1. Modulated Unit Norm Tight Frames for Compressed Sensing
Peng Zhang1, Lu Gan2, Sumei Sun3 and Cong Ling1
1 Department of Electrical and Electronic Engineering,
Imperial College London
2 College of Engineering, Design and Physics Sciences
Brunel University, United Kingdom
3 Institute for Infocomm Research, A*STAR, Singapore

2. Outline Background Proposed system
Basics of compressed sensing;
Structured random operators for compressed sensing;
Proposed system
Performance bounds;
Connection with existing systems;
Applications in convolutional compressed sensing
Compressive imaging;
Sparse channel estimation in OFDM;
Conclusions

3. Principles of Compressed Sensing (CS)
Sampling: linear, non-adaptive random projections [Candes-Romberg-Tao-2006]:
y = x
x: N×1 signal vector;
y: M×1 sampling vector (M << N);
: M×N measurement matrix;
Sparsity: x has a sparse representation under a certain transform  (DCT, wavelet);
f = x can be well approximated with only K (K < M) non-zero coefficients;
Reconstruction: nonlinear optimization;
l1 optimization;
Iterative-based methods: OMP, subspace pursuit (SP) etc.;
Now, let us a take a closer look at
In the most general case, recovery of x from y is ill-posed as it is under-determined.

4. Principles of Compressed Sensing (CS)
Restricted isometry property (RIP) [Candes-Romberg-Tao-2006]:
An M×N matrix A= is said to satisfy the RIP with Parameters (K,) if
where  represents the set of all length-N vectors with K non-zero coefficients.
Now, let us a take a closer look at
In the most general case, recovery of x from y is ill-posed as it is under-determined.

5. Fully random sampling operators
: full random matrix with independent sub-Gaussian elements
i,j follows the Gaussian or Bernoulli distributions;
Optimal bound:
Universal: applicable for any 
Limitations:
High computational cost in matrix multiplication;
Huge buffer requirement;
Difficult or even impossible to implement;

6. A wish-list for the sampling operator 

7. Randomly subsampled orthonormal system
[Candes et al ]
S: Randomly sampling operator (choose M samples uniform at random):
Q: NN Bounded unitary matrix with
Example of Q: DCT, DFT, Walsh-Hadamard matrices 7

8. Structurally random matrices
Do and Gan et al [ICASSP], 2012 [TSP]
Random Sampling
(choose M out of N)
Fast transform
FFT, WHT, DCT
Random Sign Flipping ±1
 = S F D
Fast implementation
Universal
Random sampling could be difficult! 8

9. Random Convolution (filtering)
J. Tropp, J. Romberg, H. Rauhut, F. Krahmer et al. c x y R
R : Deterministic sampling operator;
c: random sequence;
Lacks universality
=I (Identity matrix) 

10. Outline Background Proposed system Examples of potential applications
Basics of compressed sensing;
Structured random operators for compressed sensing;
Proposed system
Performance bounds;
Connection with existing systems;
Examples of potential applications
Compressive imaging;
Sparse channel estimation in OFDM;
Conclusions

11. Unit-norm tight frame
An M×N matrix U corresponds to a unit norm tight frame if
Each column vector has a unit norm:
The rows of form an orthonormal family;

12. Examples of unit norm tight frame
Partial FFT or WHT:
Partial summation operator;
Cascading of identity or Fourier matrices;
U=RF or U=RW U=
U=[I I … I] or U=[F F … F]

13. Bounded orthonormal matrix
Proposed system
The product A= can be written as
Bounded orthonormal matrix
Random Sign Flipping
Unit norm tight frame ±1 A= U D B
Many existing systems can be characterized by the above systems: random convolution, random demodulation, compressive multiplexing, random probing… 13

14. Performance bound Performance bound Main tools in the proof
Suprema of chaos processes [Krahmer et al ]
Variations and extensions
The diagonal matrix D could be constructed from any sub-Gaussian variables;
B could be a near-orthogonal (rectangular) matrix;

15. Example: Random demodulation
Random demodulation [Tropp et al.-2010]

16. Random demodulation Sampling operator: =UD, U= Sparsifying transform
: FFT matrix with column permutation
Sparsifying transform
Previous work [Tropp et al. 2010]: M≥ O(K log6N)
Our bound: M ≥ O(K log2K log2N)

17. Outline Background Proposed system
Basics of compressed sensing;
Structured random operators for compressed sensing;
Proposed system
Performance bounds;
Connection with existing systems;
Applications in convolutional compressed sensing
Compressive imaging;
Sparse channel estimation in OFDM;
Conclusions

18. Coded aperture imaging—(existing system)
Romberg et al.-2008, Marcia et al.-2009
Low resolution detector array x
Random mask D
Lens F
Lens FH
Begin with
Focus on our proposed
After that, will be presented, followed by a brief conclusion
=I
Works poorly for natural images

19. Coded aperture imaging—(existing system)
Spatially sparse, =I
Fails for this one (spectrally sparse)

20. Proposed system Sampling Operator: =RFHDF
: diagonal matrix made from the Golay sequence;
Mask based on 
Implementation: Double phase encoding [Rivenson et al. 2010]

21. Golay sequence Let a=[a0, a1, ..., aN − 1]T (an {1, -1}) and define
If a is a Golay sequence, then
for all z on the unit circle

22. Proposed system Sampling Operator: =RFHDF Sparsifying transform :
: diagonal matrix made from the Golay sequence;
U=RFH
B=F 
Sparsifying transform :
Haar Wavelet, Fourier, (Block) DCT (popular for natural images) 

23. Simulation results: Compressive imaging
Experimental setup
Test images: 256256 Lena and Hall;
Reconstruction: re-weighted BPDN [Carrillo et al.-2013]
Sampling ratio M/N=0.25

24. Simulation Results: Convolutional CS
(b)
Reconstructed images of Lena. (a) Results from conventional compressive coded aperture imaging, SNR=11.02 dB; (b) Our proposed algorithm, SNR=29.56 dB;

25. Simulation Results
(a)
(b)
Reconstructed images of Hall. (a) Results from conventional compressive coded aperture imaging, SNR=9.21 dB; (b) Our proposed system, SNR=24.62 dB;

26. Outline Background Proposed system
Basics of compressed sensing;
Structured random operators for compressed sensing;
Proposed system
Performance bounds;
Connection with existing systems;
Applications in convolutional compressed sensing
Compressive imaging;
Sparse channel estimation in OFDM;
Conclusions

27. Sparse channel estimation for OFDM system
Existing solutions: y
Low rate ADC
Meng et al.-2012:
R: deterministic sampler;
i: Variable with random phase
High Peak-to-Average Power Ratio (PAPR) after the IDFT
Li et al.-2014:
R random sampler;
i: Golay sequence;
Difficult to implement random sampling

28. Proposed system : Golay sequence  Low PAPR p(t): random signal
Random demodulation
: Golay sequence  Low PAPR
p(t): random signal
Overall Sampling Operator: =UDFFH
B=FFH U=

29. Experimental setup (OFDM)
N=1024, M=64;
Channel model
ATTC (Advanced Television Technology Center) and the Grande Alliance DTV laboratory ensemble E model.
Channel impulse response
Input signal to noise ratio (SNR): 0dB to 30dB
Reconstruction: subspace pursuit (Dai et al.-2009)

30. Simulation results for low rate OFDM channel estimation

31. Conclusions Proposed framework
A==UDB
U: unit norm tight frame, D: random diagonal matrix, B: bounded orthogonal matrix;
M ≥ O(K log2K log2N)
Improved performance bound for existing system
Random demodulation, compressive multiplexing, random probing etc.
Novel compressive sensing framework
Compressive imaging;
Sparse channel estimation for OFDM systems;