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Smart Gradient - An adaptive technique for improving gradient estimation
The Signed Cumulative Distribution Transform for 1-D signal analysis and classification
1. | Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA |
2. | Department of Biomedical Engineering, Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA |
This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [
References:
[1] |
A. Aldroubi, S. Li and G. K. Rohde, Partitioning signal classes using transport transforms for data analysis and machine learning, Sampl. Theory Signal Process. Data Anal., 19 (2021), 25pp.
doi: 10.1007/s43670-021-00009-z. |
[2] |
L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003 1–52.
doi: 10.1007/978-3-540-39189-0_1. |
[3] |
M. Arjovsky, S. Chintala and L. Bottou, Wasserstein generative adversarial networks, International Conference on Machine Learning PMLR, 2017,214–223. |
[4] |
S. Basu, S. Kolouri and G. K. Rohde,
Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry, PNAS, 111 (2014), 3448-3453.
doi: 10.1073/pnas.1319779111. |
[5] |
T. Cai, J. Cheng, N. Craig and K. Craig, Linearized optimal transport for collider events, Phys. Rev. D, 102 (2020).
doi: 10.1103/PhysRevD.102.116019. |
[6] |
T. Cai, J. Cheng, B. Schmitzer and M. Thorpe,
The Linearized Hellinger–Kantorovich Distance, SIAM J. Imaging Sci., 15 (2022), 45-83.
doi: 10.1137/21M1400080. |
[7] |
L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard,
Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal., 274 (2018), 3090-3123.
doi: 10.1016/j.jfa.2018.03.008. |
[8] |
N. Courty, R. Flamary, D. Tuia and A. Rakotomamonjy,
Optimal transport for domain adaptation, IEEE Trans. Pattern Anal. Mach. Intell., 39 (2017), 1853-1865.
doi: 10.1109/TPAMI.2016.2615921. |
[9] |
P. Embrechts and M. Hofert,
A note on generalized inverses, Math. Methods Oper. Res., 77 (2013), 423-432.
doi: 10.1007/s00186-013-0436-7. |
[10] |
B. Engquist and B. D. Froese,
Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., 12 (2014), 979-988.
doi: 10.4310/CMS.2014.v12.n5.a7. |
[11] |
B. Engquist, B. D. Froese and Y. Yang,
Optimal transport for seismic full waveform inversion, Commun. Math. Sci., 14 (2016), 2309-2330.
doi: 10.4310/CMS.2016.v14.n8.a9. |
[12] |
R. A. Fisher,
The use of multiple measurements in taxonomic problems, Ann. Eugenics, 7 (1936), 179-188.
doi: 10.1111/j.1469-1809.1936.tb02137.x. |
[13] |
W. Gangbo, W. Li, Wuchen, S. Osher and M. Puthawala, Unnormalized optimal transport, J. Comput. Phys., 399 (2019), 17pp.
doi: 10.1016/j.jcp.2019.108940. |
[14] |
S. Haker, L. Zhu, A. Tannenbaum and S. Angenent,
Optimal mass transport for registration and warping, Phys. Rev. D, 60 (2004), 225-240.
doi: 10.1023/B:VISI.0000036836.66311.97. |
[15] |
S.-W. Huang, G. K. Rohde, H.-M. Cheng and S.-F. Lin,
Discretized target size detection in electrical impedance tomography using neural network classifier, J. Nondestructive Evaluation, 39 (2020), 1-9.
doi: 10.1007/s10921-020-00723-z. |
[16] |
D. W. Kammler, A First Course in Fourier Analysis, 2nd edition, Cambridge University Press, Cambridge, 2007. |
[17] |
S. Kolouri, K. Nadjahi, U. Şimşekli, R. Badeau and G. K. Rohde, Generalized sliced Wasserstein distances, preprint, 2019, arXiv: 1902.00434. |
[18] |
S. Kolouri, S. R. Park and G. K. Rohde,
The radon cumulative distribution transform and its application to image classification, IEEE Trans. Image Process., 25 (2016), 920-934.
doi: 10.1109/TIP.2015.2509419. |
[19] |
S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev and G. K. Rohde,
Optimal mass transport: Signal processing and machine-learning applications, IEEE Signal Process. Magazine, 34 (2017), 43-59.
doi: 10.1109/MSP.2017.2695801. |
[20] |
S. Kolouri and G. K. Rohde, Transport-based single frame super resolution of very low resolution face images, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, USA, 2015.
doi: 10.1109/CVPR.2015.7299121. |
[21] |
S. Kolouri, A. B. Tosun, J. A. Ozolek and G. K. Rohde,
A continuous linear optimal transport approach for pattern analysis in image datasets, Pattern Recognition, 51 (2016), 453-462.
doi: 10.1016/j.patcog.2015.09.019. |
[22] |
S. Kolouri, Y. Zou and G. K. Rohde, Sliced Wasserstein kernels for probability distributions, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA, 2016.
doi: 10.1109/CVPR.2016.568. |
[23] |
S. Kundu, B. G. Ashinsky, M. Bouhrara, E. B. Dam and S. Demehri,
Enabling early detection of osteoarthritis from presymptomatic cartilage texture maps via transport-based learning, PNAS, 117 (2020), 24709-24719.
doi: 10.1073/pnas.1917405117. |
[24] |
S. Kundu, S. Kolouri, K. I. Erickson, A. F. Kramer, E. McAuley and G. K. Rohde,
Discovery and visualization of structural biomarkers from MRI using transport-based morphometry, NeuroImage, 167 (2018), 256-275.
doi: 10.1016/j.neuroimage.2017.11.006. |
[25] |
S. Mallat,
Group invariant scattering, Comm. Pure Appl. Math., 65 (2012), 1331-1398.
doi: 10.1002/cpa.21413. |
[26] |
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.
![]() ![]() |
[27] |
J. A. Ozolek, A. B. Tosun, W. Wang, C. Chen and S. Kolouri,
Accurate diagnosis of thyroid follicular lesions from nuclear morphology using supervised learning, Medical Image Analysis, 18 (2014), 772-780.
doi: 10.1016/j.media.2014.04.004. |
[28] |
S. R. Park, L. Cattell, J. M. Nichols, A. Watnik, T. Doster and G. K. Rohde,
De-multiplexing vortex modes in optical communications using transport-based pattern recognition, Optics Express, 26 (2018), 4004-4022.
doi: 10.1364/OE.26.004004. |
[29] |
S. R. Park, S. Kolouri, S. Kundu and G. K. Rohde,
The cumulative distribution transform and linear pattern classification, Appl. Comput. Harmon. Anal., 45 (2018), 616-641.
doi: 10.1016/j.acha.2017.02.002. |
[30] |
F. Pedregosa, et al., Scikit-Learn: Machine Learning in Python, Python package. Available from: http://jmlr.org/papers/v12/pedregosa11a.html |
[31] | |
[32] |
G. K. Rohde, et al., PyTranskit, Python package. Available from: https://github.com/rohdelab/PyTransKit. |
[33] |
H. L. Royden, Real Analysis, The Macmillan Company, New York; Collier-Macmillan Ltd., London, 1963. |
[34] |
A. H. M. Rubaiyat, K. M. Hallam, J. M. Nichols, M. N. Hutchinson, S. Li and G. K. Rohde,
Parametric signal estimation using the cumulative distribution transform, IEEE Trans. Signal Process., 68 (2020), 3312-3324.
doi: 10.1109/TSP.2020.2997181. |
[35] |
F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, Cham, 2015.
doi: 10.1007/978-3-319-20828-2. |
[36] |
M. Shifat-E-Rabbi, X. Yin, A. H. M. Rubaiyat, S. Li and S. Kolouri,
Radon cumulative distribution transform subspace modeling for image classification, J. Math. Imaging Vision, 63 (2021), 1185-1203.
doi: 10.1007/s10851-021-01052-0. |
[37] |
M. Thorpe, Introduction to Optimal Transport, 2018. Available from: https://www.math.cmu.edu/mthorpe/OTNotes. |
[38] |
M. Thorpe, S. Park, S. Kolouri, G. K. Rohde and D. Slepčev,
A transportation $L^p$ distance for signal analysis, J. Math. Imaging Vision, 59 (2017), 187-210.
doi: 10.1007/s10851-017-0726-4. |
[39] |
A. B. Tosun, O. Yergiyev, S. Kolouri, J. F. Silverman and G. K. Rohde,
Detection of malignant mesothelioma using nuclear structure of mesothelial cells in effusion cytology specimens, Cytometry Part A, 87 (2015), 326-333.
doi: 10.1002/cyto.a.22602. |
[40] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
[41] |
W. Wang, Y. Mo, J. A. Ozolek and G. K. Rohde,
Penalized Fisher discriminant analysis and its application to image-based morphometry, Pattern Recog. Lett., 32 (2011), 2128-2135.
doi: 10.1016/j.patrec.2011.08.010. |
[42] |
W. Wang, D. Slepčev, S. Basu, J. A. Ozolek and G. K. Rohde,
A linear optimal transportation framework for quantifying and visualizing variations in sets of images, Int. J. Comput. Vis., 101 (2013), 254-269.
doi: 10.1007/s11263-012-0566-z. |
[43] |
L. Zhu, Y. Yang, S. Haker and A. Tannenbaum,
An image morphing technique based on optimal mass preserving mapping, IEEE Trans. Image Process., 16 (2007), 1481-1495.
doi: 10.1109/TIP.2007.896637. |
show all references
References:
[1] |
A. Aldroubi, S. Li and G. K. Rohde, Partitioning signal classes using transport transforms for data analysis and machine learning, Sampl. Theory Signal Process. Data Anal., 19 (2021), 25pp.
doi: 10.1007/s43670-021-00009-z. |
[2] |
L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003 1–52.
doi: 10.1007/978-3-540-39189-0_1. |
[3] |
M. Arjovsky, S. Chintala and L. Bottou, Wasserstein generative adversarial networks, International Conference on Machine Learning PMLR, 2017,214–223. |
[4] |
S. Basu, S. Kolouri and G. K. Rohde,
Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry, PNAS, 111 (2014), 3448-3453.
doi: 10.1073/pnas.1319779111. |
[5] |
T. Cai, J. Cheng, N. Craig and K. Craig, Linearized optimal transport for collider events, Phys. Rev. D, 102 (2020).
doi: 10.1103/PhysRevD.102.116019. |
[6] |
T. Cai, J. Cheng, B. Schmitzer and M. Thorpe,
The Linearized Hellinger–Kantorovich Distance, SIAM J. Imaging Sci., 15 (2022), 45-83.
doi: 10.1137/21M1400080. |
[7] |
L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard,
Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal., 274 (2018), 3090-3123.
doi: 10.1016/j.jfa.2018.03.008. |
[8] |
N. Courty, R. Flamary, D. Tuia and A. Rakotomamonjy,
Optimal transport for domain adaptation, IEEE Trans. Pattern Anal. Mach. Intell., 39 (2017), 1853-1865.
doi: 10.1109/TPAMI.2016.2615921. |
[9] |
P. Embrechts and M. Hofert,
A note on generalized inverses, Math. Methods Oper. Res., 77 (2013), 423-432.
doi: 10.1007/s00186-013-0436-7. |
[10] |
B. Engquist and B. D. Froese,
Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., 12 (2014), 979-988.
doi: 10.4310/CMS.2014.v12.n5.a7. |
[11] |
B. Engquist, B. D. Froese and Y. Yang,
Optimal transport for seismic full waveform inversion, Commun. Math. Sci., 14 (2016), 2309-2330.
doi: 10.4310/CMS.2016.v14.n8.a9. |
[12] |
R. A. Fisher,
The use of multiple measurements in taxonomic problems, Ann. Eugenics, 7 (1936), 179-188.
doi: 10.1111/j.1469-1809.1936.tb02137.x. |
[13] |
W. Gangbo, W. Li, Wuchen, S. Osher and M. Puthawala, Unnormalized optimal transport, J. Comput. Phys., 399 (2019), 17pp.
doi: 10.1016/j.jcp.2019.108940. |
[14] |
S. Haker, L. Zhu, A. Tannenbaum and S. Angenent,
Optimal mass transport for registration and warping, Phys. Rev. D, 60 (2004), 225-240.
doi: 10.1023/B:VISI.0000036836.66311.97. |
[15] |
S.-W. Huang, G. K. Rohde, H.-M. Cheng and S.-F. Lin,
Discretized target size detection in electrical impedance tomography using neural network classifier, J. Nondestructive Evaluation, 39 (2020), 1-9.
doi: 10.1007/s10921-020-00723-z. |
[16] |
D. W. Kammler, A First Course in Fourier Analysis, 2nd edition, Cambridge University Press, Cambridge, 2007. |
[17] |
S. Kolouri, K. Nadjahi, U. Şimşekli, R. Badeau and G. K. Rohde, Generalized sliced Wasserstein distances, preprint, 2019, arXiv: 1902.00434. |
[18] |
S. Kolouri, S. R. Park and G. K. Rohde,
The radon cumulative distribution transform and its application to image classification, IEEE Trans. Image Process., 25 (2016), 920-934.
doi: 10.1109/TIP.2015.2509419. |
[19] |
S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev and G. K. Rohde,
Optimal mass transport: Signal processing and machine-learning applications, IEEE Signal Process. Magazine, 34 (2017), 43-59.
doi: 10.1109/MSP.2017.2695801. |
[20] |
S. Kolouri and G. K. Rohde, Transport-based single frame super resolution of very low resolution face images, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, USA, 2015.
doi: 10.1109/CVPR.2015.7299121. |
[21] |
S. Kolouri, A. B. Tosun, J. A. Ozolek and G. K. Rohde,
A continuous linear optimal transport approach for pattern analysis in image datasets, Pattern Recognition, 51 (2016), 453-462.
doi: 10.1016/j.patcog.2015.09.019. |
[22] |
S. Kolouri, Y. Zou and G. K. Rohde, Sliced Wasserstein kernels for probability distributions, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA, 2016.
doi: 10.1109/CVPR.2016.568. |
[23] |
S. Kundu, B. G. Ashinsky, M. Bouhrara, E. B. Dam and S. Demehri,
Enabling early detection of osteoarthritis from presymptomatic cartilage texture maps via transport-based learning, PNAS, 117 (2020), 24709-24719.
doi: 10.1073/pnas.1917405117. |
[24] |
S. Kundu, S. Kolouri, K. I. Erickson, A. F. Kramer, E. McAuley and G. K. Rohde,
Discovery and visualization of structural biomarkers from MRI using transport-based morphometry, NeuroImage, 167 (2018), 256-275.
doi: 10.1016/j.neuroimage.2017.11.006. |
[25] |
S. Mallat,
Group invariant scattering, Comm. Pure Appl. Math., 65 (2012), 1331-1398.
doi: 10.1002/cpa.21413. |
[26] |
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.
![]() ![]() |
[27] |
J. A. Ozolek, A. B. Tosun, W. Wang, C. Chen and S. Kolouri,
Accurate diagnosis of thyroid follicular lesions from nuclear morphology using supervised learning, Medical Image Analysis, 18 (2014), 772-780.
doi: 10.1016/j.media.2014.04.004. |
[28] |
S. R. Park, L. Cattell, J. M. Nichols, A. Watnik, T. Doster and G. K. Rohde,
De-multiplexing vortex modes in optical communications using transport-based pattern recognition, Optics Express, 26 (2018), 4004-4022.
doi: 10.1364/OE.26.004004. |
[29] |
S. R. Park, S. Kolouri, S. Kundu and G. K. Rohde,
The cumulative distribution transform and linear pattern classification, Appl. Comput. Harmon. Anal., 45 (2018), 616-641.
doi: 10.1016/j.acha.2017.02.002. |
[30] |
F. Pedregosa, et al., Scikit-Learn: Machine Learning in Python, Python package. Available from: http://jmlr.org/papers/v12/pedregosa11a.html |
[31] | |
[32] |
G. K. Rohde, et al., PyTranskit, Python package. Available from: https://github.com/rohdelab/PyTransKit. |
[33] |
H. L. Royden, Real Analysis, The Macmillan Company, New York; Collier-Macmillan Ltd., London, 1963. |
[34] |
A. H. M. Rubaiyat, K. M. Hallam, J. M. Nichols, M. N. Hutchinson, S. Li and G. K. Rohde,
Parametric signal estimation using the cumulative distribution transform, IEEE Trans. Signal Process., 68 (2020), 3312-3324.
doi: 10.1109/TSP.2020.2997181. |
[35] |
F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, Cham, 2015.
doi: 10.1007/978-3-319-20828-2. |
[36] |
M. Shifat-E-Rabbi, X. Yin, A. H. M. Rubaiyat, S. Li and S. Kolouri,
Radon cumulative distribution transform subspace modeling for image classification, J. Math. Imaging Vision, 63 (2021), 1185-1203.
doi: 10.1007/s10851-021-01052-0. |
[37] |
M. Thorpe, Introduction to Optimal Transport, 2018. Available from: https://www.math.cmu.edu/mthorpe/OTNotes. |
[38] |
M. Thorpe, S. Park, S. Kolouri, G. K. Rohde and D. Slepčev,
A transportation $L^p$ distance for signal analysis, J. Math. Imaging Vision, 59 (2017), 187-210.
doi: 10.1007/s10851-017-0726-4. |
[39] |
A. B. Tosun, O. Yergiyev, S. Kolouri, J. F. Silverman and G. K. Rohde,
Detection of malignant mesothelioma using nuclear structure of mesothelial cells in effusion cytology specimens, Cytometry Part A, 87 (2015), 326-333.
doi: 10.1002/cyto.a.22602. |
[40] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
[41] |
W. Wang, Y. Mo, J. A. Ozolek and G. K. Rohde,
Penalized Fisher discriminant analysis and its application to image-based morphometry, Pattern Recog. Lett., 32 (2011), 2128-2135.
doi: 10.1016/j.patrec.2011.08.010. |
[42] |
W. Wang, D. Slepčev, S. Basu, J. A. Ozolek and G. K. Rohde,
A linear optimal transportation framework for quantifying and visualizing variations in sets of images, Int. J. Comput. Vis., 101 (2013), 254-269.
doi: 10.1007/s11263-012-0566-z. |
[43] |
L. Zhu, Y. Yang, S. Haker and A. Tannenbaum,
An image morphing technique based on optimal mass preserving mapping, IEEE Trans. Image Process., 16 (2007), 1481-1495.
doi: 10.1109/TIP.2007.896637. |






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