KSVD - Gradient Descent Method For Compressive Sensing Optimization
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1 KSV - Gradient escent Method For Compressive Sensing Optimization Endra epartment of Computer Engineering Faculty of Engineering Bina Nusantara University
2 INTROUCTION
3 INTROUCTION
4 WHAT IS COMPRESSIVE SENSING? Candes, E.J., and Wakin, M.B., March. 2008, An Introduction to Compressive Sampling, IEEE Signal Processing Magazine., pp
5 WHAT IS COMPRESSIVE SENSING? When Sensing Meet Compression Automatically translates analog data into already compressed digital form.
6 Applications and Opportunities Of Compressive Sensing New Analog-to-igital Converters (Analog to Information)
7 COMPRESSIVE SENSING CS Theory Requires Three Aspects : 1.The desired signals/images are sparse/compressible. Need a suitable basis or sparse dictionary (Fourier, Wavelet, Overcomplete ictionary) ictionary Learning (K-SV (Singular Value ecomposition). 2. CS Matrix Requires a small mutual coherence with dictionary. 3. Reconstruction algorithms Matching / Basis Pursuit. In this paper, We used OMP (Orthogonal Matching Pursuit) & Iteratively Reweighted Least Squares (IRLS) Ell-p minimization.
8 COMPRESSIVE SENSING FRAMEWORK M 1 M N N K K 1 θ M 1 M K K 1 θ S Sparse Measurement Matrix Sparse Coefficent Equivalent ictionary y x Basis/ictionary Small Mutual Coherence Between x & M N If K N Complete (Basis) y If K N Over-Complete (ictionary)
9 PREVIOUS WORKS , avid L. onoho, Emmanuel J. Candès, Justin Romberg, and Terence Tao First Papers in CS, Using Random Matrix for CS Measurement , M. Aharon, M. Elad, and A. BrucksteinUsing KSV for esigning Overcomplete ictionaries for Sparse Representation , M. Elad Optimized CS Measurement by Reducing t-averaged Mutual Coherence , Vahid Abolghasemi, Saideh Ferdowsi, Bahador Makkiabadi and Saeid Sanei Optimized CS Measurement by Using Gradient-escent Method, Better than Elad s Method. In This Paper We Combined KSV & Gradient-escent Methods to Perform the Joint optimization of ictionary and CS Measurement Matrix.
10 OPTIMIZE MEASUREMENT MATRIX Random Gaussian Matrix that fulfill the required property of CS measurement (Incoherency & RIP) usually to be used to encode the signal. can be optimized by reducing the mutual coherence : T : max d d Equivalent ictionary,, i j,1 i, j K i close to orthonormal Gram-Matrix of Equivalent ictionary : G I min G I 2 F min t I 2 F
11 Optimized Measurement Matrix Optimized Measurement Matrix Gradient Gradient-escent Method escent Method T T T F T I I Tr I E 2 I d E E T ij 4 I E k i i T i i i i i OPTIMIZE CS MEASUREMENT MATRIX
12 min Joint Optimization of ictionary and CS Measurement Matrix Y X 2 2 X Y s.t. i, S,, F F i X Is Training Patches 0 Get by Using KSV Optimize by Using Gradient escent Method min min X I Y s.t.,, i Z Z 2 F s.t. i, i,,, eq F i 2 W 0 0 S S eq eq : W d1... d eq K Joint KSV - Gradient escent Method
13 SIMULATION METHO From Each of 30 Training-Images (481 x 321) Was Taken Randomly x 8 patches 6000 patches. These 6000 patches Were Used as Training Patches for Joint KSV - Gradient escent Method.
14 SIMULATION METHO Test Image (481 x 321)
15 RESULTS The PSNR comparison of reconstructed image from compressive sensing by using OMP for : KSV - Random, Uncoupled KSV- Gradient escent and Joint KSV-Gradient escent.
16 RESULTS The PSNR comparison of reconstructed image from compressive sensing by using (IRLS) Ell-p - Minimization for : KSV - Random, Uncoupled KSV-Gradient escent and Joint KSV- Gradient escent..
17 RESULTS KSV- Random KSV- Random OMP IRLS-ell-p Uncoupled KSV- Gradient escent Uncoupled KSV- Gradient escent
18 RESULTS Joint KSV- Gradient escent OMP IRLS-ell-p The comparison of reconstructed image for m = 15 by using OMP (left column) and IRLS ell-p - minimization (right column) where : (a) & (d) KSV-Random, (b) & (e) Uncoupled KSV- Gradient escent, (c) & (f) Joint KSV- Gradient escent.
19 CONCLUSION From the results, it showed that by optimizing measurement matrix and dictionary learning simultaneously provided the improvement of the image reconstruction from compressive sensing. Further improvement can be attempted in future work by optimizing measurement matrix and dictionary learning simultaneously based on block-sparse representations.
20 REFERENCES [1] [31] [32] Petros Boufounos, Justin Romberg and Richard Baraniuk, Compressive Sensing : Theory and Applications, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, Nevada, Apr [Online]. Available: ICASSP-mar08.pdf. [33] Jianwei Ma., ata Recovery from Compressed Measurement, School of Aerospace, Tsinghua University, Beijing. [34] E. Candès, Electrical Engineering Colloquium, University of Washington, ecember [35] Michael Elad, Optimized Projection irections for Compressed Sensing, The IV Workshop on SIP & IT Holon Institute of Technology June 20th, [36] Michael Elad, Sparse & Redundant Representation Modeling of Images, Summer School on Sparsity in Image and Signal Analysis, Holar, Iceland, August 15 20, 2010.
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