pattern chosen. For theuniform model, this corresponds to a mean modulation transfer function [7] (MTF) of about 0.75 (seeTable 3). This is explained in more detailed in Section 6 below.
It is important to note that the values plotted inFig. 11 should be treated as guidelines rather thanas absolute criteria of CTIS reconstruction quality.
Since the various CTIS designs do not sample thewavelength dimension of the data cube at the samerate, a single RMSE measure cannot provide an objective comparison across all designs. Figure 12
shows example reconstructions from a 5 × 5 CTIS design on a 512 × 512 FPA.

A.

Corner Patterns versus Full Patterns

When seeing CTIS for the first time and hearing
about the trade-off in spatial and spectral resolution
in the system [expressed by Eq. (1)], almost invariably the initial reaction is: Why not place the zeroorder into the corner of the image and allow the diffraction orders to spread across the full extent of the
focal plane array? (This configuration is shownin Fig. 6.)
Previous research has dealt with this idea [8], concluding that since the asymmetric patterns extend
the missing cone to create an even larger, skewed,
missing region within the Fourier cube, these designs
must be untenable. However, since the standard designs already have a missing region and thus sample
Fourier space nonuniformly anyway, this argument
seems weak. Do we really need to sample the Fourier
cube symmetrically in order to achieve a good reconstruction? The corner patterns sample some regionsof Fourier space which are inaccessible to the symmetric designs, so that it seems plausible that they
may have better performance when those frequencies are the important ones.
The simulation results shown in Fig. 11 (left) indicate that while the corner patterns generally perform
worse than the comparable symmetric patterns, thedifference is smaller than might be expected. And
there are cases where their performance is marginally better, such as for the Mod0-0-10 data cube model (no. 8) and the Mod8-8-8 model (no. 12)—the data
cubes where spectral resolution is more important
than spatial. For objects possessing spatial cylindrical symmetry, however, the corner patterns’ performance is disastrous, reflecting the lack ofsymmetry in their sampling of the data cube.
The excellent performance of the 3 × 3 pattern in
Fig. 11 (left) may be surprising, given that the number of measured data cube projections is only nine.
This is unfortunately an artifact of working withnoise-free data that does not translate directly into
noise-corrupted experimental systems. Because of

Fig. 11. Best RMSE for the 16 data cube models. Left, comparing the corner patterns with the symmetric full patterns; right, comparing
the outer patterns with the full patterns. The data cube models are numbered according to: (1) uniform, (2) point, (3) blob, (4) column, (5)wide column, (6) column_Mod10, (7) column_Mod20, (8) Mod0-0-10, (9) Mod5-5-0, (10) Mod8-8-0, (11) Mod5-5-5, (12) Mod8-8-8, (13)point þ Bkgd, (14) UA_Logo, (15) starfield, (16) checkerboard. In the plots above, the most important feature to notice is where the curves
diverge from one another at a given data cube model, showing that one pattern performs substantially better than the others. (Note thathigher values in the plots indicate poorer reconstruction performance.)
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Table 3.

Comparing MTF Values with the RMSE Measure for the
Two Spectrally Modulated Data Cube Models

MTF (avg)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.25
0.2
0.1
0.0

RMSE (Uniform)

RMSE (Point Source)

0%
7%
14%
21%
28%
35%
42%
49%
53%
56%
63%
≥70%

≤40%
50%
115%
122%
128%
132%
136%
138%
140%
145%
150%
≥155%

the lack of space to show this effect here, we plan toillustrate this behavior in the future.
B. Outer Patterns versus Full Patterns

Some of the projections in a given rectangular dispersion pattern have the same azimuthal projection angle and differ only in the length of spectral spread. In
a 5 × 5 dispersion pattern, the 8 projections forming
the inner ring thus share the same azimuthal angle
with 8 of the 16 projections in the outer ring of diffraction orders. The primary difference between
any two projections lying along a radial line extending from zero order is that the outer order has agreater spectral sampling, and we may wonder
whether we can replicate the data of the inner orderby merely averaging pairs of points in the outer projection along the radial direction. This implies that
there may be some redundancy in the multiplexing
and that the added information contributed by the
inner projection may be negligible. To a degree,
the simulation appears to support this idea.
To simulate this case, we can use the simulation tochoose only the outer ring of projections and ignore
the inner diffraction orders (see Fig. 13). Performing
reconstructions on the same 16 data cube models allows us to compare the outer and full patterns (theresults are given in Fig. 11, right). The comparison
shows that there is actually little difference in performance between the two. For the Mod5-5-0, Mod8-8-0,
Mod5-5-5, and Mod8-8-8 models (nos. 9–12), the outer patterns display worse performance than do thefull patterns, but for almost all of the other data cube
models, the outer patterns perform slightly better
than their full counterparts. And this comes withthe added benefit of a faster reconstruction. For
the 5 × 5 pattern, ignoring 8 of the 25 projections allows us to reduce the size of the system matrix usedin reconstruction to 17=25 ¼ 0:68 of the full size, providing a boost of 32% in reconstruction speed withno cost in reconstruction quality. In fact, likewise removing the zero-order image from the center of the
FPA data appears to have no significant effect on
the reconstruction quality, allowing an even smaller
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Fig. 12. Left, panchromatic images of three example input data
cubes. Right, panchromatic images of the corresponding reconstructed data cubes, for the models (top) Mod5-5-5, (middle) UA_Logo, and (bottom) uniform.

Fig. 13. (Color online) Outer dispersion patterns from (left) 5 × 5
and (right) 7 × 7 patterns.

system matrix (16=25 ¼ 0:64). Thus, not only may
the CTIS benefit by ignoring the inner set of diffraction orders, but when designing the disperser for agiven system, it is possible to improve the signalto-noise ratio by suppressing light going to the innerring of projections and instead to channel light into
the outer ring.

6. Specifying the Spectral Resolution of CTIS

A frustrating feature of a CTIS instrument is the factthat its spatial and spectral resolution are not constants but rather object-dependent quantities. Thatis, a CTIS instrument can demonstrate its maximum
spectral resolution when viewing a point-source object, but for a spatially uniform object the achievable
spectral resolution is far lower. Moreover, even when
considering only a single object, the resolution is nota constant for all voxels within the data cube (generally, the corners of the cube show a much higher resolution than the middle regions of the cube). The
simulation provides a convenient means of studying
these complex effects.
One way of determining the spectral resolutionof CTIS is to create a spatially pointlike object having a spectrum consisting of two monochromatic
wavelengths. As the distance between the two wavelengths is altered, we can examine the reconstructed

spectrum and use a criterion to estimate when thetwo wavelengths are successfully resolved. A more
typical use of a CTIS instrument, however, would
be to view broadband objects, in which case the above
approach is inappropriate.
For broadband objects, a convenient approach is
to use a spectral transfer function, defining a point
below which the modulations in the spectra are attenuated to such an extent that they are no longer distinguishable from the noise. This is the approach
used below. Moreover, since the resolution is object
dependent, it is probably best to define the resolution
at the two extremes mentioned above. The resolutionof any given object can then be bracketed as being
somewhere between these two.
Figure 14 shows vectorized plots of an example
input data cube, and its reconstructed counterpart, for a spatially uniform broadband object whose
spectrum is modulated (i.e., the Mod0-0-10 model).

Fig. 14. (Color online) Sections of the vectorized Mod0-0-10 data cube (of dimensions 25 × 25 × 145), where every 145 voxel section (containing 10 periods of modulation) represents the spectrum at a single spatial location in the data cube. (a) Snapshot from the ðx; yÞ ¼ ð0; 0Þ
corner of the data cube, moving to increasing values of x from left to right; (b) snapshot from the middle ðx; yÞ ¼ ð12; 12Þ region in the data
cube. Note that the flat region of the red (central) curve of (b) is the exact middle spatial location in the data cube, and that the reconstructed voxels on the left-hand side show reverse contrast.
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The dimensions of the data cube are ðx; y; λÞ ¼
25 × 25 × 145, where the 145 wavelength samples
result from the spectral spread in the 5 × 5 design’souter diagonal orders. The data cube vectorization is
performed as follows: the first vertical column in thedata cube (the spectrum corresponding to the lower
left spatial position) is extracted and then concatenated with the spectrum at the second spatial position, the third, and so on. This provides a convenientmeans of visualizing the reconstruction.
The modulation contrast decrease from the input
is evident in the reconstructed data cube, as is thechange in transfer function from a corner toward
the middle of the cube. The RMSE of the reconstruction here is 53%, which for this object corresponds toan average MTF of about 0.25. Of course, the choiceof where to declare a fringe resolvable depends on thenoise in the reconstructed data. If we sidestep noise
considerations for the moment, then we can tentatively define the above situation as being at the limitof resolvability. (The definition is arbitrary, but
provides a useful simple rule that has worked forour experimental systems.) Thus, for this data cube
model, resolvability is reached when the RMSE falls
below 53%.
The opposite extreme for determining spectral resolution is that of a spectrally modulated pointsource object. If we take a look at a section of the
input data cube and its reconstruction (Fig. 15), we
can see that the spectral modulations are largely
intact, equivalent to an MTF of about 0.95.
If we again use the criterion that the resolvability
limit is where modulations in the data cube on average experience a fringe contrast loss of no morethan 75%, then we can increase the modulation frequency in the object’s spectrum, and we find thatthe point-source spectral fringes can be resolved
up to 52 modulations across the spectrum. For use
with a 400–700 nm spectral range, this corresponds
to a spectral resolution of 5:8 nm.
Since both data cube models (uniform and point
source) use spectra that are pure sinusoids, the nor-malized RMSE values are easily compared with the
MTF (i.e., fringe contrast). A comparison between thetwo is given in Table 3. The boldface row of the tableindicates the cutoff point for resolvability. The surprisingly large RMSE numbers for the point source
are due almost entirely to the overestimation offringe contrast in the central pixel of the object.
The huge discrepancy in RMSE values for the same
fringe contrast in the signal of interest is an exampleof the limitations of the RMSE as a figure of merit.
The simulation results above are taken from a
CTIS design using a 512 × 512 detector array, giving the number of resolvable modulation periods
as 10 and 52 (i.e., 30 and 5:8 nm resolution) for
the uniform and point-source data cube models, respectively. Performing the same simulation procedure for a 2048 × 2048 detector array, we find thatthe results for choosing a MTF of 0.25 as the cutoff
give resolvable modulation periods of 30 and 200(i.e., 10 and 1:5 nm resolution), respectively. Thus,
for a 5 × 5 CTIS instrument, we can use a rule ofthumb that the point-source spectral resolution isroughly equivalent to the spectral dispersion across
3 pixels along one of the outer diagonal orders (or2 pixels across one of the other horizontal or verticalorders). Likewise, the spatially uniform object’s spectral resolution is roughly equivalent to 20 pixels
along an outer diagonal order.
The mean-MTF-based approach taken here to define spectral resolution is not only useful for defining
a rule of thumb, it is also ideally suited to establishing a design criterion for a CTICS instrument.
“CTICS” is an acronym adopted for a CTIS system
incorporating channeled spectropolarimetry, a technique whereby polarization information is encoded
into modulations of the spectrum [9]. When designing the polarization components for a CTICS system,
it is necessary to estimate the resolvable modulation
frequency in order to choose retarder thicknesses.
The rule of thumb given here for spectral resolution
can therefore replace more ad hoc means of estimating this information.

Fig. 15. (Color online) Section from the middle region of the vectorized Column_Mod10 data cube, where every 145 voxel section (containing 10 periods of modulation) represents the spectrum at a single spatial location.
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7. Summary and Comments

Here we see that a CTIS system is capable of reconstructing a data cube that is spatially uniform but
spectrally modulated. This is a data cube whose nonzero elements in the Fourier domain lie entirely
within the missing cone region. If we take the missing cone idea at face value, such a data cube could not
be reconstructed at all. This works only up to a point,of course; as the object data moves deeper into themissing cone region, the measured signal becomesincreasing attenuated and accurate reconstruction
increasingly difficult. The fact that for discrete systems the missing cone is not a region of zero sampling, but rather one where the sampled signals
are severely attenuated, has been noted before[10], but the simulation clearly confirms the concept.
Moreover, we find that for the above situation (object no. 8) the reconstruction quality is better than forsome object models whose data lie entirely outside
the missing cone (such as object no. 12). Thus, it isa mistake to think of the missing cone as the dominant mechanism for resolution loss. Rather, it ismore useful to think in terms of (1) the multiplexingof data cube voxels on the measured image and (2)the blurring introduced by the VSF emerging fromeach voxel in the measured scene.
An unexpected result from the simulation is thatthe 5 × 5 and 7 × 7 patterns appear to be improvedby deleting the inner set of diffraction orders, which
has the side benefit of speeding up the reconstructions. This provides an easy, if marginal, means ofimproving existing reconstruction speed with little
effort.
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