Analysis and Synthesis Sparse Representation Models for...
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Analysis and Synthesis Sparse Representation Models for Image Modeling
Shuhang GU
Dept. of ComputingThe Hong Kong Polytechnic University
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Outline
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Introduction
• Image Restoration and Enhancement
• Synthesis & Analysis Sparsity Models
Convolutional sparse coding for image super-resolution
• Convolutional Sparse Coding v.s. Sparse Coding
• The Proposed Method
Guided Image Enhancement via Weighted Analysis Sparsity Model
• Dependency Modeling for Guided Enhancement
• Learning dynamic guidance for guided depth enhancement
Ongoing and Future Works
• Image Separation without Training Data
• Image Restoration with Deep Denoisers
• Optimization Inspired Network Structure Design
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Introduction
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Image restoration and enhancement problems
𝑦 = 𝑥 + 𝑛 𝑦 = 𝐷(𝑘⨂𝑥) + 𝑛
𝑦 = 𝑘⨂𝑥 + 𝑛 𝑦 = 𝑀⊙ 𝑥 + 𝑛
?
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Image Inpainting
Image Denoising
Image Deconvolution
…
Image Super-resolution
Contrast Enhancement
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Synthesis & Analysis sparsity models
Synthesis representation models
Synthesis based sparse representation model assumes that a signal 𝑥can be represented as a linear combination of a small number of atoms chosen out of a dictionary 𝐷:
𝑥 = 𝐷𝛼, s.t. 𝛼 0<𝜀
5
??
.
.
.
??
A dense solution
A sparse solution
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Synthesis & Analysis sparsity models
Synthesis representation models
Synthesis based sparse representation model assumes that a signal 𝑥can be represented as a linear combination of a small number of atoms chosen out of a dictionary 𝐷:
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Synthesis & Analysis sparsity models
Analysis representation models
Analysis model generate representation coefficients by a simple multiplication operation, and assumes the coefficients are sparse:
𝑃𝑥 0<𝜀
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??
.
.
.
??
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Synthesis & Analysis sparsity models
Analysis representation models
Analysis model generate representation coefficients by a simple multiplication operation, and assumes the coefficients are sparse:
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Synthesis & Analysis sparsity models
Synthesis model
𝑚𝑖𝑛𝛼1
2𝑦 − 𝐷𝛼 𝐹
2 + 𝜓 𝛼
𝑥 = 𝐷𝛼
• Representative methods
KSVD, BM3D, LSSC, NCSR, et. al.
• Pros- Synthesis model can be more sparse
• Cons- Patch prior modeling needs aggregation
- Time consuming
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Analysis model
𝑚𝑖𝑛𝑥1
2𝑦 − 𝑥 𝐹
2 + 𝜙(𝑃𝑥)
• Representative methods
TV, wavelet methods, FRAME, FOE, CSF, TRD et. al.
• Pros- Patch divide free
- Efficient in the inference phase
• Cons- Not as sparse as synthesis model, limited capacity in modeling image prior.
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Convolutional sparse coding for image super-
resolution
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Convolutional Sparse Coding v.s. Sparse Coding
Consistency constraint
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Convolutional Sparse Coding v.s. Sparse Coding
Sparse coding
𝑚𝑖𝑛𝛼||𝑦 − 𝐷𝛼||𝐹2+𝜙(𝛼)
Convolutional sparse coding
𝑚𝑖𝑛𝒁 𝒀 − σ𝒇𝑖 ⊗𝒛𝑖 𝐹2+σ𝜑(𝒛𝑖)
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…
… … ……
Matrix Form
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Convolutional sparse coding for image SR
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N LR feature mapsM HR feature maps
Mapp. Func. M HR filters
Mapp. Func.Learning
Joint HR Filter and Mapping Function LearningLR Filter Learning
The Testing Phase
The Training Phase
N LR filters
CSC LR FilterLearning
HR FilterLearning
HR Feature MapEstimation Convolution
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Convolutional sparse coding for image SR
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N LR feature mapsM HR feature maps
Mapp. Func.Learning
Joint HR Filter and Mapping Function LearningLR Filter Learning
The Training Phase
CSC LR FilterLearning
HR FilterLearning
– LR filter training
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Convolutional sparse coding for image SR
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N LR feature mapsM HR feature maps
Mapp. Func.Learning
Joint HR Filter and Mapping Function LearningLR Filter Learning
The Training Phase
CSC LR FilterLearning
HR FilterLearning
– Joint HR filter and mapping function learning
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Convolutional sparse coding for image SR
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Mapp. Func. M HR filters
The Testing Phase
The Training Phase
N LR filters
HR Feature MapEstimation Convolution
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Convolutional sparse coding for image SR
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Optimization: SA-ADMM
The original problem can be write as:
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Convolutional sparse coding for image SR
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Optimization: SA-ADMM
SA-ADMM
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Convolutional sparse coding for image SR
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Convolutional sparse coding for image SR
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Guided Image Enhancement via Weighted
Analysis Sparsity
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Dependency Modeling
Dependent image data
Guided enhancement
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MRI PET Depth RGB
…
Single ImageEnhancement Guided Image
Enhancement
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Dependency Modeling
• Previous Arts
– 1st order method: co-different
𝑖
𝑗∈𝑆(𝑖)(𝒙𝒊 − 𝒙𝒋)
2𝜑(𝒈𝒊 − 𝒈𝒋)
𝑖
𝑗∈𝑆(𝑖)(1 − 𝜌 𝒙𝒊 − 𝒙𝒋 )𝜑(𝒈𝒊 − 𝒈𝒋)
– 2nd order method: TGV
– Other priors: Non-local mean
– Data-driven method: joint dictionary learning
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Dependency Modeling
• Weighted Analysis Sparse Representation Model
𝑖
𝑗∈𝑆(𝑖)(𝒙𝒊 − 𝒙𝒋)
2𝜑(𝒈𝒊 − 𝒈𝒋)
𝑖
𝑗∈𝑆(𝑖)1 − 𝜌 𝒙𝒊 − 𝒙𝒋 𝜑(𝒈𝒊 − 𝒈𝒋)
• How to choice 𝜌,𝜑, 𝒌?
• Highly non-convex, hard to optimize!
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Generalize the model: from one-order point –wise relationship to high-order local prior.
ො𝑥 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑋𝑓 𝒙, 𝒚 +𝜌(𝒌𝑥𝑖 ∗ 𝒙)⨀𝜑(𝒌𝑔
𝑖 ∗ 𝒈)
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Guided image enhancement via weighted analysis
sparsity
• Task-driven training of stage-wise parameters– Solving weighted analysis sparse representation model with gradient descent,
we have:
ො𝑥 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑋𝑓 𝒙, 𝒚 +𝜌(𝒌𝑥𝑖 ∗ 𝒙)⨀𝜑(𝒌𝑔
𝑖 ∗ 𝒈)
𝒙𝑡+1 = 𝒙𝑡 − 𝜏 Δ𝑓 𝒙𝑡, 𝒚 +𝒌𝑥𝑖 𝑇𝜌𝒊
′(𝒌𝑥
𝑖 ∗ 𝒙)⨀𝜑(𝒌𝑔𝑖 ∗ 𝒈)
– Stage-wise parameter training
𝒎𝒊𝒏𝒌𝑥,𝒌𝒈,𝜌,𝜑𝒍𝒐𝒔𝒔 𝒙𝑡+1 𝒙𝑡; 𝒌𝑥 , 𝒌𝒈, 𝜌, 𝜑 − 𝒙𝒈𝒕
𝒔. 𝒕. 𝒙𝑡+1 𝒙𝑡; 𝒌𝑥 , 𝒌𝒈, 𝜌, 𝜑 = 𝒙𝑡 − 𝜏 Δ𝑓 𝒙𝑡 , 𝒚 +𝒌𝑥𝑖 𝑇𝜌𝒊
′(𝒌𝑥
𝑖 ∗ 𝒙)⨀𝜑(𝒌𝑔𝑖 ∗ 𝒈)
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Guided image enhancement via weighted analysis
sparsity
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𝑰𝒏𝒊𝒕𝒊𝒂𝒍𝒊𝒛𝒂𝒕𝒊𝒐𝒏
∗ ...
...∗
𝑪𝒐𝒏𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏
...
...
...
...
𝑵𝒐𝒏𝒍𝒊𝒏𝒆𝒂𝒓𝒊𝒕𝒚 ∗
𝑹𝒆𝒈𝒖𝒍𝒂𝒓𝒊𝒛𝒂𝒕𝒊𝒐𝒏𝒕𝒆𝒓𝒎
𝑷𝒐𝒊𝒏𝒕𝒘𝒊𝒔𝒆𝑴𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒐𝒏
𝑭𝒊𝒅𝒆𝒍𝒊𝒕𝒚 𝒕𝒆𝒓𝒎
Low quality depth map
High quality color image
σ
Enhanced depth map
𝒙𝟎
𝒈
𝒙𝟏
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Guided image enhancement via weighted analysis
sparsity
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Stage-wise operation
…Stage-wise operation
𝒙𝟎 𝒙𝟏 𝒙𝑵
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• Experimental results
Guided Image Enhancement via Weighted
Analysis Sparsity
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• Experimental results
Guided Image Enhancement via Weighted
Analysis Sparsity
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Ongoing and Future Works
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• Complementary Property of
ASR and SSR
Image Separation without Training Data
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𝑚𝑖𝑛𝑢,𝑣𝑓 𝒚 − 𝒖 − 𝒗 + 𝜌𝑠 𝒖 + 𝜌𝐴(𝒗)
• Layer Separation
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• Half Quadratic Splitting
Image Restoration with Deep Denoisers
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𝑚𝑖𝑛𝑥,𝑠𝑓 𝒚, 𝒙 + 𝜆 𝒙 − 𝒔 𝐹2 + 𝜌 𝒔
𝑚𝑖𝑛𝑥𝑓 𝒚, 𝒙 + 𝜌 𝒙
Half Quadratic Splitting
𝑚𝑖𝑛𝑥𝑓 𝒚, 𝒙 + 𝜆 𝒙 − 𝒔 𝐹2 𝑚𝑖𝑛𝑠𝜆 𝒙 − 𝒔 𝐹
2 + 𝜌 𝒔
Noisy Image Residual Image
Co
nv
+ R
eLU
Co
nv
+B
No
rm+
Re
LU
Co
nv
+B
No
rm+
Re
LU
Co
nv
+B
No
rm+
Re
LU
Co
nv
Deep Denoiser
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• State-of-the-art Performance Has been Achieved by
Deep Models
– Non-blind Super-Resolution
– Gaussian Denoising
– Non-blind Deblur
• More Complex Restoration problems
– Blind Deblur, SR, Denoising?
Optimization Inspired Network Structure Design
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Related Publications and References
Related Publication• S. Gu, W. Zuo, Q. Xie, D. Meng, X. Feng, L. Zhang. "Convolutional Sparse Coding for Image
Super-resolution," In ICCV 2015.
• S. Gu, W. Zuo, S. Guo, Y. Chen, C. Chen and L. Zhang, "Learning Dynamic Guidance for Depth Image Enhancement," To appear in CVPR 2017.
• K. Zhang, W. Zuo, S. Gu and L. Zhang, "Learning Deep CNN Denoiser Prior for Image Restoration," To appear in CVPR 2017.
• S. Gu, et. al. Joint Convolutional Analysis and Synthesis Sparse Representation for Single Image Layer Separation. Submitted.
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Related Publications and References
References• Elad, M., Milanfar, P., Rubinstein, R. Analysis versus synthesis in signal priors. Inverse problems
2007.
• Buades A, Coll B, Morel JM. A non-local algorithm for image denoising. In CVPR 2005.
• M. Aharon, M. Elad, A. Bruckstein. K-svd: An algorithm for designing overcomplete dictionaries for sparse representation. TSP 2006.
• K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian. Image denoising by sparse 3-d transform-domain collaborative filtering. TIP, 2007.
• J. Marial, F. Bach, J. P once, G. Sapiro, and A. Zisserman. Non-local sparse models for image restoration. In ICCV 2009.
• W. Dong, L. Zhang, and G. Shi, “Centralized Sparse Representation for Image Restoration,” In ICCV 2011.
• Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 1992.
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Related Publications and References
References• Zhu, S.C., Wu, Y., Mumford, D.: Filters, random fields and maximum entropy (frame): Towards a
unified theory for texture modeling. IJCV 1998.
• Roth, S., Black, M.J.: Fields of experts. IJCV 2009.
• Schmidt, U., Roth, S.: Shrinkage fields for effective image restoration. In: CVPR. (2014)
• Chen, Y., Yu, W., Pock, T.: On learning optimized reaction diffusion processes for effective image restoration. In: CVPR. (2015)
• Rubinstein, Ron, Tomer Peleg, and Michael Elad. "Analysis K-SVD: A dictionary-learning algorithm for the analysis sparse model." TSP 2013.
• Zoran D, Weiss Y. From learning models of natural image patches to whole image restoration.In: ICCV 2011.
• M. D. Zeiler, D. Krishnan, G. W. Taylor, and R. Fergus. Deconvolutional networks. In CVPR, 2010.
• B. Wohlberg. Efficient convolutional sparse coding. In ICASSP, 2014.
• L. W. Zhong and J. T. Kwok. Fast stochastic alternating direction method of multipliers. In ICML,2013.
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Related Publications and References
References• J. Diebel and S. Thrun. An application of markov random fields to range sensing. In NIPS, 2005.
• B. Ham, M. Cho, and J. Ponce. Robust image filtering using joint static and dynamic guidance. In CVPR, 2015.
• D. Ferstl, C. Reinbacher, R. Ranftl, M. Ruther, and H. Bischof. Image guided depth upsamplingusing anisotropic total generalized variation. In ICCV, 2013.
• J. Park, H. Kim, Y.-W. Tai, M. S. Brown, and I. Kweon. High quality depth map upsampling for 3d-tof cameras. In ICCV, 2011.
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THANKS!
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