Abstract

Compressive measurements benefit low-light-level imaging (L3-imaging) due to the significantly improved measurement signal-to-noise ratio (SNR). However, as with other compressive imaging (CI) systems, compressive L3-imaging is slow. To accelerate the data acquisition, we develop an algorithm to compute the optimal binary sensing matrix that can minimize the image reconstruction error. First, we make use of the measurement SNR and the reconstruction mean square error (MSE) to define the optimal gray-value sensing matrix. Then, we construct an equality-constrained optimization problem to solve for a binary sensing matrix. From several experimental results, we show that the latter delivers a similar reconstruction performance as the former, while having a smaller dynamic range requirement to system sensors.

© 2016 Optical Society of America

1. Introduction

The idea of compressive imaging (CI) has been used when a large amount of data need to be acquired [1], while most of the collected data will be discarded [2]. The method is also useful when high-resolution sensor arrays are not available due to limitations of manufacturing technology, for example, in LIDAR 3D imaging [3, 4] or terahertz imaging [5]. CI is also suitable for applications where the data acquisition device is expensive, such as low-light-level imaging (L3-imaging) [6–8]. Using CI for L3-imaging takes advantage of better measurement signal-to-noise ratio (SNR). In such a system, the measurements are linear combinations of multiple pixels, consequently the measurement power is much larger than that of a single pixel.

Nevertheless, inherent challenges in CI make it difficult for implementation in many fields for real-world applications. One difficulty is its sensitivity to system calibration [9]. System calibration error can also be amplified by a system’s data processing. Another difficulty is the size of the signal space [9]. This is a challenge for computational memory allocation and computation power. Besides these difficulties, CI system’s speed also limits its applications, especially for real-time imaging. Several groups, including us, have investigated the use of GPU or FPGA to accelerate the data processing [8,10–14]. However, few have considered accelerating the data acquisition. In this paper, we develop a methodology to compute the optimized binary sensing matrices to improve the speed in the system’s measurement collection.

In compressive imaging, sensing matrix is always a crucial factor. Initially, random sensing matrices, such as matrices with elements selected from Bernoulli or Gaussian distribution, or matrices with row vectors randomly selected from a Fourier transformation matrix, have been studied. A CI system using these matrices can successfully reconstruct sparse signals, as they satisfy the restricted isometry property (RIP) [15, 16]. However, as noise increases, the performances degrades fast [17, 18]. Even without noise, given a sparse representation dictionary, searching an optimal sensing matrix is an open question for CI. Therefore, several research groups work on sensing matrix design for different applications [19,20]. A major trend focuses on the mutual coherence of the multiplication of the sensing matrix and the sparse representation dictionary [21–24]. When this coherence is small, a wide set of signals can be reconstructed successfully using the sensing matrix. Different from using mutual coherence, by analyzing the oracle estimator MSE as Gaussian noise present in system measurements, Chen et al. [25, 26] find that a good sensing matrix ought to be close to a Parseval tight frame. Based on the work in literature [21,26], Yang et al. [27] further optimize the sensing matrix combining with a matrix decomposition, thus to have the optimal reconstruction and the optimal mutual coherence performance simultaneously. Different from the works discussed above, in which the sparse representation dictionary is predefined, Duarte-Carvajalino and Sapiro [22] design the dictionary and the sensing matrix jointed from a set of training images to obtain better reconstructions. Besides the mutual coherence, several other groups use information-theoretic framework to design a sensing matrix [28–32].

Note that the design methods discussed above are for gray value sensing matrices, which has their limitation in CI system implementation. In many CI systems investigated in literature, a DMD is the spatial light modulator for measurement collection. A DMD consists of thousands of micromirrors. These mirrors flip between two angles at a predefined frequency. During one measurement collection time T0, if the mirrors use half of the time to reflect light to output, then the object light is modulated by a factor 0.5. Therefore, light modulation is based on time division in DMD. Due to this modulation methodology, for a fixed DMD working frequency f, collecting a measurement using a binary projection needs one period 1/f. However, for gray values, such as values which require 8 bit binary numbers for representation, DMD needs 256 periods to realize the light modulation. Therefore, the measurement needs much longer time(≫1/f). This slows down a CI system’s measurement collection and imaging speed. Besides this major disadvantage, using gray value projections, CI systems require more complicated system calibration process, thus potentially have larger systematical error compared to using binary projections. Because of these issues, we study binary sensing matrix design in this work.

In CI, we cannot design a sensing matrix without considering a reconstruction method. In the classical definition of CI, an object is reconstructed by solving an optimization problem leveraging object sparsity. Besides such nonlinear methods, in our previous works we also investigated the linear reconstruction Wiener operator [18,33]. The operator minimizes reconstruction mean square error (MSE). It can be written as W=RxFT(FRxFT+σ2I)1, where Rx is the object autocorrelation matrix, F is the sensing matrix, and σ2 is the noise variance. This operator does not use object sparsity for reconstruction. Its performance is slightly worse than nonlinear methods [34, 35] for sparse signals when noise is low. However, the performance difference is small [18]. When random features are collected for objects of size (80×80) with dynamic range [0,255] and sparsity K = 400 in the wavelet domain, the reconstruction RMSE (root mean square error) values are 9.21 and 6.54 using Wiener operator and a nonlinear Nowak method [36], respectively [18]. The noise standard deviation is 0.21. Same level RMSE difference was observed using PCA (Principal Component Analysis) features. Although Wiener operator does not use object sparsity, it is defined based on an object’s second order statistical parameter, the object’s autocorrelation matrix Rx. This statistical parameter is not used in the nonlinear reconstruction methods of CI. Additionally, because noise is directly used to define Wiener operator, the operator is more tolerant to noise. Thus, when noise is high or object light level is low, to obtain a better reconstruction, object sparsity is not the only object prior which can be used for reconstruction. Wiener operator presents even better reconstruction in terms of smaller RMSE. Besides these advantages, more importantly to the work discussed in this paper, the computational speed using the Wiener operator is much faster than nonlinear iterative methods, especially with GPU implementation [8,11,12]. Therefore we use the Wiener operator for sensing matrix design.

Note that, Wiener-type operators recently have been used in geophysics, and medical imaging specifically in diffuse optical tomographic imaging [37–39]. In a near-surface geophysical problem, observed data is emulated using a linear layered-earth model by representing the data as a multiplication between a matrix and a model vector [37]. To solve the geophysical problem is to understand how well the calculated data match with the observed data, and then select those well predicted data. A Wiener-type matrix, data-resolution matrix is used to do the selection. Then another Wiener-type matrix, model-resolution matrix is used to evaluate how well the layered-earth model works with the selected data. In diffuse optical tomographic imaging, the forward problem is different from CI or geophysical problem [38,39]. However, the natural logarithm of the amplitude of experimental data can still be written as a function of biological tissue’s absorption coefficients. Using the Jacobian of this function and a regularization parameter, data-resolution matrix and model-resolution matrix for this problem can be defined. Then based on data-resolution matrix, independent measurements are chosen to reduce data collection time without compromising reconstructed image quality much [38]. Model-resolution matrix is used with basic pursuit deconvolution algorithm to improve reconstructed image quality [39].

With the focuses on binary sensing matrices and Wiener operator for reconstruction, the paper is organized as follows. Before studying binary projection design, we first use maximizing measurement SNR and minimizing reconstruction MSE as the criteria to search the optimal gray value sensing matrixes in Section 2. Then based on the solutions, we design binary matrices in Section 3. Experimental results are presented in Section 4 to demonstrate the idea. Conclusions are drawn in Section 5.

2. Sensing matrix design using measurement SNR and reconstruction MSE

2.1. Sensing matrix design based on maximizing measurement SNR

We first define the mathematic model for CI. In compressive imaging, system measurements can be represented as

y=Fx+n,
where y(M×1) is the measurement vector, F=[f1f2fM]T is the sensing matrix of size (M×N), x(N×1) and n(M×1) are the object and noise vectors, respectively. The noise is assumed as additive white Gaussian noise N(0,σ2). Then the CI system measurement SNR becomes
SNRCS=E{Fx2}E{n2}=Tr{FRxFT}Mσ2.

In Eq. (2), the object autocorrelation matrix Rx=E{xxT2} can be decomposed as QDQT, where Q is an orthonormal matrix, and D is diagonal. The ith column of Q, qi, is the eigenvector of matrix R. The diagonal elements of D are sorted in descending order, such as d1d2 ≥ dN. di is the eigenvalue of R corresponding to qi. Note for natural images, all di are positive.

To obtain the optimal projection in terms of maximizing SNR, we solve the following optimization problem,

P1:maxFsubject toTr{FRxFT}FFT=I.

Using the decomposition of Rx, we rewrite Tr{FRxFT} as Tr{(FQ)D(FQ)T}. If we redefine FQ as H, then problem P1 becomes

P1a:maxHsubject toTr{HDHT}HHT=I.

The matrix H can be further written as H = [H1 H2], where H1 and H2 have sizes (M×M) and (M×(NM)).

One special case for P1 or P1a is H1H1T=I and H2 = 0. The cost function for this case is i=1Mdi. If we define QPCA=[q1q2qM], then this special case can be written as F=H1QPCAT. Note if H1 = I, then F=QPCAT is the PCA projection. We modify F slightly as

F=[q1q2qM1αqM+1α2qM+1]T
with 0 ≤ α ≤ 1, then the cost function of P1 becomes i=1Mdi+(dM+1dM)(1α2). Because dM+1 ≤ dM and 0 (1 − α2) 1, this value is no larger than i=1Mdi. Thus, we can observe that as F is driven away from the space spanned by vectors {q1q2qM}, Tr{FRxFT} will be reduced. Therefore, i=1Mdi is the largest cost function value of P1, and its solution is Fopt=AQPCAT, where A is a matrix of size (M×M) and AAT = I.

2.2. Sensing matrix design based on minimizing the reconstruction MSE

In compressive imaging, the final goal is to obtain high quality object reconstructions. Instead of improving the intermediate measurement SNR, in this subsection we study the matrix design to minimize the reconstruction error.

As discussed in Sec. 1, we use Wiener operator for object reconstruction. The Wiener operator is defined as

W=RxFT(FRxFT+σ2I)1.

Using W, the object is reconstructed as xest=Wy. Then the reconstruction MSE becomes [40]

MSECS=E{xestx2}=Tr{Rx}Tr{FRx2FT(FRxFT+σ2I)1}.

The first term in Eq. (7) is independent of F. To minimize the reconstruction error MSECS is equivalent to maximizing the second term. Thus, we define the design problem as

P2:maxFsubject toTr{FRx2FT(FRxFT+σ2I)1}FFT=I.

Due to the term (FRxFT+σ2I)1, solving P2 with a close form solution, such as using KKT theory [41], is hard. Note that, for F=AQPCAT,Tr{FRx2FT(FRxFT+σ2I)1} is j=1Mdi2/(di+σ2). Same as in Section 2.1, if we assume F=QPCAT is slightly modified as

F=[q1q2qM1αqM+1α2qM+1]T,
then we have
Tr{FRx2FT(FRxFT+σ2I)1}=j=1Mdi2di+σ2+dM2α2+dM+12(1α2)dMα2+dM+1(1α2)+σ2dM2dM+σ2=j=1Mdi2di+σ2+(1α2)[σ2(dM+12dM2)+dmdM+1(dM+1dM)](dMα2+dM+1(1α2)+σ2)(dM+σ2)j=1Mdi2di+σ2,
because 1 −α2 ∈ [0,1] and 0 ≤ dM+1 ≤ dM. Thus, the maximizer for P2 is F=AQPCAT.

Note the multiple optimal solutions to P2 provide more options for CI system implementation. A practical issue in CI is the high feature measurement dynamic range. To keep a good reconstruction performance, CI system measurement SNR needs to be large. Then, for projections such as PCA and Hadamard, the first feature collected is the summation of all object pixel values. This feature is significantly larger than others. Thus detectors having large dynamic range are required. However, as what we find in section 2.1 and 2.2, any matrix F=AQPCAT presents the optimized measurement SNR or reconstruction MSE. Therefore, we can choose one among all of these matrices, which has the most relaxed dynamic range requirement to system detectors.

3. Optimized binary sensing matrices

In Sec. 2.2, we conclude that, Fopt=AQPCAT presents the minimal MSE for any orthonomal matrix A. Note the row vectors of Fopt are M orthonomal vectors in a space ℚ, which is spanned by {q1,q2,,qM}. A matrix F presents a small MSE, if its distance from the space ℚ, defined as FAQPCATF, is small. Hence we can consider searching the optimal binary projection to minimize MSE as to search a binary matrix sgn{AQPCAT} close to ℚ. sgn() is the signum function, which has value {1,0, −1} for positive, zero, or negative input, respectively. Note that, if all elements in sgn{AQPCAT} are assumed nonzero, then the L2 norm of each sensing vector is N. To make both of the binary and the non-binary vectors having same L2 norms, we multiply N to AQPCAT. Then designing binary matrices becomes solving the following problem,

P3:minAsubject toε(A)=NAQPCATsgn(AQPCAT)F2AAT=1,
where ∥·∥F represents the Frobenius norm of a matrix. We redefine the constraint as G = 0, where G = AAT − I and the (i, j)th element of G is
Gij={k=1MAik21,i=jk=1MAikAjk,otherwise.

Using these definitions, the Lagrangian for P3 can be written as

(A,λ)=i=1Mj=1N[NAQPCATsgn(AQPCAT)]ij2+i=1Mj=iM[λ.G]ij,
where [·]ij represents the (i,j)th element of a matrix, λ is a symmetric matrix of size (M×M), and .∗ represents the element multiplication between two matrices.

P3 is an equality-constrained optimization problem. To solve P3, we use the Newton-Raphson method [41], which is specifically designed for nonlinear equality-constrained optimization problem. The algorithm can be represented using the following equations,

[AA2(a(ν),λ(ν))J(a(ν))TJ(a(ν))0][Δa(ν)Δλ(ν)]=[ε(a(ν))+J(a(ν))Tλ(ν)g(a(ν))],
and
[a(ν+1)λ(ν+1)]=[Δa(ν)Δλ(ν)]+α[a(ν)λ(ν)].

In Eq. (13) and (14), AA2 is the Hessian matrix of the Lagrangian, J is the Jacobi of the constraint for P3, and ∇ε is the gradient vector of the cost function. The dimensions of AA2, J, and ∇ε are (M2×M2), (M(M+1)2×M2), and (M2×1), respectively. Different from Eq. (12), the variables in Eq. (13) and (14) are vectors a and λ, which are obtained by lexicographically ordering the elements of A and the upper triangle part of λ. The vector a is defined as a=[A11A1MAM1AMM]T. λ is λ=[λ11λ1Mλ22λ2Mλ(M1)MλMM]T. Similarly, g is g=[g11g1Mg22g2Mg(M1)MgMM]T. In this section, the definitions of AA2, J, ∇ε are presented directly. More details about the calculation of these parameters can be found in the Appendix section.

The elements of the Hessian matrix AA2(a(ν),λ(v)) are defined as

2(a,λ)AijAij=2Nδ(ii)δ(jj)+2λiiδ(jj).

Function δ(i) is the delta function. For the Jacobi matrix of G, note that G is symmetric. Therefore only the upper triangle part of G is used to find J. We have

GijAij=δ(ii)Ajj+δ(ji)Aij.

The third parameter in the Newton-Raphson method is ε(a) with

εAij=2NAij2N[sgn(XQPCAT)QPCA]ij.

To ensure the step direction Δa(ν) is a descent direction for ε(a(v))+J(a(ν))Tλ(ν), the Hessian matrix AA2 is modified to become positive definite [41, 42]. In addition, the stepsize α in Eq. (14) is set to be small, and the number of the iterations ν is set to be large, so that ∇ε will converge to a cost value without oscillation.

4. Numerical results for fast compressive L3-imaging

As discussed in section 1, collecting features or linear combinations of object pixels will benefit system measurement SNR. Therefore, CI is well suitable for low-light-level imaging (L3-imaging) [6,7]. Figure 1 is a block-based system diagram for L3-imaging. In such a system, objects are defined into blocks. Then, for each block, multiple compressive measurements are collected using a detector. The advantage of using block-based CI (BCI) instead of CI with a single detector is to obtain high resolution object reconstructions [33].

 figure: Fig. 1

Fig. 1 A system diagram for compressive L3-imaging.

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Figure 2(a) and 2(b) present the two kinds of objects used for the experimental study [43,44]. They are low-light-level images and are rather dark, so for illustrations only we enhanced their brightness in the figures. The sizes of the two kinds of objects are (198×128) and (1056×1920). Both of them have minimum 0 and maximum 76.5, although the mean value is 45.8 for the first kind, and 33.5 for the second. The block size is set to be (32×32). M = 8 and 16 features are collected for each block. Noise is assumed to be Gaussian noise with standard deviation σ in the range [5,150]. With such noise level, the SNR range is [−13.02dB,19.24dB]. To make this SNR range having more physical meaning, we use Hamamatsu SI PIN photodiode S1223 as an example to calculate the input light power. The detector S1223 has NEP (noise equivalent power) 9.4 × 10−15W/Hz1/2. If the system bandwidth is assumed as 30MHz, then when SNR is 0dB, the input light power or the noise power is 51.486pW. In our experimental study, the lowest SNR is −13.02dB. At this SNR level, the input light power is 2.568pW. In other words, for such level of light power, directly using PIN photodiode S1223 for measurement collection, the input signal will be overwhelmed by noise. However, using a BCI system as shown in Fig. 1, we can still make the measurements, and get an object reconstructed.

 figure: Fig. 2

Fig. 2 Object examples of size (a) (196 × 128) [43,44], and (b) (1056 × 1920).

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In the first experiment, we studied seven kinds of projections, such as three Hadamard projections, two DCT projections, PCA projection QPCAT, and the projection AQPCAT. In the first Hadamard projection, Hfreq, we sort the sensing vectors based on the variation of elements’ signs. For example, the first vector has all one values. The element’s sign varies 0 time. In the second vector, the first half elements are +1, and then the rest are −1. The sign varies once. In the third vector, the element values change from +1 to −1, and then back to +1. The sign varies twice. Note that, using this sorting method, the system collects the DC value, the low frequency components, and the high frequency components of the object x sequentially. In the second Hadamard projection, Hmean, we use a training sample average to sort the sensing vectors. The sensing vectors are in the same order as the Hadamard coefficients of the sample average. In the third Hadamard projection, Hsnr, we sort the vectors using measurement SNR [7]. The two DCT for this experiment are the regular DCT, FDCT, and the DCT with sensing vectors sorted using measurement SNR, FDCTsnr. Table 1 presents the cost function ε(F) = Tr{FRxFT} for problem P1 using these projections. It can be observed, the features obtained using PCA and AQPCAT have the largest ε(F) as we discussed in Section 2.1.

Tables Icon

Table 1. The cost function ε(F) = Tr{FRxFT} in P1 when M = 32 features are obtained using PCA, DCT, and Hadamard matrices. (*The values listed in the table are ε(F)/106.)

In Fig. 3(a) and 3(b), we present the normalized reconstruction root mean square error (RMSE) vs. noise σ for the seven kinds of projections. The RMSE is defined as xestx/x. As expected, for both kinds of objects, QPCAT and AQPCAT present the same reconstruction error performance at different noise levels. These errors are smaller than the errors obtained using other kinds of projections. This is consistent with the conclusion obtained in section 2.2, that the projection AQPCAT for any orthonomal matrix A is optimal in terms of minimizing RMSE. Note that, comparing Fig. 3(a) and 3(b), we find the RMSE values for the large size object are smaller. The reason is that, for a large size object, pixel values in object blocks vary slowly, or blocks are sparse in frequency domain. Therefore, using a few feature measurements can reconstruct the block with small error.

 figure: Fig. 3

Fig. 3 Reconstruction RMSE for objects of size (a) (196×128) and (b) (1056×1920), when M = 32 Hadamard, DCT, and PCA features are collected for (32×32) object blocks.

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In the second experiment, we use the iterative algorithm discussed in section 3 to search for the optimal binary projection for the object shown in Fig. 2(b). Note that, the solution to this algorithm is a local minimizer. To find a global solution, 10M2 initializations with 4000 iterations in each trail are used, where M is the number of features, and M2 is the number of unknown variables in problem P3. For example, if M = 8, the number of initializations used to search the optimal matrix A of size (8×8) is 640. In addition, to maximize the object signal in a feature measurement, we normalize the maximum absolute value of each sensing vector to 1.

In Fig. 4(a), the 8 Hadamard vectors in Hsnr, which are selected based on measurement SNR [7], are presented. Figure 4(b) presents the optimization result, Frmse, for M = 8. It can be observed, except the all-one vector, the solution for P3 is not directly related to Hadamard projection.

 figure: Fig. 4

Fig. 4 (a) The Hadamard vectors sorted using measurement SNR, (b) the optimal binary vector set or the optimal solution for problem P3, (c) the suboptimal solution for P3 which has the minimal feature measurement dynamic range, when M is 8 for 32 × 32 blocks.

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Besides the optimal solution, we also compare the 640 local minimizers obtained by the 10M2 trials. The RMSE values obtained using these local solutions are calculated. We find 217 local minimizers including the optimal one presents very similar RMSE. The RMSE difference between these local minimizers and the optimal solution is less than 10−4. Then among these sub-optimizers, the solution having the smallest feature dynamic range, labeled as Fdyn, is presented in Fig. 4(c). The all-one vector is not among the result. Table 2 summarizes the minima, the maxima, and the dynamic ranges of features obtained using Hmean, Hsnr, FDCTsnr, PCA, Frmse, and Fdyn. It is clear that, Fdyn needs about 30% less dynamic range than the other five projections.

Tables Icon

Table 2. The dynamic ranges of M = 8 features obtained using different sensing matrices. (*The values listed in the table are the actual values divided by 104.)

Figure 5(a) presents the reconstruction errors using Hmean, Hsnr, FDCTsnr, QPCAT, Frmse, and Fdyn vs. noise σ. It can be observed that Frmse presents even smaller RMSE than PCA, although it is binary. The reason is, after normalizing each vector having maximum 1, a system using Frmse can collect more object light into a measurement than using QPCAT. Thus system measurement SNR and reconstruction RMSE are improved. From the figure we can also observe that Fdyn presents similar RMSE values as Frmse. As a comparison, we also present the reconstruction RMSE for the standard Hadamard and random binary projections in Fig. 5(b). It is clear that, both projections have larger reconstruction errors. To investigate the reason that the optimized binary projections work well, we calculate the Fourier transform of each projection vector. Figure 6 shows the normalized FFT of the vectors in the standard Hadamard, Hsnr, and Fdyn matrices. The FFT for Frmse is similar to Fdyn. From the figure, it is clear that, Fdyn will collect object information at more frequencies. Hsnr will collect at fewer frequencies, while the standard Hadamard collects at the fewest. Thus the reconstruction using Fdyn has the smallest RMSE.

 figure: Fig. 5

Fig. 5 When M = 8 features are collected for 32×32 blocks, the normalized RMSE using (a) the sorted Hadamard, DCT, PCA, Frmse, and Fdyn matrices, and (b) the standard Hadamard, random binary, and Frmse matrices.

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 figure: Fig. 6

Fig. 6 When M = 8 features are collected for 32×32 blocks, the normalized Fourier transforms of the vectors in (a) the standard Hadamard matrix, (b) Hsnr, and (c) Fdyn.

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The experiment is repeated for M = 16. Sensing matrices Frmse and Fdyn are shown in Fig. 7(a) and 7(b). The reconstruction RMSE for Hmean, Hsnr, FDCTsnr, QPCAT, Frmse, and Fdyn are plotted as functions of noise in Fig. 7(c). Once again, the optimal and the suboptimal sensing matrices present very similar reconstruction performance. We also calculate the feature dynamic range values. We find the feature dynamic range is reduced 28% from 7.47 × 104 for Hsnr to 5.39×104 for Fdyn. Figure 7(d) and 7(e) presents the normalized FFT of the vectors in Hsnr and Fdyn matrices. Once again, we observe that Fdyn will collect more object information at more frequencies. Thus the system using this matrix has better reconstruction performance.

 figure: Fig. 7

Fig. 7 (a) The optimal binary vector set or the optimal solution for problem P3, (b) the suboptimal solution for P3 which has the minimal feature measurement dynamic range, (c) the normalized RMSE for sorted Hadamard, DCT, PCA matrices, Frmse, and Fdyn, and the normalized Fourier transforms of the vectors in (d) Hsnr and (e) Fdyn, when M is 16 for 32 × 32 blocks.

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Figure 8 presents 4 reconstructions using different imaging systems and projections. Figure 8(a) is the average of 16 conventional imaging measurements. (b), (c), and (d) are the reconstructions obtained using the sensing matrix Hmean, Hsnr, and Fdyn, where M = 16 measurements are collected, and the noise level is σ = 148.67. To make the reconstruction more visually pleasant, we increase the intensity 3 times. It is clear, all reconstructions using CI have smaller RMSE compared with the conventional result. The RMSE values for the conventional imaging, Hmean, Hsnr, and Fdyn are 0.7312, 0.0733, 0.0645 and 0.0635, respectively. Note that, the reconstructions using Hsnr, and Fdyn are very similar to each other, and are close to the original. To have a better observation, Fig. 9 shows two magnified detail parts of both reconstructions. Figure 9(a) and 9(b) are for Hsnr. Figure 9(c) and 9(d) are for Fdyn. It can be observed that, the reconstruction using Fdyn has slightly smoother reconstruction with less block effect compared with the reconstruction using Hsnr.

 figure: Fig. 8

Fig. 8 The object reconstruction using (a) conventional imaging, (b) Hmean, (c) Hsnr, and (d) Fdyn, when M = 16 features are collected for 32 × 32 blocks and σ is 148.67.

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 figure: Fig. 9

Fig. 9 The object reconstructions (a)&(b) using Hsnr, and (c)&(d) using Fdyn when M = 16 features are collected for 32 × 32 blocks and σ is 148.67.

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In the next experiment, we add a bias to each projection to make it nonnegative. Then the maximum of the obtained projection is normalize to 1. For the PCA and DCT projections, the vector values are quantized to 128 levels. As discussed in the beginning of the paper, CI systems using gray value projections, require longer time to make measurements. In the time to collect a measurement with a 8-bit projection, a system can make 128 measurements using binary vectors. Therefore, binary projections benefit a CI system imaging speed. Figure 10 presents the reconstruction RMSE with these projections. We get the same conclusions as in Fig. 7(c). The RMSE values for Frmse and Fdyn are almost same. They are smaller than the RMSE values using other projections. Among the rest four projections, Hsnr has the smallest reconstruction at almost all noise levels. Table 3 summarizes the maximal feature values or the dynamic ranges use all six projections. Once again, we can see Fdyn has the smallest dynamic range.

 figure: Fig. 10

Fig. 10 Normalized RMSE for different sets of nonnegative sensing vectors when M = 16 features are collected for 32 × 32 block.

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Tables Icon

Table 3. The dynamic ranges of M = 16 features obtained using different nonnegative matrices. (*The values listed in the table are the actual values divided by 104.)

In the last set of experiments, we studied the performance of Frmse and Fdyn with sparse objects. The object blocks in Fig. 2(b) are transformed into the wavelet domain using the 4-level wavelet transformation matrix. Except the largest 64 coefficients in each block, the rest are set to zero. Using this sparse object, we repeat the experiment as in Fig. 10. Besides the six matrices we discussed in last several experiments, wavelet and curvelet projections are also studied. The wavelet projection matrix is for 4 level transformation. Both kinds of projection vectors are sorted using measurement SNR. These vectors are also added with a bias to make them nonnegative, and then quantized to 128 discrete levels. Figure 11 presents the reconstruction RMSE vs. noise σ. We observe similar trend as in Fig. 10, except that the RMSE values are smaller because the object sparsity benefits the reconstruction. For wavelet and curvelet transforms, the reconstruction error are larger than other projections at most noise levels. The reason is that, both transformation vectors are spatial localized. They are suitable to extract object’s local information and make the transformation sparse. However, this property is not helpful in compressive low-light-level imaging, in which more light from an object are required to surpass detector noise. Between wavelet and curvelet transforms, the performance of the latter is worse. This is caused by the non-orthogonality among the transformation vectors.

 figure: Fig. 11

Fig. 11 Normalized RMSE for different sets of nonnegative sensing vectors when M = 16 features are collected for 32 × 32 blocks. The object blocks are sparse in the wavelet domain. The sparsity is K = 64.

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In the last experiment, we make the object further sparse with K = 4 for each block. The sparse object is shown in Fig. 12(a). Then we use a nonlinear method, fast linearized Bregman iterative algorithm, to reconstruct the object. In Fig. 12(b), 16 random binary vectors are used to make the feature measurements. In Fig. 12(c), Fdyn is used. The noise level in the experiment is σ = 148.67. We can observe that the reconstruction using Fdyn is visually better. The RMSE for Fig. 12(b) is 0.0623, while the RMSE for 12(c) is 0.0268. The designed binary sensing matrix works well for the sparse object. Because it is more structured in Fourier domain, it works better than random projections. Additionally, we also present the reconstruction using Wiener operator in Fig. 12(d) when Fdyn is used. The performance is even better than using the linearized Bregman iterative algorithm. As we discussed in the beginning of the paper, although Wiener operator does not use sparsity for reconstruction, it incorporates object second order statistical parameter, object autocorrelation matrix Rx. It is more suitable when system measurements have much noise. The RMSE value obtained using Wiener operator is 0.0131 for the sparse object.

 figure: Fig. 12

Fig. 12 (a) The original object which is sparse in the wavelet domain. The sparsity for each 32 × 32 block is K = 4. The object reconstructions using (b) the fast linearized Bregman iterative method with a binary random matrix, (c) the fast linearized Bregman iterative method with Fdyn, and (d) Wiener operator with Fdyn, when M = 16 features are collected for each block and σ is 148.67.

Download Full Size | PPT Slide | PDF

Table 4 summarizes the measurement collection time and the object reconstruction time using different kinds of sensing matrices and linear/nonlinear reconstruction methods. We assume TI Discovery 4100 development tool set is used to make measurements. Then the DMD working frequency is set as 32kHz. With this working frequency, for 16 binary features, we need 0.5ms to make the measurements. For 128-level gray sensing matrix, the time becomes 64ms. Using the linear reconstruction method, Wiener operator, 12ms is required to obtain the reconstruction. Using the nonlinear method, fast linearized Bregman iterative algorithm, the reconstruction time is more than 100s. Thus, to make the system imaging speed faster, binary sensing matrix and Wiener operator should be used.

Tables Icon

Table 4. The (measurement collection time, reconstruction time) for different kinds of sensing matrices and linear/nonlinear methods. 16 feature are collected for each block. The DMD working frequency is set as 32kHz.

5. Conclusions

In this paper, we study to design binary sensing matrices for fast compressive L3-imaging. Two optimization problems, maximizing measurement SNR and minimizing RMSE, are solved first. We find the solutions to both problems are matrices AQPCAT, where A is orthonomal. Based on the observation, a binary matrix design problem is defined as to search a matrix sgn(AQPCAT). Then from the experimental study, we find the optimal solution presents similar reconstruction as PCA, or even better than PCA when the maximum absolute values of projection vectors are normalized to 1. More interestingly, many suboptimal binary matrices are found to have almost the same reconstructions as the optimal. Among them, we can choose the matrix presenting the least measurement dynamic range to relax the requirement to CI sensors.

We focus on Gaussian noise in this work. When the light from object is extremely low, the noise becomes Poisson. For this case, the compressive imaging idea will still help the system’s reconstruction performance, because system measurement SNR will be improved using BCI. However, the linear Wiener operator and the nonlinear reconstruction algorithms used for CI need to be modified for Poisson noise. Besides Gaussian noise, in a physical system there are other kinds of detector noise, long-term drift error, and systematic errors from stray-light. Different kinds of detectors or imaging elements have different noise. For photomultipliers, photoconductive and photovoltaic detectors, shot noise, generation-recombination noise or g-r noise, thermal noise, and 1/f noise are the four main factors. Shot noise matches Poisson noise probability density distribution. When the signal frequency is not very high, g-r noise can be modeled as white noise. Thermal noise generally is white Gaussian noise. 1/f noise can be ignored in most cases when a signal frequency is higher than 1kHz. For different kinds of noise, using BCI, we can always have larger detector signal. When the signal power is larger, Poisson noise becomes Gaussian noise. For long-term drift, except calibrating the imaging system, using the optimized binary sensing matrix will shorten the measurement collection time. Thus it will help the system to deal with such kind of error. Stray-light will cause detector collecting undesired background or scattered light. For active imaging, modulating the illumination source and then preprocessing the detected signal using a bandpass filter will suppress stray-light in a system.

We study fast compressive L3-imaging in this paper. Thus, we use binary sensing matrices to make measurements and linear Wiener operator for object reconstruction. However, for applications in which imaging speed is not restricted and object light is not low or noise is not high, gray-value sensing matrices and nonlinear reconstruction methods have their advantages. To further improve the reconstruction performance of fast compressive L3-imaging, in the future we will investigate linear methods including object sparsity as an prior.

Appendix - the calculation of AA2(a(ν),λ(ν)), J, and ε(a)

As discussed in section 3, to solve problem P3, parameters AA2(a(ν),λ(ν)), J, and ε(a) need to be calculated.

To find AA2(a(ν),λ(ν)), we first calculate ε(a). Note that the cost function of P3, ε(a), can be written as ε(a)=i=1Mj=1N[NAQPCATsgn(AQPCAT)]ij2. Taking the derivative of ε(a) respects to Ai j, we have

εAij=2k=1N[N[AQPCAT]iksgn([AQPCAT]ik)][N[QPCAT]jkδ([AQPCAT]ik)[QPCAT]jk]

We assume AQPCAT nonzero. Therefore, ∂ε/∂Aij becomes

εAij=2k=1N[N[AQPCAT]iksgn([AQPCAT]ik)]N[QPCAT]jk=2N[AQPCATQPCA]ij2N[sgn(AQPCAT)QPCA]ij=2N[A]ij2N[sgn(AQPCAT)QPCA]ij,
where AQPCATQPCA=1.

To define the Hessian matrix, we need to find 2/∂Aij∂Aij. If we represent the second term of Eq. (12) as ελ, then 2/∂Aij∂Ai′j′ = ∂2ε/∂Aij∂Ai′j′ +2ελ/∂Aij∂Ai′j′. Using the result of Eq. (19), we have

2εAijAij=2Nδ(ii)δ(jj)2NAij[k=1Nsgn([AQPCAT]ik)[QPCA]kj]=2Nδ(ii)δ(jj)2Nδ(ii)k=1Nδ([AQPCAT]ik)[QPCAT]ik[QPCA]kj=2Nδ(ii)δ(jj)
with the assumption that AQPCAT is nonzero.

The derivative of ελ respective to Aij is calculated as

ελAij=k,k=1M[λ.*Aij(AAT1)]kk=k,k=1MλkkGkkAij

Using the definition of Gkk presented in section 3, we have

CkkAij=δ(ki)Akj+δ(ki)Akj.

Then ∂ελ/∂Aij is

ελAij=k,k=1Mλkk(δ(ki)Akj+δ(ki)Akj)=k=1M2λikAkj
and ∂2ελ/∂Aij∂Ai′j′ is
2ελAijAij=(k=1M2λikAkj)Aij=2λiiδ(jj)

Therefore, with Eq. (20) and (24), 2L/∂ Ai j∂ Ai j becomes

2AijAij=2εAijAij+2ελAijAij=2Nδ(ii)δ(jj)+2λiiδ(jj)

Note Eq. (22) defines the Jacobi for the constraint of Problem P3. Then, we have all of the parameters AA2, J, and ∇ε defined by Eq. (25), (19), and (22), respectively.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Project 61307022, and by the Ministry of Education of China, under project 20131101120025.

References and links

1. X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015). [CrossRef]  

2. Y. Kashter, O. Levi, and A. Stern, “Optical compressive change and motion detection,” Appl. Opt. 51(13): 2491–2496 (2012). [CrossRef]   [PubMed]  

3. A. Kirmani, A. Colaço, F. N. C. Wong, and V. K. Goyal, “Exploiting sparsity in time-of-flight range acquisition using a single time-resolved sensor,” Opt. Express 19(22): 21485–21507 (2011). [CrossRef]   [PubMed]  

4. G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21(20): 23822–23837 (2013). [CrossRef]   [PubMed]  

5. Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27: (7)1638–1646 (2010). [CrossRef]  

6. J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013). [CrossRef]  

7. J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.

8. J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014). [CrossRef]  

9. M. E. Gehm and D. J. Brady, “Compressive sensing in the EO/IR,” Appl. Opt. 54(8): C14–C22 (2015). [CrossRef]   [PubMed]  

10. D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12): 14871–14884 (2014). [CrossRef]   [PubMed]  

11. D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014). [CrossRef]  

12. P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

13. P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012). [CrossRef]  

14. G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012). [CrossRef]  

15. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346(9): 589–592 (2008). [CrossRef]  

16. R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008). [CrossRef]  

17. Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).

18. M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46(22): 5293–5303 (2007). [CrossRef]   [PubMed]  

19. Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011). [CrossRef]  

20. L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011). [CrossRef]  

21. M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55(12): 5695–5702 (2007). [CrossRef]  

22. J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009). [CrossRef]  

23. J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010). [CrossRef]  

24. G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013). [CrossRef]  

25. W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012). [CrossRef]  

26. W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013). [CrossRef]  

27. A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013). [CrossRef]  

28. D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.

29. A. Ashok, L. Huang, and M. A. Neifeld, “Information optimal compressive sensing: static measurement design,” J. Opt. Soc. Am. A 30: (5): 831–853 (2013). [CrossRef]  

30. W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012). [CrossRef]  

31. Y. Gu and N. A. Goodman, “Compressed sensing kernel design for radar range profiling,” in Proceedings of 2013 IEEE Radar Conference (RADAR) (IEEE, 2013), 1–5.

32. A. Mahalanobis and M. A. Neifeld, “Optimizing measurements for feature-specific compressive sensing,” Appl. Opt. 53(26): 6108–6118 (2014). [CrossRef]   [PubMed]  

33. J. Ke and E. Y. Lam, “Object reconstruction in block-based compressive imaging,” Opt. Express 20(20): 22102–22117 (2012). [CrossRef]   [PubMed]  

34. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007). [CrossRef]  

35. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007). [CrossRef]   [PubMed]  

36. J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006). [CrossRef]  

37. J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008). [CrossRef]  

38. D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012). [CrossRef]   [PubMed]  

39. J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014). [CrossRef]  

40. J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005). [CrossRef]  

41. R. Baldick, Applied Optimization: Formulation and Algorithms for Engineering Systems(Cambridge University, 2006). [CrossRef]  

42. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

43. A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

44. A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009). [CrossRef]  

References

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  • |

  1. X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
    [Crossref]
  2. Y. Kashter, O. Levi, and A. Stern, “Optical compressive change and motion detection,” Appl. Opt. 51(13): 2491–2496 (2012).
    [Crossref] [PubMed]
  3. A. Kirmani, A. Colaço, F. N. C. Wong, and V. K. Goyal, “Exploiting sparsity in time-of-flight range acquisition using a single time-resolved sensor,” Opt. Express 19(22): 21485–21507 (2011).
    [Crossref] [PubMed]
  4. G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21(20): 23822–23837 (2013).
    [Crossref] [PubMed]
  5. Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27: (7)1638–1646 (2010).
    [Crossref]
  6. J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013).
    [Crossref]
  7. J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.
  8. J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
    [Crossref]
  9. M. E. Gehm and D. J. Brady, “Compressive sensing in the EO/IR,” Appl. Opt. 54(8): C14–C22 (2015).
    [Crossref] [PubMed]
  10. D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12): 14871–14884 (2014).
    [Crossref] [PubMed]
  11. D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
    [Crossref]
  12. P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).
  13. P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
    [Crossref]
  14. G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
    [Crossref]
  15. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346(9): 589–592 (2008).
    [Crossref]
  16. R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
    [Crossref]
  17. Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).
  18. M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46(22): 5293–5303 (2007).
    [Crossref] [PubMed]
  19. Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
    [Crossref]
  20. L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
    [Crossref]
  21. M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55(12): 5695–5702 (2007).
    [Crossref]
  22. J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009).
    [Crossref]
  23. J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
    [Crossref]
  24. G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
    [Crossref]
  25. W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
    [Crossref]
  26. W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
    [Crossref]
  27. A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
    [Crossref]
  28. D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.
  29. A. Ashok, L. Huang, and M. A. Neifeld, “Information optimal compressive sensing: static measurement design,” J. Opt. Soc. Am. A 30: (5): 831–853 (2013).
    [Crossref]
  30. W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
    [Crossref]
  31. Y. Gu and N. A. Goodman, “Compressed sensing kernel design for radar range profiling,” in Proceedings of 2013 IEEE Radar Conference (RADAR) (IEEE, 2013), 1–5.
  32. A. Mahalanobis and M. A. Neifeld, “Optimizing measurements for feature-specific compressive sensing,” Appl. Opt. 53(26): 6108–6118 (2014).
    [Crossref] [PubMed]
  33. J. Ke and E. Y. Lam, “Object reconstruction in block-based compressive imaging,” Opt. Express 20(20): 22102–22117 (2012).
    [Crossref] [PubMed]
  34. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007).
    [Crossref]
  35. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007).
    [Crossref] [PubMed]
  36. J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006).
    [Crossref]
  37. J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
    [Crossref]
  38. D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012).
    [Crossref] [PubMed]
  39. J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
    [Crossref]
  40. J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005).
    [Crossref]
  41. R. Baldick, Applied Optimization: Formulation and Algorithms for Engineering Systems(Cambridge University, 2006).
    [Crossref]
  42. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).
  43. A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).
  44. A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
    [Crossref]

2015 (2)

M. E. Gehm and D. J. Brady, “Compressive sensing in the EO/IR,” Appl. Opt. 54(8): C14–C22 (2015).
[Crossref] [PubMed]

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

2014 (6)

J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
[Crossref]

D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12): 14871–14884 (2014).
[Crossref] [PubMed]

D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
[Crossref]

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

A. Mahalanobis and M. A. Neifeld, “Optimizing measurements for feature-specific compressive sensing,” Appl. Opt. 53(26): 6108–6118 (2014).
[Crossref] [PubMed]

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

2013 (6)

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
[Crossref]

A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
[Crossref]

A. Ashok, L. Huang, and M. A. Neifeld, “Information optimal compressive sensing: static measurement design,” J. Opt. Soc. Am. A 30: (5): 831–853 (2013).
[Crossref]

J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013).
[Crossref]

G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21(20): 23822–23837 (2013).
[Crossref] [PubMed]

2012 (7)

Y. Kashter, O. Levi, and A. Stern, “Optical compressive change and motion detection,” Appl. Opt. 51(13): 2491–2496 (2012).
[Crossref] [PubMed]

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
[Crossref]

D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012).
[Crossref] [PubMed]

J. Ke and E. Y. Lam, “Object reconstruction in block-based compressive imaging,” Opt. Express 20(20): 22102–22117 (2012).
[Crossref] [PubMed]

2011 (3)

A. Kirmani, A. Colaço, F. N. C. Wong, and V. K. Goyal, “Exploiting sparsity in time-of-flight range acquisition using a single time-resolved sensor,” Opt. Express 19(22): 21485–21507 (2011).
[Crossref] [PubMed]

Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
[Crossref]

L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
[Crossref]

2010 (2)

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27: (7)1638–1646 (2010).
[Crossref]

J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
[Crossref]

2009 (2)

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009).
[Crossref]

A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
[Crossref]

2008 (3)

J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
[Crossref]

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346(9): 589–592 (2008).
[Crossref]

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

2007 (4)

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46(22): 5293–5303 (2007).
[Crossref] [PubMed]

M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55(12): 5695–5702 (2007).
[Crossref]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007).
[Crossref]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007).
[Crossref] [PubMed]

2006 (1)

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006).
[Crossref]

2005 (2)

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005).
[Crossref]

A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

Ashok, A.

Bai, H.

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Baldick, R.

R. Baldick, Applied Optimization: Formulation and Algorithms for Engineering Systems(Cambridge University, 2006).
[Crossref]

Baraniuk, R.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

Baraniuk, R. G.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Bellasi, D. E.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007).
[Crossref] [PubMed]

Brady, D.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

Brady, D. J.

Burg, A.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Calderbank, R.

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Candès, E. J.

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346(9): 589–592 (2008).
[Crossref]

Cao, Z.

J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
[Crossref]

Carin, L.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Carson, W. R.

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Chang, H. S.

Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).

Chang, L.

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Chen, M.

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Chen, W.

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
[Crossref]

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
[Crossref]

Chin, S.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Chung, S. H.

A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

Colaço, A.

Dao, M.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Davenport, M.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

Dehghani, H.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

DeVore, R.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

Donoho, D. L.

D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.

Duarte-Carvajalino, J. M.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009).
[Crossref]

Elad, M.

M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55(12): 5695–5702 (2007).
[Crossref]

Eldar, Y. C.

L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
[Crossref]

Etienne-Cummings, R.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Felber, N.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007).
[Crossref] [PubMed]

Freeman, W. T.

Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).

Gehm, M. E.

Gilbert, A. C.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007).
[Crossref]

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Goodman, N. A.

Y. Gu and N. A. Goodman, “Compressed sensing kernel design for radar range profiling,” in Proceedings of 2013 IEEE Radar Conference (RADAR) (IEEE, 2013), 1–5.

Goyal, V. K.

Gu, Y.

Y. Gu and N. A. Goodman, “Compressed sensing kernel design for radar range profiling,” in Proceedings of 2013 IEEE Radar Conference (RADAR) (IEEE, 2013), 1–5.

Haupt, J.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006).
[Crossref]

Hou, Z.

A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
[Crossref]

Howell, J. C.

Howland, G. A.

Huang, L.

Huang, Y.

Javanmard, A.

D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.

Kaeslin, H.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Kang, J. U.

Karkala, D.

D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012).
[Crossref] [PubMed]

Kashter, Y.

Ke, J.

D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
[Crossref]

J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
[Crossref]

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013).
[Crossref]

J. Ke and E. Y. Lam, “Object reconstruction in block-based compressive imaging,” Opt. Express 20(20): 22102–22117 (2012).
[Crossref] [PubMed]

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46(22): 5293–5303 (2007).
[Crossref] [PubMed]

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005).
[Crossref]

J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.

Kirmani, A.

Lam, E. Y.

Levi, O.

Li, G.

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Li, P.

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

Llull, P.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

Lum, D. J.

Maechler, P.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Mahalanobis, A.

Maleki, A.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Miller, R. D.

J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
[Crossref]

Montanari, A.

D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Neifeld, M. A.

Ng, A. Y.

A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
[Crossref]

A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

Nguyen, D. T.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Nowak, R.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006).
[Crossref]

Orchard, G.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Petropulu, A. P.

Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
[Crossref]

Pi, Y.

J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
[Crossref]

Pogue, B. W.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

Poor, H. V.

Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
[Crossref]

Posch, C.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Prakash, J.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

Rodrigues, M. R. D.

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
[Crossref]

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
[Crossref]

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Rosenblum, K.

L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
[Crossref]

Sapiro, G.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009).
[Crossref]

Saxena, A.

A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
[Crossref]

A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

Stenner, M. D.

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005).
[Crossref]

Stern, A.

Studer, C.

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

Sui, D.

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
[Crossref]

J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
[Crossref]

Sun, M.

A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
[Crossref]

Suo, Y.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Tran, T. D.

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Tropp, J. A.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007).
[Crossref]

Tsai, T.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

Wakin, M.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

Ware, M. R.

Wassell, I. J.

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
[Crossref]

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
[Crossref]

Wei, P.

J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
[Crossref]

D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
[Crossref]

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013).
[Crossref]

J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.

Weiss, Y.

Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).

Wong, F. N. C.

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Xia, J.

J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
[Crossref]

Xu, D.

Xu, J.

J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
[Crossref]

Xu, Y.

J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
[Crossref]

Xu, Z.

Yalavarthy, P. K.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012).
[Crossref] [PubMed]

Yang, A.

A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
[Crossref]

Yang, D.

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Yu, Y.

Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
[Crossref]

Yuan, X.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

Zelnik-Manor, L.

L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
[Crossref]

Zhang, J.

A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
[Crossref]

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

Zhang, X.

J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.

Zhu, R.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

Zhu, Z.

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Advances in Neural Information Processing Systems (1)

A. Saxena, S. H. Chung, and A. Y. Ng, “Learning depth from single monocular images,” Advances in Neural Information Processing Systems 18: 1161–1168 (2005).

Appl. Opt. (4)

Comptes Rendus Mathematique (1)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346(9): 589–592 (2008).
[Crossref]

Constructive Approximation (1)

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28(3): 253–263 (2008).
[Crossref]

EURASIP Journal on Advances in Signal Processing (1)

J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” EURASIP Journal on Advances in Signal Processing 2010, 43 (2010).
[Crossref]

IEEE J. Emerg. Sel. Topic Circuits Syst. (2)

P. Maechler, C. Studer, D. E. Bellasi, A. Maleki, A. Burg, N. Felber, H. Kaeslin, and R. G. Baraniuk, “VLSI design of approximate message passing for signal restoration and compressive sensing,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 579–590 (2012).
[Crossref]

G. Orchard, J. Zhang, Y. Suo, M. Dao, D. T. Nguyen, S. Chin, C. Posch, T. D. Tran, and R. Etienne-Cummings, “Real time compressive sensing video reconstruction in hardware,” IEEE J. Emerg. Sel. Topic Circuits Syst. 2(3): 604–615 (2012).
[Crossref]

IEEE J. Sel. Topics Signal Process. (1)

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Topics Signal Process. 9(6): 964–976 (2015).
[Crossref]

IEEE Signal Process. Lett. (1)

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “On the use of unit-norm tight frames to improve the average mse performance in compressive sensing applications,” IEEE Signal Process. Lett. 19(1): 8–11 (2012).
[Crossref]

IEEE Trans. Image Process. (2)

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7): 395–1408 (2009).
[Crossref]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12): 2992–3004 (2007).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (2)

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52(9): 4036–4048 (2006).
[Crossref]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12): 4655–4666 (2007).
[Crossref]

IEEE Trans. Med. Imag. (1)

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imag. 33(4): 891–901 (2014).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Saxena, M. Sun, and A. Y. Ng, “Make3D: Learning 3D scene structure from a single still image,” IEEE Trans. Pattern Anal. Mach. Intell. 30(5): 824–840 (2009).
[Crossref]

IEEE Trans. Signal Process. (5)

W. Chen, M. R. D. Rodrigues, and I. J. Wassell, “Projection design for statistical compressive sensing: A tight frame based approach,” IEEE Trans. Signal Process. 61(8): 2016–2029 (2013).
[Crossref]

G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” IEEE Trans. Signal Process. 61(11): 2887–2898 (2013).
[Crossref]

Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Trans. Signal Process. 59(11): 5338–5352 (2011).
[Crossref]

L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process. 59(9): 4300–4312 (2011).
[Crossref]

M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55(12): 5695–5702 (2007).
[Crossref]

J. Opt. Soc. Am. A (2)

Journal of Communications (1)

A. Yang, J. Zhang, and Z. Hou, “Optimized sensing matrix design based on parseval tight frame and matrix decomposition,” Journal of Communications 8(7): 456–462 (2013).
[Crossref]

Medical Physics (1)

D. Karkala and P. K. Yalavarthy, “Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography,” Medical Physics 39(8): 4715–4725 (2012).
[Crossref] [PubMed]

Opt. Express (4)

Proc. SPIE (5)

D. Sui, J. Ke, and P. Wei, “Implementing two compressed sensing algorithms on GPU,” Proc. SPIE 9273, 92730J (2014).
[Crossref]

P. Li, J. Ke, D. Sui, and P. Wei, “Linear bregman algorithm implemented in parallel GPU,” Proc. SPIE 9622, 962216 (2014).

J. Ke and P. Wei, “Using compressive measurement to obtain images at ultra low-light-level,” Proc. SPIE 8908, 89081O (2013).
[Crossref]

J. Ke, D. Sui, and P. Wei, “Fast object reconstruction in block-based compressive low-light-level imagin,” Proc. SPIE 9301, 930136 (2014).
[Crossref]

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” Proc. SPIE 5817, 7–12 (2005).
[Crossref]

Pure and Applied Geophysics (1)

J. Xia, R. D. Miller, and Y. Xu, “Data-resolution matrix and model-resolution matrix for rayleigh-wave inversion using a damped least-squares method,” Pure and Applied Geophysics 165(7): 1227–1248 (2008).
[Crossref]

SIAM Journal on Imaging Sciences (1)

W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM Journal on Imaging Sciences 5(4): 1185–1212 (2012).
[Crossref]

Other (6)

Y. Gu and N. A. Goodman, “Compressed sensing kernel design for radar range profiling,” in Proceedings of 2013 IEEE Radar Conference (RADAR) (IEEE, 2013), 1–5.

D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in Proceedings of IEEE International Symposium on Information Theory Proceedings (IEEE, 2012), 1231–1235.

Y. Weiss, H. S. Chang, and W. T. Freeman, “Learning compressed sensing,” In Snowbird Learning Workshop, Allerton, CA (2007).

J. Ke, P. Wei, X. Zhang, and E. Y. Lam, “Block-based compressive low-light-level imaging,” in Proceedings of IEEE International Conference on Imaging Systems and Techneques (IEEE2013), 311–316.

R. Baldick, Applied Optimization: Formulation and Algorithms for Engineering Systems(Cambridge University, 2006).
[Crossref]

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

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Figures (12)

Fig. 1
Fig. 1 A system diagram for compressive L3-imaging.
Fig. 2
Fig. 2 Object examples of size (a) (196 × 128) [43,44], and (b) (1056 × 1920).
Fig. 3
Fig. 3 Reconstruction RMSE for objects of size (a) (196×128) and (b) (1056×1920), when M = 32 Hadamard, DCT, and PCA features are collected for (32×32) object blocks.
Fig. 4
Fig. 4 (a) The Hadamard vectors sorted using measurement SNR, (b) the optimal binary vector set or the optimal solution for problem P3, (c) the suboptimal solution for P3 which has the minimal feature measurement dynamic range, when M is 8 for 32 × 32 blocks.
Fig. 5
Fig. 5 When M = 8 features are collected for 32×32 blocks, the normalized RMSE using (a) the sorted Hadamard, DCT, PCA, Frmse, and Fdyn matrices, and (b) the standard Hadamard, random binary, and Frmse matrices.
Fig. 6
Fig. 6 When M = 8 features are collected for 32×32 blocks, the normalized Fourier transforms of the vectors in (a) the standard Hadamard matrix, (b) Hsnr, and (c) Fdyn.
Fig. 7
Fig. 7 (a) The optimal binary vector set or the optimal solution for problem P3, (b) the suboptimal solution for P3 which has the minimal feature measurement dynamic range, (c) the normalized RMSE for sorted Hadamard, DCT, PCA matrices, Frmse, and Fdyn, and the normalized Fourier transforms of the vectors in (d) Hsnr and (e) Fdyn, when M is 16 for 32 × 32 blocks.
Fig. 8
Fig. 8 The object reconstruction using (a) conventional imaging, (b) Hmean, (c) Hsnr, and (d) Fdyn, when M = 16 features are collected for 32 × 32 blocks and σ is 148.67.
Fig. 9
Fig. 9 The object reconstructions (a)&(b) using Hsnr, and (c)&(d) using Fdyn when M = 16 features are collected for 32 × 32 blocks and σ is 148.67.
Fig. 10
Fig. 10 Normalized RMSE for different sets of nonnegative sensing vectors when M = 16 features are collected for 32 × 32 block.
Fig. 11
Fig. 11 Normalized RMSE for different sets of nonnegative sensing vectors when M = 16 features are collected for 32 × 32 blocks. The object blocks are sparse in the wavelet domain. The sparsity is K = 64.
Fig. 12
Fig. 12 (a) The original object which is sparse in the wavelet domain. The sparsity for each 32 × 32 block is K = 4. The object reconstructions using (b) the fast linearized Bregman iterative method with a binary random matrix, (c) the fast linearized Bregman iterative method with Fdyn, and (d) Wiener operator with Fdyn, when M = 16 features are collected for each block and σ is 148.67.

Tables (4)

Tables Icon

Table 1 The cost function ε(F) = Tr{FRxFT} in P1 when M = 32 features are obtained using PCA, DCT, and Hadamard matrices. (*The values listed in the table are ε(F)/106.)

Tables Icon

Table 2 The dynamic ranges of M = 8 features obtained using different sensing matrices. (*The values listed in the table are the actual values divided by 104.)

Tables Icon

Table 3 The dynamic ranges of M = 16 features obtained using different nonnegative matrices. (*The values listed in the table are the actual values divided by 104.)

Tables Icon

Table 4 The (measurement collection time, reconstruction time) for different kinds of sensing matrices and linear/nonlinear methods. 16 feature are collected for each block. The DMD working frequency is set as 32kHz.

Equations (26)

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y = F x + n ,
S N R CS = E { F x 2 } E { n 2 } = Tr { FR x F T } M σ 2 .
P 1 : max F subject to Tr { FR x F T } FF T = I .
P 1 a : max H subject to Tr { HDH T } HH T = I .
F = [ q 1 q 2 q M 1 α q M + 1 α 2 q M + 1 ] T
W = R x F T ( FR x F T + σ 2 I ) 1 .
M S E CS = E { x est x 2 } = Tr { R x } Tr { FR x 2 F T ( FR x F T + σ 2 I ) 1 } .
P 2 : max F subject to Tr { FR x 2 F T ( FR x F T + σ 2 I ) 1 } FF T = I .
F = [ q 1 q 2 q M 1 α q M + 1 α 2 q M + 1 ] T ,
Tr { FR x 2 F T ( FR x F T + σ 2 I ) 1 } = j = 1 M d i 2 d i + σ 2 + d M 2 α 2 + d M + 1 2 ( 1 α 2 ) d M α 2 + d M + 1 ( 1 α 2 ) + σ 2 d M 2 d M + σ 2 = j = 1 M d i 2 d i + σ 2 + ( 1 α 2 ) [ σ 2 ( d M + 1 2 d M 2 ) + d m d M + 1 ( d M + 1 d M ) ] ( d M α 2 + d M + 1 ( 1 α 2 ) + σ 2 ) ( d M + σ 2 ) j = 1 M d i 2 d i + σ 2 ,
P 3 : min A subject to ε ( A ) = N AQ P C A T sgn ( AQ PCA T ) F 2 AA T = 1 ,
G i j = { k = 1 M A i k 2 1 , i = j k = 1 M A i k A j k , otherwise .
( A , λ ) = i = 1 M j = 1 N [ N AQ PCA T sgn ( AQ PCA T ) ] i j 2 + i = 1 M j = i M [ λ . G ] i j ,
[ AA 2 ( a ( ν ) , λ ( ν ) ) J ( a ( ν ) ) T J ( a ( ν ) ) 0 ] [ Δ a ( ν ) Δ λ ( ν ) ] = [ ε ( a ( ν ) ) + J ( a ( ν ) ) T λ ( ν ) g ( a ( ν ) ) ] ,
[ a ( ν + 1 ) λ ( ν + 1 ) ] = [ Δ a ( ν ) Δ λ ( ν ) ] + α [ a ( ν ) λ ( ν ) ] .
2 ( a , λ ) A i j A i j = 2 N δ ( i i ) δ ( j j ) + 2 λ i i δ ( j j ) .
G i j A i j = δ ( i i ) A j j + δ ( j i ) A i j .
ε A i j = 2 N A i j 2 N [ sgn ( XQ PCA T ) Q PCA ] i j .
ε A i j = 2 k = 1 N [ N [ AQ PCA T ] i k s g n ( [ AQ PCA T ] i k ) ] [ N [ Q PCA T ] j k δ ( [ AQ PCA T ] i k ) [ Q PCA T ] j k ]
ε A i j = 2 k = 1 N [ N [ AQ PCA T ] i k s g n ( [ AQ PCA T ] i k ) ] N [ Q PCA T ] j k = 2 N [ AQ PCA T Q PCA ] i j 2 N [ s g n ( AQ PCA T ) Q PCA ] i j = 2 N [ A ] i j 2 N [ s g n ( AQ PCA T ) Q PCA ] i j ,
2 ε A i j A i j = 2 N δ ( i i ) δ ( j j ) 2 N A i j [ k = 1 N s g n ( [ AQ PCA T ] i k ) [ Q PCA ] k j ] = 2 N δ ( i i ) δ ( j j ) 2 N δ ( i i ) k = 1 N δ ( [ AQ PCA T ] i k ) [ Q PCA T ] i k [ Q PCA ] k j = 2 N δ ( i i ) δ ( j j )
ε λ A i j = k , k = 1 M [ λ . * A i j ( AA T 1 ) ] k k = k , k = 1 M λ k k G k k A i j
C k k A i j = δ ( k i ) A k j + δ ( k i ) A k j .
ε λ A i j = k , k = 1 M λ k k ( δ ( k i ) A k j + δ ( k i ) A k j ) = k = 1 M 2 λ i k A k j
2 ε λ A i j A i j = ( k = 1 M 2 λ i k A k j ) A i j = 2 λ i i δ ( j j )
2 A i j A i j = 2 ε A i j A i j + 2 ε λ A i j A i j = 2 N δ ( i i ) δ ( j j ) + 2 λ i i δ ( j j )

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