SINE
x
Super-Resolution of Complex Exponentials from Modulations with Unknown Waveforms
Dehui Yang, Gongguo Tang, and Michael B. Wakin
Department of Electrical Engineering and Computer Science, Colorado School of Mines
s(x)
SIgnals and NEtworks
1
Subspace Model and Atomic Norm Minimization
A subspace model for g j
1. A simple example
I
Rewrite the observation
y(n) =
J
X
cj a(τj )
H
H
e n b n hj
j=1
=
* J
X
+
cj hj a(τj )
H
H
, b n en
j=1
=:
H
X o , b n en
we use CVX to solve the optimization problem (SDP)
I set N = 64, J = 3 and K = 4, randomly generate the locations of
1
J spikes on [0, 1) under the minimum separation condition ∆τ = N
I build B with entries generated randomly from the standard
Gaussian distribution
I hj is also generated using i.i.d. real Gaussian random variables and
is then normalized
I
1
i2π(−2M )τ
i2π(2M )τ T
where a(τ ) = e
.
··· 1 ··· e
1. Super-resolution with unknown point spread functions
cj δ(t − τj ) ∗ gτj (t)
j=1
3D single-molecule microscopy
I non-stationary blind deconvolution of seismic data
I
2. Parameter estimation in radar imaging
y(t) =
J
X
k
e st i m at e of g 1
2
1
0.6
10
20
30
40
S amp l e i n d e x
50
60
2
g2
1.5
e st i m at e of g 2
1
0.5
0.4
0
0.2
dual p olynomial
locations of spikes
0
0
0.2
0.4
0.6
0.8
10
20
30
40
S amp l e i n d e x
50
60
2
g3
1.5
e st i m at e of g 3
1
0.5
0
1
10
20
30
40
S amp l e i n d e x
50
60
τ
2. Phase transition
Number of s amples N = 64
Dimens ion of s ubs pace K = 4
16
k
1
1
90
14
We solve
minimize kXkA
H
subject to y(n) = hX, bnen i, n = −2M, · · · , 2M.
Number of s pikes J = 4
1
0.8
12
0.6
10
8
0.4
6
4
0.2
90
80
0.8
70
60
0.6
50
0.4
40
30
0.2
20
80
0.8
70
60
0.6
50
0.4
40
30
0.2
20
2
10
5
cj e
i2πνj t
x(t − µj )
j=1
New Model
Consider the observation model:
J
X
−i2πnτj
y(n) =
cj e
g j (n), n = −2M, . . . , 2M.
j=1
Given samples {y(n)}, the goal is to
I super-resolve {τj }
I recover {cj }
I recover samples of the unknown waveforms
g j (n)
This problem is severely ill-posed
I number of samples N := 4M + 1
I number of unknowns JN + 2J
email:
ck ,τk ,khk k2=1
g1
dyang@mines.edu
P2M
10
0
15
2
K : Dimens ion of s ubs pace
i2πnτ
bn as the dual polynomial with
Denoting q(τ ) = n=−2M λ(n)e
λ being the dual optimizer, {τj } are localized by selecting out the
corresponding values of τ such that kq(τ )k2 = 1.
Main Result
If the following conditions
1
1. ∆τ = mink6=j |τk − τj | ≥ M , M ≥ 64,
2. bn are i.i.d.
samples
from
a
distribution
F
satisfying
H
2
i) E bb = I K ; ii) max1≤p≤K |b(p)| ≤ µ for b ∈ F,
3. hj drawn i.i.d. from the uniform distribution on the
K−1
complex unit sphere CS
,
are satisfied, then there exists
some
some
C
such
that
M JK
2 MK
M ≥ CµJK log
log
δ
δ
is sufficient to guarantee that we can recover X o with
probability at least 1 − δ.
10
4
6
0
8
2
J : Number of s pikes
4
6
0
8
K : Dimens ion of s ubs pace
3. A practical example
I
set J = 3 and generate the locations of {τj } uniformly at random
1
between 0 and 1 under the minimum separation ∆τ = M
1
√
I g j (n) are samples of the Gaussian waveform gJ (t) =
2πσJ2
2
σJ
e
t2
− 2
2σJ
∈ [0.1, 1]
with unknown variance
I B is a rank-5 approximation of the dictionary D g , where
D g = g σJ =0.1 g σJ =0.11 g σJ =0.12 · · · g σJ =1
0.4
Waveforms of columns of B
1
B (: , 1)
B (: , 2)
B (: , 3)
B (: , 4)
B (: , 5)
0.3
0.2
0.8
kq(τ )k 2
y(t) =
J
X
Lift the non-convex problem into a convex program
Define the atomic norm associated with the set of atoms
H
K×1
A = ha(τ ) : τ ∈ [0, 1), khk2 = 1, h ∈ C
as
kXkA = inf {t > 0 : X ∈ tconv(A)}
(
)
X
X
H
=
inf
|ck | : X =
ck hk a(τk )
.
3
0
J : Number of s pikes
Motivation
0.8
kq(τ )k 2
I
Magn i t u d e
I single-molecule
N : Number of s amples
H
K×1
g j = Bhj , B = b−2M , · · · , b2M , bn ∈ C
Magn i t u d e
microscopy
I medical imaging
I radar imaging
I astronomy
I
Magn i t u d e
Enhancing the resolution limit of sensing systems
Numerical Simulations
N : Number of s amples
Super-Resolution
0.1
0
0.6
0.4
−0.1
0.2
dual p olynomial
locations of spikes
−0.2
−0.3
−0.5
0
0.5
0
0
0.2
0.4
0.6
0.8
1
τ
Reference
Y. Chi, “Guaranteed blind sparse spikes deconvolution via lifting and
convex optimization,” arXiv preprint arXiv:1506.02751, 2015.