Non-Local Compressive Sampling Recovery
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Non-Local Compressive Sampling Recovery
Xianbiao Shu1, Jianchao Yang2, Narendra Ahuja1
1{xshu2,n-ahuja}@[email protected]
2 1
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 2
Outline• Introduction to Compressive Sampling (CS)
• Overview of CS recovery methods
• Proposed recovery method----Non-Local CS (NLCS)
• Experimental results
• Summary
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 3
Introduction• Nyquist Sampling
– Map one object point to one image point (one measurement) – The sampling rate , which is very challenging in many applications:
• To reduce the sampling rate– Low-Resolution Sampling (LRS), but it suffers loss of information– Compressive Sampling (CS)[Donoho06]
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Heavy encoding burden before transmission
Long sensing timein medical imaging (e.g. MRI)
High-resolution sensor in Infrared thermal imaging
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Introduction
– If , then this L0-norm problem has unique solution.
– If , then L0-norm solution = L1-norm solution.
Random matrix:
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Restricted Isometry Property [Candes08]
If there exists , s.t.
holds for all -sparse signals
Then, this sampling matrix satisfies RIP condition at isometry constant
Problem Formulation
where is -sparse signal is sampled data is random sampling matrix
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 5
Image Compressive Sampling
• Sensing – Random sampling in single-pixel camera [Duarte08]
multiple the image with a 2D random matrix
exactly satisfies RIP condition but costly– Fourier sampling in MRI [Lustig07]
also satisfies RIP condition and efficient– Circulant sampling [Romberg09][Yin10] convolute the image with a random kernel
efficient and the same performance as random sampling
we use circulant sampling in this paper.
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 6
Image Compressive Sampling• 2DCS (state-of-the-art recovery) [Lustig07][Duarte08][Ma08]
– Sparsity in -transform domain, e.g. wavelet sparsity– Total Variation (TV), i.e., gradient sparsity
It still requires a high sampling rate for real-word images.
• RIP condition:
When sampling rate r=M/N=25%
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2DCS (30.04dB) LRS (26.29dB)
Whichone
isbetter?
Cameraman
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 7
Proposed Method• Spatial correlation in CS, which is motivated by
– Temporal correlation in video CS [Shu11]
– Non-local mean methods (BM3D) [Egiazarian07] in image restoration
• Non-local CS (NLCS)
denotes an image
denotes n patch groups
contains the patch coordinates in i-th group
is the 3D stack of patches in i-th group
– L1-norm based TV[Shu10][Li10]
– Non-Local Sparsity (NLS).
patch correlation within the same group
wavelet sparsity to keep image sharpness
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 8
• Non-Local Sparsity (NLS) measures– Non-Local Wavelet Sparsity (NLWS): sum of 3D wavelet sparsity
– Non-Local Joint Sparsity (NLJS): sum of Joint Sparsity (JS) [Baron05]:
Proposed Method
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NLWS
NLJS
2D wavelet
Common patch
Sparse errorstack
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 9
Iterative Algorithm• “Chicken-and-egg” problem: recover image and • Two iterative steps:
– Non-local grouping– Non-local recovery
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 10
Non-Local Grouping• Patch grouping on image
– Divide the image into reference patches– For each , search for up to best matched patches in its neighborhood
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 11
• Non-local joint sparsity (NLJS)
• Lagrangian Function auxiliary variables
Lagrangian multipliers
• Alternating-Direction-Method-of-Multipliers (ADMM) [Boyd11]
Non-Local Recovery Algorithm
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 12
• Lagrangian Function
• Three iterative steps of ADMM– Estimate of auxiliary variable
– Joint reconstruction of image
– Update Lagragian multipliers
Non-Local Recovery Algorithm
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B
g1
g2
I
R
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 13
Experimental Results• Barbara (r=20%) (PSNR)
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Ground truth
NLJS_ideal(32.12dB)
2DCS (24.85dB) NLWS (28.06dB) NLJS (30.60dB)
2DCS error map NLWS error map NLJS error map
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 14
Experimental Results
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1 2 3 4 5 6 7 8
NLWS NLJS
SamplingRate r=20%
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 15
Experimental Results
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1 2 3 4 5 6 7 8
On average, NLWS-2DCS = 2.56dB NLJS-2DCS = 3.80dB
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 16
Cameraman(r =20%)
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Ground truth 2DCS (28.52dB)
NLWS (30.10dB) NLJS (31.02dB)
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 17
Cameraman(r =25%)
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Ground truth 2DCS (30.04dB)
NLWS (30.10dB) NLJS (32.32dB)LRS (26.29dB)
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 18
Brain (r =20%)
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Ground truth 2DCS (28.54dB)
NLWS (29.83dB) NLJS (30.91dB)
Ground truth
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 19
Train Station (r =10%)
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Ground truth 2DCS (29.33dB)
NLWS (34.35dB) NLJS (36.07dB)
Required sampling rate=10%
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 20
Summary• Non-Local Compressive Sampling (NLCS) recovery
– NLCS exploits a new prior knowledge---nonlocal spatial correlation,
in addition to conventional local piecewise smoothness and wavelet sparsity– Two non-local sparsity measures (NLJS >NLWS) is proposed to seek spatial
correlation in compressive sampling.– An efficient algorithm is given to recover the image and its patch grouping in
NLCS by iterating between non-local grouping and non-local recovery.– NLCS significantly improves the recovery quality and reduces the sampling rate
to a practical level (e.g. 10%)
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 21
Sources/Modalities Used in My Research• Dell WorkStation• Matlab Simulation • MRI dataset• Common test images
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Non-Local Compressive Sampling Recovery X. Shu, J. Yang, N. Ahuja 22
Reference• [Baron05]: D. Baron, M. B.Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk. Distributed compressed sensing. 2005
• [Donoho06]: D. Donoho. Compressed sensing. IEEE Transaction on Information Theory, 2006.
• [Egiazarian07]: K. Egiazarian, A. Foi, and V. Katkovnik. Compressed sensing image reconstruction via recursive spatially adaptive filtering. In ICIP, 2007.
• [Lustig07]: M. Lustig, D. Donoho, J. Santos, and J. Pauly. Compressed sensing MRI. IEEE Sig. Proc. Magazine, 2007.
• [Candes08]: E. J Candes. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique 346.9 (2008): 589-592.
• [Duarte08]: M. F. Duarte, M. A. Davenport, D. Takhar, and et al. Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine, 25(2):83–91, 2008.
• [Ma08]: S. Ma, W. Yin, Y. Zhang, and A. Chakraborty. An efficient algorithm for compressed MR imaging using total variation and wavelets. In CVPR, 2008.
• [Romberg09]: J. Romberg. Compressive sensing by random convolution. SIAM Journal on Imaging Science, 2009.
• [Li10]: C. Li, W. Yin, and Y. Zhang. TVAL3: TV minimization by Augmented Lagrangian and ALternating direction Algorithms, 2010.
• [Shu10]: X. Shu and N. Ahuja. Hybrid compressive sampling via a new total variation TVL1. In ECCV, 2010.
• [Shu11]: X. Shu and N. Ahuja. Imaging via three-dimensional compressive sampling (3DCS). In ICCV, 2011.
• [Boyd11]: S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1):1122, 2011.
• [Yin10] W. Yin, S. P. Morgan, J. Yang, and Y. Zhang. Practical compressive sensing with toeplitz and circulant matrices. Rice University CAAM Technical Report TR10-01, 2010.
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Thanks
And
Questions?
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