The proposed method for blur detection and identification is based on the extrema that has been extracted by scanning the image horizontally and vertically. Furthermore, this has been tested only on Gaussian, HM, and OOF. In this case, 1D extrema was only considered. However, scanning directions can be diagonal and 2D extrema can also be extracted. Ad-ditionally, motion blurs may be in different directions. Thus, the effects of different scanning method and motion direction to the accuracy of the proposed method is still unknown.
For degraded images, the proposed cost function has been extensively tested on the assumption that the blurring function is invariant. Further
investigations are required on its performance when a variant blur is under consideration. In this case, the method for determining the RPSF must be modified and its accuracy in representing the PSF must also be determined.
During the deconvolution process, iterative methods are inherently time consuming and computationally expensive. Some preconditioning methods can be employed to decrease the processing time but this is still a less explored area.
The iterative nature of the deconvolution technique in this work will naturally yield several images. At a certain iteration or condition, ring-ing and color artifacts become more pronounced. Explorring-ing more efficient methods for accurately detecting and monitoring these distortions can in-crease the quality of the estimated images. This is not only applicable to image reconstruction but to other areas of study in image processing as well. In this work, the estimated images has been compared using pixel variance and maxima kurtosis. An aspect that has not been investigated yet is the relationship between the statistical moments of pixel and extrema values for different types of blurs. Results from this have potential use on the comparison of images that is independent of unblurred images.
Selection of Regularization Parameters
Like many regularization problems, the reconstruction method in this disser-tation also requires for the determination of regularization parameters. Sev-eral methods have been proposed such as the discrepancy principle, L-curve, generalized corss validation, and statistical approach [63]. Other techniques uses the Kullback-Leibler distance [80] or generalized singular value decom-position [81]. For the multidimensional case, the L-curve method has been extended to L-hypersurface [82] and applied to the determination of the parameters for the different wavelet decomposition scales [83]. However, these are computationally expensive, time consuming, and mostly for one parameter only. This section discusses a method for the selection of the three regularization parameters that is required by the proposed cost func-tion in Chapter 4. The concept will follow that in [84, 85], which makes the regularization parameter selection process less complicated. The idea is
133
based on the fact that the partial derivatives of the cost function are zero assuming that we are given the correct values of f and h. Thus equations (4.25) and (4.27) will become:
HTHf −HTg+λTf(ε)f = 0 (A.1)
and
FTF h−FTg+αTh(ε)h+β(h−r) = 0, (A.2) respectively. The resulting equations are overdetermined system of linear equations (OSLE). In general, an OSLE with E linear equations and N unknowns can be expressed as:
c=Aχ (A.3)
wherecis a vector of given values withc∈ ℜE×1,Ais a matrix of coefficients with A ∈ ℜE×N, and χ is a vector of unknown variables with χ ∈ ℜN×1. Using minimum sum of squared error approximation [86], the solution can be determined by:
χ= (ATA)−1ATc. (A.4)
Based on equation (A.1),λ can then be computed by:
λ= (ATfAf)−1ATfcf (A.5)
where Af = −Tf(ε)f and cf = HTHf −HTg. Similarly, α and β can be solved from equation (A.2):
[ α β ]T = (AThAh)−1AThch (A.6)
where Ah =−[Th(ε)h (h−r)] and ch =FTF h−FTg.
In practical applications, the correct values off andhare unknown. The estimated values, ˆf and ˆh, may be utilized in lieu of the unknown correct values then equations (A.5) and (A.6) can be used to compute approximate values of the regularization parameters. In this case, parameter tuning is done iteratively with a stopping criterion which may be based on maximum PSNR; a fixed iteration count; or minimum cost function among others.
During regularization, estimated values are initially set thus solving for the parameters based only on these will yield results with inferior quality. A better approach is to compute the parameters after every AM iteration, i.e., outside the CG loop. The proper values will be selected after a few iterations or it may also be updated until the desired number of iterations is reached. Aside from number of iterations, selection may also be achieved in combination with some other criteria like PSNR of the estimated image, value of the cost function, estimated PSF error, among others.
[1] S. Esedoglu. Blind deconvolution of bar code signals. Inverse Problems, 20(1):121–135, February 2004.
[2] S. Yahyanejad and J. Strom. Removing motion blur from barcode images. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) 2010, pages 41–46, San Francisco, CA, June 2010.
[3] R. Choksi and Y. van Gennip. Deblurring of one dimensional bar codes via total variation energy minimisation. SIAM J. on Imaging Sciences, 3-4:735–764, 2010.
[4] T. H. Li and K. S. Lii. A joint estimation approach for two-tome image deblurring by blind deconvolution. IEEE Transactions on Image Processing, 11(8):847–858, August 2002.
[5] C. H. Chu, D. N. Yang, and M. S. Chen. Extracting barcodes from a camera-shaken image on camera phones. In Proceedings of the IEEE International Conference on Multimedia and Expo 2007, pages 2062–
2065, Beijing, July 2007.
137
[6] R. Choksi, Y. van Gennip, and A. Oberman. Anisotropic total variation regularizedl1-approximation and denoising/deblurring of 2D bar codes.
Inverse Problems and Imaging, 2010. submitted.
[7] C. H. Chu, D. N. Yang, Y. L. Pan, and M. S. Chen. Stabilization and extraction of 2D barcodes for camera phones. Multimedia Systems, pages 1–21, October 2010.
[8] R. Cappelli, A. Lumini, D. Maio, and D. Maltoni. Fingerprint image reconstruction from standard templates. IEEE Transactions on Pat-tern Analysis and Machine Intelligence, 29(9):1489–11503, September 2007.
[9] B. J. Kang and K. R. Park. Real-time image restoration for iris recog-nition systems.IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 37(6):1555–1566, December 2007.
[10] F. Xin, Z. Qi, L. Dequn, and Z. Ling. Face image restoration based on statistical prior and image blur measure. In Proceedings of the IEEE International Conference on Multimedia and Expo (ICME 2003), volume 3, pages III–297–300, July 2003.
[11] C. H. Chu, D. N. Yang, and M. S. Chen. Single image deblurring for a real-time face recognition system. In Proceedings of the 36th Annual Conference on IEEE Industrial Electronics Society (IECON 2010), pages 1185–1192, Glendale, AZ, November 2010.
[12] M. Nishiyama, A. Hadid, H. Takeshima, J. Shotton, T. Kozakaya, and O. Yamaguchi. Facial deblur inference using subspace analysis for
recognition of blurred faces. IEEE Transactions on Pattern Analysis and Machine Intelligence, PP(99):1–1, November 2010.
[13] M. Jerian, S. Paolino, F. Cervelli, S. Carrato, A. Mattei, and L. Garo-fano. A forensic image processing environment for investigation of surveillance video. Forensic Science International, 167(2):207–212, 2007.
[14] A. Jalobeanu, L. Blanc-Feraud, and J. Zerubia. Satellite image deblur-ring using complex wavelet packets. International Journal of Computer Vision, 51(3):205–217, 2003.
[15] M. Trimeche, D. Paliy, M. Vehvilainen, and V. Katkovnik. Multichan-nel image deblurring of raw color components. In Proceedings of the SPIE, volume 5674, pages 169–178, 2005.
[16] H. Tong, M. Li, H. Zhang, and C. Zhang. Blur detection for digital images using wavelet transform. In Proceedings of the IEEE Inter-national Conference on Multimedia and Expo (ICME2004), volume 1, pages 17–20, Taipei, Taiwan, June 2004.
[17] I. Aizenberg, C. Butakoff, V. Karnaukhov, N. Merzlyakov, and O. Milukova. Blurred image restoration using the type of blur and blur parameters identification on the neural network. InProceedings of the SPIE, volume 4667.
[18] I. Aizenberg, C. Butakoff, V. Karnaukhov, N. Merzlyakov, and O. Milukova. Type of blur and blur parameters identification
us-ing neural network and its application to image restoration. In Pro-ceedings of the International Conference on Artificial Neural Networks (ICANN2002), pages 1231–1236, Madrid, Spain, August 2002.
[19] I. Aizenberg, D. Paliy, C. Moraga, and J. Astola. Blur identification using neural network for image restoration. InProceedings of the Inter-national Conference 9th Fuzzy Days, pages 441–455, Dortmund, Ger-many, September 2006.
[20] I. Aizenberg, D. V. Paliy, J. M. Zurada, and J. T. Astola. Blur iden-tification by multilayer neural network based on multivalued neurons.
IEEE Trans. on Neural Networks, 19(5):883–898, May 2008.
[21] J. Biemond, R. L. Lagendijk, and R. M. Mersereau. Iterative methods for image deblurring. Proceedings of the IEEE, 78(5):856–883, May 1990.
[22] D. Kundur and D. Hatzinakos. Blind image deconvolution.IEEE Signal Processing Magazine, 13(3):43–64, May 1996.
[23] D. Kundur and D. Hatzinakos. Blind image deconvolution revisited.
IEEE Signal Processing Magazine, 13(6):61–63, November 1996.
[24] M. R. Banham and A. K. Katsaggelos. Digital image restoration.IEEE Signal Processing Magazine, 14(2):24–41, March 1997.
[25] R. L. Lagendijk and J. Biemond. Basic methods for image restoration and identification. In A. Bovik, editor, Handbook of Image & Video
Processing, chapter 3.5, pages 167–181. Elsevier Academic Press, Or-lando, FL, second edition, 2005.
[26] T. E. Bishop, S. D. Babacan, B. Amizic, A. K. Katsaggelos, T. Chan, and R. Molina. Blind image deconvolution: Problem formulation and existing approaches. In P. Campisi and K. Egiazarian, editors, Blind Image Deconvolution: Theory and Applications, chapter 1, pages 1–41.
Taylor & Francis Group, 2007.
[27] O. V. Michailovich and D. R. Adam. Deconvolution of medical images from microscopic to whole body images. In P. Campisi and K. Egiazar-ian, editors, Blind Image Deconvolution: Theory and Applications, chapter 5, pages 169–237. Taylor & Francis Group, 2007.
[28] Y. He, K. H. Yap, L. Chen, and L. P. Chau. A soft MAP framework for blind super-resolution image reconstruction. Image and Vision Com-puting, 27(4):364–373, March 2009.
[29] N. P. Galatsanos, M. N. Wernick, A. K. Katsaggelos, and R. Molina.
Multichannel image recovery. In A. Bovik, editor, Handbook of Image
& Video Processing, chapter 3.7, pages 203–217. Elsevier Academic Press, Orlando, FL, second edition, 2005.
[30] E. P. Simoncelli. Statistical modeling of photographic images. In A. Bovik, editor, Handbook of Image & Video Processing, chapter 4.7, pages 431–441. Elsevier Academic Press, Orlando, FL, second edition, 2005.
[31] F. Rooms, A. Pizurica, and W. Philips. Estimating image blur in the wavelet domain. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP2002), volume 4, pages IV–4190, Orlando, FL, May 2002.
[32] P. Marziliano, F. Dufaux, S. Winkler, and T. Ebrahimi. A no-reference perceptual blur metric. In Proceedings of the IEEE International Conference on Image Processing (ICIP2002), volume 3, pages 57–60, Rochester, NY, September 2002.
[33] P. Marziliano, F. Dufaux, S. Winkler, and T. Ebrahimi. Perceptual blur and ringing metrics: application to JPEG2000. Signal Processing:
Image Communication, 19:163–172, 2004.
[34] Y. C. Chung, J. M. Wang, R. R. Bailey, S. W. Chen, and S. L. Chang.
A non-parametric blur measure based on edge analysis for image pro-cessing applications. In Proceedings of the IEEE International Con-ference on Cybernetics and Intelligent Systems (CIS2004), volume 1, pages 356–360, Singapore, December 2004.
[35] R. Ferzli, L. J. Karam, and J. Caviedes. A robust image sharpness metric based on kurtosis measurement of wavelet coefficients. In Inter-national Workshop on Video Processing and Quality Metrics for Con-sumer Electronics, January 2005.
[36] D. Li, R. M. Mersereau, and S. Simske. Blur identification based on kurtosis minimization. In Proceedings of the IEEE International
Con-ference on Image Processing (ICIP2005), pages I–905–8, Genoa, Italy, September 2005.
[37] L. Chen and K. H. Yap. Efficient discrete spatial techniques for blur support identification in blind image deconvolution.IEEE Transactions on Signal Processing, 54(4):1557–1562, 2006.
[38] A. E. Savakis and H. J. Trussell. Blur identification by residual spectral matching. IEEE Transactions on Image Processing, 2:141–151, 1993.
[39] A. A. Bhutta and H. Foroosh. Blind blur estimation using low rank approximation of cepstrum. InProceedings of the International Confer-ence on Image Analysis and Recognition (ICIAR2006), pages I:94–103, Portugal, September 2006.
[40] S. Wu, Z. Lu, E. P. Ong, and W. Lin. Blind image blur identification in cepstrum domain. In Proceedings of the IEEE International Confer-ence on Computer Communications and Networks, pages 1166–1171, Honolulu, HI, August 2007.
[41] M. M. Chang, A. M. Tekalp, and A. T. Erdem. Blur identification using the bispectrum. IEEE Transactions on Signal Processing, 39(10):2323–
2325, 1991.
[42] A. E. Savakis and R. L. Easton Jr. Blur identification based on higher order spectral nulls. In Proceedings of the SPIE Conference on Image Restoration), volume 2302, pages 168–177, San Diego, California, July 1994.
[43] S. Mallat and S. Zhong. Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelli-gence, 14(7):710–732, July 1992.
[44] E. J. Stollnitz, A. D. DeRose, and D. H. Salesin. Wavelets in com-puter graphics: a primer 1.IEEE Computer Graphics and Applications, 15(3):76–84, May 1995.
[45] R. O. Duda and P. E. Hart. Use of the Hough transformation to detect lines and curves in pictures. Comm ACM, 15(1):11–15, January 1972.
[46] R. C. Gonzales and R. E. Woods. Digital Image Processing. Pearson Education, USA, third edition, 2008.
[47] I. Aizenberg and C. Moraga. Multilayer feedforward neural net-work based on multi-valued neurons (MLMVN) and a backpropagation learning algorithm. Soft Computing, 11(2):169–183, January 2007.
[48] P. Campisi, A. Neri, S. Colonnese, G. Panci, and G. Scarano. Blind image deconvolution using Bussgang techniques: applications to image deblurring and texture synthesis. In P. Campisi and K. Egiazarian, ed-itors,Blind Image Deconvolution: Theory and Applications, chapter 2, pages 43–93. Taylor & Francis Group, 2007.
[49] A. Jalobeanu, J. Zerubia, and L. Blanc-Feraud. Bayesian estimation of blur and noise in remote sensing imaging. In P. Campisi and K. Egiazar-ian, editors, Blind Image Deconvolution: Theory and Applications, chapter 6, pages 239–275. Taylor & Francis Group, 2007.
[50] E. Pantin, J. L. Starck, and F. Murtagh. Deconvolution and blind deconvolution in astronomy. In P. Campisi and K. Egiazarian, editors, Blind Image Deconvolution: Theory and Applications, chapter 7, pages 277–316. Taylor & Francis Group, 2007.
[51] M. K. Ng and R. J. Plemmons. Blind deconvolution and structured matrix computations with applications to array imaging. In P. Campisi and K. Egiazarian, editors, Blind Image Deconvolution: Theory and Applications, chapter 10, pages 377–422. Taylor & Francis Group, 2007.
[52] A. K. Katsaggelos and C. J. Tsai. Iterative image restoration. In A. Bovik, editor, Handbook of Image & Video Processing, chapter 3.9, pages 235–252. Elsevier Academic Press, Orlando, FL, second edition, 2005.
[53] J. Besag. On the statistical analysis of dirty pictures. Journal of the Royal Statistical Society, B-48:259–302, 1986.
[54] G. Ayers and J. Dainty. Iterative blind deconvolution method and its applications. Optics Letters, 13:547–549, 1988.
[55] L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.
[56] T. F. Chan and C. K. Wong. Total variation blind deconvolution. IEEE Transactions on Image Processing, 7(3):370–375, March 1998.
[57] T. F. Chan and C. K. Wong. Convergence of the alternating min-imization algorithm for blind deconvolution. Linear Algebra and its Applications, 316(1–3):259–285, September 2000.
[58] Z. Wang and A. C. Bovik. Modern Image Quality Assessment. Morgan
& Claypool, USA, 2006.
[59] R. M. Chong and T. Tanaka. Image extrema analysis and blur detec-tion with identificadetec-tion. InProceedings of the International Conference on Signal Image Technology and Internet Based System (SITIS2008), volume 1, pages 320–326, Bali, Indonesia, December 2008.
[60] R. M. Chong and T. Tanaka. Detection and classification of invariant blurs. IEICE Transactions on Fundamentals, E92-A(12):3313–3320, December 2009.
[61] T. M. Cover and P. E. Hart. Nearest neighbor pattern classification.
IEEE Transactions on Information Theory, 13(1):21–27, January 1967.
[62] S. Theodoridis and K. Koutroumbas. Pattern Recognition. Elsevier Academic Press, USA, third edition, 2006.
[63] W. C. Karl. Regularization in image restoration and reconstruction. In A. Bovik, editor,Handbook of Image & Video Processing, chapter 3.6, pages 183–202. Elsevier Academic Press, Orlando, FL, second edition, 2005.
[64] Y. Yang, N. P. Galatsanos, and H. Stark. Projection-based blind de-convolution. J. Opt. Soc. Am. A, 11(9):2401–2409, 1994.
[65] Y. He, K. H. Yap, L. Chen, and L. P. Chau. A novel hybrid model framework to blind color image deconvolution. IEEE Transactions on Systems, Man, and Cybernetics–Part A: Systems and Humans, 38(4):867–880, July 2008.
[66] C. R. Vogel. Computational Methods for Inverse Problems, chap-ter 8, pages 129–150. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002.
[67] A. Brook, R. Kimmel, and N. A. Sochen. Variational restoration and edge detection for color images. Journal of Mathematical Imaging and Vision, 18:247–268, 2003.
[68] C. R. Vogel and M. E. Oman. Fast, robust total variation - based reconstruction of noisy, blurred images. IEEE Transactions on Image Processing, 7(6):813–824, June 1998.
[69] T. Chan, S. Esedoglu, F. Park, and A. Yip. Total variation im-age restoration: overview and recent developments. In N. Paragios, Y. Chen, and O. Faugeras, editors, Handbook of Mathematical Mod-els in Computer Vision, chapter 2, pages 17–31. Springer Science + Business Media, Inc., New York, NY, 2006.
[70] T. F. Chan, G. H. Golub, and P. Mulet. A nonlinear primal-dual method for total variation-based image restoration. SIAM Journal on Scientific Computing, 20(6):1964–1977, November 1999.
[71] P. Blomgren and T. F. Chan. Color TV: total variation methods for restoration of vector-valued images. IEEE Transactions on Image Pro-cessing, 7(3):304–309, March 1998.
[72] L. Bar, A. Brook, N. Sochen, and N. Kiryati. Deblurring of color images corrupted by impulsive noise.IEEE Transactions on Image Processing, 16(4):1101–1111, April 2007.
[73] T. Chan and W. Chiu-Kwong. Total variation blind deconvolution.
IEEE Transactions on Image Processing, 16(4):1101–1111, April 2005.
[74] V. Katkovnik, K. Egiazarian, and J. Astola. Blind multiframe image deconvolution using anisotropic spatially adaptive filtering for denois-ing and regularization. In P. Campisi and K. Egiazarian, editors,Blind Image Deconvolution: Theory and Applications, chapter 3, pages 95–
139. Taylor & Francis Group, 2007.
[75] R. M. Chong and T. Tanaka. Blur identification based on maxima locations for color image restoration. InProceedings of the International Conference on Multimedia and Ubiquitous Engineering (MUE2010), Cebu, Philippines, August 2010.
[76] R. M. Chong and T. Tanaka. Detection of motion blur direction based on maxima locations for blind deconvolution. In Proceedings of the IS&T/SPIE Electronic Imaging, January 2011. to be published.
[77] R. M. Chong and T. Tanaka. Maxima exploitation for reference blur-ring function in motion deconvolution. IEICE Transactions on Funda-mentals, E94-A(3), March 2011. to be published.
[78] R. M. Chong and T. Tanaka. Motion blur identification using maxima locations for blind color image restoration. Journal of Convergence. to be published.
[79] H. Hu and G. de Haan. Low cost robust blur estimator. In Proceedings of the IEEE International Conference on Image Processing (ICIP2006), pages 617–620, Atlanta, GA, October 2006.
[80] A.M. Urmanov, A.V. Gribok, H. Bozdogan, J. W. Hines, and R.E.
Uhrig. Information complexity-based regularization parameter selec-tion for soluselec-tion of ill condiselec-tioned inverse problems. Inverse Problems, 18:L1–L9, 2002.
[81] R. A. Renaut, I. Hnetynkova, and J. Mead. Regularization param-eter estimation for large-scale Tikhonov regularization using a priori information. Computational Statistics and Data Analysis, 54(12):3430–
3445, December 2010.
[82] M. Belge, M. E. Kilmer, and E. L. Miller. Simultaneous multiple reg-ularization parameter selection by means of the L-hypersurface with applications to linear inverse problems posed in the wavelet domain.
In Proceedings of the SPIE: Bayesian Inference Inverse Problems, July 1998.
[83] M. Belge, M. E. Kilmer, and E. L. Miller. Wavelet domain image restoration with adaptive edge-preserving regularization. IEEE Trans-actions on Image Processing, 9(4):597–608, 2000.
[84] Y.L. You and M. Kaveh. A regularization approach to joint blur iden-tification and image restoration. IEEE Transactions on Image Process-ing, 5(3):416–428, March 1996.
[85] F. Sroubek and J. Flusser. Multichannel blind iterative image restora-tion. IEEE Transactions on Image Processing, 12(9):1094–1106, September 2003.
[86] J. A. Cadzow. Minimumℓ1,ℓ2, andℓ∞ norm approximate solutions to an overdetermined system of linear equations. Digital Signal Process-ing, 12(4):524–560, October 2002.
International Journal Papers
1. R.M. Chong and T. Tanaka, Maxima exploitation for reference blur-ring function in motion deconvolution,IEICE Transactions on Funda-mentals of Electronics, Communications and Computer Sciences, vol.
E94–A, no. 3, March 2011 (to be published).
2. R.M. Chong and T. Tanaka, Motion blur identification using maxima locations for blind color image restoration, Journal of Convergence, vol. 1, no. 1, pp. 49 – 56, December 2010.
3. R.M. Chong and T. Tanaka, Detection and classification of invariant blurs, IEICE Transactions on Fundamentals of Electronics, Commu-nications and Computer Sciences, E92 - A, pp. 3313 - 3320, December 2009.
151
International Conference Papers
1. R.M. Chong and T. Tanaka, Detection of motion blur direction based on maxima locations for blind deconvolution. In IS&T/SPIE Elec-tronic Imaging 2011, California, United States, January 23 - 27, 2011 (to be published).
2. R.M. Chong and T. Tanaka, Blur identification based on maxima lo-cations for color image restoration. In International Conference on Multimedia and Ubiquitous Engineering (MUE2010), Cebu, Philip-pines, August 11 - 13, 2010.
3. R.M. Chong and T. Tanaka, Image extrema analysis and blur detec-tion with identificadetec-tion. In International Conference on Signal-Image Technology & Internet-Based Systems (SITIS2008), Bali, Indonesia, November 30 - December 3, 2008.
Co-authored International Conference Papers
1. T. Tanaka, R. Miyamoto, and R.M. Chong, Super-resolution based on blind deconvolution using similarity of power spectra. In ACM/IEEE International Conference on Distributed Smart Cameras, Como, Italy, August 30 - September 2, 2009.
2. T. Tanaka, K. Makino, R. Miyamoto, and R.M. Chong, Blind image deconvolution based on similarity measure of power spectra. In RISP International Workshop on Nonlinear Circuits and Signal Processing,