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Probabilistic interpretation of the Calderón problem
Image segmentation with dynamic artifacts detection and bias correction
1. | University of California, Los Angeles, Department of Mathematics, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095-1555, USA |
2. | University of California, Los Angeles, California NanoSystems Institute (CNSI), 570 Westwood Plaza, Building 114, Los Angeles, CA 90095, USA |
3. | University of California, Los Angeles, Department of Chemistry and Biochemistry, 607 Charles E. Young Drive, Los Angeles, CA 90095, USA |
4. | University of California, Los Angeles, Department of Materials Science and Engineering, 410 Westwood Plaza, Los Angeles, CA 90095, USA |
5. | University of California, Los Angeles, Department of Mathematics, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095-1555, USA |
Region-based image segmentation is well-addressed by the Chan-Vese (CV) model. However, this approach fails when images are affected by artifacts (outliers) and illumination bias that outweigh the actual image contrast. Here, we introduce a model for segmenting such images. In a single energy functional, we introduce 1) a dynamic artifact class preventing intensity outliers from skewing the segmentation, and 2), in Retinex-fashion, we decompose the image into a piecewise-constant structural part and a smooth bias part. The CV-segmentation terms then only act on the structure, and only in regions not identified as artifacts. The segmentation is parameterized using a phase-field, and efficiently minimized using threshold dynamics.
We demonstrate the proposed model on a series of sample images from diverse modalities exhibiting artifacts and/or bias. Our algorithm typically converges within 10-50 iterations and takes fractions of a second on standard equipment to produce meaningful results. We expect our method to be useful for damaged images, and anticipate use in applications where artifacts and bias are actual features of interest, such as lesion detection and bias field correction in medical imaging, e.g., in magnetic resonance imaging (MRI).
References:
[1] |
A. Ayvaci, M. Raptis and S. Soatto,
Sparse occlusion detection with optical flow, International Journal of Computer Vision, 97 (2012), 322-338.
doi: 10.1007/s11263-011-0490-7. |
[2] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran and S. Osher,
Fast global minimization of the active contour/snake model, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[3] |
T. Brox and D. Cremers,
On local region models and a statistical interpretation of the piecewise smooth mumford-shah functional, International Journal of Computer Vision, 84 (2008), 184-193.
doi: 10.1007/s11263-008-0153-5. |
[4] |
V. Caselles, R. Kimmel and G. Sapiro,
Geodesic active contours, Int. J. Comput. Vis., 22 (1995), 61-79.
doi: 10.1109/ICCV.1995.466871. |
[5] |
T. F. Chan and L. A. Vese,
Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[6] |
T. Chan and W. Zhu,
Level set based shape prior segmentation, , in CVPR 2005, IEEE, 2 (2005), 1164-1170.
doi: 10.1109/CVPR.2005.212. |
[7] |
T. F. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of image segmentation and denoising models, SIAM Journal on Applied Mathematics, 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[8] |
T. F. Chan, B. Sandberg and L. A. Vese,
Active contours without edges for vector-valued images, Journal of Visual Communication and Image Representation, 11 (2000), 130-141.
doi: 10.1006/jvci.1999.0442. |
[9] |
S. A. Claridge, W.-S. Liao, J. C. Thomas, Y. Zhao, H. H. Cao, S. Cheunkar, A. C. Serino, A. M. Andrews and P. S. Weiss,
From the bottom up: Dimensional control and characterization in molecular monolayers, Chem. Soc. Rev., 42 (2013), 2725-2745.
doi: 10.1039/C2CS35365B. |
[10] |
D. Cremers, S. J. Osher and S. Soatto,
Kernel density estimation and intrinsic alignment for shape priors in level set segmentation, International Journal of Computer Vision, 69 (2006), 335-351.
doi: 10.1007/s11263-006-7533-5. |
[11] |
D. Cremers, N. Sochen and C. Schnörr,
A multiphase dynamic labeling model for variational recognition-driven image segmentation, International Journal of Computer Vision, 66 (2006), 67-81.
doi: 10.1007/s11263-005-3676-z. |
[12] |
S. Esedoglu and F. Otto,
Threshold dynamics for networks with arbitrary surface tensions, Communications on Pure and Applied Mathematics, 68 (2015), 808-864.
doi: 10.1002/cpa.21527. |
[13] |
S. Esedoglu and Y. H. R. Tsai,
Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384.
doi: 10.1016/j.jcp.2005.05.027. |
[14] |
V. Estellers and S. Soatto,
Detecting occlusions as an inverse problem, Journal of Mathematical Imaging and Vision, 54 (2016), 181-198.
doi: 10.1007/s10851-015-0596-6. |
[15] |
V. Estellers, D. Zosso, R. Lai, S. Osher, J.-P. Thiran and X. Bresson,
Efficient algorithm for level set method preserving distance function, IEEE Transactions on Image Processing, 21 (2012), 4722-4734.
doi: 10.1109/TIP.2012.2202674. |
[16] |
P. Filzmoser, R. G. Garrett and C. Reimann,
Multivariate outlier detection in exploration geochemistry, Computers & Geosciences, 31 (2005), 579-587.
doi: 10.1016/j.cageo.2004.11.013. |
[17] |
B. K. Horn,
Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.
doi: 10.1016/0146-664X(74)90022-7. |
[18] |
M. Jung, M. Kang and M. Kang,
Variational image segmentation models involving non-smooth data-fidelity terms, Journal of Scientific Computing, 59 (2013), 277-308.
doi: 10.1007/s10915-013-9766-0. |
[19] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[20] |
R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel,
A variational framework for Retinex, International Journal of Computer Vision, 52 (2003), 7-23.
|
[21] |
E. H. Land,
The retinex, American Scientist, 52 (1964), 247-264.
|
[22] |
E. H. Land,
The Retinex theory of color vision, Scientific American, 237 (1977), 108-128.
|
[23] |
E. H. Land and J. J. McCann,
Lightness and Retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.
|
[24] |
C. Li, C.-Y. Kao, J. C. Gore and Z. Ding,
Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing, 17 (2008), 1940-1949.
doi: 10.1109/TIP.2008.2002304. |
[25] |
C. Li, C. Xu, C. Gui and M. D. Fox,
Level set evolution without re-initialization: A new variational formulation, in IEEE CVPR, IEEE, 1 (2005), 430-436.
|
[26] |
C. Li, C. Xu, C. Gui and M. D. Fox,
Distance regularized level set evolution and its application to image segmentation, IEEE Transactions on Image Processing, 19 (2010), 3243-3254.
doi: 10.1109/TIP.2010.2069690. |
[27] |
C. Li, C. Xu, K. M. Konwar and M. D. Fox, Fast distance preserving level set evolution for
medical image segmentation, in 2006 9th International Conference on Control, Automation,
Robotics and Vision, IEEE, 2006, 1–7.
doi: 10.1109/ICARCV.2006.345357. |
[28] |
F. Li, S. Osher, J. Qin and M. Yan, A multiphase image segmentation based on fuzzy
membership functions and L1-norm fidelity, J. Sci. Comput., 69 (2016), 82–106, URL
http://arxiv.org/abs/1504.02206.
doi: 10.1007/s10915-016-0183-z. |
[29] |
W. Ma and S. Osher,
A TV Bregman iterative model of Retinex theory, Inverse Problems
and Imaging (IPI), 6 (2012), 697-708.
doi: 10.3934/ipi.2012.6.697. |
[30] |
R. Madani, A. Bourquard and M. Unser, Image segmentation with background correction
using a multiplicative smoothing-spline model, in 2012 9th IEEE International Symposium
on Biomedical Imaging (ISBI), IEEE, 2012, 186–189.
doi: 10.1109/ISBI.2012.6235515. |
[31] |
B. Merriman, J. K. Bence and S. Osher,
Diffusion Generated Motion by Mean Curvature, Technical report, UCLA CAM Report 92-18, 1992. |
[32] |
B. Merriman, J. K. Bence and S. J. Osher,
Motion of multiple junctions: A level set approach, Journal of Computational Physics, 112 (1994), 334-363.
doi: 10.1006/jcph.1994.1105. |
[33] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Archive for Rational Mechanics and Analysis, 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[34] |
J.-M. Morel, A.-B. Petro and C. Sbert,
A PDE formalization of Retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.
doi: 10.1109/TIP.2010.2049239. |
[35] |
D. Mumford and J. Shah,
Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[36] |
J. Nocedal and S. J. Wright,
Numerical Optimization, 2nd edition, Springer, Berlin, 2006. |
[37] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[38] |
M. Prastawa, E. Bullitt, S. Ho and G. Gerig,
A brain tumor segmentation framework based on outlier detection., Medical Image Analysis, 8 (2004), 275-283.
doi: 10.1016/j.media.2004.06.007. |
[39] |
M. Rousson and N. Paragios, Shape Priors for Level Set Representations, in ECCV 2002 (eds.
A. Heyden, G. Sparr, M. Nielsen and P. Johansen), vol. 2351 of Lecture Notes in Computer
Science, Springer Berlin Heidelberg, Berlin, Heidelberg, 2002, 78–92.
doi: 10.1007/3-540-47967-8_6. |
[40] |
S. Ruuth,
Efficient algorithms for diffusion-generated motion by mean curvature, Journal of Computational Physics, 144 (1998), 603-625.
doi: 10.1006/jcph.1998.6025. |
[41] |
Y. van Gennip, N. Guillen, B. Osting and A. L. Bertozzi,
Mean curvature, threshold dynamics, and phase field theory on finite graphs, Milan Journal of Mathematics, 82 (2014), 3-65.
doi: 10.1007/s00032-014-0216-8. |
[42] |
L. A. Vese and T. F. Chan,
A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293.
|
[43] |
L. Wang, C. Li, Q. Sun, D. Xia and C.-Y. Kao,
Active contours driven by local and global intensity fitting energy with application to brain MR image segmentation, Computerized medical imaging and graphics, 33 (2009), 520-531.
doi: 10.1016/j.compmedimag.2009.04.010. |
[44] |
P. S. Weiss,
Functional molecules and assemblies in controlled environments: Formation and measurements, Accounts of Chemical Research, 41 (2008), 1772-1781.
doi: 10.1021/ar8001443. |
[45] |
M. Yan,
Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting, SIAM Journal on Imaging Sciences, 6 (2013), 1227-1245.
doi: 10.1137/12087178X. |
[46] |
Y. Yang, C. Li, C. -Y. Kao and S. Osher, Split bregman method for minimization of regionscalable fitting energy for image segmentation, in Advances in Visual Computing, vol. 6454
of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2010, 117–128. |
[47] |
D. Zosso, G. Tran and S. Osher, A unifying Retinex model based on non-local differential operators in IS & T/SPIE Electronic Imaging (ed. C. A. Bouman), 8657 (2013), 865702.
doi: 10.1117/12.2008839. |
[48] |
D. Zosso, G. Tran and S. J. Osher,
Non-local Retinex—a unifying framework and beyond, SIAM J. Imaging Sciences, 8 (2015), 787-826.
doi: 10.1137/140972664. |
show all references
References:
[1] |
A. Ayvaci, M. Raptis and S. Soatto,
Sparse occlusion detection with optical flow, International Journal of Computer Vision, 97 (2012), 322-338.
doi: 10.1007/s11263-011-0490-7. |
[2] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran and S. Osher,
Fast global minimization of the active contour/snake model, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[3] |
T. Brox and D. Cremers,
On local region models and a statistical interpretation of the piecewise smooth mumford-shah functional, International Journal of Computer Vision, 84 (2008), 184-193.
doi: 10.1007/s11263-008-0153-5. |
[4] |
V. Caselles, R. Kimmel and G. Sapiro,
Geodesic active contours, Int. J. Comput. Vis., 22 (1995), 61-79.
doi: 10.1109/ICCV.1995.466871. |
[5] |
T. F. Chan and L. A. Vese,
Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[6] |
T. Chan and W. Zhu,
Level set based shape prior segmentation, , in CVPR 2005, IEEE, 2 (2005), 1164-1170.
doi: 10.1109/CVPR.2005.212. |
[7] |
T. F. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of image segmentation and denoising models, SIAM Journal on Applied Mathematics, 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[8] |
T. F. Chan, B. Sandberg and L. A. Vese,
Active contours without edges for vector-valued images, Journal of Visual Communication and Image Representation, 11 (2000), 130-141.
doi: 10.1006/jvci.1999.0442. |
[9] |
S. A. Claridge, W.-S. Liao, J. C. Thomas, Y. Zhao, H. H. Cao, S. Cheunkar, A. C. Serino, A. M. Andrews and P. S. Weiss,
From the bottom up: Dimensional control and characterization in molecular monolayers, Chem. Soc. Rev., 42 (2013), 2725-2745.
doi: 10.1039/C2CS35365B. |
[10] |
D. Cremers, S. J. Osher and S. Soatto,
Kernel density estimation and intrinsic alignment for shape priors in level set segmentation, International Journal of Computer Vision, 69 (2006), 335-351.
doi: 10.1007/s11263-006-7533-5. |
[11] |
D. Cremers, N. Sochen and C. Schnörr,
A multiphase dynamic labeling model for variational recognition-driven image segmentation, International Journal of Computer Vision, 66 (2006), 67-81.
doi: 10.1007/s11263-005-3676-z. |
[12] |
S. Esedoglu and F. Otto,
Threshold dynamics for networks with arbitrary surface tensions, Communications on Pure and Applied Mathematics, 68 (2015), 808-864.
doi: 10.1002/cpa.21527. |
[13] |
S. Esedoglu and Y. H. R. Tsai,
Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384.
doi: 10.1016/j.jcp.2005.05.027. |
[14] |
V. Estellers and S. Soatto,
Detecting occlusions as an inverse problem, Journal of Mathematical Imaging and Vision, 54 (2016), 181-198.
doi: 10.1007/s10851-015-0596-6. |
[15] |
V. Estellers, D. Zosso, R. Lai, S. Osher, J.-P. Thiran and X. Bresson,
Efficient algorithm for level set method preserving distance function, IEEE Transactions on Image Processing, 21 (2012), 4722-4734.
doi: 10.1109/TIP.2012.2202674. |
[16] |
P. Filzmoser, R. G. Garrett and C. Reimann,
Multivariate outlier detection in exploration geochemistry, Computers & Geosciences, 31 (2005), 579-587.
doi: 10.1016/j.cageo.2004.11.013. |
[17] |
B. K. Horn,
Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.
doi: 10.1016/0146-664X(74)90022-7. |
[18] |
M. Jung, M. Kang and M. Kang,
Variational image segmentation models involving non-smooth data-fidelity terms, Journal of Scientific Computing, 59 (2013), 277-308.
doi: 10.1007/s10915-013-9766-0. |
[19] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[20] |
R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel,
A variational framework for Retinex, International Journal of Computer Vision, 52 (2003), 7-23.
|
[21] |
E. H. Land,
The retinex, American Scientist, 52 (1964), 247-264.
|
[22] |
E. H. Land,
The Retinex theory of color vision, Scientific American, 237 (1977), 108-128.
|
[23] |
E. H. Land and J. J. McCann,
Lightness and Retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.
|
[24] |
C. Li, C.-Y. Kao, J. C. Gore and Z. Ding,
Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing, 17 (2008), 1940-1949.
doi: 10.1109/TIP.2008.2002304. |
[25] |
C. Li, C. Xu, C. Gui and M. D. Fox,
Level set evolution without re-initialization: A new variational formulation, in IEEE CVPR, IEEE, 1 (2005), 430-436.
|
[26] |
C. Li, C. Xu, C. Gui and M. D. Fox,
Distance regularized level set evolution and its application to image segmentation, IEEE Transactions on Image Processing, 19 (2010), 3243-3254.
doi: 10.1109/TIP.2010.2069690. |
[27] |
C. Li, C. Xu, K. M. Konwar and M. D. Fox, Fast distance preserving level set evolution for
medical image segmentation, in 2006 9th International Conference on Control, Automation,
Robotics and Vision, IEEE, 2006, 1–7.
doi: 10.1109/ICARCV.2006.345357. |
[28] |
F. Li, S. Osher, J. Qin and M. Yan, A multiphase image segmentation based on fuzzy
membership functions and L1-norm fidelity, J. Sci. Comput., 69 (2016), 82–106, URL
http://arxiv.org/abs/1504.02206.
doi: 10.1007/s10915-016-0183-z. |
[29] |
W. Ma and S. Osher,
A TV Bregman iterative model of Retinex theory, Inverse Problems
and Imaging (IPI), 6 (2012), 697-708.
doi: 10.3934/ipi.2012.6.697. |
[30] |
R. Madani, A. Bourquard and M. Unser, Image segmentation with background correction
using a multiplicative smoothing-spline model, in 2012 9th IEEE International Symposium
on Biomedical Imaging (ISBI), IEEE, 2012, 186–189.
doi: 10.1109/ISBI.2012.6235515. |
[31] |
B. Merriman, J. K. Bence and S. Osher,
Diffusion Generated Motion by Mean Curvature, Technical report, UCLA CAM Report 92-18, 1992. |
[32] |
B. Merriman, J. K. Bence and S. J. Osher,
Motion of multiple junctions: A level set approach, Journal of Computational Physics, 112 (1994), 334-363.
doi: 10.1006/jcph.1994.1105. |
[33] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Archive for Rational Mechanics and Analysis, 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[34] |
J.-M. Morel, A.-B. Petro and C. Sbert,
A PDE formalization of Retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.
doi: 10.1109/TIP.2010.2049239. |
[35] |
D. Mumford and J. Shah,
Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[36] |
J. Nocedal and S. J. Wright,
Numerical Optimization, 2nd edition, Springer, Berlin, 2006. |
[37] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[38] |
M. Prastawa, E. Bullitt, S. Ho and G. Gerig,
A brain tumor segmentation framework based on outlier detection., Medical Image Analysis, 8 (2004), 275-283.
doi: 10.1016/j.media.2004.06.007. |
[39] |
M. Rousson and N. Paragios, Shape Priors for Level Set Representations, in ECCV 2002 (eds.
A. Heyden, G. Sparr, M. Nielsen and P. Johansen), vol. 2351 of Lecture Notes in Computer
Science, Springer Berlin Heidelberg, Berlin, Heidelberg, 2002, 78–92.
doi: 10.1007/3-540-47967-8_6. |
[40] |
S. Ruuth,
Efficient algorithms for diffusion-generated motion by mean curvature, Journal of Computational Physics, 144 (1998), 603-625.
doi: 10.1006/jcph.1998.6025. |
[41] |
Y. van Gennip, N. Guillen, B. Osting and A. L. Bertozzi,
Mean curvature, threshold dynamics, and phase field theory on finite graphs, Milan Journal of Mathematics, 82 (2014), 3-65.
doi: 10.1007/s00032-014-0216-8. |
[42] |
L. A. Vese and T. F. Chan,
A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293.
|
[43] |
L. Wang, C. Li, Q. Sun, D. Xia and C.-Y. Kao,
Active contours driven by local and global intensity fitting energy with application to brain MR image segmentation, Computerized medical imaging and graphics, 33 (2009), 520-531.
doi: 10.1016/j.compmedimag.2009.04.010. |
[44] |
P. S. Weiss,
Functional molecules and assemblies in controlled environments: Formation and measurements, Accounts of Chemical Research, 41 (2008), 1772-1781.
doi: 10.1021/ar8001443. |
[45] |
M. Yan,
Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting, SIAM Journal on Imaging Sciences, 6 (2013), 1227-1245.
doi: 10.1137/12087178X. |
[46] |
Y. Yang, C. Li, C. -Y. Kao and S. Osher, Split bregman method for minimization of regionscalable fitting energy for image segmentation, in Advances in Visual Computing, vol. 6454
of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2010, 117–128. |
[47] |
D. Zosso, G. Tran and S. Osher, A unifying Retinex model based on non-local differential operators in IS & T/SPIE Electronic Imaging (ed. C. A. Bouman), 8657 (2013), 865702.
doi: 10.1117/12.2008839. |
[48] |
D. Zosso, G. Tran and S. J. Osher,
Non-local Retinex—a unifying framework and beyond, SIAM J. Imaging Sciences, 8 (2015), 787-826.
doi: 10.1137/140972664. |






I | M × N | CV | CV+X | CV+B | CV+XB | |||||||||
i [1] | t [s] | i [1] | t [s] | i [1] | t [s] | i [1] | t [s] | |||||||
1 | 78 × 119 | 11 | 0.02 | 20 | 0.03 | 11 | 0.05 | 11 | 0.05 | |||||
2 | 75 × 79 | 14 | 0.02 | 16 | 0.02 | 65 | 0.24 | 66 | 0.25 | |||||
3 | 96 × 127 | 22 | 0.06 | 45 | 0.12 | 19 | 0.11 | 35 | 0.21 | |||||
4 | 110 × 111 | 15 | 0.04 | 30 | 0.08 | 71 | 0.41 | 100 | 0.58 | |||||
5 | 131 × 103 | 33 | 0.11 | 33 | 0.12 | 30 | 0.26 | 36 | 0.32 | |||||
6 | 124 × 184 | 11 | 0.05 | 18 | 0.07 | 55 | 0.45 | 26 | 0.23 |
I | M × N | CV | CV+X | CV+B | CV+XB | |||||||||
i [1] | t [s] | i [1] | t [s] | i [1] | t [s] | i [1] | t [s] | |||||||
1 | 78 × 119 | 11 | 0.02 | 20 | 0.03 | 11 | 0.05 | 11 | 0.05 | |||||
2 | 75 × 79 | 14 | 0.02 | 16 | 0.02 | 65 | 0.24 | 66 | 0.25 | |||||
3 | 96 × 127 | 22 | 0.06 | 45 | 0.12 | 19 | 0.11 | 35 | 0.21 | |||||
4 | 110 × 111 | 15 | 0.04 | 30 | 0.08 | 71 | 0.41 | 100 | 0.58 | |||||
5 | 131 × 103 | 33 | 0.11 | 33 | 0.12 | 30 | 0.26 | 36 | 0.32 | |||||
6 | 124 × 184 | 11 | 0.05 | 18 | 0.07 | 55 | 0.45 | 26 | 0.23 |
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