Spectral imaging is a technique that generates a spatial
map of spectral variation, making it a useful tool in many applications
including environmental remote sensing, military target discrimination,
astrophysics and biomedical optics. When imaging a scene, a spectral
imager produces a two-dimensional spatial array of vectors which
represents the spectrum at each pixel location. The resulting
three-dimensional dataset containing the two spatial dimensions and one
spectral dimension is known as the datacube.
Many different techniques for spectral imaging have been
developed over the years. Whiskbroom, pushbroom and tunable filter imagers
are all conceptually simple spectral imager designs. These instruments
capture a one- or two-dimensional subset of the datacube, and thus require
the temporal scanning of the remaining dimension(s) to obtain the complete
datacube. Furthermore, they have poor light collection efficiency for
incoherent sources, resulting in a poor signal-to-noise ratio (SNR).
Multiplex spectral imagers including Fourier and Hadamard transform based
instruments are designed to address the light throughput problem, but
still require some form of scanning, making it difficult to use them for
spectral imaging of non-static scenes.
Tomographic approaches have also produced major advances.
Mooney et al. developed a direct-view design that maximizes the light
gathering efficiency by not requiring any spatial filter, such as a slit.
With this design, the source is viewed through a rotating dispersive
element. Measurements are taken at different rotation angles. These
measurements are projective measurements through the datacube that can be
tomographically reconstructed. While the light gathering efficiency of
such an instrument is high, the geometry of the system limits the range of
angles over which projections are made. As a result of the Fourier slice
theorem, this results in an unsampled region in the Fourier space, a
problem known as the ``missing cone problem'', because the unsampled
region is a conical volume in the Fourier domain representation of the
datacube. The computed tomography imaging spectrometer (CTIS)
system is a static, snapshot instrument that captures multiple projections
of the datacube at once. These capabilities make the CTIS instrument ideal
for spectral imaging of transient scenes. However, the instrument requires
a large focal plane area and also suffers from the missing cone problem.
The CASSI revolution
An important characteristic shared by all the designs
described above is that the total number of measurements they generate is
greater than or equal to the total number of elements in the reconstructed
datacube. In contrast, our group has introduced the idea of compressive
spectral imaging, an approach to spectral imaging that intentionally
generates fewer measurements than elements in the reconstructed datacube.
We utilize the power of compressed sensing ideas (to be described below)
to solve our underdetermined problem by relying on a crucial property of
natural scenes, namely that they tend to be sparse on some multiscale
basis. To achieve compressive spectral imaging, our group has developed a
new class of imagers dubbed the coded aperture snapshot spectral imager (CASSI).
CASSI instruments utilize a coded aperture and one or more dispersive
elements to modulate the optical field from a scene. A detector captures a
two-dimensional, multiplexed projection of the three-dimensional datacube
representing the scene. The nature of the multiplexing performed depends
on the relative position of the coded aperture and the dispersive
element(s) within the instruments.
Dual Disperser CASSI (DD-CASSI)
| Associated publication: M.E. Gehm,
R. John, D.J. Brady, R.M. Willett, and T.J. Schulz, "Single-shot compressive spectral imaging with a dual-disperser
architecture," Optics Express, October 2007. |
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| The source is imaged through two
sequentially dispersive arms arranged in opposition so that the
dispersion in the second arm cancels the dispersion introduced by the
first arm. A coded aperture is placed between the two arms. Recovery
of the datacube from the detector
measurement is performed using an expectation-maximization method designed
for hyperspectral images. Such a design applies spatially-varying,
spectral filter functions with narrow features. Through these filters, the
detector measures a projective measurement of the datacube in the spectral
domain. In essence, the DD-CASSI sacrifices spatial information to gain
spectral information about the datacube. Spectral information from each
spatial location in the scene is multiplexed over a localized region on
the detector. A useful property of the design is that the measurement
resembles the scene, making it easy to focus the camera on objects in the
scene. This also makes it possible to perform local block processing of
the detector data to generate smaller datacubes of subsets of the entire
scene. |
| DD-CASSI Experimental Prototype |
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Instrument Characteristics
- Spectral range of 520-590 nm with a bandpass filter placed at
the input to remove out-of-band stray light.
- Aperture code based on an order 192 S-matrix.
- Equilateral prism used for dispersion as required dispersion is
small and a low dispersion grating produces undesirable overlapping
orders.
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| DD-CASSI Spectral Image Reconstructions |
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Detector measurement of a ping pong ball illuminated by
a 532 nm source |
Sum through the reconstructed datacube over the
wavelength dimension |
Intensity as a function of wavelength through the
reconstructed ball |
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Detector measurement of a scene consisting of three
lemons |
Sum through the reconstructed datacube over the
wavelength dimension |
Intensity as a function of wavelength through each of
the three fruits |
Single Disperser CASSI (SD-CASSI)
| Associated publication:
A. Wagadarikar, R. John, R. Willett, and D.J. Brady, "Single disperser design for
compressive, single-snapshot spectral imaging," Proc. SPIE 6714
(2007). |
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In working with the dual disperser, we realized
that we could build a simpler spectral imaging instrument. We called
this new design the SD-CASSI instrument, short for a single disperser
coded aperture snapshot imaging spectrometer. The instrument
essentially consists of an imaging lens that images the scene on to
the aperture code and a pair of relay lenses that relay the image from
the plane of the aperture code to the detector through a dispersive
element placed between them.
Like the DD-CASSI, the SD-CASSI does not directly
measure each voxel in the desired three-dimensional datacube. It
collects a small number (relative to the size of the datacube) of
coded measurements and a sparse reconstruction method is used to
estimate the datacube from the noisy projections. The instrument
disperses spectral information from each spatial location in the scene
over a large area across the detector. Thus, spatial and spectral
information from the scene is multiplexed on the detector, implying
that the null space of the sensing operation of the SD-CASSI is
different from that of the DD-CASSI. Also, a raw measurement of a
scene on the detector rarely reveals spatial structure of the scene
and makes block processing more challenging.
The SD-CASSI removes the Computed Tomographic
Imaging Spectrometer (CTIS) constraints of measuring multiple
projections of the datacube and using a large focal plane array.
Essentially, just one spectrally-dispersed projection of the datacube that
is spatially modulated by the aperture code over all
wavelengths is sufficient to reconstruct the entire spectral datacube. |
| SD-CASSI Experimental Prototype |
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Instrument Characteristics
- Spectral range of 540-640 nm with a bandpass filter placed at the
input to remove out-of-band stray light.
- Aperture code based on an order 192 S-matrix.
- Equilateral prism used for dispersion as required dispersion is
small and a low dispersion grating produces undesirable overlapping
orders.
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SD-CASSI Spectral Image Reconstructions |
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A scene consisting of a ping pong ball illuminated
with a 543 nm green laser and a 632 nm red laser. |
Detector measurement of the scene consisting of the
ping pong ball. Spatio-spectral multiplexing is clearly evident as we
see two wavelength dependent, aperture code modulated images of the
ball separated by the prism. |
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Spatial content of the scene in each of 30 spectral channels between 540
nm and 640 nm. The eyes and smile on the ball are clearly evident. As
the ball was illuminated with 2 laser sources, it would ideally be
present only in channels 3 and in one of channels 27 or 28. The
additional image of the ball is visible because the instrument suffers
from a wavelength dependent horizontal stretch (an anomorphic
distortion). |
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Spectral intensity through a point on the ping pong
ball illuminated by the 543 nm and 632 nm laser. |
DD-CASSI vs. SD-CASSI
Since the DD-CASSI only multiplexes spectral information
in the datacube, it cannot reconstruct the spectrum of a point source
object. On the other hand, the SD-CASSI can reconstruct the spectrum of a
point source, provided that the source spatially maps to an open element
on the coded input aperture. This implies that for reconstructions
demanding high spatial resolution with less stringent demands on spectral
resolution, the DD-CASSI instrument should be the compressive spectral
imager of choice. On the other hand, where spectral resolution is more
critical than spatial resolution in the datacube, the SD-CASSI instrument
should be chosen. The SD-CASSI has the additional benefit of requiring
fewer optical elements, making optical alignment much easier. The table
below summarizes the key differences between the DD-CASSI and SD-CASSI
instruments.
| DD-CASSI |
SD-CASSI |
- 9 optical elements
- More difficult to align
- No spatial multiplexing,
fewer observations
- Cannot spectrally resolve
point sources
- Block processing possible
- Instrument of choice for high
spatial resolution but lesser spectral resolution
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- 6 optical elements
- Easier to align
- More spatial multiplexing
- May not spatially resolve
point sources
- Block processing more
challenging
- Instrument of choice for high
spectral resolution but lesser spatial resolution
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