Signal Processing - September 2016 - 99

in the original papers [19]-[27], but rather
grating or prism at the rate (dx/dm) . A simThe coded apertureso that their light paths are equivalent. For
ple single disperser coded aperture system
based undersampling
DD-CASSI, the original implementation in
integrates along the wavelength dimension
systems employ different
[20] has two dispersers to realize the dison detection, basically taking tomographic
sampling strategies
persion and pixel-wise focusing (i.e., all the
projections along this dimension. Due to
according to their optical
spectra of a single point passed through the
the coded aperture, however, features along
configurations and exploit mask focus on a single pixel), but its diathis dimension are modulated to improve
gram in Figure 4 has only one disperser
the coherence of the forward model relastatistical properties
(grating) to achieve the same focusing
tive to simple tomographic projections. The
of multispectral data,
by just tuning the location of the spatial
rate of modulation is easily determined by
which leads to different
modulator (mask) and the image sensor. By
considering the number of independent
sensing performance
representing systems with different kinds
code features observed at each detection
in terms of spectral
of modulation (i.e., point-wise coding and
point. A given code feature is shifted spasheared coding) and imaging (pixel-wise
tially by C (dx/dm) , where C is the separeconstruction quality.
focusing and dispersed imaging) using simration between the shortest and longest
ilar optical paths in Figure 4, the intrinsic
wavelength observed. Therefore, integrating
differences between the four systems are revealed. As shown
along a single wavelength channel, the number of independent
in Figure 4, the PMVIS, SD-CASSI, and DD-CASSI systems
wavelength coding elements observed is N = (Cdx/Tdm) . The
only use a single mask to modulate the input light. The main
spectral resolution is C/N = T dm/dx. For a grating of period
difference between them is the placement of the mask. Both
F
,
(
d
m
/
dx
)
=
(
L
/
F
)
.
With
L imaged with a lens of focal length
PMVIS and SD-CASSI place the mask on the imaging plane,
L = 3 microns and F = 3 cm, for example, a code feature
leading to point-wise coding (i.e., all the spectra of a single
of size 10 microns yields a spectral resolution of 10 nm, corpoint are either passed through or blocked by the mask), while
responding to 30-40 spectral features over the visible range.
the DD-CASSI places the mask in front of the image plane,
Better spectral resolution can be obtained with faster gratings
which leads to a spectrally sheared coding (i.e., the 3-D code
or longer focal lengths, but a multiplexing level of 30-40 is
is generated by stacking the same 2-D code with different
already fairly aggressive for snapshot imaging. Multiform
offsets). In contrast, the sensor of DD-CASSI is put on the
integration methods will likely be necessary for more heavily
image plane to achieve pixel-wise focusing, while PMVIS and
multiplexed systems.
SD-CASSI place the sensor behind the imaging plane, which
We see, therefore, that coded apertures present a simple and
leads to dispersed imaging (i.e., spectra of a single point disstraightforward mechanism for complex spectral filter implepersed to a set of pixels). As for 3-D-CASSI, two masks are
mentation. In addition, depending on the implementation, they
utilized to achieve both the spatial and spectral modulation
have reasonably local kernels that allow spatially separable
simultaneously, and the sensor is put on the focus plane to
data cube estimation.
ensure pixel-wise focusing.
Even within the family of coded aperture spectral imagers, numerous design choices may be considered for code
implementation, dispersive elements, and sensing. Since we
Prism-mask video imaging spectrometry
cannot comprehensively consider all design choices, here
PMVIS [26], [27] straightforwardly acquires the spectra of
we focus on comparing the coherence of the forward model
scene points with the aid of a prism and utilizes a mask with
for several model systems based upon compressive coded
uniformly distributed holes that prevent overlaps of the
aperture designs proposed and demonstrated over the past
decade. We specifically do not consider implications of static
codes implemented on slides versus dynamic codes impleΓ
mented using spatial light modulators. While spatial light
∆
modulators suffer scatter and numerical aperture limitations
dx
not found with static codes, we hope that the reader will find
Γ
dλ
our comparisons without detailed physical implementations
sufficiently compelling to postpone full consideration of
practical issues.
The coded aperture-based undersampling systems employ
Γ dx
different sampling strategies according to their optical conN=
∆ dλ
figurations and exploit statistical properties of multispectral
x
Γ
dx
data, which leads to different sensing performance in terms of
∆λ =
=∆
N
dλ
spectral reconstruction quality. Figure 4 displays diagrams of
λ
four undersampling multispectral cameras. It is worth noting
that to facilitate comparison, the diagrams of the systems are
Figure 3. A diagram of the relationship between the spectral resolution
drawn not according to the physical configurations proposed
and the code feature size.
IEEE SIgnal ProcESSIng MagazInE

September 2016

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