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Compressive Spectral Image Sensing, Processing, and Optimization
1. Compressive Spectral Image
Sensing and Optimization
Gonzalo R. Arce
Charles Black Evans Professor
University of Delaware
Newark, Delaware, USA 19716
Distinguished Lecturer Series
Aristotle University of Thessaloniki
March 14, 2014
2. Optical imaging and spectroscopy discovers the characteristics of scenes and
materials by capturing EM radiation in the 0.01 to 10000 nm spectrum window.
Sensitive not only to spatial and spectral information of a scene, but also to
polarization, tomographic, angular, and even chemical composition.
Multidimensional imaging provides dimension preserving mappings
y = Hf,
where H is the “forward model“, characterizing the focal plane data.
Optical coding shapes H, under some criteria, to match the computational tools
of inverse problems to significantly improve the overall imaging performance.
MURA Hadamard SPECT
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 2 / 50
3. THE SPECTRAL IMAGING PROBLEM?
Push broom spectral imaging: Traditional approach, expensive, low
sensing speed, senses N × N × L voxels
Optical Filters; Again senses N × N × L voxels; limited by number of
colors
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 3 / 50
4. Compressive Spectral Imaging (CASSI), New revolutionary method, CS
makes a significant difference, senses only N2
N × N × L
Coded apertures are
the only variable
element.
Coded Apertures are
the key elements in
CASSI.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 4 / 50
5. WHY IS THIS IMPORTANT?
Remote sensing and surveillance
Visible, NIR, SWIR
Devices are challenging in NIR and SWIR: cost, size,
resolution, cooling
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 5 / 50
6. WHY IS THIS IMPORTANT?
Medical Imaging: Vascular tissue imaging, angiography,
contrast agent
paint restoration
Compressive Spectral Imaging
Reduce sensing complexity
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 6 / 50
7. Introduction
Compressive sensing introduced by Donoho†, Candès‡, Tao,
Romberg...
Measurements are given by y = Φx
y
xΦ
M x 1
Measurements
M x N
Sampling Operator
N x 1
Sparse Signal
A sparse solution x is recovered from y by solving the inverse
problem
ˆx = min
x
x 1 s.t. y = Φx.
†
Donoho. IEEE Trans. on Information Theory. December 2006.
‡
Candès, Romberg and Tao. IEEE Trans. on Information Theory. April 2006.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 7 / 50
8. Introduction
Measurements are given by y = Φx
y ΨΦ α
x
A sparse solution α is recovered from y by solving the inverse
problem
ˆα = min
α
α 1 s.t. y = ΦΨα.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 8 / 50
9. Introduction
Datacube
f = Ψθ
Compressive Measurements
g = HΨθ + w
Underdetermined system of equations
ˆf = Ψ{min
θ
g − HΨθ 2 + τ θ 1}
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 9 / 50
12. Undetermined system of equations: N × M × L Unknowns and
N(M + L − 1) Equations.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 12 / 50
13. Computational Model
A single shot compressive measurement across the FPA:
Gnm =
L−1
i=0
Fi(n+m)mTi(n+m) + win
F is the N × M × L datacube
T is the binary code aperture
w is the sensing noise
In vector form, the FPA measurement can be written as
g = Hf + w
H accounts for the aperture code and the dispersive element.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 13 / 50
14. CASSI Multishot Matrix Model
g0
g1
...
gk−1
=
H0
H1
...
Hk−1
f, (1)
g = Hf, (2)
where H ∈ {0, 1}N(M+L−1)K×NML
.
Multi-shot coding done by using multiple coded apertures or a
Digital-Micromirror-Device (DMD)
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 14 / 50
15. Matrix CASSI representation g = Hf
Data cube:
N × N × L
Spectral bands: L
Spatial resolution:
N × N
Sensor size
N × (N + L − 1)
V=N(N+L-1)
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 15 / 50
16. Coded Aperture Optimization: Sensing and
Reconstruction
Restricted Isometry Property in CASSI
Given
g = HΨ
A
θ with |θθθ| = S
The RIP of the CASSI matrix A is defined as the smallest constant δs
such that
(1 − δs) ||θθθ||2
2 ≤ ||Aθθθ||2
2 ≤ (1 + δs) ||θθθ||2
2, (3)
where
δs = max
T⊂[n],|T|≤S
λmax A|T||T| − I (4)
A|T||T| = AT
|T|A|T|, A|T| is a m × |T| matrix whose columns are equal to
|T| columns of the CASSI matrix A, and λmax (.) denotes the largest
eigenvalue.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 16 / 50
17. Let the entries of ΨΨΨ be Ψj,k and let the columns of ΨΨΨ be [ψψψ0, . . . ,ψψψn−1].
The entries of A|T| can be written as
A|T| jk
= (HψψψΩk
)j = hT
j ψψψΩk
=
L−1
r=0
ti
j−rN
Ψj+r(N ),Ωk
for j = 0, . . . , m − 1, k = 0, . . . , |T| − 1, where i = j/V , N = N2 − N,
and Ωk ∈ {0, . . . , n − 1}. The entries of A|T||T| can be expressed as
A|T||T| jk
=
K−1
i=0
V−1
=0
L−1
r=0
L−1
u=0
ti
−rN
ti
−uN
Ψ +rN ,Ωj
Ψ +uN ,Ωk
(5)
for j, k = 0, . . . , |T| − 1.
Note that the coded aperture products ti
−rN
ti
−uN
determine the
eigenvalues of A|T||T|, and consequently they determine the constant
δs in the RIP.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 17 / 50
18. Boolean Coded Apertures
An optimal ensemble of four 64 × 64 boolean coded apertures.
Each spatial
coordinate in the
ensemble contains
only one 1-value
entry.
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19. Bernoulli Coded Apertures
An ensemble of four 64 × 64 Bernoulli coded apertures.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 19 / 50
20. Performance of coded apertures
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21. Original datacube Boolean (40.4dB) Unsigned grayscale (31.2dB)
Binary (27.7dB) Hadamard (27.7dB) Signed grayscale (22.7dB)
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22. Original datacube Boolean Unsigned grayscale
Binary Hadamard Signed grayscale
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23. Coded Aperture Optimization: Spectral Selectivity
UAV sensor requirements depend on flight duration, range,
altitude, etc.
Need: Hyperspectral imaging that dynamically adapts to optimal
spectral bands.
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25. ((a)) Original ((b)) 12 Random Codes ((c)) 9 Optimal Codes
The resulting spectral data cubes are shown as they would be viewed by a Stingray
F-033C CCD Color Camera.
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26. ((a)) Random Code ((b)) Original ((c)) Optimal Code
((d)) Random Codes ((e)) Optimized Codes
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27. ((a)) Random ((b)) Optimized
((c)) ((d))
Differences between the original and the reconstructed 3rd
spectral channel (479nm)
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 27 / 50
28. ((a)) Original ((b)) 12 Random Codes ((c)) 12 Optimized Codes
The resulting spectral data cubes are shown as they would be viewed by a Stingray
F-033C CCD Color Camera.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 28 / 50
29. Coded Aperture Optimization: Image Classification
Goal: classification of a spectral scene using
Compressive measurements
Optimal code aperture
Sparsity signal model
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30. Given the FPA measurements g, every test pixel fi belongs to one of
the P known classes
{H(1)
, H(2)
, ..., H(P)
}
0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spectrum Band
Glutamine
0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spectrum Band
Histidine
0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spectrum Band
Isoleucine
If a pixel fi ∈ H(k), its spectral profile lies in a low-dimensional
subspace spanned by the training samples: {s
(k)
j }j=1,...,Np
fi ≈ [s
(k)
1 , s
(k)
2 , ..., s
(k)
Nk
][α
(k)
1 , α
(k)
2 , ..., α
(k)
Np
]T
= S(k)
α(k)
where, α(k) is a sparse vector.Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 30 / 50
31. Sparsity Model
Combining all class sub-dictionaries
fi ≈ [S(1)
, ..., S(k)
, ..., S(P)
][α(1)
, ..., α(k)
, ..., α(P)
]T
= Sα
Ideally, if fi ∈ H(k), then
α(j) = 0; ∀j = 1, ..., P; j = k.
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 31 / 50
36. NEW FAMILY OF CODED APERTURES
Boolean Spectrally Selective
Super-resolution Colored
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37. Colored coded aperture model
Colored coded aperture is a color filter array
Each entry is a wavelength selective color filter
3D Mask model has the same dimensions than the objective discrete data cube
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 37 / 50
38. Linear dispersion and focal plane array integration
Linear shifting operation
Focal plane array (FPA) projections
The number of pixels of the FPA
detector is N(N + L − 1) N2
L
(size of the spectral data cube)
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40. 4 Colors Random Code
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 40 / 50
41. Restricted Isometry Property of Colored CASSI
A = HΨΨΨ, fff = ΨΨΨθθθ, ΨΨΨ = W ⊗ ΨΨΨ2D
Definition
(1 − δs) ||θθθ||2
2 ≤ ||Aθθθ||2
2 ≤ (1 + δs) ||θθθ||2
2,
δs = max
T ⊂[N2L],|T |≤S
||AT
|T |A|T | − I||2
2,
A|T |, |T | columns of A indexed by the set T
δs = max
T ⊂[N2L],|T |≤S
λmax A|T ||T | − I
A|T | ir
= hiψψψΩr
=
L−1
k=0
t i
k mi −kN
Ψmi +k(N ),Ωr
(hi )j =
t i
kj i− i V−kj N
, if i − i V = j − kj N
0, otherwise,
A = HΨΨΨ2D
A = H W ⊗ ΨΨΨ2D
Gonzalo R. Arce () Compressive Spectral Image Sensing Oct., 2013 41 / 50