Signal Processing - September 2016 - 104

Table 1. The spectral sensing coherence information between the
sensing matrices of different systems and the learned bases.

40
35

PSNR/db

PMVIS: 17.25 db
SD-CASSI: 20.90 db
DD-CASSI: 27.21 db
3-D-CASSI: 27.30 db

SD-CASSI

DD-CASSI

3-D-CASSI

PMVIS

K-SVD

0.7920

0.7787

0.7737

0.8148

PCA

0.7048

0.6432

0.6663

0.7251

25
20
15
10
5

PMVIS

SD-CASSI

DD-CASSI 3-D-CASSI

Figure 8. PSNR comparison for image reconstruction with the four types
of undersampling systems on a set of 50 multispectral images. Crosses
of different colors mark the average PSNR for the different methods. The
PNSR value for each individual multispectral image is also plotted, as blue
dots, to illustrate the statistical distribution of the reconstruction accuracy.
Except for PMVIS in which linear interpolation is used for reconstruction,
we use the ADMM [46] to compute the reconstruction results.

database images. The inputs of the four systems are generated
by sampling the multispectral images according to the corresponding sensing matrices described in the section "Undersampling Systems." Image reconstruction is performed using a
widely employed algorithm-the alternating direction method
of multipliers (ADMM) [46]-except for PMVIS which simply
employs linear interpolation (as it cannot be solved by ADMM
directly because of its special sampling scheme). ADMM is
widely used in image reconstruction and has shown superior
performance. It is worth noting that the choice of the algorithm
may affect the reconstruction accuracy, but the ranking of the
results does not change.
In testing PMVIS, we use an image down-sampling rate
of 0.3%, as is the case in the prototype camera [27]. Theoretically, in PMVIS systems, a minimal down-sampling rate of
1/X (where X is the number of spectral channels) is needed
to prevent overlaps between the spectra of different samples.
The current prototypes are not well calibrated, so the downsampling rate may potentially be improved in the future.

(a)

Table 1 presents the spectral sensing coherence information values computed between the sensing matrices of the
four types of the undersampling systems and the three kinds
of bases. Note that since hybrid PMVIS [30], [31] and hybrid
CASSI [32] each obtain two snapshots, they are omitted in this
analysis for an even comparison. Multiple snapshot systems
are discussed in the section "Evaluation of Undersampling
Systems." For the coded aperture-based systems, binary codes
randomly generated by the Bernoulli distribution, with the
same probability p (x = 1) = 0.5, are applied.
seconds
Specifically, the codes of SD-CASSI and DD-CASSI are
derived by shifting and stacking the randomly generated 2-D
patterns. As for 3-D-CASSI, the code is generated directly in
3-D space. Both the K-SVD and PCA bases are learned from
the database.
The 3-D-CASSI system has the most complex modulation
and achieves the best spectral sensing coherence information on the overcomplete dictionary learned by the K-SVD
algorithm. However, for the PCA bases, DD-CASSI provides
the best spectral sensing coherence information. For both of
the bases, the coherences of DD-CASSI and 3-D-CASSI are
very close, which indicates comparable quality of their sensing matrices. It is shown in Figure 7 that DD-CASSI and
3-D-CASSI also perform comparably on hyperspectral image
reconstruction accuracy, which is consistent with the theorem
discussed in the section "Understampling Systems."
Aside from DD-CASSI and 3-D-CASSI, the coherence values of the other systems have a consistent ranking on both the
PCA and K-SVD bases, which suggests that the relative quality of sensing matrices is not greatly affected by the bases, if
they represent the sparse structure of the data well. This is also
indicated by the reconstruction results in Figure 8.
The reconstruction performance of the four undersampling
systems is displayed for the 610-nm channel of an example
image in Figure 9. The result of PMVIS exhibits blocking

(b)

(c)

(d)

Figure 9. A comparison of reconstructed results for the four undersampling systems. All of the results are shown at the 610-nm channel. (a) PMVIS
PSNR = 16.6845 db, (b) SD-CASSI PSNR = 18.0859 db, (c) DD-CASSI PSNR = 29.8178 db, and (d) 3-D-CASSI PSNR = 32.5659 db.
104

IEEE SIgnal ProcESSIng MagazInE

September 2016

Table of Contents for the Digital Edition of Signal Processing - September 2016