Abstract

Compressed Sensing is a new exciting eld which challenges the celebrated Nyquist Theorem. It proposes that signals which are sparse in some transform domain can be reconstructed accurately with KlogN samples where K is the sparsity of the signal in the Transform Domain and N is the length of the original signal. By utilizing the compressed sensing notion of obtaining signals, the complexity of the design of the sensor (camera) is reduced. This could practically materialize the eld of sensor networks. Whenever a scene is captured using a still video camera, background subtracted images latch on to the relevant new information in each frame. Hence compressed sensing applied to background subtracted images is an important issue to be handled. Hence, background subtracted images are compressed sensed and reconstructed using three dierent algorithms L1 norm minimization, Weighted L1 norm minimization and Orthogonal Pursuit Matching and the eciency and the complexities involved in these processes are analysed in this project.

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Introduction

Every scene obtained by a still video camera (single pixel camera) consists of moving and stationary objects. The stationary objects form the background and the moving objects comprise the foreground. In every new frame obtained, the additional information carried by the frame is some sense captured by the foreground or the background subtracted image. Hence, instead of transmitting the entire image, it would be almost sucient if the background subtracted image is transmitted assuming the receiver knows the background information. Hence, compressed sensing background subtracted images is an important issue as they can be reconstructed using much fewer samples of the original signal. This is intuitively expected as background subtracted images are more sparse and can be compressed to a much greater extent and this compressability can be transferred to the sensor by sampling the signal at a much lower rate. The primary reference paper in this project where background subtracted images are reconstructed is [5].

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Compressed Sensing

Compressive sensing is a non adaptive sampling technique which requires the signal to be compressible (to be sparse in some domain, transform domain) to be taken advantage of. It also requires the measurement functions(sampling functions) to be incoherent to the transform domain. When these conditons are satised a sparse signal can be recovered using much smaller sampling rate than one suggested by Nyquist Theorem. Compressed sensing cameras have several applications in astronomy, MRI imaging, sensor networks where conventional cameras have inherent limitations. Hence, the reconstruction of compressed sensed images is an important problem to be handled. In this project three dierent reconstruction algorithms are analysed - L1 norm minimization, Weighted L1 norm minimization and Orthogonal Pursuit Matching to recover the original signal from compressed sensed images and the performance of these techniques are compared. A owchart of compressed sensing is shown in gure below

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Background subtracted images are inherently sparse. Hence, the transform Ψ can be considered to be the identity matrix I. So, the reconstruction algorithms fall into the simple category of sparse signal recovery.

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Implementation Details

A vehicle monitoring video http://i2lwww.ira.uka.de/image_sequences/ using a still video camera was the prototype on which compressed sensing and reconstruction was performed. First the background was estimated using simple median operation and median ltering over a small segment of the video. Then, the background subtracted image was obtained for a frame selected randomly and reconstruction of this frame was the problem under analysis. The measurement functions used for compressed sensing are noiselets i.e randomly generated 1s and +1s. The measurement matrix is similar to an incomplete Walsh-Hadamard Transform matrix. After sampling(compressed sensing) the background subtracted image, three reconstruction algorithms were used to reconstruct the original image. Two softwares, SEDUMI and CVX were installed in MATLAB to perform convex optimization. The performances of the reconstruction algorithms were analysed in the following sections

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L1 norm minimization

L1 norm minimization is a very standard technique for sparse signal recovery suggested in the preliminary papers [2, 3, 4] on compressed sensing. The selective advantage of L1 norm optimization over L2 norm optimization is that it equinorm surface i.e recovery signal search direction is more oriented towards more sparser directions though we pay in terms of computational complexity. The following convex minimization is implemented minimize

||x||1

subject to ϕf = yk

This technique produces a sparse solution and the simulation results on a 50Ö50 resized image is shown below

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(1)

Better reconstruction was obtained when the number of samples to reconstruct the image is increased. Compressed sensing sampling theorem suggests a rate of around 1300 samples to almost exactly reconstruct the image. But, suboptimal reconstructions using 400 - 1200 samples was analysed and for 1200 samples nearly perfect reconstruction was obtained.

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Weighted L1 norm reconstruction

This method tries to emulate L0 norm reconstruction using L1 norm reconstruction. In this technique instead of minimizing ||x||1 , ||W x||1 should be minimized, subject to the same constraints as L1 norm in 1. The weight matrix is intelligently chosen to get better performance than L1- norm minimization. Here W was chosen to be a diagonal 1 matrix with the diagonal entries wii to be xo,i where x0 is the original signal. As the original signal is M-sparse only M elements in W matrix are non-zero and the rest of the values are innity in the main diagonal. Intuitively, minimizing ||W x||1 makes sure that the rest of the values other than the M non-zero values in x go to zero and hence a M sparse reconstruction is obtained. But, this technique has a glaring assumption about the prior knowledge of the original signal. Hence to converge to this technique, x is estimated using L1 norm minimization and then W matrix is computed using this estimate and the process is repeated. A good analysis of this techique is presented in [7].

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In this simulation, the original image (resized 50×50 image) is reconstructed by compressed sensing the background subtracted image using 1200 samples using Weighted L1 norm minimization(5 iterations). It was observed that Weighted L1 norm minimization gave a lower Mean square error compared to L1 norm minimization, but not a signicantly astonishing improvement. The Weighted L1 norm minimization only gives a very good reconstruction after a lot of iterations. A comparision of the MSE of L1 and Weighted L1 techniques is shown below (MSE is dened as the square of the frobenius norm of the dierence between the original and the nal reconstructed image).

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Orthogonal Pursuit Matching

This is a greedy algorithm which is less computationally complex compared to L1 norm minimization because it makes use of L2 norm minimization instead. The basic idea is to split the measurement matrix into columns φ = [φ1 φ2 ... φN ]. Now φf = yk is solved. Here, only M values in φcontribute to the reconstruction of f . Next, the best M φi which contribute yk is estimated. The key idea here is we consider yk as a weighted linear combination M φi . We estimate the best φi by correlating it with the output yk and reconstruct the signal fˆ at each stage using L2 norm reconstruction until we nd the best M φi . At each stage we also estimate the the correspinding fˆi The complete algorithm is presented in [6].

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In this simulation the background subtracted image (size - 50×50) was reconstructed using Orthogonal Matching Pursuit algorithm assuming the sparsity of the image to be 100,300 and 500 for a sampling rate of 1200 samples. The reconstruction accuracy increased as higher sparsity was assumed. It should be noted that the actual sparsity of the background subtracted image was around 500.

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Comparision of Reconstruction Algorithms

L1 norm and Weighted L1 norm minimization reconstruction techniques are computationally more complicated as they try to minimize L1 norm which has a computational complexity of O(N 3 ) using linear programming. But, the critical parameter that aected the project was storage complexity. When images of size greater than 50×50 were compressed sensed, Matlab was out of memory. The problem is that the reconstruction algorithms try to reduce computational complexity by increasing storage complexity when they use techniques like dynamic programming. One plausible reason for this issue in this project was that Sedumi and CVX use dynamic programming to solve L1 norm minimization. But Orthogonal Matching pursuit uses L2 norm minimization at each stage. This reduces storage complexity issues, but computational complexity was still an issue because of repeated L2 norm minimization. But one pays for gain in storage complexity of Orthogonal Matching Pursit for the accuarcy of reconstruction as it is a pretty suboptimal method and an estimate of the sparsity of the original signal is required. In terms of accuarcy of reconstruction, Weighted L1 performed better than L1 as expected. The relation below summarizes this discussion. With respect to Computational complexity : Weighted L1 > L1 ≈ Orthogonal Pursuit Matching With respect to Storage complexity : Weighted L1 ≈ L1 > Orthogonal Pursuit Matching With respect to Reconstruction accuracy : Weighted L1 > L1 > Orthogonal Pursuit Matching

Conclusion In this project the reconstruction of compressed sensed background subtracted images was acheived using three different reconstruction algorithms - L1 norm minimization, Weighted L1 norm minimization and Orthogonal Pursuit 6

Matching. A detailed analysis and comparision of all the three algorithms was performed. The primary issue in the project was the storage complexity of the reconstruction algorithms and hence compressed sensing could only be applied to images of very small sizes. Finally, can be concluded that these reconstruction algorithms work reliably for recovering background subtracted images which are inherently sparse.

References [1] Emmanuel Candès, Compressive Sampling. (Int. Congress of Mathematics, 3, pp. 1433-1452, Madrid, Spain,2006 ) [2] Richard Baraniuk, Compressive sensing. (IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007) [3] Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. (IEEE Signal Processing Magazine, 25(2), pp. 21 - 30, March 2008) [High-resolution version] [4] Justin Romberg, Imaging via compressive sampling. (IEEE Signal Processing Magazine, 25(2), pp. 14 - 20, March 2008) [5] Volkan Cevher, Aswin Sankaranarayanan, Marco Duarte, Dikpal Reddy, Richard Baraniuk, and Rama Chellappa, Compressive sensing for background subtraction. (European Conf. on Computer Vision (ECCV), Marseille, France, October 2008) [6] Joel Tropp and Anna Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. (IEEE Trans. on Information Theory, 53(12) pp. 4655-4666, December 2007) [7] Emmanuel Candès, Michael Wakin, and Stephen Boyd, Enhancing sparsity by reweighted ell-1 minimization. (Preprint, 2008)

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