# A Weighted Generalized Coherence Approach for Sensing Matrix Design

###### Abstract

As compared to using randomly generated sensing matrices, optimizing the
sensing matrix w.r.t. a carefully designed criterion is known to lead to
better quality signal recovery given a set of compressive measurements. In
this paper, we propose generalizations of the well-known mutual coherence
criterion for optimizing sensing matrices starting from random initial
conditions. We term these generalizations as bi-coherence or tri-coherence and
they are based on a criterion that discourages any one column of the sensing
matrix from being close to a sparse linear combination of other columns. We
also incorporate training data to further improve the sensing matrices through
weighted coherence, weighted bi-coherence, or weighted tri-coherence criteria,
which assign weights to sensing matrix columns as per their importance. An
algorithm is also presented to solve the optimization problems. Finally, the
effectiveness of the proposed algorithm is demonstrated through empirical
results.

Index Terms—  coherence, compressed sensing, prior information, weights,
sensing matrix design

##  1 Introduction

Compressed sensing is a signal processing technique for efficiently acquiring
and reconstructing a signal $\boldsymbol{x}\in\mathbb{R}^{n}$, by finding
solutions to under-determined linear systems [1]. For reconstruction, the
sparsity or compressibility of signals in a dictionary $\boldsymbol{\Psi}$
(typically a discrete cosine transform (DCT) or a wavelet transform) is
utilized. Such signals can be recovered from $m\ll n$ (noisy) linear
projections of the form $\boldsymbol{y}\triangleq\boldsymbol{\Phi
x}+\boldsymbol{\eta}$ (where $\boldsymbol{\Phi}$ is a $m\times n$ sensing
matrix and $\boldsymbol{\eta}$ is a noise vector with $m$ elements), with
rigorous theoretical guarantees at even sub-Nyquist sampling rates [2][3]. The
quality of the reconstruction depends on the choice of the sensing matrix
$\boldsymbol{\Phi}$ and on how sparse the signal is in $\boldsymbol{\Psi}$.
Successful recovery with theoretical guarantees is possible with efficient
algorithms like Basis pursuit which minimizes the $\ell_{1}$ norm of the
sparse coefficient vector ($\boldsymbol{\theta}$) such that
$\|\boldsymbol{y}-\boldsymbol{\Phi\Psi\theta}\|_{2}\leq\epsilon$ where
$\|\boldsymbol{\eta}\|_{2}<\epsilon$ [1]. One well-studied sufficient
condition on $\boldsymbol{\Phi}$ for guaranteeing good quality reconstructions
is the Restricted Isometric Property (RIP) [4]. It can also be shown that
matrices drawn from Gaussian or Bernoulli distributions satisfy RIP with high
probability [1].

At the same time, it has been shown that optimizing $\boldsymbol{\Phi}$ w.r.t.
a carefully designed ‘matrix quality measure’ leads to better-than-random
recovery. [5, 6, 7, 8, 9, 10, 11, 12, 13]. However, computing the Restricted
Isometric Constant (RIC), which characterizes the RIP, is known to be
computationally prohibitive. Hence, other measures have been proposed, such as
mutual coherence [4] or average mutual coherence as they are computationally
much more efficient. [14] also shows that improving the mutual coherence
improves a worst case bound on the reconstruction error. The mutual coherence
is defined as
$\mu_{max}\triangleq\|\boldsymbol{A}^{T}\boldsymbol{A}-\boldsymbol{I}\|_{\infty}$
wherein the columns of $\boldsymbol{A}\triangleq\boldsymbol{\Phi\Psi}$ are
taken to be unit-normalized. Further, to relax the $\ell_{\infty}$ norm,
approaches such as minimizing the average mutual coherence
$\mu_{avg}\triangleq\|\boldsymbol{A}^{T}\boldsymbol{A}-\boldsymbol{I}\|_{F}$
have also been proposed [6, 7, 9].

To further enhance the quality of reconstruction, some approaches have also
focused on incorporating prior information while designing the sensing matrix.
For example, in the work proposed in [15], a weighted average mutual coherence
minimization is proposed. The approach is used to reconstruct video frames
wherein the the recovered image from the previous frame measurement is used to
create weights for the subsequent frame to assign relative importance values
to different columns of the matrix $\boldsymbol{A}$. A piecewise-linear
estimator (PLE) with a Gaussian Mixture Model (GMM) prior was introduced in
[16], which was extended to the Statistical Compressive Sensing (SCS)
framework [17]. [18] evolved a design method based on the minimum mean squared
error (MMSE) criterion, imposing priors on signal (for natural images) and
using noise negative log-likelihood.

While most of the methods in literature have focused on optimizing the sensing
matrix based on certain bounds, [12] provides an approach to subsample from
the signal/image via the identification of modularity and submodularity
structures. Besides Restricted Isometric Property and mutual coherence, the
$\ell_{1}-\ell_{\infty}$ measure was proposed in [19]. The
$\ell_{1}-\ell_{\infty}$ measure provides tighter reconstruction bounds than
the coherence while being computationally easier to calculate than the
Restricted Isometric Constant. It is defined as
$\omega_{2}(\boldsymbol{Q},s)\triangleq\textrm{min}_{\boldsymbol{z}:\|\boldsymbol{z}\|_{1}/\|\boldsymbol{z}\|_{\infty}\leq
s}\|\boldsymbol{Qz}\|_{2}/\|\boldsymbol{z}\|_{\infty}$ where
$\boldsymbol{Q}\triangleq\boldsymbol{A}^{T}\boldsymbol{A}$ and $s$ is a
sparsity-parameter.

In this paper we propose a generalization of the mutual coherence criterion by
proposing our bi-coherence and tri-coherence approaches. Our proposed criteria
are easier to optimize on, as compared to [19] as well as the RIC. We also
take into account training data to further improve our sensing matrices
through our proposed weighted coherence, weighted bi-coherence and weighted
tri-coherence criteria. Finally, we propose algorithms to solve the proposed
maximization problems starting with random binary or random $U(0,1)$ sensing
matrices.

##  2 Acquisition Model

We consider designing sensing matrices for the block single pixel camera
acquisition model as described in [20]. In this architecture, compressive
measurements are acquired independently for each non-overlapping image block
of size $n\times n$. Consider there are ($m\ll n^{2}$) measurements available
per block. Thus, our measurements $\boldsymbol{y_{i}}$ corresponding to the
$i^{\textrm{th}}$ block $\boldsymbol{x_{i}}$ can be written as
$\boldsymbol{y_{i}}=\boldsymbol{\Phi}\boldsymbol{x_{i}}+\boldsymbol{\eta_{i}}$
where $\boldsymbol{x_{i}}$ denotes the $i^{\textrm{th}}$ vectorized block and
$\boldsymbol{\eta_{i}}$ denotes the measurement noise vector containing
elements drawn i.i.d. from $\mathcal{N}(0,\sigma^{2})$. Furthermore, we
consider that the elements of the sensing matrix $\boldsymbol{\Phi}$ are drawn
from either a Bernoulli $\\{0,1\\}$ or a $U(0,1)$ distribution.

##  3 Proposed Approach

We propose generalized coherence based approaches for matrix design. Coherence
is a matrix quality measure that discourages any one column
$\boldsymbol{A_{i}}$ of the matrix from being too similar to any other column
$\boldsymbol{A_{j}}$ [21, Fig. 1, Page 3]. The main intuition behind our
approach is to discourage any column vector of the sensing matrix from being
accurately expressible as a _sparse linear combination_ of other column
vectors. Optimizing on such a generalized coherence criterion ensures that
during sparse recovery, the picking of the non-zero atoms becomes easier. This
is because the sensing matrix column corresponding to that atom would now be
hard to replace using other column vectors. Based on this idea, we propose bi-
coherence and tri-coherence approaches.

###  3.1 Bi-coherence

The bi-coherence criterion is based on the idea that any column vector of the
matrix $\boldsymbol{A}=\boldsymbol{\Phi\Psi}$ should not be able to be written
as a linear combination of any other two column vectors of $\boldsymbol{A}$.
Thus, the following loss function results:

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j}^{n}\textrm{argmin}_{\alpha(i,j,k)}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k}}\alpha(i,j,k)\|_{2}^{2},$
(1)

where $\boldsymbol{A_{i}}$ denotes the $i^{\textrm{th}}$ column of
$\boldsymbol{A}$, $\boldsymbol{A_{j,k}}$ is the $n\times 2$ matrix formed by
concatenating the $j^{\textrm{th}}$ and $k^{\textrm{th}}$ columns of
$\boldsymbol{A}$, and $\alpha(i,j,k)$ is a $2\times 1$ coefficient vector. The
minimization part can be solved by assigning
$\alpha(i,j,k)=(\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{j,k}})^{-1}\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{i}}$.
Thus, the following optimization problem results:

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k}}(\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{j,k}})^{-1}\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}.$
(2)

###  3.2 Tri-coherence

Similar to bi-coherence, the tri-coherence criterion would discourage that any
column vector of the matrix $\boldsymbol{A}$ from being accurately expressible
as a linear combination of any other three column vectors of $\boldsymbol{A}$,
giving rise to the following problem:

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j,l>k}\textrm{argmin}_{\alpha(i,j,k,l)}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k,l}}\alpha(i,j,k,l)\|_{2}^{2},$

where $\boldsymbol{A_{j,k,l}}$ is the $n\times 3$ matrix formed by
concatenating the $j^{\textrm{th}}$, $k^{\textrm{th}}$ and $l^{\textrm{th}}$
columns of $\boldsymbol{A}$. Following steps similar to bi-coherence, we get

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j,l>k}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k,l}}(\boldsymbol{A_{j,k,l}}^{T}\boldsymbol{A_{j,k,l}})^{-1}\boldsymbol{A_{j,k,l}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}.$
(3)

###  3.3 Relationship with the average coherence minimization criteria

Similar to bi-coherence and tri-coherence, if we have a single column vector
$\boldsymbol{A}_{j}$, we will have the following loss function,

$\displaystyle\boldsymbol{A}=$
$\displaystyle\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i}\textrm{argmin}_{\alpha(i,j)}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j}}\alpha(i,j)\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j}}(\boldsymbol{A_{j}}^{T}\boldsymbol{A_{j}})^{-1}\boldsymbol{A_{j}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i}\|\boldsymbol{A_{i}}\|^{2}-\frac{(\boldsymbol{A_{j}}^{T}\boldsymbol{A_{i}})^{2}}{\|\boldsymbol{A_{j}}\|^{2}}$
(4)

For the case of unit normalized columns of $\boldsymbol{A}$, the resulting
expression is same as minimizing average coherence.

###  3.4 Weights based approach using training data

We also propose weighted coherence based approaches for cases in which
training data is available. The main idea is that certain frequencies in the
DCT or wavelet basis contribute more to the signal than others. Hence, we
assign weights to emphasize certain groups of column vectors over other column
vectors. For this, we compute the mean absolute value of the coefficients at
the $l^{\textrm{th}}$ frequency (in a chosen $\boldsymbol{\Psi}$) over a
training set, and set it to be equal to the weight $W_{l}$. Thus,
$\boldsymbol{W}$ is a vector of weights where each element represents the
weight assigned to a certain frequency. Using the weights we propose weighted
coherence, weighted bi-coherence and weighted tri-coherence.
1) Weighted Coherence

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i}^{n}\textrm{argmin}_{\alpha(i,j)}\sqrt{\boldsymbol{W}(i)\cdot\boldsymbol{W}(j)}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j}}\alpha(i,j)\|_{2}^{2}$
(5)

where $\boldsymbol{W}$ is the weight vector obtained from training data.
Following steps similar to the unweighted approaches, we get

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i}^{n}\sqrt{\boldsymbol{W}(i)\cdot\boldsymbol{W}(j)}\|\boldsymbol{A_{i}}-\boldsymbol{A_{j}}(\boldsymbol{A_{j}}^{T}\boldsymbol{A_{j}})^{-1}\boldsymbol{A_{j}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}.$
(6)

2) Weighted bi-Coherence

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j}(\boldsymbol{W}(i)\cdot\boldsymbol{W}(j)\cdot\boldsymbol{W}(k))^{\frac{1}{3}}$
$\displaystyle\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k}}(\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{j,k}})^{-1}\boldsymbol{A_{j,k}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}.$
(7)

3) Weighted tri-Coherence

$\displaystyle\boldsymbol{A}=\textrm{argmax}_{\boldsymbol{A}}\sum_{i,j>i,k>j,l>k}$
$\displaystyle(\boldsymbol{W}(i)\cdot\boldsymbol{W}(j)\cdot\boldsymbol{W}(k)\cdot\boldsymbol{W}(l))^{\frac{1}{4}}$
$\displaystyle\|\boldsymbol{A_{i}}-\boldsymbol{A_{j,k,l}}(\boldsymbol{A_{j,k,l}}^{T}\boldsymbol{A_{j,k,l}})^{-1}\boldsymbol{A_{j,k,l}}^{T}\boldsymbol{A_{i}}\|_{2}^{2}.$
(8)

###  3.5 Optimizing over the matrices

For the binary matrices, we propose Alg. 1. Here $L(.)$ denotes the loss
function. We iterate over the entire sensing matrix and check whether
switching the binary value at any entry improves the loss at every location of
the matrix. This is repeated for several epochs. Convergence is achieved when
the increase in the loss function for successive epochs is less than some
convergence parameter $\delta$. For optimizing over continuous-valued matrices
(starting with a random $U(0,1)$ initial condition), we use projected gradient
ascent. While updating the matrix, we first perform a gradient ascent step and
then project the resultant matrix on the space of matrices (unit normalized
columns) with entries confined to $[0,1]$.

take $\boldsymbol{\phi}$ as random binary matrix;

while  _convergence_ do

for  _$i=1:m$_ do

for  _$j=1:n$_ do

$l_{1}=L(\boldsymbol{\phi})$;

$\boldsymbol{\phi}(i,j)=1-\boldsymbol{\phi}(i,j)$;

$l_{2}=L(\boldsymbol{\phi})$;

if  _$l_{2}\leq l_{1}$_ then

$\boldsymbol{\phi}(i,j)=1-\boldsymbol{\phi}(i,j)$;

end if

end for

end for

end while

Algorithm 1 Binary matrix optimization

##  4 Evaluations

We perform reconstructions on the compressive measurements from cropped images
of the Berkley dataset (image size = $160\times 240$). The patch size is taken
to be $10\times 10$ and iid noise from $\mathcal{N}(0,4)$ is added to the
compressive measurements. The DCT basis is used for sparse representation. A
block-based sparse reconstruction is preferred instead of the Rice single
pixel approach [22] owing to the large image size. Reconstruction is performed
on 100 test images while the weights are generated using 200 training images
(with no overlap with the test images). The convergence parameter $\delta$ is
chosen to be 0.001. The basis pursuit algorithm is used for reconstruction
using the SPGL solver [23, 24]. with the noise parameter
$\epsilon\geq\|\boldsymbol{\eta}\|_{2}$ chosen using cross-validation.

###  4.1 Results for continuous-valued sensing matrices

Here, we present results with continuous-valued sensing matrices optimized
based on our proposed coherence, bi-coherence, weighted coherence and weighted
bi-coherence criteria. We can see from the SSIM values in Fig. 1 that the
weighted versions perform better than the ones without training data with
weighted bi-coherence performing the best. Reconstructed images are presented
in Figs. 2, 3, 4 and 5. The results with weighted bicoherence typically show
less block artifacts and better edge reconstruction than with others.

Fig. 1: SSIM comparison of different coherence based methods on continuous-
valued matrices Fig. 2: $m=40,n^{2}=100$; Left to right top to bottom: ground
truth image, reconstructions with random uniform (PSNR 27.31), coherence
(27.46), double coherence (28), weighted coherence (28.93) and weighted double
coherence (29.25). Fig. 3: $m=40,n^{2}=100$; Left to right top to bottom:
ground truth image, reconstructions with random uniform (PSNR 24.42),
coherence (24.61), double coherence (25.06), weighted coherence (25.84) and
weighted double coherence (26.24). Fig. 4: $m=50,n^{2}=100$; Left to right top
to bottom: ground truth image, reconstructions with random uniform (PSNR
23.39), coherence (23.51), double coherence (23.9), weighted coherence (24.78)
and weighted double coherence (25.31). Fig. 5: $m=70,n^{2}=100$; Left to right
top to bottom: ground truth image, reconstructions with random uniform (PSNR
32.05), coherence (32.14), double coherence (32.57), weighted coherence
(33.55) and weighted double coherence (34.77).

###  4.2 Comparison for binary matrices

In this section, we first present results with non data-driven techniques such
as by maximizing coherence, bi-coherence, tri-coherence and the
$\ell_{1}-\ell_{\infty}$ measure from [19]. The hyperparameter $s$ for the
$\ell_{1}-\ell_{\infty}$ measure is chosen empirically, by observing the range
of values of $\|\boldsymbol{\theta}\|_{1}/\|\boldsymbol{\theta}\|_{\infty}$ on
a training set of image patches (here $\boldsymbol{\theta}$ is the vector of
transform coefficients of a patch). The average is observed to be 2 while the
99 percentile is around 5. Hence, we choose the best $s$ within this range.
The average SSIM values are plotted in Fig. 6 and image results are presented
in Figs. 7, 8, 9, 10 and 11, showing best results with bi-coherence or tri-
coherence.

Fig. 6: SSIM comparison of different non-data driven matrix optimization
methods on binary matrices Fig. 7: $m=20,n^{2}=100$; Left to right top to
bottom: ground truth image, reconstructions with random binary (PSNR 22.65),
$l_{1}-l_{\infty}$ (23.47), coherence (26.71), double coherence (26.77) and
triple coherence (27.3). Fig. 8: $m=30,n^{2}=100$; Left to right top to
bottom: ground truth image, reconstructions with random binary (PSNR 23.42),
$l_{1}-l_{\infty}$ (25.41), coherence (27.68), double coherence (29.76) and
triple coherence (29.21). Fig. 9: $m=40,n^{2}=100$; Left to right, top to
bottom: ground truth image, reconstructions with random binary (PSNR 22.53),
$l_{1}-l_{\infty}$ approach (23), coherence (25.02), double coherence (25.73)
and triple coherence (25.77). Fig. 10: $m=50,n^{2}=100$; Left to right, top to
bottom: ground truth image, reconstructions with random binary (PSNR 20.37),
$l_{1}-l_{\infty}$ approach (20.95), coherence (23.47), double coherence
(24.44) and triple coherence (24.44). Fig. 11: $m=60,n^{2}=100$; Left to
right, top to bottom: ground truth image, reconstructions with random binary
(PSNR 26.65), $l_{1}-l_{\infty}$ approach (26.96), coherence (29.65), double
coherence (30.19) and triple coherence (29.94).

We now present results on binary matrices, optimized by maximizing coherence,
double coherence, triple coherence, weighted coherence and weighted double
coherence. The SSIM values are plotted in Fig. 12. We can see that the
weighted versions perform better than the ones without training data. A few
image results are presented in Figs. 13, 14, 15 and 16.

Fig. 12: SSIM comparison of different coherence based methods on binary
matrices Fig. 13: $m=30,n^{2}=100$; From left to right, top to bottom: ground
truth image, reconstruction with random binary (21.73), coherence (24.79),
double coherence (25.57), triple coherence (25.94), weighted coherence (26.2)
and weighted double coherence (26.41). Fig. 14: $m=40,n^{2}=100$; Left to
right, top to bottom: ground truth image, reconstructions with random binary
(PSNR 30.21), coherence (33.78), double coherence (35.22), triple coherence
(35), weighted coherence (35.33) and weighted double coherence (35.52). Fig.
15: $m=50,n^{2}=100$; Left to right, top to bottom: ground truth image,
reconstructions with random binary (PSNR 21.52), coherence (24.61), double
coherence (25.35), triple coherence (25.31), weighted coherence (25.26) and
weighted double coherence (25.2). Fig. 16: $m=60,n^{2}=100$; Left to right,
top to bottom: ground truth image, reconstructions with random binary (PSNR
22.08), coherence (24.95), double coherence (25.83), triple coherence (25.64),
weighted coherence (25.82) and weighted double coherence (25.6).

##  5 Conclusion and Future work

In this paper, we have proposed a new set of quality measures for sensing
matrix design, showing improved quality of reconstruction compared to random
matrices or optimizing over measures such as the average coherence of the
$\ell_{1}-\ell_{\infty}$ measure. Furthermore, the expressions for our quality
measures are available in closed form. Hence, these measures can be optimized
using gradient-based methods, which is not easily possible in case of the
$\ell_{1}-\ell_{\infty}$ measure. Likewise, the RIC is also computationally
expensive as it requires computing singular values of many sub-matrices and is
difficult to optimize over, due to lack of availability of closed-form
expressions. One could argue that our proposed design criterion could be
computationally expensive if we move further from tri-coherence (i.e. increase
the number of column vectors for the linear combination). However, even for
the $\ell_{1}-\ell_{\infty}$ measure, the optimal hyperparameter $s-1$ was
found to be around 3 to 4, for image patch reconstruction problems. We have
also modified our quality measures to the case where training data are
available, producing weighted quality measures. Besides potentially improving
encoder design in practical imaging systems, another direct application of our
work could be in the field of encoder design for group testing [25]. Exploring
theoretical performance guarantees for the proposed measures is another avenue
for future research.

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