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...a few landmark papers. The Kalman filter. The classical solution to estimate a sequence of signals satisfying (1) or, in the control terminology, the state of a dynamical system, is the Kalman filter =-=[41]-=-. The Kalman filter is an online algorithm that is least-squares optimal when the model is linear, i.e., fk ({x[i]}k−1i=0 ) = Fx[k], and the sequence {ǫ[k]} is Gaussian and independent across time. Se...

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...n this context if the foreground pixels, i.e., those associated to a moving object, occupy a small area in each frame. Assuming the background image is known beforehand, compressed sensing techniques =-=[31]-=-, [32] such as ℓ1-norm minimization allow reconstructing each foreground. This not only reconstructs the original frame (if we add the reconstructed foreground to the known background), but also perfo...

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... Rn is assumed known and is the so-called prior or side information: a vector similar to the vector that we want to reconstruct, say x⋆. Note that if we set β = 0 in (2), we obtain basis pursuit (BP) =-=[38]-=-, a well-known sparse reconstruction problem at the core of compressed sensing [31], [32]. Problem (2) generalizes BP by integrating the side information w. The work in [36], [37] shows that, if w has...

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... context if the foreground pixels, i.e., those associated to a moving object, occupy a small area in each frame. Assuming the background image is known beforehand, compressed sensing techniques [31], =-=[32]-=- such as ℓ1-norm minimization allow reconstructing each foreground. This not only reconstructs the original frame (if we add the reconstructed foreground to the known background), but also performs ba...

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.... Applications Many problems require estimating a sequence of signals from a sequence of measurements satisfying the model in (1). These include classification and tracking in computer vision systems =-=[10]-=-, [11], radar tracking [12], dynamic MRI [13], [14] and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction [16]. Background subtracti...

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...ptimal when the model is linear, i.e., fk ({x[i]}k−1i=0 ) = Fx[k], and the sequence {ǫ[k]} is Gaussian and independent across time. Several extensions are available when these assumptions do not hold =-=[42]-=-–[44]. The Kalman filter and its extensions, however, are inapplicable to our scenario, as they do not easily integrate the additional knowledge that the state is sparse. Dynamical sparse signal recon...

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...rices (random variables) Ak, generated in steps 3 and 12. Define the event Si as “perfect reconstruction at time i.” Since we assume that ŝ1 and ŝ2 are larger than the true sparsity 13 of x[1] and x=-=[2]-=-, there holds [56] P(Si) ≥ 1− exp [ − 1 2 (mi −√mi)2 ] ≥ 1− exp [ − 1 2 (m−√m)2 ] , (22) for i = 1, 2, where the second inequality is due to mi ≥ m and 1− exp(−(1/2)(x−√x)2) being an increasing functi...

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...een applied, for example, in video surveillance [17], [18], traffic monitoring [19], [20], and medical imaging [21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], =-=[24]-=-, most of them assume access to full frames and, thus, are inapplicable in compressive video sensing [25]– [30], a technology used in cameras where sensing is expensive. In particular, devices such as...

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...ed the magnitude of its components by 30%. This, according to the theory in [36], improves the quality of the side information. To solve BP in the reconstruction of the first two frames we used SPGL1 =-=[69]-=-, [70].10 To solve the ℓ1-ℓ1 minimization problem (5) in the reconstruction of the remaining frames we used DECOPT [71], [72].11 Recall that Algorithm 1 assumes noiseless measurements. To illustrate i...

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... has been applied, for example, in video surveillance [17], [18], traffic monitoring [19], [20], and medical imaging [21], [22]. Although there are many background subtraction techniques, e.g., [11], =-=[23]-=-, [24], most of them assume access to full frames and, thus, are inapplicable in compressive video sensing [25]– [30], a technology used in cameras where sensing is expensive. In particular, devices s...

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...ation Figure 2. Scheme of motion-compensated extrapolation. We use the motion between matching blocks in ẑ[k− 2] and ẑ[k− 1] to create an estimate e[k] of frame z[k]. motion-compensated predictions =-=[58]-=-–[60]. Our technique is illustrated in Fig. 2. In the first stage, we perform forward block-based motion estimation between the reconstructed frames ẑ[k−2] and ẑ[k−1]. The block matching algorithm i...

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...−1i=1 ). Although described for a noiseless scenario, the algorithm easily adapts to the noisy scenario, as discussed later. Such adaptation is essential on a real system, e.g., a single-pixel camera =-=[26]-=-. A. Algorithm description The algorithm consists of two parts: the initialization, where the first two signals x[1] and x[2] are reconstructed using BP, 5and the online estimation, where the remainin...

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...tees that enable our scheme for computing of the number of measurements online. Robust PCA. A technique that has been successfully applied to perform background subtraction is Robust PCA (RPCA) [24], =-=[50]-=-, [51]. RPCA decomposes a data matrix into the sum of a sparse and a low-rank matrix. In the context of background subtraction, a column of the data matrix corresponds to a video frame, which is decom...

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... the quantities ξ and h, which depend on the signs of the entries of x⋆ and w−x⋆, but not on their magnitudes. When w is such that h is small, the bound in (7) is much smaller than the one for BP3 in =-=[56]-=-: m ≥ 2s log (n s ) + 7 5 s+ 1 . (8) Namely, [56] establishes that if (8) holds and if A ∈ Rm×n has i.i.d. Gaussian entries with zero mean and variance 1/m then, with probability similar to the one in...

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...s require estimating a sequence of signals from a sequence of measurements satisfying the model in (1). These include classification and tracking in computer vision systems [10], [11], radar tracking =-=[12]-=-, dynamic MRI [13], [14] and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction [16]. Background subtraction is a key task for detect...

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...tion [16]. Background subtraction is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], [18], traffic monitoring [19], =-=[20]-=-, and medical imaging [21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in compressiv...

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...work in [45], [46] addresses the same problem using a different approach: it integrates an estimate of the signal’s support into the sparse reconstruction scheme using a technique known as ModifiedCS =-=[47]-=-. Related work that also assumes a fixed number of measurements includes [48], which uses approximate belief propagation, and [49], which integrates sparsity knowledge into a Kalman filter via a pseud...

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...knowledge that the state is sparse. Dynamical sparse signal reconstruction. Some prior work incorporates signal structure, such as sparsity, into online sparse reconstruction procedures. For example, =-=[3]-=-–[5] adapts a Kalman filter to estimate a sequence of sparse signals. Roughly, we have an estimate of the signal’s support at each time instant and use the Kalman filter to compute the (nonzero) signa...

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...aging [21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in compressive video sensing =-=[25]-=-– [30], a technology used in cameras where sensing is expensive. In particular, devices such as the single-pixel camera [25]–[30] acquire compressive measurements from images using few 1If x[k] is not...

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...ed number of measurements includes [48], which uses approximate belief propagation, and [49], which integrates sparsity knowledge into a Kalman filter via a pseudo-measurement technique. The works in =-=[6]-=-, [7] and [8] propose online algorithms named GROUSE and PETRELS, respectively, for estimating signals that lie on a low-dimensional subspace. Their model can be seen as a particular case of (1), wher...

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...mputer vision systems [10], [11], radar tracking [12], dynamic MRI [13], [14] and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction =-=[16]-=-. Background subtraction is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], [18], traffic monitoring [19], [20], and...

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... compressive background subtraction [16]. Background subtraction is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], =-=[18]-=-, traffic monitoring [19], [20], and medical imaging [21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus,...

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...half-pel accuracy and considers a block size of γ × γ pixels and a search range of ρ pixels. The required interpolation for half-pel motion estimation is performed using the 6-tap filter of H.264/AVC =-=[61]-=-. In addition, we use the ℓ1norm (or sum of absolute differences, SAD) as error metric. The resulting motion vectors are then spatially smoothed by applying a weighted vector-median filter [62]. The f...

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...construction. VI. EXPERIMENTAL RESULTS We applied the scheme described in the previous section to several video sequences.4 The sequences are described in Table I and were obtained from [65]5, [34]6, =-=[66]-=-7, [18]8, and [68]9. The table shows the acronyms we refer each sequence by, their source, the frame numbers and resolution we used, and a sample image from each sequence. We performed two types of ex...

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...[21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in compressive video sensing [25]– =-=[30]-=-, a technology used in cameras where sensing is expensive. In particular, devices such as the single-pixel camera [25]–[30] acquire compressive measurements from images using few 1If x[k] is not spars...

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...l when the model is linear, i.e., fk ({x[i]}k−1i=0 ) = Fx[k], and the sequence {ǫ[k]} is Gaussian and independent across time. Several extensions are available when these assumptions do not hold [42]–=-=[44]-=-. The Kalman filter and its extensions, however, are inapplicable to our scenario, as they do not easily integrate the additional knowledge that the state is sparse. Dynamical sparse signal reconstruc...

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...he sparse reconstruction scheme using a technique known as ModifiedCS [47]. Related work that also assumes a fixed number of measurements includes [48], which uses approximate belief propagation, and =-=[49]-=-, which integrates sparsity knowledge into a Kalman filter via a pseudo-measurement technique. The works in [6], [7] and [8] propose online algorithms named GROUSE and PETRELS, respectively, for estim...

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...integrates an estimate of the signal’s support into the sparse reconstruction scheme using a technique known as ModifiedCS [47]. Related work that also assumes a fixed number of measurements includes =-=[48]-=-, which uses approximate belief propagation, and [49], which integrates sparsity knowledge into a Kalman filter via a pseudo-measurement technique. The works in [6], [7] and [8] propose online algorit...

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...ng a sequence of signals from a sequence of measurements satisfying the model in (1). These include classification and tracking in computer vision systems [10], [11], radar tracking [12], dynamic MRI =-=[13]-=-, [14] and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction [16]. Background subtraction is a key task for detecting and tracking o...

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...e x[k] has a sparse representation in a linear, invertible transform.1 An aspect that distinguishes our problem from other recursive signal reconstruction problems, such as the ones addressed in [2]– =-=[9]-=-, is that the number of measurements mk varies at each iteration and has to be computed recursively. A. Applications Many problems require estimating a sequence of signals from a sequence of measureme...

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...64/AVC [61]. In addition, we use the ℓ1norm (or sum of absolute differences, SAD) as error metric. The resulting motion vectors are then spatially smoothed by applying a weighted vector-median filter =-=[62]-=-. The filtering improves the spatial coherence of the resulting motion field by removing outliers (i.e., motion vectors that are far from the true motion field). Assuming linear motion between ẑ[k − ...

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...es, making it a batch algorithm: all frames are processed together, not online as in our algorithm. There are, however, online extensions of RPCA, for example, the stochastic optimization approach of =-=[53]-=-, and an algorithm called Prac-ReProCS [54]. The online algorithm in [53] is shown to converge to the same solution as the batch RPCA, but it requires access to full frames. Prac-ReProCS [54] is also ...

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...alled Prac-ReProCS [54]. The online algorithm in [53] is shown to converge to the same solution as the batch RPCA, but it requires access to full frames. Prac-ReProCS [54] is also backed up by theory =-=[55]-=- and can handle compressive measurements. It requires a training sequence of background images, which is akin to our assumption of knowing the background image, and works under the assumption that the...

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...satisfying the model in (1). These include classification and tracking in computer vision systems [10], [11], radar tracking [12], dynamic MRI [13], [14] and several tasks in wireless sensor networks =-=[15]-=-. Our application focus, however, is compressive background subtraction [16]. Background subtraction is a key task for detecting and tracking objects in a video sequence and it has been applied, for e...

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...ubtraction [16]. Background subtraction is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], [18], traffic monitoring =-=[19]-=-, [20], and medical imaging [21], [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in comp...

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...EXPERIMENTAL RESULTS We applied the scheme described in the previous section to several video sequences.4 The sequences are described in Table I and were obtained from [65]5, [34]6, [66]7, [18]8, and =-=[68]-=-9. The table shows the acronyms we refer each sequence by, their source, the frame numbers and resolution we used, and a sample image from each sequence. We performed two types of experiments, Type I ...

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...by-product [16]. With the exception of [33], [34], most approaches to compressive video sensing and compressive background subtraction assume a fixed number of measurements for all frames [16], [25], =-=[27]-=-–[30], [35]. If this number is too small, reconstruction fails. If it is too large, reconstruction succeeds, but at the cost of unnecessary measurements in some or all frames. The work in [33], [34] a...

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... Figure 2. Scheme of motion-compensated extrapolation. We use the motion between matching blocks in ẑ[k− 2] and ẑ[k− 1] to create an estimate e[k] of frame z[k]. motion-compensated predictions [58]–=-=[60]-=-. Our technique is illustrated in Fig. 2. In the first stage, we perform forward block-based motion estimation between the reconstructed frames ẑ[k−2] and ẑ[k−1]. The block matching algorithm is per...

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...strating the 2Although Gaussian matrices are hard to implement in practical systems, they have optimal performance. There are, however, other more practical matrices with a similar performance, e.g., =-=[39]-=-, [40]. 3performance of our algorithm are shown in section VI; and section VII concludes the paper. The appendix contains the proofs of our results. II. RELATED WORK There is extensive literature on r...

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...measurements includes [48], which uses approximate belief propagation, and [49], which integrates sparsity knowledge into a Kalman filter via a pseudo-measurement technique. The works in [6], [7] and =-=[8]-=- propose online algorithms named GROUSE and PETRELS, respectively, for estimating signals that lie on a low-dimensional subspace. Their model can be seen as a particular case of (1), where each map fk...

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... are processed together, not online as in our algorithm. There are, however, online extensions of RPCA, for example, the stochastic optimization approach of [53], and an algorithm called Prac-ReProCS =-=[54]-=-. The online algorithm in [53] is shown to converge to the same solution as the batch RPCA, but it requires access to full frames. Prac-ReProCS [54] is also backed up by theory [55] and can handle com...

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...mber of measurements includes [48], which uses approximate belief propagation, and [49], which integrates sparsity knowledge into a Kalman filter via a pseudo-measurement technique. The works in [6], =-=[7]-=- and [8] propose online algorithms named GROUSE and PETRELS, respectively, for estimating signals that lie on a low-dimensional subspace. Their model can be seen as a particular case of (1), where eac...

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...d. This not only reconstructs the original frame (if we add the reconstructed foreground to the known background), but also performs background subtraction as a by-product [16]. With the exception of =-=[33]-=-, [34], most approaches to compressive video sensing and compressive background subtraction assume a fixed number of measurements for all frames [16], [25], [27]–[30], [35]. If this number is too smal...

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... is Laplacian [58], [63]. In our model, that corresponds to each ǫ[k] in (1a) being Laplacian. We assume each ǫ[k] is independent from the matrix of measurements Ak. Model for ǫ[k]. As in [58], [63], =-=[64]-=- (and references therein), we assume that ǫ[k] is independent from ǫ[l], for k 6= l, and that the entries of each ǫ[k] are independent and have zero-mean. The probability distribution of ǫ[k] is then ...

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...nificantly reduce the number of measurements. B. Overview of our approach and contributions Overview. Our approach to adaptive-rate signal reconstruction is based on the recent theoretical results of =-=[36]-=-, [37]. These characterize the performance of sparse reconstruction schemes in the presence of side information. The scheme we are most interested in is ℓ1-ℓ1 minimization: minimize x ‖x‖1 + β‖x− w‖1 ...

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...r compute) the number of measurements; indeed, the number of measurements is assumed constant along time. Also assuming the support varies slowly and using a fixed number of measurements, the work in =-=[45]-=-, [46] addresses the same problem using a different approach: it integrates an estimate of the signal’s support into the sparse reconstruction scheme using a technique known as ModifiedCS [47]. Relate...

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...hat enable our scheme for computing of the number of measurements online. Robust PCA. A technique that has been successfully applied to perform background subtraction is Robust PCA (RPCA) [24], [50], =-=[51]-=-. RPCA decomposes a data matrix into the sum of a sparse and a low-rank matrix. In the context of background subtraction, a column of the data matrix corresponds to a video frame, which is decomposed ...

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...in Section VI. Note that the conditions under which our algorithm performs well differ from the majority of prior work. For example, the algorithms in [3], [4], [6]–[8], [16], [33], [34], [48], [49], =-=[57]-=- work well when the sparsity pattern of x[k] varies slowly between consecutive time instants. Our algorithm, in contrast, works well when the quality parameters ξk and hk and also the sparsity sk vary...

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...ng the 2Although Gaussian matrices are hard to implement in practical systems, they have optimal performance. There are, however, other more practical matrices with a similar performance, e.g., [39], =-=[40]-=-. 3performance of our algorithm are shown in section VI; and section VII concludes the paper. The appendix contains the proofs of our results. II. RELATED WORK There is extensive literature on reconst...

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...ing perfect reconstruction. VI. EXPERIMENTAL RESULTS We applied the scheme described in the previous section to several video sequences.4 The sequences are described in Table I and were obtained from =-=[65]-=-5, [34]6, [66]7, [18]8, and [68]9. The table shows the acronyms we refer each sequence by, their source, the frame numbers and resolution we used, and a sample image from each sequence. We performed t...

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... magnitude of its components by 30%. This, according to the theory in [36], improves the quality of the side information. To solve BP in the reconstruction of the first two frames we used SPGL1 [69], =-=[70]-=-.10 To solve the ℓ1-ℓ1 minimization problem (5) in the reconstruction of the remaining frames we used DECOPT [71], [72].11 Recall that Algorithm 1 assumes noiseless measurements. To illustrate its ext...

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...rmation. To solve BP in the reconstruction of the first two frames we used SPGL1 [69], [70].10 To solve the ℓ1-ℓ1 minimization problem (5) in the reconstruction of the remaining frames we used DECOPT =-=[71]-=-, [72].11 Recall that Algorithm 1 assumes noiseless measurements. To illustrate its extension to noisy measurements, i.e., y[k] = Akx[k] + ηk, with ‖ηk‖2 ≤ σk, we also applied it to the case where the...

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...equence of signals from a sequence of measurements satisfying the model in (1). These include classification and tracking in computer vision systems [10], [11], radar tracking [12], dynamic MRI [13], =-=[14]-=- and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction [16]. Background subtraction is a key task for detecting and tracking objects...

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...raction is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], [18], traffic monitoring [19], [20], and medical imaging =-=[21]-=-, [22]. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in compressive video sensing [25]– [30]...

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...w-rank matrix). In RPCA, both the foreground and background are unknown, and the latter may vary slowly across time. Also, it either requires access to full frames or, in the case of compressive RPCA =-=[52]-=-, each measurement may contain information from several different frames, e.g., the first and last frames, making it a batch algorithm: all frames are processed together, not online as in our algorith...

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...secutive frames is approximately constant. In practice, the background may change during the operation of the algorithm due to, for example, illumination change or camera movement. As in video coding =-=[59]-=-–[61], this indicates a scene cut, which can be detected by our algorithm via a dramatic increase in the measurement rate. In that case, the algorithm should take enough measurements to reconstruct an...

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...dation under grants SNF 200021-132548, SNF 200021-146750, and SNF CRSII2-147633. Part of this work was presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing, 2015 =-=[1]-=-. J. Mota and M. Rodrigues are with the Department of Electronic and Electrical Engineering, University College London, UK. E-mail: {j.mota,m.rodrigues}@ucl.ac.uk. N. Deligiannis is with the Departmen...

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...ications Many problems require estimating a sequence of signals from a sequence of measurements satisfying the model in (1). These include classification and tracking in computer vision systems [10], =-=[11]-=-, radar tracking [12], dynamic MRI [13], [14] and several tasks in wireless sensor networks [15]. Our application focus, however, is compressive background subtraction [16]. Background subtraction is ...

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...n is a key task for detecting and tracking objects in a video sequence and it has been applied, for example, in video surveillance [17], [18], traffic monitoring [19], [20], and medical imaging [21], =-=[22]-=-. Although there are many background subtraction techniques, e.g., [11], [23], [24], most of them assume access to full frames and, thus, are inapplicable in compressive video sensing [25]– [30], a te...

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...s not only reconstructs the original frame (if we add the reconstructed foreground to the known background), but also performs background subtraction as a by-product [16]. With the exception of [33], =-=[34]-=-, most approaches to compressive video sensing and compressive background subtraction assume a fixed number of measurements for all frames [16], [25], [27]–[30], [35]. If this number is too small, rec...

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...ute) the number of measurements; indeed, the number of measurements is assumed constant along time. Also assuming the support varies slowly and using a fixed number of measurements, the work in [45], =-=[46]-=- addresses the same problem using a different approach: it integrates an estimate of the signal’s support into the sparse reconstruction scheme using a technique known as ModifiedCS [47]. Related work...

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... background. D. Reconstruction guarantees for Laplacian modeling noise It is well known that the noise produced by a motioncompensated prediction module, as the one just described, is Laplacian [58], =-=[63]-=-. In our model, that corresponds to each ǫ[k] in (1a) being Laplacian. We assume each ǫ[k] is independent from the matrix of measurements Ak. Model for ǫ[k]. As in [58], [63], [64] (and references the...

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...n. To solve BP in the reconstruction of the first two frames we used SPGL1 [69], [70].10 To solve the ℓ1-ℓ1 minimization problem (5) in the reconstruction of the remaining frames we used DECOPT [71], =-=[72]-=-.11 Recall that Algorithm 1 assumes noiseless measurements. To illustrate its extension to noisy measurements, i.e., y[k] = Akx[k] + ηk, with ‖ηk‖2 ≤ σk, we also applied it to the case where the Hall ...

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... µ+ t− 1 ∣∣A, D) = 1− P(h− µ ≤ −µ ∣∣A, D)− P(h− µ ≥ t− 1 ∣∣A, D) (32) ≥ 1− exp [ − 2µ 2∣∣Σ∣∣ ] − exp [ − 2(t− 1) 2∣∣Σ∣∣ ] , (33) where the last step, explained below, is due to Hoeffding’s inequality =-=[73]-=-. Note that once this step is proven, (33) together with (27) and (31) give (17), proving the theorem. Proof of step (32)-(33). Hoeffding’s inequality states that if {Zj}Lj=1 is a sequence of independ...