In the recent literature, an Acoustic Single-Pixel Imager has been successfully developed for source localization in a two-dimensional waveguide. Source bearing angle estimation was carried out by applying sparse recovery techniques on sensor measurements taken over different imaging screens. This paper shows that the Mutual Coherence of the sensing matrix can be used as a metric to predict the source localization capability of the single-pixel imaging system. In particular, this paper's analysis focuses on the sparsity of open cells within the imaging screen and the number of imaging screens used to maximize the probability of correct detection over varying levels of source sparsity. In this work, a simulation environment to demonstrate how the mutual coherence of the sensing matrix correlates with source localization performance over source sparsity, sparsity of open screen cells, and number of measurements used for sparse recovery is developed. The analysis shows that the leading factor in source localization performance gains is primarily from the number of imaging screens used to measure the acoustic wave-field.

In this paper, mutual coherence is shown to be a useful metric for predicting source localization performance as a function of screen selection for the acoustic single-pixel imager in the additive noise-free case. In particular, mutual coherence is computed for random open/closed screen cell distributions with varying levels of open cell sparsity. It is shown that for a given screen cell sparsity level, the location of open cells within the screen has a negligible effect on the mutual coherence and thus minimal change in source localization performance of the acoustic single-pixel imager. Furthermore, it is demonstrated that the mutual coherence of the single-pixel imaging system can be pre-computed for a given open cell sparsity level and an arbitrary number of measurements. Therefore, the mutual coherence of the imaging system can be used to infer source localization performance a priori and subsequently optimize the selection of physical imaging screens to provide improved acoustic source localization. Recently, an acoustic single-pixel imaging system has been developed for source localization and imaging.1 In that work, Rogers et al. designed and experimentally validated the use of an acoustic single-pixel imager for acoustic source localization in a two-dimensional (2D) waveguide. There, the authors fabricated a prototype system based on a waveguide design where a passive imaging screen consisting of open/closed cells was used to steer acoustic pressure waves to a single sensor microphone array. The screens open/closed cell patterns followed a 2-bit quantized Discrete Cosine Transform (DCT) pattern parameterized by the screen index and cell position. In that work, four orthogonal DCT screens were used to localize a single source and the statistical Restricted Isometry Coefficient (RIC) was used as a metric to characterize the reliability of sparse recovery. However, the choice of the set of deterministic screens was somewhat ad hoc and not based in any optimality. It is commonly accepted in practice that random sensing matrices will satisfy RIC, ensuring sparse recovery.2,3 However, the sensing matrices assumed in the single pixel framework are not truly random due to the deterministic structure of the imaging screen and acoustic signal propagation model. Furthermore, it is not clear what effect the number of distinct screens used to gather measurements and the sparsity of open screen cells have on the l1 minimization process used to estimate bearing angles of sources within the waveguide.

The theoretical foundation for the concept of the single-pixel imager comes from acoustic wave propagation and passive sensor array signal processing. In particular, in passive sensing, a sensor array enables spatial filtering which can be used to estimate the relative bearing of an acoustic source. The most common sensor array geometries used in practice are arrays with uniformly spaced elements having inter-element spacings of half the source wavelength of interest. The half-wavelength element spacing criteria is known as spatial Nyquist sampling and when sensor arrays are designed with this constraint, spatial aliasing can be avoided.4 In the literature, attention has been placed on the design of sparse sensor array configurations that greatly reduce the number of sensor elements over that of a uniform array.5–8 The design of sparse arrays is primarily concerned with how to place sensors within the desired aperture such that array processing techniques can be developed to mitigate the negative effects of under-sampling the aperture. These techniques rely on methods to reduce the impact of side-lobes and grating lobes on source localization performance.4,9 The concept of sparsity has also been applied to source localization via compressive sensing techniques. In this framework, a constraint on signal sparsity enables the application of convex optimization to recover source bearings from the sensor array channel data. This approach has been shown to outperform conventional array processing methods in array resolution in situations where arrivals are coherent, and in low signal-to-noise ratio signal regimes.10,11 In addition to source sparsity, sparse sensor array configurations have been considered in the compressing sensing paradigm.12–16 In the work presented in this paper, we bring together the concepts from passive array processing and compressive sensing to analyze the performance of the single-pixel imaging system.

The one-dimensional (1D) acoustic single-pixel imaging system consists of a passive imaging screen, transmit source array, and a single receive sensor element enclosed in a waveguide. Figure 1 shows a top-down view of the waveguide showing the transmit and receive sensor geometries. In the design of the simulation experiments presented in this work, a non-reflective boundary along all edges of the waveguide is assumed. In addition, we assume all transmitting sources are in the far-field of the imaging screen. These modeling simplifications allow us to develop a first-order relationship between the distribution of open/closed screen cells, the number of independent measurements used in sparse recovery, and the overall source localization performance of the single-pixel imager. In future works, we will consider reflection and diffraction properties inherent to the imaging system.

FIG. 1.

(Color online) Single-Pixel Imager Simulation Environment: The simulation environment consists of a waveguide, imaging screen, transmit source array, and single receive sensor element (single-pixel). This example shows a source sparsity level of K = 3 and open cell sparsity level P = 6.

FIG. 1.

(Color online) Single-Pixel Imager Simulation Environment: The simulation environment consists of a waveguide, imaging screen, transmit source array, and single receive sensor element (single-pixel). This example shows a source sparsity level of K = 3 and open cell sparsity level P = 6.

Close modal

The imaging screen is a 1 × N cell array consisting of open/closed screen cells. Let P denote the number of open screen elements among the N possible positions, and assume that P < N. The parameter, P, represents the sparsity level among the open screen cells for a given imaging screen. The receive sensing element collects acoustic signals emitted from sources that propagate through the waveguide and are then occluded by the closed cells of the imaging screen. The position of the receive sensor is rb (m) from the imaging screen. The transmit grid (Tx-grid) denotes the spatial location of potential sources in the far-field of the imaging screen. The Tx-grid consists of a total of M source locations parametrized by the bearing angle, { θ 1 , θ 2 , , θ M } , relative to the center of the imaging screen. The Tx-grid is a distance of ra (m) from the imaging screen, where r a > D 2 / 8 λ . The quantity D 2 / 8 λ represents an approximation to the far-field boundary, where λ is the source wavelength and D is the aperture length of the imaging screen.17 Let K denote the number of active sources elements among the total possible M sources locations, where K < M. Here the parameter K indicates the sparsity level corresponding to the number of active sources at a given time instant.

Recently, it was demonstrated that the diffracted signal observed at the single receive element can be used to reliably estimate source bearing angle with a limited number of measurements collected from a distinct set imaging screens.1 In that work, sensing matrix mutual coherence is analyzed using a simplified model, based on Rogers et al.,1 that assumes the received signal can be represented by a sum of point sources emanating from each screen opening. In the current work, we explore how open screen cell sparsity, the number of independent measurements, and source sparsity interact as seen through sensing matrix mutual coherence. In the remainder of this section, we mathematically define the single-pixel imager and describe how we design the imaging screen classes considered in this work.

Using the imaging framework described above and shown in Fig. 1, our objective is to use the acoustic signal collected at the receive sensor to reconstruct the source bearing angle distribution exhibited on the Tx-grid. In 1D source localization, the far-field source position is parameterized by the bearing angle θ. Suppose K sources are transmitting a common narrowband signal with frequency f0 (Hz) and the source bearing angles are constrained to lie on a uniform grid consisting of M possible positions distributed over the range θ 0 θ θ 0 . To estimate the source locations, we exploit source sparsity and define a compressive sensing model that utilizes a limited number of linear single sensor measurements. A compressive sensing model for this situation can be defined in the following manner. Let the additive noise-free single sensor measurement, y C 1 × 1 be given by

(1)

where the sensing matrix A is defined by A = h Ψ and x ( 0 , 1 ) M × 1 is a M × 1 ones and zeros column vector indicating which source is active on Tx-grid. The sensing matrix A is composed of two matrices, one that models the distribution of open/closed cells of the imaging screen, h, and the other that models the far-field propagation of the acoustic waves arriving at the imaging screen, Ψ . The imaging screen h ( 0 , 1 ) 1 × N is a 1 × N dimensional vector that has entries 1 or 0 depending on if a screen cell is open or closed. The matrix Ψ is an N × M matrix comprised of replica vectors representing the phase delay that a plane wave imparts over the cells of the imaging screen. The ith column of Ψ represents the plane-wave phase delay of a signal emanating from an active source at bearing angle θi relative to the imaging screen. The propagation matrix Ψ is defined as

(2)

where each column is given by

(3)

where c is the speed of sound in air and { d 1 , d 2 , , d N 1 } are the inter-cell distances measured from the center of the left most cell to the center of each subsequent screen cell. In this measurement model, the action of x on Ψ selects the position of active sources on the Tx-grid and the action of h on the product Ψ x sums the plane-wave phase delays associated with open screen cells of the imaging screen. The product h Ψ x produces a 1D measurement y that represents the signal received at the single-pixel sensor element. In the work presented here, the diffraction effects after the imaging screen are neglected, making rb = 0 in Fig. 1. In future work, these diffraction effects will be considered.

In this work, we analyze how the number of linear measurements from independent imaging screens impacts the source localization capability of the single-pixel imaging system. In addition, we compare source localization results from classes of imaging screens with varying levels of open cell sparsity. When distinct imaging screens are used to obtain individual single-pixel measurements, the compressive sensing formulation described in Eq. (1) becomes y = Ax. The compressive sensing measurement is now vector-valued where y C L × 1 and the parameter L represents the number of distinct imaging screens used in the imaging process. Thus, each entry of y represents an independent measurement of the acoustic wave-field inside the waveguide from a distinct imaging screen. In this situation, the sensing matrix A is written as A = H Ψ with

(4)

where each row of H represents an independent 1 × N imaging screen vector. As in the 1D case, the mathematical model used to represent the single-pixel imaging system is defined by the matrix product A x = H Ψ x . However, in practice, the imaging screen would be swapped out for different screen realizations and acoustic measurements would be concatenated to produce the vector-valued measurement y.

When L < M, the measurement equation represents a linear system of underdetermined equations. Under conditions imposed on the structure of the sensing matrix and the sparsity of the source vector, l1 inversion techniques can be applied to recover a sparse estimate of the source distribution vector x ̂ from the measurement vector y. We assume that the source location vector x is K-sparse where only K of M entries of x are non-zero with, K M . In recovering the sparsity constrained vector x, it is possible to estimate the source location distribution for the far-field acoustic sources in the waveguide. The signal recovery l1 minimization problem is given by

(5)

In general, sparse recovery theory was developed under the minimization of the l0-norm of x; however, this problem is known to be NP-hard.18 It is well known that by minimizing x with respect to the l1-norm, the optimization problem described in Eq. (5) is convex and the l1-norm is a sparsity promoting norm which serves as a proxy to the l0-norm. To guarantee a reliable solution to Eq. (5) the sensing matrix must satisfy the Restricted Isometry Property (RIP)

(6)

which provides a measure of the contraction or dilation of x under A. However, evaluating the RIP condition for a given sensing matrix A of modest dimensions has been shown to be NP-hard.19,20 In our setting, the dimensions of A are L × M and verifying RIP requires an exhaustive search over ( M K ) subspaces. When M is small, this could be possible, but once M 15 , RIP verification becomes computationally challenging. In light of this, we use the mutual coherence of the sensing matrix as a metric to examine the source recovery performance of different deterministic sensing matrices.

The matrix H ( 0 , 1 ) L × N is a collection of one and zero row vectors that represent the open/closed cell distribution of individual imaging screens. Each row of the screen matrix H is used to produce independent measurements of the wave-field within the waveguide. In this analysis, the number of screen cells is chosen to be constant at N = 75 and the number of measurements L is allowed to run over the range 1 L N + 10 . In addition, we consider three distinct screen element classes where imaging screens used to form H are drawn. In particular, we consider a class of orthogonal imaging screens, I 1 ; where only one screen cell is open, we consider a class of non-orthogonal sparsity constrained imaging screens, S P , where exactly P screen cells are open and drawn at random from the collection of N possible cell positions, and we consider a class of imaging screens where screen cells are chosen open or closed completely at random, R , with no restriction on the number of open cells per screen realization. The space of sparsity constrained screens are drawn for the sparsity levels P = 2 , 10 , 30 , 50 , 70 . These classes of screens allow for the design of a simulation protocol to assess how the distribution of open cells within the imaging screen impacts source localization performance. In addition, the outcome of the analysis drawn from these simulations can lead to simpler imaging screen selection that can be realized with low computational complexity in a physical single-pixel imaging system.

Figure 2 shows examples of screen matrices H ( 0 , 1 ) 10 × 75 for each of the classes considered in this work. The screens show a collection of L = 10 imaging screens per class, where each screen is composed of N = 75 cells. When the source frequency is f 0 = 10 kHz , the total aperture of the screen is L = 1.26 m . Figure 2(a) shows a set of orthogonal screens where only one screen cell per screen is open. Figures 2(b) and 2(c) show a set of screens where each screen is constrained to contain exactly ten open cells and a set of screens where open cells per screen are chosen at random, respectively. Choosing cells to be open or closed at random implies that the number of open cells vary within the imaging screen set, { h i } i = 1 L . Figure 2(d) shows a histogram over the number of open cells for imaging screen drawn from R . On average, the number of open screen cells is approximately 38, while some imaging screen realizations have as little as 20 open cells or as many as 55 open cells. Not shown in this figure are the constrained imaging screen sets with 2, 30, 50, or 70 open cells per screen that are also considered in this work. Considering these classes of imaging screens allows for a simple approach to evaluating source localization performance over the space of imaging screens for a given aperture.

FIG. 2.

(Color online) Example Imaging Screens Matrices H: Imaging screen realizations from different screen classes. The screen class I 1 is constrained to consist orthogonal screens with an open cell sparsity level of P = 1, while S P is only constrained by sparsity level P > 1. The class R consists of imaging screen whose open/closed cell positions are chosen completely at random. (a) I 1 1 open cell per imaging screen. (b) S 10 , ten open cells per imaging screen. (c) R , open cells chosen completely random selection per imaging screen. (d) Distribution of the number of open cells for imaging screens drawn from R .

FIG. 2.

(Color online) Example Imaging Screens Matrices H: Imaging screen realizations from different screen classes. The screen class I 1 is constrained to consist orthogonal screens with an open cell sparsity level of P = 1, while S P is only constrained by sparsity level P > 1. The class R consists of imaging screen whose open/closed cell positions are chosen completely at random. (a) I 1 1 open cell per imaging screen. (b) S 10 , ten open cells per imaging screen. (c) R , open cells chosen completely random selection per imaging screen. (d) Distribution of the number of open cells for imaging screens drawn from R .

Close modal

Theoretical bounds on recovery accuracy have been developed over different classes of sensing matrices. It has been shown that when the sensing matrix A is a random matrix that has Independent Identically Distributed (IID) entries drawn from Gaussian or Bernoulli distribution, the RIP constraint will be satisfied with high probability, which enables reliable signal recovery.2,3 In addition, when the sensing matrices are deterministic, Calderbank et al.21 has shown that by imposing a group structure under point-wise multiplication across the columns of the sensing matrix, imposing an orthogonality and sum constraint on the rows of the sensing matrix, and imposing a bounded energy constraint over the sensing matrix, a large class of deterministic matrices satisfy the RIP constraint with high probability. Examples of these type of deterministic sensing matrices are derived from Discrete Chirp waveform, Reed-Muller Codes, or Fourier ensembles.22 While these classes of sensing matrices lead to guarantees on sparse recovery performance, they do not always fit real-world modeling constraints. To address these issues, an approach for compressive sensing that does not require the RIP constraint to be satisfied has been investigated. Candés and Plan23 showed that sparse recovery is possible when the sensing matrix follows an incoherence property as well as an isometry property while not requiring RIP to hold. Zhang24 developed RIP-free recovery and stability criteria that depends only on the null space of the sensing matrix. In the work presented in this paper, we focus on deterministic sensing matrix designs that are constrained by the acoustic plane-wave model and the single pixel imaging process. As a result of this sensing phenomena, the degrees of freedom in the design of the sensing matrix are parameterized by open/closed cell positions of the imaging screen matrix H, and the number of measurements L used for sparse recovery. Our approach to assess the source localization performance of the single-pixel imaging system is to statistically evaluate the mutual coherence of the sensing matrix A as a function of the imaging screen matrix, H, the number of independent measurements, L, and over different levels of source sparsity, K.

The mutual coherence is a measure that has been used to quantify the performance of sensing matrices for sparse recovery.25 The mutual coherence of a matrix is defined as

(7)

where a i are the columns of the sensing matrix A. The mutual coherence metric has been related to the restricted isometry constant and a well-established lower-bound has been developed that can be used for assessing the quality of the mutual coherence of a given sensing matrix.25,26 The lower-bound (Welch Bound) on mutual coherence is of particular interest and for A C L × M is given as

(8)

which can be used to compare with empirical values of mutual coherence. In practical terms, if the mutual coherence is small, then the columns of A are nearly orthogonal and reliable sparse recovery can be expected.

In recent literature, mutual coherence has been used in the assessment and design of sensing matrices. Obermeier and Martinez-Lorenzo27 developed an algorithm for the design of sensing matrices used in electromagnetic imaging based on the minimization of mutual coherence. These authors applied their work to determine sensor positions of an imaging system whose sensing matrix would exhibit minimal mutual coherence and showed the optimized sensor configuration outperformed other sensor distributions over the same imaging task. Similarly, Xu et al.28 developed methods to minimize the mutual coherence between the projection matrix and signal dictionary that leads to improved sparse recovery. Zhang et al.29 proposed a class of sparse binary sensing matrices based on the photograph Low-Density Parity-Check (LDPC) codes and analyzed recovery performance as a function of mutual coherence. In that work, it was shown that there is a direct correlation between low mutual coherence and the probably of exact recovery. Similarly, in this paper, we determine the correlation between the mutual coherence of our sensing matrices and the probability of correct detection realized through the single-pixel imaging system.

In the imaging simulations presented here, we perform a Monte Carlo (MC) analysis to determine the operating conditions under which optimal source localization can be expected. The MC process iterates over the sparsity level K of the source vector, the sparsity level of the imaging screen P, and the number of measurements used to estimate the source vector x from the measurement vector y , L. In particular, we consider an imaging screen that has 75 screen cells which spans an aperture of D = 74 ( λ / 2 ) where each cell has a width of λ / 2 , with λ defined as the source wavelength. In these experiments, all far-field sources are transmitting a common narrowband continuous wave (cw) tone at 10 KHz in air producing a source wavelength of λ = c / f = 343 ( m / s ) / 10 kHz = 0.0343 ( m ) . Under this source wavelength, the aperture of the imaging screen becomes D = 1.26 ( m ) . Since the source Tx-grid is assumed in the far-field of the imaging screen, the parameter ra is assumed to be r a > D 2 / 8 λ = 5.86 ( m ) , where D 2 / 8 λ is an approximation of the far-field boundary.17 In our simulations, the receive pixel is positioned at the center of the imaging screen making rb = 0. While the dimensions of the waveguide induced by the size of the imaging screen aperture are large, the statistical information gained from this simulation environment can be used in further studies for smaller scale systems. In particular, the source parameters are consistent with work by Rogers et al.;1 however, we consider a larger aperture imaging screen to allow for clearer delineations of source localization operating performance regions.

The primary screen design parameter is the number of open cells within the screen aperture for a given imaging screen. Evaluating all possible configurations is computationally infeasible, and we therefore rely on the random selection of screens but place a constraint on the number of open cells per screen realization. To accomplish this type of random sampling for a given open cell sparsity level P, we draw P samples from a list of integers over the range [ 1 , N ] without replacement. In this sense, each realization of an imaging screen h with sparsity level P will contain exactly P non-zero values. By repeating this process, we can draw L imaging screens of open sparsity level P with near-zero probability of repeating a given screen. The classes of screen considered in this work are discussed in Sec. II B. The source sparsity range we consider in this work represents a range from 2 to 70 far-field sources. The number of measurements L used to produce an estimate of the source location vector, x ̂ spans a range from 2 L 85 . With these simulation constraints, we can statistically determine how many screen cells should be open and how many measurements need to be taken to maximize the localization performance for a given number of far-field acoustic sources.

The source location vector x is a M × 1 vector that describes the angular positions of far-field sources. When x is K-sparse, then only K of the entries of x are non-zero and the vector indexes of those entries indicate the bearing angles, θi, of active sources in the far-field, where i { 1 , 2 , 3 , , K } . Given a particular K-sparse vector x, we can define the subset of entries containing one and zero entries as x On and x Off , respectively. To determine, after l1-inversion, if a source is present at a particular source grid point, we can threshold the amplitudes of entries observed in x ̂ , derived from Eq. (5) corresponding to the on and off source subsets. The decision criteria taken over the amplitudes is as follows: Correct Detection when x ̂ On ( i ) τ , Missed Detection when x ̂ On ( i ) < τ , and False Alarm when x ̂ Off ( i ) τ . Thus, for a given entry of x ̂ , we can classify the corresponding point on the source grid as active or inactive and determine if the active entries are true sources or false alarms.

Figure 3 shows an example source vector x and source estimate vector x ̂ where each entry of these vectors represents an angular position on the uniform source grid defined over [ θ 0 , θ 0 ] . In this example, we have a total of seven far-field sources with angular position indicated by the position of the non-zero entries in x. The source estimate vector x ̂ contains entries that take values according to the l1-inversion process, which are inherently not restricted to take values of 0 or 1. Once the decision threshold τ is applied to the amplitudes of x ̂ , we can classify each entry of x ̂ as either a correct detection, missed detection, or false alarm. Over the subset of active source positions, x ̂ On , we see that amplitudes of five of seven sources lie above the decision threshold and can be classified as correct detections, while two fall below the decision threshold and are classified as missed detections. Over the subset of source positions, x ̂ Off , two amplitudes exceed the decision threshold and are classified as false alarms, while the remaining amplitudes in this subset fall below the threshold, implying a source is not present.

FIG. 3.

(Color online) Source and Source Estimate Distributions: The l1 inversion process returns a vector of signal amplitudes representing an estimate of the likelihood that a source is transmitting a signal from a given angle relative to the center of the imaging screen. FA and MD represent false alarms and missed detection resulting from the l1 inversion process, respectively.

FIG. 3.

(Color online) Source and Source Estimate Distributions: The l1 inversion process returns a vector of signal amplitudes representing an estimate of the likelihood that a source is transmitting a signal from a given angle relative to the center of the imaging screen. FA and MD represent false alarms and missed detection resulting from the l1 inversion process, respectively.

Close modal

The threshold detector is used to determine the probability of correct detection as a function of source sparsity, K, and the number of measurements, L, used to produce a source vector estimate. The probability of correct detection is computed by taking the ratio of the number of correct detection over the total number of active sources for a given sparsity level, K. Over the MC iterations for a class of screens and number of measurements L, we report the average probability of correct detection for each sparsity level K.

For each source sparsity level K, a total of 125 source vector realizations x are used for these simulations and the bearing angles are restricted to lie between [ 90 ° , 90 ° ] . The source vector realization sets are common to all MC iterations over the number of measurements and screen classes ( I 1 , S P , and R ). For a given number of measurements L and imaging screen class, 500 realization of the imaging screen matrix H are applied in sparse recovery. In addition, the threshold used in classifying recovery source locations is set to τ = 0.9. This choice of threshold allows for consistent recovery results that empirically maximize the number of correct detections while minimizing the number of false alarms.

Figure 4 shows the probability of correct detection and the mutual coherence of the sensing matrix A as a function of the number of measurements, L, used in sparse recovery. Figures 4(a)4(c), and 4(e) show the probability of correct detection over the measurement range 2 L 85 [the number rows in H, shown in Eq. (4)] where each curve represents the average probability of correct detection for a given class of imaging screens with different sparsity levels P, where P = 1, 2, 10, 30, 50, 70. In this figure, we consider three distinct source sparsity levels, K = 4, K = 10, and K = 50. From these figures, the PD curves are nearly identical for each class of screens suggesting that the number of open cells per screen has minimal impact on sparse recovery. Figures 4(b), 4(d), and 4(f) show the corresponding mutual coherence over the number of measurements, L, used in sparse recovery for the same sparsity levels as those discussed above. This set of figures highlights how the mutual coherence can be used as a sparse recovery performance metric. In particular, as the source sparsity increases from K = 4 to K = 10, the mutual coherence in each screen class increases. This suggests that the recovery performance is expected to be poorer for K = 10 than K = 4. Similarly, we have the same results when sparsity increases from K = 10 to K = 50. However, in this case, the mutual coherence reaches its maximum of 1 implying that recovery performance is expected to be unreliable. Figures 4(a), 4(c), and 4(e) support this interpretation as the probability of correct detection deteriorates as source sparsity increases.

FIG. 4.

(Color online) Mutual Coherence and Probability of Correct Detection (PD): For a given sparsity level K, the regions corresponding to high mutual coherence correspond to poor PD, while the regions of low mutual coherence correspond to favorable PD. (a) Source Sparsity Level K = 4. (b) Source Sparsity Level K = 4. (c) Source Sparsity Level K = 10. (d) Source Sparsity Level K = 10. (e) Source Sparsity Level K = 50. (f) Source Sparsity Level K = 50.

FIG. 4.

(Color online) Mutual Coherence and Probability of Correct Detection (PD): For a given sparsity level K, the regions corresponding to high mutual coherence correspond to poor PD, while the regions of low mutual coherence correspond to favorable PD. (a) Source Sparsity Level K = 4. (b) Source Sparsity Level K = 4. (c) Source Sparsity Level K = 10. (d) Source Sparsity Level K = 10. (e) Source Sparsity Level K = 50. (f) Source Sparsity Level K = 50.

Close modal

Figure 5 shows a comparison of the probability of correct detection and mutual coherence for imaging screens with open/closed patterns chosen, from R , completely at random over a source sparsity range of 2 K 70 . Here, we examine the relationship between sparse recovery performance and mutual coherence when sparse recovery is performed with L = 7, 10, 15, 30 measurements. In each of these figures, when the mutual coherence is low, the probability of correct detection is high; when the mutual coherence is high, the probability of correct detection is low. In addition, these figures show that as the number of measurements, L, increases sparse recovery improves allowing for high probability of correct detection as the number of far-field sources increases. These figures also indicate how the probability of correct detection does not converge to zero as the performance deteriorates, but instead stabilizes near a fixed point. Over the subset of active source entries, xOn, the proportion of those entries that produce a sparse recovery response amplitude above the decision threshold converges approximately to a state that is proportional to the rank of the corresponding sensing matrix. This is attributed to a breakdown in the l1 inversion process where the number of active sources exceeds the rank of the sensing matrix. This implies that desirable sparse recovery can be achieved when the number of active far-field sources is less than the number of measurements used to estimate the source location distribution.

FIG. 5.

(Color online) Mutual Coherence and Probability of Correct Detection: For a given number of measurements L, the mutual coherence of the sensing matrix A inversely tracks the probability of correct detection. This suggests that mutual coherence can be used to predict sparse recovery performance as a function of source sparsity. In addition, for a given set of measurement L, the probability of correct detection converges, with respect to source sparsity K, to a quantity proportional to the rank of the sensing matrix A, namely L/100. (a) No. Measurements L = 7. (b) No. Measurements L = 10. (c) No. Measurements L = 15. (d) No. Measurements L = 30.

FIG. 5.

(Color online) Mutual Coherence and Probability of Correct Detection: For a given number of measurements L, the mutual coherence of the sensing matrix A inversely tracks the probability of correct detection. This suggests that mutual coherence can be used to predict sparse recovery performance as a function of source sparsity. In addition, for a given set of measurement L, the probability of correct detection converges, with respect to source sparsity K, to a quantity proportional to the rank of the sensing matrix A, namely L/100. (a) No. Measurements L = 7. (b) No. Measurements L = 10. (c) No. Measurements L = 15. (d) No. Measurements L = 30.

Close modal

In these figures, we have shown how the mutual coherence inversely correlates with the probability of correct detection as a function of source sparsity, the number of measurements used for sparse recovery, and the sparsity level of open screen cells. This analysis suggests that, in the additive noise-free case, the leading factor to ensure accurate source localization is the availability of unique measurements which exceed the sparsity level of the source signal.

In this paper, we analyzed the sparse recovery performance of far-field source localization based on a noise-free acoustic single-pixel imaging system. The single-pixel imager is designed as a system consisting of a single receive sensor that collects acoustic waves traveling within a waveguide obstructed by an imaging screen. The imaging screen is a collection of open/closed cells that span a fixed aperture across the width of the waveguide. The cells obstruct the acoustic pressure field created by the far-field source that can be exploited by sparse recovery techniques. In this work, we performed a statistical analysis over the probability of correct detection as a function of three imaging design parameters. Our results suggest that the mutual coherence of the sampling matrix can serve as a useful metric to assess the expected quality of sparse recovery. In addition, we have found that the sparsity level of open screen cells has a minimal impact on source localization performance. Instead, the leading factor in attributing to sparse recovery performance is the number of independent measurements used in estimating the source position vector x. Using these results, we can design an experiment on a physical waveguide, similar to that conducted by Rogers et al.,1 where we can experimentally verify our findings.

This research was supported by the Office of Naval Research.

1.
J. S.
Rogers
,
C. A.
Rohde
,
M. D.
Guild
,
C. J.
Naify
,
T. P.
Martin
, and
G. J.
Orris
, “
Demonstration of acoustic source localization in air using single pixel compressive imaging
,”
J. Appl. Phys.
122
(
21
),
214901
(
2017
).
2.
E. J.
Candes
and
T.
Tao
, “
Near-optimal signal recovery from random projections: Universal encoding strategies?
,”
IEEE Trans. Inf. Theory
52
(
12
),
5406
5425
(
2006
).
3.
R.
Baraniuk
,
M.
Davenport
,
R.
DeVore
, and
M.
Wakin
, “
A simple proof of the restricted isometry property for random matrices
,”
Construct. Approx.
28
(
3
),
253
263
(
2008
).
4.
H. L.
Van Trees
,
Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory
(
Prentice Hall
,
Englewood Cliffs, NJ
,
2002
).
5.
H.
Unz
, “
Linear arrays with arbitrarily distributed elements
,”
IEEE Trans. Antennas Propag.
8
(
2
),
222
223
(
1960
).
6.
A.
Moffet
, “
Minimum-redundancy linear arrays
,”
IEEE Trans. Antennas Propag.
16
(
2
),
172
175
(
1968
).
7.
S.
Holm
, “
Sparse and irregular sampling in array processing
,” in
Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing
, Istanbul, Turkey (June 5–9,
2000
), pp.
3850
3853
.
8.
P.
Vaidyanathan
and
P.
Pal
, “
Sparse sensing with co-prime samplers and arrays
,”
IEEE Trans. Signal Process.
59
(
2
),
573
586
(
2011
).
9.
H.
Oraizi
and
F.
Fallahpout
, “
Nonuniformly spaced linear array design for the specified beamwidth/sidelobe level or specified directivity/sidelobe level with coupling consideration
,”
Prog. Electromagn. Res.
4
,
185
209
(
2008
).
10.
P.
Gerstoft
,
A.
Xenaki
,
C. F.
Mecklenbrker
, and
E.
Zochmann
, “
Multiple snapshot compressive beamforming
,” in
Proceedings of the 2015 49th Asilomar Conference on Signals, Systems and Computers
, Pacific Grove, CA (November 8–11,
2015
), pp.
1774
1778
.
11.
G. F.
Edelmann
and
C. F.
Gaumond
, “
Beamforming using compressive sensing
,”
J. Acoust. Soc. Am.
130
(
4
),
EL232
EL237
(
2011
).
12.
Z.
Tan
and
A.
Nehorai
, “
Sparse direction of arrival estimation using co-prime arrays with off-grid targets
,”
IEEE Signal Process. Lett.
21
(
1
),
26
29
(
2014
).
13.
Z.
Tan
,
Y. C.
Eldar
, and
A.
Nehorai
, “
Direction of arrival estimation using co-prime arrays: A super resolution viewpoint
,”
IEEE Trans. Signal Process.
62
(
21
),
5565
5576
(
2014
).
14.
Y. D.
Zhang
,
S.
Qin
, and
M. G.
Amin
, “
DOA estimation exploiting coprime arrays with sparse sensor spacing
,” in
Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
, Florence, Italy (May 4–9,
2014
), pp.
2267
2271
.
15.
E.
BouDaher
,
Y.
Jia
,
F.
Ahmad
, and
M. G.
Amin
, “
Multi-frequency co-prime arrays for high-resolution direction-of-arrival estimation
,”
IEEE Trans. Signal Process.
63
(
14
),
3797
3808
(
2015
).
16.
C. F.
Gaumond
and
G. F.
Edelmann
, “
Sparse array design using statistical restricted isometry property
,”
J. Acoust. Soc. Am.
134
(
2
),
EL191
EL197
(
2013
).
17.
A. J.
Weiss
and
B.
Friedlander
, “
Range and bearing estimation using polynomial rooting
,”
IEEE J. Oceanic Eng.
18
(
2
),
130
137
(
1993
).
18.
E. J.
Candes
and
T.
Tao
, “
Decoding by linear programing
,”
IEEE Trans. Inf. Theory
51
(
12
),
4203
4215
(
2005
).
19.
A. M.
Tillmann
and
M. E.
Pfetsch
, “
The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing
,”
IEEE Trans. Inf. Theory
60
(
2
),
1248
1259
(
2014
).
20.
P.
Koiran
and
A.
Zouzias
, “
On the certification of the restricted isometry property
,” arXiv:1211.0665 (
2011
).
21.
R.
Calderbank
,
S.
Howard
, and
S.
Jafarpour
, “
Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property
,”
IEEE J. Selected Topics Signal Process.
4
(
2
),
358
374
(
2010
).
22.
T. L.
Nguyen
and
Y.
Shin
, “
Deterministic sensing matrices in compressive sensing: A survey
,”
Sci. World J.
2013
,
192795
(
2013
).
23.
E. J.
Candés
and
Y.
Plan
, “
A probabilistic and ripless theory of compressed sensing
,”
IEEE Trans. Inf. Theory
57
(
11
),
7235
7254
(
2011
).
24.
Y.
Zhang
, “
Theory of compressive sensing via ℓ1-minimization: A non-rip analysis and extensions
,”
J. Oper. Res. Soc. China
1
(
1
),
79
105
(
2013
).
25.
S.
Foucart
and
H.
Rauhut
,
A Mathematical Introduction to Compressive Sensing
(
Birkhäuser
,
Basel, Switzerland
,
2013
), Vol.
1
.
26.
L.
Welch
, “
Lower bounds on the maximum cross correlation of signals (corresp.)
,”
IEEE Trans. Inf. Theory
20
(
3
),
397
399
(
1974
).
27.
R.
Obermeier
and
J. A.
Martinez-Lorenzo
, “
Sensing matrix design via mutual coherence minimization for electromagnetic compressive imaging applications
,”
IEEE Trans. Comput. Imaging
3
(
2
),
217
229
(
2017
).
28.
J.
Xu
,
Y.
Pi
, and
Z.
Cao
, “
Optimized projection matrix for compressive sensing
,”
EURASIP J. Adv. Signal Process.
2010
(
1
),
560349
(
2010
).
29.
J.
Zhang
,
G.
Han
, and
Y.
Fang
, “
Deterministic construction of compressed sensing matrices from protograph LDPC codes
,”
IEEE Signal Processing Lett.
22
(
11
),
1960
1964
(
2015
).