Two-Dimensional FRI Signal Reconstruction Using
Blind Deconvolution
Aniruddha Adiga∗ , Satish Mulleti∗ , Prasad Sudhakar† , Chandra Sekhar Seelamantula∗
∗ Department
of Electrical Engineering
Indian Institute of Science, Bangalore, India 560012
Email:{aniruddha, satish.mulleti}@ee.iisc.ernet.in, chandra.sekhar@ieee.org
† Medical Image Analysis Lab
GE Global Research, Bangalore, India 560066
Email: firstname.lastname@ge.com
Abstract—Finite-rate-of-innovation (FRI) signal model, which
has found usage in a wide range of applications, requires that the
continuous-time form of the underlying template signal be known.
In this paper, we consider the situation where the template is
unknown but comes from a parametric family, and only its
sparse linear combination is involved in signal generation. Given
the sampled version of such a signal, we estimate the model
parameters (the coefficients of linear combination as well as
the template signal) by formulating it as a blind deconvolution
problem. Our two-stage alternating `p − `2 minimization scheme
exploits the excitation sparsity and parametric form of the
template signal for estimation. The excitation locations are superresolved using a sub-pixel estimation step. Experimental results
show that even in the presence of noise, the proposed method
can estimate the model parameters with high accuracy.
I. I NTRODUCTION
In some imaging applications, the underlying scene consists
of isolated point excitations that are sparse. Examples include
biological imaging where the fluorophores are tagged [1],
astronomical imaging [2], quantitative optics [3], etc. The
locations of the point excitations are crucial to the study and
hence they have to be estimated with high accuracy. Most
imaging systems have finite-sized apertures, which limit the
resolution of the images. Mathematically, the aperture acts as
a blurring optical filter, which can be modeled by a point
spread function (PSF). When a scene is blurred by the PSF
of the imaging device, the point excitations are blurred and
the information about their true locations is lost. Moreover,
when the measurements are digitized, it further undergoes
spatial sampling thereby increasing the spatial ambiguity of
the excitation locations. Thus, it becomes essential to undo the
blurring of the PSF in order to make quantitative observations
about the image. Baboulaz and Dragotti [4] showed that the
point source excitation convolved with a known PSF is a
finite-rate-of-innovation (FRI) signal and showed that a high
resolution signal reconstruction is possible from low-resolution
images.
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
A. Related Work
Vetterli et al. [5] first introduced the FRI signal model, characterized by a finite number of parameters per unit interval:
f (t) =
L
X
a` h(t − t` ),
(1)
`=1
where {a` , t` }L
`=1 are unknown and the pulse h(t) is known.
Typical pulse shapes considered are Dirac impulses, differential Dirac impulses, piecewise polynomials, nonuniform
splines, [5] and arbitrary known pulses [6]. To sample and
efficiently reconstruct FRI signals, sampling kernels such as
the sinc, Gaussian [5], kernels that satisfy generalized StrangFix conditions [7], [8], causal exponential kernels [9], [10],
and sum-of-sincs kernel [6], [11] have been considered. The
reconstruction relies on high-resolution spectral estimation
tools such as the annihilating filter method [12], matrix pencil
[13], algebraically coupled matrix pencil method (ACMP)
[14], etc. FRI method has been extended to reconstruct a
sum of 2-D Dirac impulses, lines and polygons [15], 2-D
polynomials, n-dimensional Diracs and convex polytopes [4],
step edges [16], etc.
The FRI signal in (1) can be alternatively written as f (t) =
L
X
h(t)∗
a` δ(t−t` ). Hence, estimating {a` , t` }L
`=1 from f (t)
`=1
with a known h(t) can be seen as a deconvolution problem.
In situations where h(t) is unknown, f (t) may not be an FRI
signal. Matusiak and Eldar proposed a multichannel sampling
approach to reconstruct signals composed of finite-duration
unknown pulses [17].
B. Our Contribution
We consider 2-D FRI signals of the form
f (x, y) =
L
X
a` g(x − x` , y − y` ),
(2)
`=1
where the pulse g(x, y) is parametric and the parameters are
unknown. Such signals are practically more relevant than a
completely known pulse case. For example, in an imaging
system g(x, y) represents the PSF and the excitation signal is
modeled as
L
X
a` δ(x − x` , y − y` ). Based on the acquisition
`=1
setup, we may know that the shape of the PSF resembles
a 2-D Gaussian but that its covariance matrix is unknown.
Moreover, we may have access only to the samples of f (x, y).
We propose a blind-deconvolution-based two-stage alternating
minimization approach to estimate the parameters of the
sampled 2-D FRI signal by assuming a functional prior on
the PSF and exploiting sparsity of the excitation signal. As
we are given a sampled signal, (x` , y` )L
`=1 are estimated to
the nearest locations on the sampling grid.
In applications where precise location estimates are required
(e.g., optical deflectometry [3]), we propose a local gradient
descent stage around the on-grid estimates, using the estimated
PSF, to obtain off-grid locations. As a first step towards generic
parametric kernels, we only consider Gaussian kernels in this
paper.
II. P ROBLEM S TATEMENT AND P ROPOSED M ETHOD
Consider the 2-D FRI signal
f (x, y) =
L
X
a` g(x − x` , y − y` ),
(3)
`=1
1
−1
T
where
h 2 g(x, y)
i = exp − 2 [x y]Σ [x y] , with Σ =
σx ρσx σy
. In (3), {a` , x` , y` }L
`=1 , σx , σy and ρ are the
ρσx σy σy2
unknown parameters and the model order L is known. Given
uniform samples f (nTx , mTy ), (n, m) ∈ Z × Z of f (x, y),
with known Tx and Ty , the goal is to estimate the unknown
parameters.
A. Blind Deconvolution
The FRI signal given in (3) is a sum of L scaled Gaussian
pulses placed at different locations. Assuming linearity and
shift-invariance, we write
f (x, y) = g(x, y) ∗
L
X
a` δ(x − x` , y − y` ) .
(4)
`=1
|
{z
e(x,y)
In most practical systems, we have access only to the
samples of f (x, y), which is represented as,
f (nTx , mTy ) = g(x, y)∗
L
X
`=1
.
x=nTx ,y=mTy
(5)
When the impulses lie on the sampling grid, that is (x` =
n` Tx , y` = m` Ty ), (5) can be written as
f (nTx , mTy ) = g(nTx , mTy )∗
L
X
a` δ((n−n` )Tx , (m−m` )Ty ).
`=1
(6)
We consider the above model for deconvolution and develop
an alternating projections based algorithm for estimating the
samples of g(x, y) and e(x, y). We assume a normalized
sampling grid, that is, Tx = Ty = 1.
B. 2-D Alternating `p − `2 Projections Algorithm
As the deconvolution problem is ill-posed, we need to
impose priors on the unknown parameters. For instance, in
the signal model considered in (6), it is known that e(x, y) is
a point source excitation consisting of L Dirac deltas in two
dimensions, where L M × N and thus, e(n, m) is sparse.
We also assume that the PSF (filter) is Gaussian. An alternating
`p − `2 projections algorithm (ALPA) is presented in [23]
for deconvolution of 1-D signals (speech, to be specific),
wherein the algorithm is developed with the assumption that
the excitation is sparse with no particular model for the filter.
However, it is assumed that the filter is predictable from the
measurement. We develop an algorithm on similar lines as
ALPA for the 2-D deconvolution problem.
Ideally, the problem of factorizing a given signal into
excitation and filter is posed as a joint optimization problem,
which is hard to solve. Hence, the approach taken is to first
estimate one of the signals keeping the other fixed and then
to switch their roles. The process of estimation is refined
iteratively until a suitable stopping condition is satisfied.
The model in (6) is represented in matrix form as
}
We observe that the parameters σx , σy and ρ are encoded in
g(x, y), while {a` , x` , y` }L
`=1 are encoded in e(x, y). Hence,
the problem at hand is to obtain a suitable decomposition from
which the parameters can be estimated. Due to the structure
of (4), the decomposition can be posed as a deconvolution
problem. Since both pulse and excitation are not known, it
becomes a specific case of blind deconvolution.
The problem of blind deconvolution arises in several image
processing applications, and a review of available techniques
is presented in [18], [19]. In many situations such as starfield imaging [20], fluorescence microscopy [1], neural signal
processing [21], the excitation is inherently sparse. Since blind
deconvolution involves estimation of two unknown quantities,
the joint estimation is a hard problem to solve (non-convex)
and hence alternating estimation schemes are a viable alternative [22].
a` δ(x − x` , y − y` )
y = He,
(7)
where H denotes a block Toeplitz matrix associated with
the PSF H ∈ RN ×N , e and y are the vectorized versions
of the excitation E ∈ RM ×M and measurement Y ∈
R(M +N −1)×(M +N −1) , respectively. To begin with, we start
with the estimation of a sparse e, wherein a cost function
is formulated to incorporate trade-off between the sparsity of
the signal and least-squares fit of the approximation with the
signal. As for the sparsity, `p −norms have been extensively
employed as sparsity-promoting regularizers [24], [25], [26]
and hence the optimization problem for estimation of e is
written as,
e = arg min ks − Hek22 + λkekpp ,
e
(8)
where 0 ≤ p ≤ 1. For the case p = 1, the problem (8)
is called LASSO [27] or basis pursuit denoising [28]. The
ability of this formulation in producing sparse solution has
been extensively studied in the literature. Also, availability of
solvers for the convex non-differentiable `1 −norm has made
it a popular sparsity promoting functional. Ideally, one would
like to solve for the case where p = 0, but solving such a
cost function is NP-hard. The possibility of using 0 < p < 1
for obtaining sparser results is discussed in [20], [29], [24],
[25]. The intuition can be drawn by considering a unit ball
in R2 for kekp . As p → 0, the unit ball approaches the unit
ball surface of the indicator function of `0 −norm (indicator
function indicates whether an element of vector is non-zero or
not). In probabilistic modeling of data, point-source excitations
given in (5), where L M × N , can be modeled using
low order generalized p−Gaussian (gpG) probability density
function as shown for star-field images in [20]. It can be shown
that, with this assumption, the maximum a posteriori estimate
of e from measurements corrupted by AWGN is a vector that
minimizes the cost function, which is similar to the function
in (8).
A problem with `p −norms for 0 < p < 1 is that they
are non-convex and pose difficulty in optimization. Also, the
optimization could lead to local minima. However, a mathematically tractable majorization-minimization (MM)-based
algorithm is presented in [26], where the `p -norm is majorized
with weighted `2 . It is shown that, iteratively solving the new
cost function will lead to the minimizer. We take a similar
approach and modify the cost in (8) as,
ẽ(i) = arg min ks − Hek22 + λeT W(i) e.
e
(9)
The PSF is initialized with a 2-D Gaussian of arbitrary variance. A procedure to chose λ is given in [30]. The quadratic
cost in (9) is differentiable and the solution for the ith iteration
is
−1
ẽ(i) = HT H + λW(i)
HT s.
(10)
In order to ensure that the entries of ẽ do not become small,
we normalize it to have unit energy. The new excitation is
(i)
denoted as e. W(i) is a diagonal matrix with entries wjj =
p−2
(i−1) ej
. This choice of weights results in ill-conditioned
solution of least-squares problem:
h̃(i) = arg min kY − E(i) hk22 ,
h
(13)
and then fitting a 2-D Gaussian to this H̃ (h̃ rearranged to get
a H̃ ∈ N × N ) to get H(i) . A new H(i) matrix is constructed
from H(i) and plugged into (9). The modified cost function is
ẽ(i+1) = arg min ks − H(i) ek22 + λeT W(i+1) e.
e
(14)
The process of alternating between the equivalent models is
repeated until a suitable stopping criterion is met. The criterion
we use in ALPA to stop is when ke(i) −e(i−1) k22 ≤ Γ, where Γ
is a predetermined threshold. Alternatively, one could run the
algorithm for a prefixed number of iterations. The procedure
is presented in Algorithm 1.
The locations are estimated by thresholding the sparse excitation and picking only those locations above the threshold.
Algorithm 1 ALPA: Algorithm to estimate sparse excitation
and PSF
Input: measurement vector Y
Outputs: excitation E(i) and filter H(i)
Stopping criteria: ke(i) − e(i−1) k2 ≤ T or i = fixed number of iterations R.
Initialization: i = 0
1. Choose size of PSF = L.
1. H(0) = M × M filter.
2. H(0) = Block Toeplitz(H(0) ).
3. W(0) = IN ×N .
4. λ = 1.
while stopping criterion false do
Step 1: Excitation optimization step:
ẽ(i) = arg min ks − H(i−1) ek22 + λeT W(i) e.
e
Step
Step
Step
Step
2:
3:
4:
5:
Energy normalization: e(i) = ẽ(i) /kẽ(i) k.
Rearrange e(i) into matrix E(i) .
Construct E(i) = Block Toeplitz(e(i) ).
Filter optimization step:
h̃(i) = arg min kY − E(i) hk22 .
h
Fit a 2D Gaussian function to reshaped h̃(i) to get H(i) .
Step 6: Update H(i) = Block Toeplitz(H(i) ).
end while
(i−1)
matrices when ej
≈ 0. Hence, in practice, a small positive
regularization constant is added to the weights:
p/2−1
(i)
(i−1) 2
wjj = (ej
) +
.
(11)
C. Gradient Descent for Obtaining Off-Grid Estimates
A method to chose is given in [26].
In the next step, we use this estimate of e to compute an
approximation to h. Using commutativity of convolution, (6)
is expressed equivalently as
Let τ̂ ` = (x̂` , ŷ` ) and Σ̂` , 1 ≤ ` ≤ L denote the on-grid
locations and the covariances of the Gaussian provided by
ALPA, and let g(τ̂ k , Σ̂k ) be the 2-D Gaussian with covariance
Σ̂k centered at τ̂ k . To estimate the off-grid
PL location τ̃ ` =
(x̃` , ỹ` ), we build a new signal ŝ` = s − k=1k6=` g(τ̂ k , Σ̂k )
and solve
s = E(i) h(i) ,
τ̃ ` = arg min kŝ` − g(τ , Σ̂` )k22 ,
(12)
where E(i) is the block Toeplitz convolution matrix constructed from E(i) and h is a vectorized version of PSF. We
determine h(i) to minimize the energy of the error between
approximation and signal with the constraint that H is a 2-D
Gaussian. We solve this by first obtaining the pseudo-inverse
τ
(15)
by a gradient descent algorithm with initialization τ = τ̂ ` .
This is repeated for each ` = 1, 2, · · · , L. The new signal
ŝ` ensures that contribution from only the `th Gaussian
shape is considered for computing the energy in (15), thereby
improving the accuracy of the off-grid estimates.
III. S IMULATION R ESULTS
A. Monte Carlo Analysis
In order to determine the performance in noise of the
algorithm a deconvolution experiment was carried out for
SNRs varying from 0 dB to 50 dB in steps of 5 dB, with
500 noise realizations for each SNR. The results are shown
in Figure 3. It is observed that, for most cases the Euclidean
distance between (xl , yl ) and (x̃l , ỹl ) is below the half-sample
error and by using gradient-descent, the error can be reduced
further.
B. Sensitivity to Proximity
The ability of our approach to recover the locations accurately is also dependent on how far apart the Gaussians are
separated in space. In the noiseless case, with two isotropic
Gaussians (σx = σy = σ), by varying the distance between
them along one axis, we experimentally found that the method
fails to resolve them when the Euclidean distance between
the two peaks falls below 2σ. In such situations, the filter is
trivially estimated to be the signal itself.
0.5
y
y
0
5
10
15
−5
0.5
5
10
0
5
x
(a)
10 15
15
−5
0
−4
5
x
(b)
10 15
y
0.4
2
2
0.6
0
0.4
2
0.2
0
x
(c)
0.8
−2
0.6
0
4
−4 −2
0
0
−4
0.8
−2
y
1
−5
0
4
−4 −2
4
0.2
0
x
(d)
2
4
−4
0
2
4
−4 −2
0
0.8
0.6
y
−2
y
The algorithm is validated on synthetic data shown in
Figure 1(a), which consists of three Gaussians located at
(xl , yl ) = (2.6, 4.0), (5.2, 1.6), (7.3, 7.0), with corresponding
amplitudes al = {1.0, 0.8, 0.75}. The standard deviations
σx = 1.2 and σy = 0.8; ρ = 0.48.
As for the parameters of the algorithm, p = 0.1 was
considered and λ = 1 was found to be optimal based on
experimentation. As an initial estimate of the filter, a Gaussian
with σx = σy = 0.6 and ρ = 0 was considered. The
algorithm was validated on noisy data at two SNR levels
of 20 dB and 5 dB and results are illustrated in Figures 1
and 2. As discussed previously, the deconvolution algorithm
can only estimate with a precision of ±T /2 about the actual
location, however, the estimates are refined using the gradientdescent method. In the former case, the on-grid estimates
were estimated as (x̂l , ŷl ) = (2.5, 4.0), (5.0, 1.5), (7.5, 7),
which is within the half-sample error, and by using gradientdescent algorithm the estimates were refined to obtain
(x̃l , ỹl ) = (2.59, 4.01), (5.13, 1.57), (7.38, 6.99). The amplitude and correlation parameter ρ were estimated as â =
{1, 0.77, 0.68} and ρ̂ = 0.49, respectively. As for the
SNR = 5 dB case, the on-grid estimates were found to
be (x̂l , ŷl ) = (2.5, 4.0), (5.0, 1.5), (7.5, 7); however, by using gradient-descent algorithm, the estimates got refined to
(x̃l , ỹl ) = (2.61, 4.03), (5.14, 1.56), (7.43, 6.99). The amplitude and correlation parameter ρ were estimated as â =
{1, 0.74, 0.68} and ρ̂ = 0.46, respectively.
The signal-to-reconstruction error ratio for 20 dB and 5 dB
cases were 19 dB and 18.5 dB, respectively. The closeness
in signal-to-reconstruction error ratio indicates the denoising
ability of ALPA. Although, no explicit denoising is applied
in the algorithm, the sparsity constraint provides implicit
denoising of excitation while fitting a Gaussian over the noisy
filter estimate denoises the filter.
1
−5
0.4
0.2
0.6
0.4
5
0.2
10
0
0
5
10
0 2 4
x
x
(e)
(f)
Fig. 1. Deconvolution results (SNR = 20 dB). (a) Signal consisting of three
Gaussian functions located at (2.6, 4.0), (5.2, 1.6) and (7.3, 7) with σx =
1.2, σy = 0.8 and amplitudes {1, 0.8, 0.75}, (b) noisy signal, (c) original
filter, (d) filter used as initialization for algorithm, (e) estimated filter with
σ̂x = 1.21, σ̂y = 0.79, and (f) estimated excitation (on the grid) with Dirac
impulses at (2.5, 4.0), (5, 1.5), (7.5, 7.0) and amplitudes as {1, 0.77, 0.68}.
Using gradient-descent optimization the subpixel estimates obtained as (2.59,
4.01), (5.13, 1.57) and (7.38, 6.99).
IV. C ONCLUSION
We considered the estimation of FRI signal parameters from
its sampled version by posing it as a blind deconvolution
problem and proposed an alternating minimization approach
called ALPA. The method provides estimates of the filter and
also the on-grid locations of the excitation. A gradient-descent
stage was used to obtain the sub-pixel excitation locations.
The method demonstrates that by imposing proper structure
on the unknowns, they can be recovered even in high noise
conditions. With robust estimation of signal parameters, the
algorithm also inherently performs denoising.
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