IRJET- K-SVD: Dictionary Developing Algorithms for Sparse Representation of Signal
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This document discusses the K-SVD algorithm, which is a dictionary learning method for sparse representations of signals. It alternates between sparse coding to represent signals using the current dictionary, and dictionary updating to learn a new dictionary from the signal representations. The algorithm generalizes the K-means clustering process. The document provides details on the methodology of K-SVD, including its basic steps and application to synthetic and real image signals. It is concluded that K-SVD learns dictionaries that better represent given signal groups compared to alternative methods.
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 3009
in sequence manner and permit to changing the respective coefficients. Once we modify the search column according to K-
means then ready to procced to next generalization of the K-mean
We designed dictionary for performance of K-SVD in synthetic andreal imageapplication. In recentdaysK-SVDalgorithmhave
more demand, especially when the dimensions of the dictionary increase, or the number of training signals becomes large.
Initially apply the K-SVD algorithm on synthetic signals method, for checking is this algorithm occupies the original dictionary
D that originating from the data and to compare its results with relatedalgorithms.Generationofthedata trainthatdescribesa
random matrix (referred to later as the generating dictionary) of size 20x50wasgeneratedwithuniformlydistributedentries.
Each column was normalized to a unit l^2-norm. Then, 1500 data signals of dimension 20 were produced, each of them made
by a linear combination of three various dictionary atoms, with uniformly distributed coefficients inrandomandindependent
locations. White Gaussian noise with varying signal-to-noise ratio (SNR) was added to the resulting data signals. Applying the
K-SVD, the dictionary was initialized with data signals. The coefficients were found using OMP with a fixed number of three
coefficients. The maximum number of iterations was set to 80. Comparison to other reportedworksweimplementedtheMOD
algorithm and applied it on the same data, using OMP with a fixed number of three coefficientsandinitializinginthesame way.
We executed the MOD algorithm for a total number of 80 iterations. We also executed the MAP-based algorithm of Kreutz-
Delgado. This algorithm was executed as is, thereforeusingFOCUSSasitsdecompositionmethod.Here,again,a maximumof80
iterations were allowed.
3. CONCLUSION
According to work we proposed the problem of generating and using overcomplete dictionaries. We developed an algorithm
the K-SVD for training an overcomplete dictionary which is better for group of given signals. From this we generalized K-
means algorithm, implement designed for solving a same but related problem. We also proved that dictionary which foundby
K-SVD. On that performance we will apply for both synthetic and real image in different applications. We used all this
technology for filling in missing pixels andcompressionandoutperformsalternatives suchasthenon-decimatedHaarandover
complete or unitary DCT.
REFERENCES
1. K. Engan, S. O. Aase, and J. H. Hakon-Husoy, “Method of optimal directions for frame design,”in IEEE Int.Conf.
Acoust.,Speech, Signal Process., 1999, vol. 5, pp. 2443–2446.
2. K. Engan, B. D. Rao, and K. Kreutz-Delgado, “Frame design using focus with method of optimal directions (mod),”in
Norwegian Signal Process. Symp., 1999,vol.65-69.[42]J. F.MurrayandK.Kreutz-Delgado,“Animprovedfocuss-based
learning algorithm for solving sparse linear inverse problems,” in IEEE Int. Conf. Signals, Syst. Comput., 2001, vol.
4119-53.
3. A.J. BellandT.J. Sejnowski, “An information maximization approach to blind separation and blind
deconvolution,”NeuralComp.,vol.7,no. 6, pp. 1129–1159, 1996.
4. K. Kreutz-Delgado and B. D. Rao, “FOCUSS-baseddictionarylearningalgorithms,”inWaveletApplicationsinSignal and
Image Process. VIII, 2000, vol. 4119
5. S. Lesage, R.Gribonval,F.Bimbot,andL.Benaroya,“Learning unions of orthonormal bases with thresholded singular
value decomposition,” in IEEE Int. Conf. Acoust., Speech, Signal Process., 2005.
6. S. Sardy, A. G. Bruce, and P. Tseng, “Block coordinate relaxation methods for nonparametric signal denoising with
wavelet dictionaries,” J. Comp. Graph. Statist., vol. 9, pp. 361–379, 2000.
7. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy.
Statist. Soc., ser. B, vol. 39, no. 1, pp. 1–38, 1977.
8. K. Engan, S. O. Aase, and J. H. Husøy, “Multi-frame compression: Theory and design,” EURASIP Signal Process., vol. 80,
no. 10, pp. 2121–2140, 2000.
9. K. Kreutz-Delgado, J.F. Murray, B.D. Rao, K. Engan, T. Levant’s. Sejnowski, “Dictionary learning algorithms for sparse
representation,” Neural Comp ., vol. 15, no. 2, pp. 349–396, 2003.
10. J.A. Troop, “Topics in sparse approximation,” Ph.D. dissertation, Univ. of Texas at Austin, Austin, 2004.
11. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Norwell, MA: Kluwer Academic, 1991.](https://image.slidesharecdn.com/irjet-v6i6610-191216060440/85/IRJET-K-SVD-Dictionary-Developing-Algorithms-for-Sparse-Representation-of-Signal-3-320.jpg)
IRJET- K-SVD: Dictionary Developing Algorithms for Sparse Representation of Signal
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 3007 K-SVD: Dictionary Developing Algorithms for Sparse Representation of Signal Snehal Patil G.1, Prof.A.G. Patil2 1P.G. Student, Department of Electronics & Telecommunication Engineering, Padmabhooshan Vasantraodada Patil Institute of Technology, Budhgaon 2Assistant Professor, Department of Electronics & Telecommunication Engineering, Padmabhooshan Vasantraodada Patil Institute of Technology, Budhgaon ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - Now days trend in study of sparse representation of signals. In spare representation having overcomplete dictionary that contain prototype signal-atoms, signals are elaborated by sparse linear combinations of these atoms. There are many applications in sparse representation whichincludecompression, consistency in inverseproblems, featureextraction, and more. We concentrated basically on study purpose of pursuit algorithms that decompose signalswithrespectto agivendictionaryD. Weare developing method the K-SVD algorithm generalizing the K-means clustering process. K-SVD is a mathematical method that develop the algorithm alternates between sparse coding data using the dictionary Dandapplyprocessforupdatingthedictionary atoms to get the correct data. After updated dictionary columns which is combined with an update of the sparse representations. The developed K-SVD algorithm is adaptable. It can also work with any type of pursuit method. Key Words: Basis pursuit, dictionary, FOCUSS, K-means, K-SVD, matching pursuit. 1. INTRODUCTION Sparse representations using learned dictionariesDarebeingmorehelpful withsuccessinvarious typesofdata processingand machine learning applications. The accessibility of large amount of training data necessitates the development of suitable, robust and better dictionary learning algorithms. For algorithmic stabilityandgeneralizationofdictionarylearningalgorithms we are using two cases: 1. Complete: a system {yi} in X is complete if every element in X can be arbitrarily well in norm by linear combinations of elements in {yi}. 2. Overcomplete: if removal of an element from the system {yi} results in a complete system. The arbitrary approximation in norm can be thought as a representation somehow. K-SVD algorithm for studying dictionaries D. We explained its development and analysis, and formalized applications to establish its usability and the advantage of trained dictionaries D. Diversities of the K-SVD algorithm for learning structural constrained dictionaries are also showcased. Out of those constraints are the non-negativity of the dictionary and shift invariance property. K-SVD deals with development of a state-of-the art image denoising algorithm. This case study is important as it nourishes the message that the general model of sparsity and redundancy, alongwithfitteddictionariesasalso used here, it is the good practical applications in image processing. 1.1 Sparse Representation of signal Let us consider the overcomplete dictionary matrix D ∈ Rn×K that include K prototype signal atoms for columns, {dj} K j=1, a signal, here y ∈Rn can be represented as a linear combination of these atoms. That present the y = Dx, or y ≈ Dx, ||y−Dx|| p ≤ ℇ. The vector x ∈ R^k contains the representation coefficients of the signal y. In some method, for measurement of the deviation we are using the l^p -norms for p = 1,2 and ∞. Here we are focusing on the case of p = 2. If n < K and D is a full-rank matrix, several alternative methods are available for the representationproblem.Thesolutionwithnonzerocoefficientsiscertainlyan applicable for representation. This sparsest representation is the solution of either (P0) min ||x||0 for finding approximating solutions have been extensively investigated and indeed, several effective decomposition algorithms are available. 2. Methodology K-SVD Algorithm two steps, one is Sparse Coding that contain producing sparse representations matrix X, given the current dictionary D. Another one is Dictionary Update D include updatingdictionaryatoms,giventhecurrentsparsesrepresentations.
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 3008 2.1K-SVD frame Fig: Basic Frame of K-SVD 2.2 Basic Algorithm of K-SVD 1. Input: Signal set Y, initial dictionaries D0, target sparsity S, number of iteration L. 2. Output: Dictionary D and sparse matrix X such that Y=DX. 3. Init: set D =D0 4. for n=1, ……...L do 5. subject to ||x||0 ≤ S 6. for j=1, …. K do 7. dj=0 8. I= {indices of the signals in Y whose representations use dj} 9. E=Yi=D Xi=Yi- 10.{d, g}= || 11.dj =d; 12.Xj, I= 13.end for 14.end for Initially, we set D and to search the better coefficient matrix. According to given data searchingtheoptimal isdifficult.Byusing pursuit method, we will calculate coefficients, it can provide a solution data with a fixed. When that step is done, next step is showing to find for a best dictionary. This task updates only one column at a time, fixing all columns in rather than one and searching a new column and new values for that coefficients will helpful for better decreases the MSE. We change thecolumns
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 3009 in sequence manner and permit to changing the respective coefficients. Once we modify the search column according to K- means then ready to procced to next generalization of the K-mean We designed dictionary for performance of K-SVD in synthetic andreal imageapplication. In recentdaysK-SVDalgorithmhave more demand, especially when the dimensions of the dictionary increase, or the number of training signals becomes large. Initially apply the K-SVD algorithm on synthetic signals method, for checking is this algorithm occupies the original dictionary D that originating from the data and to compare its results with relatedalgorithms.Generationofthedata trainthatdescribesa random matrix (referred to later as the generating dictionary) of size 20x50wasgeneratedwithuniformlydistributedentries. Each column was normalized to a unit l^2-norm. Then, 1500 data signals of dimension 20 were produced, each of them made by a linear combination of three various dictionary atoms, with uniformly distributed coefficients inrandomandindependent locations. White Gaussian noise with varying signal-to-noise ratio (SNR) was added to the resulting data signals. Applying the K-SVD, the dictionary was initialized with data signals. The coefficients were found using OMP with a fixed number of three coefficients. The maximum number of iterations was set to 80. Comparison to other reportedworksweimplementedtheMOD algorithm and applied it on the same data, using OMP with a fixed number of three coefficientsandinitializinginthesame way. We executed the MOD algorithm for a total number of 80 iterations. We also executed the MAP-based algorithm of Kreutz- Delgado. This algorithm was executed as is, thereforeusingFOCUSSasitsdecompositionmethod.Here,again,a maximumof80 iterations were allowed. 3. CONCLUSION According to work we proposed the problem of generating and using overcomplete dictionaries. We developed an algorithm the K-SVD for training an overcomplete dictionary which is better for group of given signals. From this we generalized K- means algorithm, implement designed for solving a same but related problem. We also proved that dictionary which foundby K-SVD. On that performance we will apply for both synthetic and real image in different applications. We used all this technology for filling in missing pixels andcompressionandoutperformsalternatives suchasthenon-decimatedHaarandover complete or unitary DCT. REFERENCES 1. K. Engan, S. O. Aase, and J. H. Hakon-Husoy, “Method of optimal directions for frame design,”in IEEE Int.Conf. Acoust.,Speech, Signal Process., 1999, vol. 5, pp. 2443–2446. 2. K. Engan, B. D. Rao, and K. Kreutz-Delgado, “Frame design using focus with method of optimal directions (mod),”in Norwegian Signal Process. Symp., 1999,vol.65-69.[42]J. F.MurrayandK.Kreutz-Delgado,“Animprovedfocuss-based learning algorithm for solving sparse linear inverse problems,” in IEEE Int. Conf. Signals, Syst. Comput., 2001, vol. 4119-53. 3. A.J. BellandT.J. Sejnowski, “An information maximization approach to blind separation and blind deconvolution,”NeuralComp.,vol.7,no. 6, pp. 1129–1159, 1996. 4. K. Kreutz-Delgado and B. D. Rao, “FOCUSS-baseddictionarylearningalgorithms,”inWaveletApplicationsinSignal and Image Process. VIII, 2000, vol. 4119 5. S. Lesage, R.Gribonval,F.Bimbot,andL.Benaroya,“Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE Int. Conf. Acoust., Speech, Signal Process., 2005. 6. S. Sardy, A. G. Bruce, and P. Tseng, “Block coordinate relaxation methods for nonparametric signal denoising with wavelet dictionaries,” J. Comp. Graph. Statist., vol. 9, pp. 361–379, 2000. 7. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Statist. Soc., ser. B, vol. 39, no. 1, pp. 1–38, 1977. 8. K. Engan, S. O. Aase, and J. H. Husøy, “Multi-frame compression: Theory and design,” EURASIP Signal Process., vol. 80, no. 10, pp. 2121–2140, 2000. 9. K. Kreutz-Delgado, J.F. Murray, B.D. Rao, K. Engan, T. Levant’s. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comp ., vol. 15, no. 2, pp. 349–396, 2003. 10. J.A. Troop, “Topics in sparse approximation,” Ph.D. dissertation, Univ. of Texas at Austin, Austin, 2004. 11. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Norwell, MA: Kluwer Academic, 1991.