Compressive Sensing for Signal Recovery

Introduction

One of the central tenets of signal processing is the Nyquist/Shannon sampling theory: the

number of samples needed to reconstruct a signal without error is dictated by its

bandwidth – the length of the shortest interval which contains the support of the spectrum

of the signal under study. The classical theory of C. Shannon says that the recovery of the

original signal from the large samples with optimal loss, whereas Compressive Sensing (CS)

gives the provision of selecting from small samples using the sparsity and incoherence of

the signal. The area of CS was initiated in 2006 by two groundbreaking papers, namely [1]

by Donoho and [2] by Cand`es, Romberg, and Tao. In recent times, only after eight years, an

abundance of theoretical aspects of compressed sensing are explored in more than

Thousand articles. Moreover, this methodology is currently extensively utilized by applied

mathematicians, computer scientists, and engineers for a variety of applications in

astronomy, biology, medicine, radar, and seismology, to name a few.

The literature already contains a well-developed theory of sampling, which we summarize below. Although algorithmic work is progressing, the state of knowledge is less than complete. We assert that a practical signal reconstruction algorithm should have all of the following properties. • It should accept samples from a variety of sampling schemes.

• It should succeed using a minimal number of samples.

• It should be robust when samples are contaminated with noise.

• It should provide optimal error guarantees for every target signal.

• It should offer provably efficient resource usage. From this perspective, we propose to develop new algorithms for the signal recovery of the

sparse matrix based on CS.

State of the Art in CS

CS is a framework that enables the recovery of a sparse signal from few of its measurements by exploiting the sparsity as the prior knowledge of the original signal. Famous applications of CS include computational photography and seismic data processing, in which the original signal and its measurements are linearly related, in most cases, by the Fourier and/or wavelet transformations. In general, these relationships are expressed by dense matrices. Accordingly, much effort has been made in analyzing the performance [1, 6-8] and developing practical algorithms for the signal recovery [9-12] for the density matrix based CS. Moreover, Pruente [13] developed an algorithm on the basis of

belief propagation (BP) [14-15] in conjunction with using a mixture of two finite variance Gaussians as a sparse prior. The computational cost of the algorithm increases only linearly with respect to the signal size.

The Compressed Sensing Problem

Compressive sensing techniques generally deal with incomplete linear systems of the type y = Ax (1) where the 𝑁-dimensional coefficient vector x describes the signal, the 𝑀-dimensional vector y collects the measurements obtained by a linear sensor and A is an 𝑀 × 𝑁 matrix - the sensing matrix - characterizing how the coefficient vector is mapped to the measurements. The vectors and matrices may be real or complex valued. Compressive sensing assumes that 𝑀 < 𝑁, hindering that Eq. 1 can simply be inverted to reconstruct x from the measurements. A vector x is called S-sparse if at most S of its coefficients are unequal to zero, and compressible if ∥x − x(S)∥ decreases quickly to zero with growing S, when x(S) denotes the best S-sparse approximation of x. There is a variety of algorithms aiming to reconstruct x from the deterministic measurements y = Ax or the noisy measurements y = Ax + n under the assumption that x is sparse. Well known are the Basis Pursuit, minimizing the 𝑙1-norm ∥ x ∥1 subject to y = Ax, the Basis Pursuit Denoising [3] minimizing ∥ x ∥1 subject to ∥ y − Ax ∥2 ≤ 𝜎, or equivalently:

∥ x ∥1+𝜆∥ y − Ax ∥2, the Orthogonal Matching Pursuit (OMP)[4] and the Compressive Sampling Matched Pursuit (CoSaMP) [5]. For the mode of operation of these algorithms, the reader is referred to literature. It has been shown that under certain conditions on the sensing matrix exact (for the noise-free case) or robust (for the noisy case) reconstruction is guaranteed, if the number 𝑀 of measurements is larger or equal to an expression depending on 𝑁, S and a quantity describing a specific property of the sensing matrix. One of these measures is the Restricted Isometry Constant given by the minimum number 𝛿 with

(1−𝛿) ∥ x ∥22 ≤∥ Ax ∥2

2≤ (1 − δ) ∥ x ∥22

for all S-sparse vectors x, assuring that measurements Ax and Ax′ are different, if only the S-sparse vectors x and x’ are sufficiently separated. Since this Restricted Isometry Property (RIP) is difficult to prove for a concrete matrix, a theory has been developed regarding classes of statistical sensing matrices, showing that e.g. RIP is fulfilled with a probability close to 1. For instance it was shown that for Gaussian or Rademacher random matrices

will have the (S, 𝛿)-RIP with high probability, if M = 𝒪 (𝑆 log

𝑁

𝑆

𝛿2). For one type of statistical

sensing matrices generated by selecting 𝑀 rows of the complete 𝑁 × 𝑁 Fourier matrix at random.

Signal Recovery using compressive sensing

In applications, most signals of interest contain scant information relative to their ambient dimension, but the classical approach to signal acquisition ignores this fact. We usually collect a complete representation of the target signal and process this representation to sieve out the actionable information. Then we discard the rest. Contemplating this ugly inefficiency, one might ask if it is possible instead to acquire compressive samples. In other words, is there some type of measurement that automatically winnows out the information from a signal? Incredibly, the answer is sometimes yes.

Compressive sampling refers to the idea that, for certain types of signals, a small number of nonadaptive samples carries sufficient information to approximate the signal well. Research in this area has two major components: Sampling: How many samples are necessary to reconstruct signals to a specified precision? What type of samples? How can these sampling schemes be implemented in practice? Reconstruction: Given the compressive samples, what algorithms can efficiently construct a signal approximation?

Fund Requirement

Detailed year wise break-up for the Project budget should be given as follows:

Degree I st Year II nd Year III st Year

Research Scientist

Research Associate

Research Fellows

(M.Sc/B.Tech degree)

Rs. 2,16,000

Rs.2,16,000

Rs.2,16,000

Supporting Technical Staff

(Diploma holders)

Total : Rs.6,48,000/-

I st Year II nd Year III rd Year Total

Softwares &

Equipments

Rs.5,00,000

Rs.3,00000

Rs. 2,00000

Rs.10,00,000/-

I st Year II nd Year III rd Year Total

Consumables and

Supplies

Rs. 20,000

Rs. 10,000

Rs. 10,000

Rs. 50,000/-

I st Year II nd Year III rd Year Total

Travel Rs. 30,000

Rs. 30,000

Rs. 30,000

Rs. 90,000/-

Grand Total :

17,88,000/-

Available institutional Facilities & Expertise:

a) PCs and a server are also available. Apart from these the University will provide Office

Space, Internet, Electricity and Library Support.

b) The Applied Sciences department has faculty members in the field of Image Processing,

Data Analysis, Cryptography, Mathematical Modelling and Functional analysis etc.

Reference [1] D. L. Donoho, “Compressed sensing” IEEE Trans. Inform. Theory, vol. 52, pp. 1289–

1306, 2006. [2] E. Cand`es, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal

reconstruction from highly incomplete Fourier information” IEEE Trans. Inform. Theory”, vol. 52, pp. 489-509, 2006.

[3] S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,”, IEEE

Transactions on Signal Processing, vol. 41, no. 12, pp. 3397–3415, 1993. [4] D. Needell, J. Tropp, and R. Vershynin, “Greedy signal recovery review,” 42nd

Asilomar Conference on Signals, Systems and Computers, pp. 1048–1050, 2008. [5] J. H. G. Ender, “On compressive sensing applied to radar,” Journal of Signal

Processing, vol. 90, Issue 5, pp. 1402–1414, 2010.

[6] E. J. Cand`es and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203–4215, 2005.

[7] S. Rangan, A. K. Fletcher and V. K. Goyal, “Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing” in Proc. Neural Information Processing Systems Conf., vol. 22, Vancouver, CA, Dec. 2009, pp. 1545–155, Dec 2009.

[8] Y. Kabashima, T. Wadayama and T. Tanaka, “A typical reconstruction limit for compressed sensing based on Lp-norm minimization,” Journal of Statistical Mechanics, vol. 2009.

[9] T. Blumensath and M.E. Davies, “Iterative Thresholding for Sparse Approximations,” The Journal of Fourier Analysis and Applications, vol.14, no 5, pp. 629–654, Dec. 2008.

[10] A. Maleki and D. L. Donoho, “Optimal iterative thresholding algorithms,” in Proc. of signal Processing with Adaptive Sparse Representations, SPARS 09, 2009.

[11] D. L. Donoho, A. Maleki and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci., vol. 106, no. 45, pp. 18914-18919, Nov. 2009.

[12] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Info. Theory, vol. 53, no. 12, pp. 4655–4666, 2007.

[13] L. Pruente, “Application of compressed sensing to SAR/GMTI-data,” 8th European Conference on Synthetic Aperture Radar (EUSAR), pp. 1–4, 2010.

[14] Y. Yu, A. Petropulu, and H. Poor, “CSSF MIMO radar: Compressive-sensing and step-frequency based MIMO radar,” IEEE Transactions on Aerospace and Electronic Systems, vol. 48, no. 2, pp. 1490–1504, 2012.

[15] S. Gogineni and A. Nehorai, “Target estimation using compressive sensing for distributed MIMO radar,” Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers (ASILOMAR 2010), pp. 793–797, 2010.

ITM University Millimeter Wave Research Centre

(IUMRC)

Participating Department:

Department of Applied Sciences, School of Engineering and Technology

ITM University, Gurgaon

Participating faculty Members:

1. Prof. A. K. Yadav, Head Department of Applied Sciences

2. Prof. Narayana Swamy Chandramowliswaran

3. Dr. Sangeet Srivastava

4. Dr. Gaurav Gupta

5. Dr. Ravindra Kishore Bisht

Detailed Profile of the Team Members

Department of Applied Sciences

Prof. A K Yadav Professor and HOD (Applied Sc.), Postdoctorate, North Carolina State University, USA, PhD, MSc, IIT Delhi, GATE, NET As a professor in Mathematics, Professor Yadav is interested in research in Image Processing, and Mathematical Modelling and Simulation. Prof. Yadav has been awarded two UGC sponsored research projects (one major and one minor). Prof. Yadav has published more than 35 research papers in journals and conferences of international repute. He has visited several universities abroad in pursuit of research and academic excellence. In 2009, He received a prestigious grant of

$170,000 from Hewlett Packard in 2009 for Innovation in Engineering Education and an award from the Indo-US Collaboration of Engineering Education (IUCEE) for his leadership role in enhancing engineering education. Prof. Yadav is currently guiding three PhD scholars and he is Co-Principal Investigator of a DRDO sponsored project on Phase Retrieval Algorithms. Prof. Yadav is a life member of the Indian Society for Technical Education and the Indian Meteorological Society. Prof. Narayanaswamy Chandramowliswaran PhD, IIT DELHI MSc, Annamalai University CSIR – UGC JRF (NET) Before joining ITMU in July 2014, Prof. Chandramowliswaran was working at VIT University and Kanchi Deemed University as a professor. He has published several research papers in Cryptography in reputed international journals including those published by Taylor & Francis. Prof. Chandramowliswaran has 18 years of teaching and research experience.

Dr. Gaurav Gupta Assistant Professor (Sr Scale) PhD, HNB GarhwalUniversity, Uttarakhand MSc, Gurukul Kangri University, Haridwar

Dr Gaurav Gupta's research focus includes Computer Vision and Image Processing. Before joining ITMU, he was working at the Greater Noida Institute of Technology (GNIT). Dr Gupta was also engaged in an ISRO-funded research project at the Department of Mathematics, IIT Roorkee, for more than three years as a Junior Research Fellow. He has also worked in a DST-funded project on the Gangotri Glacier at the National Institute of Hydrology (NIH), Roorkee, as a Project Officer. He has published 14 research papers in international journals and conferences. Currently, he is guiding four PhD scholars of the Department of Computer Science & IT, and Department of EECE. He has also guided four MTech and three BTech dissertations.

Dr. Sangeet Srivastava Assistant Professor (Sel. Grade) PhD, IIT Delhi Visiting Scientist (Meteo-France), Postdoctorate (NTU, Taiwan) MSc, IIT Kanpur Dr Sangeet Srivastava is keenly interested in research on Climate Change and Climate Modelling. He has visited several universities abroad for research and academic excellence. The Young Scientist Award was conferred on him by the DST (Government of India), and he has contributed towards many projects on climate change in India. Recently, he co-authored a book, Climate Change and Disease Dynamics in India, which was presented to the Vice-President of India on 29 May 2012.

Dr. Ravindra Kishor Bisht Assistant Professor PhD, MSc, Kumaun University, Nainital

Dr Ravindra Kishor Bisht’s PhD thesis was titled, ‘A Study on Chaotic Dynamical Systems, Fixed Points and Computer Simulation of Fractals’. He joined ITM University in July 2014. He specializes in the area of Metric Fixed Point Theory. He has 23 research publications to his credit. Dr Bisht is a life member of the International Academy of Physical Sciences.

Research Publications of the Team Members

1. H. Singh, A.K. Yadav, S. Vashisth and K. Singh, Fully-phase image encryption using double random-

structured phase masks in gyrator domain, Applied Optics, 2014. IF=1.649 (in Press).

2. S. Vashisth, H. Singh, A. K. Yadav, K. Singh, Image encryption using fractional Mellin transform, structured phase filters, and phase retrieval. Optik, Vol.125, 5309–5315.2014. IF=0.769.

3. S Vashisth, H Singh, AK Yadav, K Singh, Devil’s Vortex Phase Structure as Frequency Plane Mask for Image Encryption Using the Fractional Mellin Transform, International Journal of Optics, Vol. 2014. Article ID 728056, 9 pages, 2014.

4. J. Biswas, E. Upadhyay, M.Nayak and A.K. Yadav (2011) An Analysis of ambient air quality conditions over Delhi, India from 2004 to 2009. Accepted for publication in Atmospheric and Climate Sciences.

5. Krishan Kumar, A.K. Yadav, H. Hassan, M. P. Singh and V. K. Jain (2004) Forecasting Daily Maximum Surface Ozone Concentrations in Brunei Darussalam - an ARIMA Modeling Approach. Accepted for publication in Journal of Air and Waste Management Association, 54, No.7, 809-814.

6. Maithili Sharan, Manish Modani and A.K. Yadav (2003) Atmospheric dispersion: an overview of mathematical modeling framework. Journal of the Proceeding of Indian National Science Academy, 69, A, No. 6, 725-744.

7. A.K. Yadav, S. Raman and D.D.S. Niyogi (2003) A note on the estimation of eddy diffusivity and dissipation length in low winds over a tropical urban terrain, PAGEOPH, 160, 395-404.

8. K.N. Mehta and A.K. Yadav (2003) A non-Gaussian two-dimensional dispersion model with concentration dependent wind and diffusivity profiles. Indian Journal of Pure and Applied Mathematics, 34(6), 963-972.

9. A.K. Yadav, Krishan Kumar, Awg Makarimi bin Hj Awg Kasim, M.P. Singh, S.K. Parida and Maithili Sharan (2003) Visibility and incidence of respiratory diseases during the 1998 have episode in Brunei Darussalam. PAGEOPH, 160, 265-277

10. Maithili Sharan, Anil Kumar Yadav and Manish Modani (2002) Simulation of short-range diffusion experiments in low wind convective conditions. Atmos. Environ, 36, 1901-1906.

11. A.K. Yadav and K.N. Mehta (2000) Sensitivity of plume descriptors of a Gaussian plum mode to deposition and source elevation, II Nuovo Cimento, Vol 023C, issue 03, 251-262.

12. Maithili Sharan, Anil Kumar Yadav (1999) Accounting for the source strength in the solution of the diffusion equation: Alternative Mathematical Formulations. Atmos. Environ, 33, No. 8, 1327-1330.

13. Maithili Sharan and Anil Kumar Yadav (1998) Simulation of diffusion experiments under light wind, stable conditions by a variable K-theory model, Atmos. Environ., 32, No. 20, 3481-3492.

14. Maithili Sharan, Anil Kumar Yadav and M.P. Singh (1996) Plume dispersion simulation using a mathematical model based on coupled plume segment and Gaussian puff approaches. J. Appl. Meteoro., 35, No. 10, 1625-1631.

15. Anil Kumar Yadav, Sethu Raman and Maithili Sharan (1996) Surface layer turbulence spectra and eddy dissipation during low winds in tropics. Bound.- Layer Meteoro., 79, 205-224.

16. Maithili Sharan, M.P. Singh and Anil Kumar Yadav (1996) Mathematical model for atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmos. Environ., 30, 1137-1145.

17. Maithili Sharan, Anil Kumar Yadav, M.P. Singh, P. Agarwal and S. Nigam (1996) A mathematical model for the dispersion of air pollutants in low wind conditions. Atmos. Environ., 30, 2595-2606.

18. Anil Kumar Yadav and Maithili Sharan (1996) Statistical evaluation of sigma schemes for estimating dispersion in low wind conditions. Atmos. Envorn., 30, 2595-2606.

19. Anil Kumar Yadav, Maithili Sharan and M.P. Singh (1996) Atmospheric dispersion in low wind conditions. Proceedings of the First World Congress of Nonlinear Analysts, p.3567-3593, Tampa, Florida, August 19-26, 1992; Editior V. Lashmikantham, Walter de Gruyter, Berlin, New York.

20. Maithili Sharan, Anil Kumar Yadav and M.P. Singh (1995) Comparison of sigma schemes for estimating air pollutant dispersion in low winds. Atmos. Environ., 29, 2051-2059.

21. P. Agarwal, Anil Kumar Yadav, Amita Gulati, Sethu Raman, Suman Rao, M.P. Singh, S. Nigam and Neerja Reddy (1995) Surface layer turbulence processes in low windspeeds over land. Atmos. Environ., 29, 2089-2098.

22. NarayanSwami Chandramowliswaran, Authenticated Key Distribution using given set of Primes for Secret Sharing” in Journal of Systems Sciences and Control Engineering (Taylor and Francis) (Paper Accepted).

23. NarayanSwami Chandramowliswaran, Secret Key Distribution Technique using Theory of Numbers” in Italian Journal of Pure and Applied Mathematics, No. 32, 2014 (paper accepted).

24. NarayanSwami Chandramowliswaran, Secure Schemes for Secret Sharing and Key Distribution using Pell’s Equation” in International Journal of Pure and Applied Mathematics Volume 85 No. 5 2013, 933-937.

25. NarayanSwami Chandramowliswaran, Authenticated Multiple Key Distribution using Simple Continued Fraction” in International Journal of of Pure and Applied Mathematics 87 No. 2 2013, 349-354.

26. NarayanSwami Chandramowliswaran, Tree Generation Theorems”, KYOTO International Conference on Computational Geometry and Graph Theory – in honor of Jin Akiyama and Vasek Chvatal on their 60th birthdays, June 11-15, 2007, Kyoto, Japan.

27. NarayanSwami Chandramowliswaran, Arithmetical Properties of Tree Generation Codes and

Algorithm to generate all tree codes for a given number of edges” KYOTO International Conference on Computational Geometry and Graph Theory – in hon or of Jin Akiyama and Vasek Chvatal on their 60th birthdays, June 11-15, 2007, Kyoto, Japan.

28. R. P. Pant and Ravindra K. Bisht, Common fixed points of pseudo compatible mappings, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (RACSAM, Springer), Vol. 108, pp. 475-481, 2014. IF- 0.69.

29. Ravi P. Agarwal, Ravindra K. Bisht and Naseer Shahzad, A comparison of various noncommuting conditions in metric fixed point theory and their applications, Fixed Point Theory and Applications (Springer), Vol. 2014, Issue 1, pp. 1-33, 2014. IF- 2.49.

30. J. R. Morales, E. M. Rojas and Ravindra K. Bisht, Common fixed points for pairs of mappings with variable contractive parameters, Abstract and Applied Analysis, Vol. 2014, Article ID 209234, pp. 1-7, 2014. IF- 1.27.

31. Vyomesh Pant and Ravindra K. Bisht, A new continuity condition and fixed point theorems with applications, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (RACSAM, Springer), Vol. 108, pp. 653-668, 2014. IF- 0.69.

32. Ravindra K. Bisht and Naseer Shahzad, Faintly compatible mappings and common fixed points, Fixed Point Theory and Applications (Springer), Vol. 2013, Issue 1, pp. 1-9, 2013. IF- 2.49.

33. Ravindra K. Bisht and V. Rakocevic, Some notes on PD-operator pairs, Mathematical Communications, Vol. 18, pp. 441-445, 2013. IF-0.45.

34. M. Hanmandlu, Shaveta Arora, Gaurav Gupta and Latika Singh, “Information Set based Color Image

Enhancement”, Submitted to IEEE Transactions on Image Processing.

35. Gaurav Gupta and Manoj Kumar" An Iterative Marching with Correctness Criterion algorithm for Shape from Shading under Oblique Light Source" Advances in Intelligent Systems and Computing (Springer Series), Volume 236, pp 535-546.

36. Sanjay Rawat, Gaurav Gupta, Balasubramanian Raman, M.S. Rawat, “Digital Watermarking based

Stereo Image Coding”, Contemporary Computing Communications in Computer and Information Science (Springer Series), Volume 94, 2010, pp 435-445.

37. Gaurav Gupta, Balasubramanian Raman, M. S. Rawat, Rama Bhargava, and B. Gopala Krishna, “Stereo Matching for 3D Building Reconstruction”, Advances in Computing, Communication and Control Communications in Computer and Information Science (Springer Series), Volume 125, 2011, pp 522-529.

38. M. Hanmandlu, S. Arora, Gaurav Gupta, ; Latika Singh, “A Novel Optimal Fuzzy Color Image Enhancement using Particle Swarm Optimization”, In IEEE proceedings of Sixth International Conference on Contemporary Computing (IC3), Page: 41 – 46, 8-10 Aug. 2013, Noida, India.

39. Gaurav Gupta, R. Balasubramanian and Rama Bhargava, “Reconstruction of 3D Plane using Min-Max Approach”, International Journal of Recent Trends in Engineering (Academy Publishers), Vol. 2, No. 1, pp. 174 – 178, 2009.

40. Gaurav Gupta, R. Balasubramanian, Rama Bhargava, B. Gopala, M.S. Rawat, “Region Growing Stereo Matching Method for 3D Building Reconstruction” International Journal of Computational Vision and Robotics (IJCVR), Vol. 2, No. 1, 2011.