Abstract

Current OCT devices provide three-dimensional (3D) in-vivo images of the human retina. The resulting very large data sets are difficult to manually assess. Automated segmentation is required to automatically process the data and produce images that are clinically useful and easy to interpret. In this paper, we present a method to segment the retinal layers in these images. Instead of using complex heuristics to define each layer, simple features are defined and machine learning classifiers are trained based on manually labeled examples. When applied to new data, these classifiers produce labels for every pixel. After regularization of the 3D labeled volume to produce a surface, this results in consistent, three-dimensionally segmented layers that match known retinal morphology. Six labels were defined, corresponding to the following layers: Vitreous, retinal nerve fiber layer (RNFL), ganglion cell layer & inner plexiform layer, inner nuclear layer & outer plexiform layer, photoreceptors & retinal pigment epithelium and choroid. For both normal and glaucomatous eyes that were imaged with a Spectralis (Heidelberg Engineering) OCT system, the five resulting interfaces were compared between automatic and manual segmentation. RMS errors for the top and bottom of the retina were between 4 and 6 μm, while the errors for intra-retinal interfaces were between 6 and 15 μm. The resulting total retinal thickness maps corresponded with known retinal morphology. RNFL thickness maps were compared to GDx (Carl Zeiss Meditec) thickness maps. Both maps were mostly consistent but local defects were better visualized in OCT-derived thickness maps.

© 2011 OSA

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2010 (5)

2009 (4)

T. Fabritius, S. Makita, M. Miura, R. Myllyla, and Y. Yasuno, “Automated segmentation of the macula by optical coherence tomography,” Opt. Express 17, 15659–15669 (2009).
[CrossRef] [PubMed]

A. Mishra, A. Wong, K. Bizheva, and D. A. Clausi, “Intra-retinal layer segmentation in optical coherence tomography images,” Opt. Express 17, 23719–23728 (2009).
[CrossRef]

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

2007 (1)

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (2)

2004 (3)

N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthamol. Vis. Sci. 45, 2606–2612 (2004).
[CrossRef]

X.-R. Huang, H. Bagga, D. S. Greenfield, and R. W. Knighton, “Variation of peripapillary retinal nerve fiber layer birefringence in normal human subjects,” Invest. Ophthamol. Vis. Sci. 45, 3073–3080 (2004).
[CrossRef]

2003 (1)

S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer-Verlag, 2003).

2001 (1)

P. Viola and M. Jones, “Rapid object detection using a boosted cascade of simple features,” IEEE CVPR 1, 511–518 (2001).

1995 (2)

V. N. Vapnik, The Nature of Statistical Learning Theory (Springer-Verlag, 1995).

C. Cortes and V. N. Vapnik, “Support-vector networks,” Mach. Learn. 20, 273–297 (1995).
[CrossRef]

1987 (1)

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes - Active Contour Models,” Int. J. Comput. Vision 1, 321–331 (1987).
[CrossRef]

1982 (1)

V. N. Vapnik, Estimation of Dependences Based on Empirical Data (Springer-Verlag, 1982).

1964 (1)

M. Aizerman, E. Braverman, and L. Rozonoer, “Theoretical foundations of the potential function method in pattern recognition learning,” Autom. Rem. Control 25, 821–837 (1964).

Abramoff, M. D.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Aizerman, M.

M. Aizerman, E. Braverman, and L. Rozonoer, “Theoretical foundations of the potential function method in pattern recognition learning,” Autom. Rem. Control 25, 821–837 (1964).

Akkin, T.

Araie, M.

Bagga, H.

X.-R. Huang, H. Bagga, D. S. Greenfield, and R. W. Knighton, “Variation of peripapillary retinal nerve fiber layer birefringence in normal human subjects,” Invest. Ophthamol. Vis. Sci. 45, 3073–3080 (2004).
[CrossRef]

Bizheva, K.

Bouma, B.

Braverman, E.

M. Aizerman, E. Braverman, and L. Rozonoer, “Theoretical foundations of the potential function method in pattern recognition learning,” Autom. Rem. Control 25, 821–837 (1964).

Burns, T. L.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Cense, B.

Chan, K.

Chan, R.

Chang, K.-W.

Y.-W. Chang, C.-J. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Lin, “Training and testing low-degree polynomial data mappings via linear SVM,” J. Mach. Learn. Res. 11, 1471–1490 (2010).

Chang, Y.-W.

Y.-W. Chang, C.-J. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Lin, “Training and testing low-degree polynomial data mappings via linear SVM,” J. Mach. Learn. Res. 11, 1471–1490 (2010).

Chen, T.

Chen, T. C.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthamol. Vis. Sci. 45, 2606–2612 (2004).
[CrossRef]

Chiu, S. J.

Choi, S. S.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Clausi, D. A.

Cortes, C.

C. Cortes and V. N. Vapnik, “Support-vector networks,” Mach. Learn. 20, 273–297 (1995).
[CrossRef]

Crabb, D. P.

de Boer, J.

de Boer, J. F.

E. C. Lee, J. F. de Boer, M. Mujat, H. Lim, and S. H. Yun, “In vivo optical frequency domain imaging of human retina and choroid,” Opt. Express 14, 4403–4411 (2006).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthamol. Vis. Sci. 45, 2606–2612 (2004).
[CrossRef]

Drexler, W.

Duker, J. S.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Fabritius, T.

Farsiu, S.

Friberg, T. R.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Fujimoto, J. G.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Fukuma, Y.

Fuller, A. R.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Gabriele, M. L.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Garvin, M. K.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Garway-Heath, D. F.

Greenfield, D. S.

X.-R. Huang, H. Bagga, D. S. Greenfield, and R. W. Knighton, “Variation of peripapillary retinal nerve fiber layer birefringence in normal human subjects,” Invest. Ophthamol. Vis. Sci. 45, 3073–3080 (2004).
[CrossRef]

Gregori, G.

Hamann, B.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Hangai, M.

Hermann, B.

Ho, T.

Hofer, B.

Hood, D. C.

Hsieh, C.-J.

Y.-W. Chang, C.-J. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Lin, “Training and testing low-degree polynomial data mappings via linear SVM,” J. Mach. Learn. Res. 11, 1471–1490 (2010).

Huang, X.

Huang, X.-R.

X.-R. Huang, H. Bagga, D. S. Greenfield, and R. W. Knighton, “Variation of peripapillary retinal nerve fiber layer birefringence in normal human subjects,” Invest. Ophthamol. Vis. Sci. 45, 3073–3080 (2004).
[CrossRef]

Ishikawa, H.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Izatt, J. A.

Jiao, S.

Jones, M.

P. Viola and M. Jones, “Rapid object detection using a boosted cascade of simple features,” IEEE CVPR 1, 511–518 (2001).

Joo, C.

Kagemann, L.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Kajic, V.

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes - Active Contour Models,” Int. J. Comput. Vision 1, 321–331 (1987).
[CrossRef]

Kim, J.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Knighton, R.

Knighton, R. W.

X.-R. Huang, H. Bagga, D. S. Greenfield, and R. W. Knighton, “Variation of peripapillary retinal nerve fiber layer birefringence in normal human subjects,” Invest. Ophthamol. Vis. Sci. 45, 3073–3080 (2004).
[CrossRef]

Lee, E. C.

Li, X. T.

Lim, H.

Lin, C.-J.

Y.-W. Chang, C.-J. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Lin, “Training and testing low-degree polynomial data mappings via linear SVM,” J. Mach. Learn. Res. 11, 1471–1490 (2010).

Makita, S.

Marshall, D.

Mishra, A.

Miura, M.

Mujat, M.

Myllyla, R.

Nassif, N.

Nicholas, P.

Osher, S.

S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer-Verlag, 2003).

Paragios, N.

S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer-Verlag, 2003).

Park, B.

Park, B. H.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthamol. Vis. Sci. 45, 2606–2612 (2004).
[CrossRef]

Pierce, M.

Pierce, M. C.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthamol. Vis. Sci. 45, 2606–2612 (2004).
[CrossRef]

Povazay, B.

Puliafito, C.

Raza, A. S.

Reisman, C. A.

Ringgaard, M.

Y.-W. Chang, C.-J. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Lin, “Training and testing low-degree polynomial data mappings via linear SVM,” J. Mach. Learn. Res. 11, 1471–1490 (2010).

Rosin, P. L.

Rozonoer, L.

M. Aizerman, E. Braverman, and L. Rozonoer, “Theoretical foundations of the potential function method in pattern recognition learning,” Autom. Rem. Control 25, 821–837 (1964).

Russell, S. R.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Schlottmann, P. G.

Schuman, J. S.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Sonka, M.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Sung, K. R.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Tearney, G.

Terzopoulos, D.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes - Active Contour Models,” Int. J. Comput. Vision 1, 321–331 (1987).
[CrossRef]

Tomidokoro, A.

Toth, C. A.

Townsend, K. A.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Vapnik, V. N.

V. N. Vapnik, The Nature of Statistical Learning Theory (Springer-Verlag, 1995).

C. Cortes and V. N. Vapnik, “Support-vector networks,” Mach. Learn. 20, 273–297 (1995).
[CrossRef]

V. N. Vapnik, Estimation of Dependences Based on Empirical Data (Springer-Verlag, 1982).

Viola, P.

P. Viola and M. Jones, “Rapid object detection using a boosted cascade of simple features,” IEEE CVPR 1, 511–518 (2001).

Wang, Z.

Werner, J. S.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Wiley, D. F.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes - Active Contour Models,” Int. J. Comput. Vision 1, 321–331 (1987).
[CrossRef]

Wollstein, G.

H. Ishikawa, J. Kim, T. R. Friberg, G. Wollstein, L. Kagemann, M. L. Gabriele, K. A. Townsend, K. R. Sung, J. S. Duker, J. G. Fujimoto, and J. S. Schuman, “Three-dimensional optical coherence tomography (3d-oct) image enhancement with segmentation-free contour modeling c-mode,” Invest. Ophthamol. Vis. Sci. 50, 1344–1349 (2009).
[CrossRef]

Wong, A.

Wu, X.

M. K. Garvin, M. D. Abramoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, “Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images,” IEEE Trans. Med. Imaging 28, 1436–1447 (2009).
[CrossRef] [PubMed]

Yang, Q.

Yasuno, Y.

Yoshimura, N.

Yun, S.

Yun, S. H.

Zawadzki, R. J.

R. J. Zawadzki, A. R. Fuller, D. F. Wiley, B. Hamann, S. S. Choi, and J. S. Werner, “Adaptation of a support vector machine algorithm for segmentation and visualization of retinal structures in volumetric optical coherence tomography data sets,” J. Biomed. Opt. 12, 041206 (2007).
[CrossRef] [PubMed]

Zhu, H.

Autom. Rem. Control (1)

M. Aizerman, E. Braverman, and L. Rozonoer, “Theoretical foundations of the potential function method in pattern recognition learning,” Autom. Rem. Control 25, 821–837 (1964).

IEEE CVPR (1)

P. Viola and M. Jones, “Rapid object detection using a boosted cascade of simple features,” IEEE CVPR 1, 511–518 (2001).

IEEE Trans. Med. Imaging (1)

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Figures (9)

Fig. 1
Fig. 1

Overview of the feature calculation and classification process. Every A-line is processed to produce averages and gradients at different scales, thereby transforming every pixel into a feature vector. The classifier calculates the label based on the feature vector, resulting in a labeled A-line.

Fig. 2
Fig. 2

Features calculated for each pixel on an A-line, at different scales. The A-line is displayed in grayscale in the background of each graph. The features are defined by averages (red lines) and gradients (green lines).

Fig. 3
Fig. 3

Graphical representation of the features for each pixel in a B-scan. The feature vector for each pixel consists of one matching pixel from each of the processed B-scans.

Fig. 4
Fig. 4

(a) SLO image of the retina indicating the position of a B-scan (red line) and the total scan area (blue square). (b) OCT B-scan acquired along the red line of Fig. 4(a). (c) Reconstructed en-face image based on 193 B-scans.

Fig. 5
Fig. 5

Manually (left) and automatically (right) labeled B-scan of a normal eye. This eye was excluded from the training data when obtaining the automatic segmentation.

Fig. 6
Fig. 6

Manually (left) and automatically (right) labeled B-scan of a glaucomatous eye. No glaucomatous eye was included in the training data.

Fig. 7
Fig. 7

Thickness maps produced after segmentation. The top row shows the results for a normal eye; the second and third row show the results for glaucomatous eyes. The first column shows the en-face reconstruction, the second column shows the full retinal thickness, the third column shows the RNFL thickness and the last column shows the thickness as assessed by a GDx (dark, blue colors correspond to a thin RNFL and warm, red colors correspond to a thick RNFL). Edges of local defects are indicated by red arrows.

Fig. 8
Fig. 8

Integrated OCT data just above the RPE, showing (shadows of) retinal vessels, for a normal (left) and a glaucomatous (right) eye.

Fig. 9
Fig. 9

Integrated OCT data just below the RPE, showing choroidal vasculature (and remnants of retinal vessels), for a normal (left) and a glaucomatous (right) eye.

Tables (2)

Tables Icon

Table 1 Localization Errors of Automatically Segmented Interfaces Compared to Manual Segmentations*

Tables Icon

Table 2 Processing Times (Standard Deviation) of the Classification (Including Calculating the Features) and Regularization Steps

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

g 0 ( z ) = f ( z ) .
g d ( z ) = 1 2 d z = 1 2 d 1 1 + 2 d 1 f ( z + z ) .
h 0 ( z ) = f ( z + 1 ) f ( z ) = g 0 ( z + 1 ) g 0 ( z )
h d ( z ) = g d ( z + 2 d 1 ) g d ( z 2 d 1 ) .
x ( z ) = [ g 0 ( z ) , h 0 ( z ) , g 1 ( z ) , h 1 ( z ) , , g d ( z ) , h d ( z ) ] .
s ( x ) = w , x + b
s ( x ) = i = 1 N α i y i x i , x + b ,
s ( x ) = i = 1 N α i y i K ( x i , x ) + b .
K ( x i , x j ) = ϕ ( x i ) , ϕ ( x j ) .
s ( x ) = i = 1 N α i y i K ( x i , x ) + b = i = 1 N α i y i ϕ ( x i ) , ϕ ( x ) + b = w , ϕ ( x ) + b ,
ϕ ( x ) = ( 1 , 2 x 1 , , 2 x n , x 1 x 2 , , x n 1 x n , x 1 2 , , x n 2 )
ϕ t = F | ϕ | .
ϕ t = α κ | ϕ | β L ( x ) | ϕ | ,

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