Efficient Deterministic Compressed Sensing for Images with Chirps and Reed-Muller Sequences Kang-Yu (Connie) Ni, Arizona State University, joint with Somantika Datta, Prasun Mahanti, Svetlana Roudenko, Douglas Cochran

Compressed  Sensing:  Overview  

!

" x = y

!

" sensing matrixy data/measurementsx sparse signal

!

n " N n "1

!

N "1

want to recover!

1.  : k-sparse 2.  : RIP (Restricted Isometry Property), e.g. random matrices 3.  Practical reconstruction algorithms, e.g. l1  minimization

*Candes, Romberg, Tao 2006 and Donoho 2006

!

x

!

"

k < n << N

Sensing  Matrix:  Determinis7c  Approach  •  Why deterministic sensing?

–  Explicit reconstruction algorithm –  Efficient storage –  Smaller error in reconstruction

•  Existing works (1d signals) –  DeVore – via finite fields –  Chirp matrices – Applebaum, Howard, Searle, Calderbank –  2nd-order RM sequences – Howard, Calderbank, Searle, Jafarpour –  Indyk, Iwen, Herman, … see http://dsp.rice.edu/cs

•  Need a method suitable for images

Sta7s7cal  Restricted  Isometry  Property                    is -StRIP if for k-sparse , holds with probability exceeding 1-

!

(1"#) x 22$ %x 2

2$ (1+#) x 2

2

!

"

!

"

!

(k, ", #)

!

x " RN

*Calderbank, Howard, Jafarpour, 2010

CS  with  Chirps  and  RM  Sequences  •  2nd-order Reed-Muller functions

!

"P ,b (a) =12m

i2bT a+ Pa( )T a

a, b#$ 2m binary vectors of length m

P : m % m binary symmetric matrix

!

"r,m (l) =1ne2#in ml+ 2#i

n rl 2

r, m, l$% n

•  discrete chirp signal

 **  Howard  et  al.  

•  Hadamard matrix

   •  Reed-Muller matrix (P is zero-diagonal)

   

m=2! 1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1

•  Fourier matrix

•  chirp matrix

n=3!

 *  Applebaum  et  al.  

!

" = [UP1UP2

! UP2m(m#1) /2

]

2m X 2m (m+1)/2!

!

! " # # # $ # # #

!

" = [Ur1Ur2! Urn

]

n X n2!

!

! " # # # $ # # #

Pros: Outperform MP in recon error and computational complexity –  MP –  det CS with chirp

Cons: Not suited for images 256 × 256 image with 10% sparsity k = 6,554, N = 65,536 –  rule of thumb n ≈ 22,670,

but n ×N = image not sparse enough –  least squares problem becomes too large

!

n > k log2(1+ N /k)

Chirp  and  RM  Reconstruc7on  Algorithms  

2m X 2m (m+1)/2

!

O(knN)

!

O kn2 logn( )

!

Nn

=6553622670

" 2.89

Results  n / N Image,  Sparsity   n / k noiselets   chirp   RM  25%   Brain,  7%   3.6   25.2  dB  

 123  dB    

119  dB    

12.5%   Vessel,  5%   2.5   10.1  dB    

49.9  dB    

10.6  dB    

6.25%   Man,  2.38%   2.6   14.5  dB    

112  dB    

109  dB    

!

" = U1 U2 U3 U4[ ]

!

" = U1 U2 U3 U4 U5 U6 U7 U8[ ]

!

" = U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16[ ]

Reconstruc7on  Algorithm   0. Perform initial approximation and then get residual Repeat 1 - 3 until residual is sufficiently small

1  Detect support 2  Determine coefficients 3  Get residual

!

y0 = y " A˜ z

Construc7on  of  Sensing  Matrices  •  N = image size (expl: 512 X 512 = 218 ) •  n = N / 4 (expl: 216 ) •  Sensing matrix: •  Satisfy the Statistical Restricted Isometry Property

!

" = [ U1 #U2 U3 #U4 ]

0.  Ini7al  Best  Approxima7on  of  Solu7on  •  Detection of the “bulk” of a signal •  Based on energy of wavelets concentrate on upper-left region

!

y ="x = [ U1 U2 U3 U4 ] =U1x1 +U2x2 +U3x3 +U4x4

!

U1*y =U1

*U1x1 +U1*U2x2 +U1

*U3x3 +U1*U4x4 " x1

!

x1

x2

x3

x4

"

#

$ $ $ $

%

&

' ' ' '

Acknowledgement  •  This work was partially supported by NSF-DMS FRG grant #0652833, NSF-DUE

#0633033, ONR-BRC grant #N00014-08-1-1110 •  Robert Calderbank, Sina Jafarpour, Stephen Howard, Stephen Searle – discussions

of their work in deterministic CS. Justin Romberg – advice about noiselets and l1  algorithms. Jim Pipe – guidance about medical imaging, providing MRI images

Email: kangyu.ni@gmail.com

2.  Determine  Coefficients  by  Least  Squares

!

A z = y

solved by LSQR [Paige & Saunders]  

!

˜ z = argminz

A z " y2

!

Ut = DvtU1

!

n " k

!

SNR(dB) = 10 log10 || xactual ||2 || xactual " xrecon ||2[ ]

•  Hard-threshold to obtain a set of locations, denoted by •  Let , the initial approx. is

!

U1*y

!

"

!

A =U1 "

!

˜ z = A*y

1.  Detect  Support  by  DCFT  (or  DCHT)  •  From

•  Update

•  Let !

w(t, l) =n

DFT y0(l)vt (l){ }, t =1, 2, 3, 4, vt =1st column of Ut

!

" = "# locations associated with d largest w(t, l){ }

!

A ="#

Conclusion  •  Extend the utility of CS using deterministic matrices •  Demonstrate a method that supports imaging applications

Ongoing works: •  Investigate more natural formulations for multi-dimensional signals •  Exploit deterministic CS with a priori knowledge on signals

noiselets: random noiselet measurements* with l1 minimization** *Candes and Romberg, sparsity and incoherence in compressive sampling **Zhang, Yang, and Yin, YALL1: Your ALgorithms for L1

!

y = " x + µ

!

µ : noise with standard deviation "

n / N Image   σ noiselets   chirp   RM  25%   Brain   0  

0.05  0.1  

23.4  dB  16.5  dB  12.5  dB  

28.4  dB  25.2  dB  21.3  dB  

25.7  dB  24.9  dB  20.9  dB  

25%   Vessel   0  0.05  0.1  

12.0  dB      6.9  dB      2.6  dB  

14.1  dB  12.4  dB      9.9  dB  

13.4  dB  12.3  dB      9.1  dB  

25%   Man   0  0.05  0.1  

20.0  dB  16.2  dB  12.7  dB  

23.2  dB  22.5  dB  20.2  dB  

22.6  dB  21.7  dB  19.4  dB  

!

x : compressible (not sparsified)

!

1 1 1

1 e2"i3 1#1 e

2"i3 2#1

1 e2"i3 1#2 e

2"i3 2#2

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