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Spherical CNNs
2018/5/26 ICLR @PFN

Kohta Ishikawa [@_kohta]
!1
• S2 CNN 

• CNN building block Cross-correlation (Convolution)


• Fourier
!2
Fourier Correlation
• Fourier
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
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Filter Signal
Translation
!3
Fourier Correlation
• Fourier
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
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!4
S2-Correlation?
• 

• Translation?

•
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
<latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">AAAEF3icpVPNi9NAFJ9N/FjrV1ePXgaL2xRpSRZBEYRFLx5XsbsLnW6ZTCbtkMkHMxNpCfkvvPivePGgiFe9+d84maRgmy578EHg8d7v471Hxs84k8p1/+xZ9rXrN27u3+rcvnP33v3uwYNTmeaC0DFJeSrOfSwpZwkdK6Y4Pc8ExbHP6Zkfvan6Zx+pkCxNPqhVRqcxnicsZAQrXZodWEMnhEgqLCDKJBs4ywE8fPkKIpaoGYqxWvh+8b40TWfZh0OoAaHOBhDBQBcQ6hzugNe97eoCq6JSKp1IC9CLgkX9taohRZeQQs1oCMt+jbzE2UjscBpqq2XZRq/BtUOrX21hfJ0IPoVay9gbZ+cIZWwH5f8GCChXeG12ldOm0bDib1/rymk3NC4KFOD5nIp6mJbYrNtzR64J2E68JumBJk5m3d8oSEke00QRjqWceG6mpgUWihFOyw7KJc0wifCcTnSa4JjKaWH+6xI+0ZUAhqnQX6Kgqf7LKHAs5Sr2NbLaSG73quKu3iRX4YtpwZIsVzQhtVGYc6hSWD0SGDBBieIrnWAimJ4VkgUWmCj9lDr6CN72yu3k9GjkuSPv3bPe8evmHPvgEXgMHOCB5+AYvAUnYAyI9cn6Yn2zvtuf7a/2D/tnDbX2Gs5DsBH2r79N6EoP</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">AAAEF3icpVPNi9NAFJ9N/FjrV1ePXgaL2xRpSRZBEYRFLx5XsbsLnW6ZTCbtkMkHMxNpCfkvvPivePGgiFe9+d84maRgmy578EHg8d7v471Hxs84k8p1/+xZ9rXrN27u3+rcvnP33v3uwYNTmeaC0DFJeSrOfSwpZwkdK6Y4Pc8ExbHP6Zkfvan6Zx+pkCxNPqhVRqcxnicsZAQrXZodWEMnhEgqLCDKJBs4ywE8fPkKIpaoGYqxWvh+8b40TWfZh0OoAaHOBhDBQBcQ6hzugNe97eoCq6JSKp1IC9CLgkX9taohRZeQQs1oCMt+jbzE2UjscBpqq2XZRq/BtUOrX21hfJ0IPoVay9gbZ+cIZWwH5f8GCChXeG12ldOm0bDib1/rymk3NC4KFOD5nIp6mJbYrNtzR64J2E68JumBJk5m3d8oSEke00QRjqWceG6mpgUWihFOyw7KJc0wifCcTnSa4JjKaWH+6xI+0ZUAhqnQX6Kgqf7LKHAs5Sr2NbLaSG73quKu3iRX4YtpwZIsVzQhtVGYc6hSWD0SGDBBieIrnWAimJ4VkgUWmCj9lDr6CN72yu3k9GjkuSPv3bPe8evmHPvgEXgMHOCB5+AYvAUnYAyI9cn6Yn2zvtuf7a/2D/tnDbX2Gs5DsBH2r79N6EoP</latexit>
?
!5
SO(3) S2-Translation
• = R , = 3 = SO(3)
!6
S2
• 

• Haar
Z
S2
f(x)dx =
1
4⇡
Z 2⇡
0
d↵
Z ⇡
0
sin d f(↵, )
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!7
Z
S2
f(Rx)dx =
Z
S2
f(x)dx, R 2 SO(3)
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parameterization
ZYZ-Euler angle SO(3) parameterization
S2 parameterization α β
S2-Correlation
• Correlation dx S2 

•
!8
(f ? )(R) :=
Z
S2
(R 1
x)f(x)dx
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(f ? )(R) :=
Z
S2
X
k
k(R 1
x)f(x)dx
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CNN
Fourier
• Fourier =
!9
Fourier
Fourier Fourier
ga : x 7! x + a<latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit>
gR : x 7! Rx<latexit sha1_base64="h1Wjuu1mYc0aeGk8KJnN8Y7JchU=">AAAB+3icbVBNSwMxEJ31s9avtR69BIvgqeyKoHgqevFYi/2AdlmyadqGJtklyUrL0r/ixYMiXv0j3vw3pu0etPXBwOO9GWbmRQln2njet7O2vrG5tV3YKe7u7R8cukelpo5TRWiDxDxW7QhrypmkDcMMp+1EUSwiTlvR6G7mt56o0iyWj2aS0EDggWR9RrCxUuiWBmEd3aAx6gqcaBOj+jh0y17FmwOtEj8nZchRC92vbi8mqaDSEI617vheYoIMK8MIp9NiN9U0wWSEB7RjqcSC6iCb3z5FZ1bpoX6sbEmD5urviQwLrScisp0Cm6Fe9mbif14nNf3rIGMySQ2VZLGon3Jkf5wFgXpMUWL4xBJMFLO3IjLEChNj4yraEPzll1dJ86LiexX/4bJcvc3jKMAJnMI5+HAFVbiHGjSAwBie4RXenKnz4rw7H4vWNSefOYY/cD5/AGtrk2M=</latexit><latexit sha1_base64="h1Wjuu1mYc0aeGk8KJnN8Y7JchU=">AAAB+3icbVBNSwMxEJ31s9avtR69BIvgqeyKoHgqevFYi/2AdlmyadqGJtklyUrL0r/ixYMiXv0j3vw3pu0etPXBwOO9GWbmRQln2njet7O2vrG5tV3YKe7u7R8cukelpo5TRWiDxDxW7QhrypmkDcMMp+1EUSwiTlvR6G7mt56o0iyWj2aS0EDggWR9RrCxUuiWBmEd3aAx6gqcaBOj+jh0y17FmwOtEj8nZchRC92vbi8mqaDSEI617vheYoIMK8MIp9NiN9U0wWSEB7RjqcSC6iCb3z5FZ1bpoX6sbEmD5urviQwLrScisp0Cm6Fe9mbif14nNf3rIGMySQ2VZLGon3Jkf5wFgXpMUWL4xBJMFLO3IjLEChNj4yraEPzll1dJ86LiexX/4bJcvc3jKMAJnMI5+HAFVbiHGjSAwBie4RXenKnz4rw7H4vWNSefOYY/cD5/AGtrk2M=</latexit><latexit sha1_base64="h1Wjuu1mYc0aeGk8KJnN8Y7JchU=">AAAB+3icbVBNSwMxEJ31s9avtR69BIvgqeyKoHgqevFYi/2AdlmyadqGJtklyUrL0r/ixYMiXv0j3vw3pu0etPXBwOO9GWbmRQln2njet7O2vrG5tV3YKe7u7R8cukelpo5TRWiDxDxW7QhrypmkDcMMp+1EUSwiTlvR6G7mt56o0iyWj2aS0EDggWR9RrCxUuiWBmEd3aAx6gqcaBOj+jh0y17FmwOtEj8nZchRC92vbi8mqaDSEI617vheYoIMK8MIp9NiN9U0wWSEB7RjqcSC6iCb3z5FZ1bpoX6sbEmD5urviQwLrScisp0Cm6Fe9mbif14nNf3rIGMySQ2VZLGon3Jkf5wFgXpMUWL4xBJMFLO3IjLEChNj4yraEPzll1dJ86LiexX/4bJcvc3jKMAJnMI5+HAFVbiHGjSAwBie4RXenKnz4rw7H4vWNSefOYY/cD5/AGtrk2M=</latexit><latexit sha1_base64="h1Wjuu1mYc0aeGk8KJnN8Y7JchU=">AAAB+3icbVBNSwMxEJ31s9avtR69BIvgqeyKoHgqevFYi/2AdlmyadqGJtklyUrL0r/ixYMiXv0j3vw3pu0etPXBwOO9GWbmRQln2njet7O2vrG5tV3YKe7u7R8cukelpo5TRWiDxDxW7QhrypmkDcMMp+1EUSwiTlvR6G7mt56o0iyWj2aS0EDggWR9RrCxUuiWBmEd3aAx6gqcaBOj+jh0y17FmwOtEj8nZchRC92vbi8mqaDSEI617vheYoIMK8MIp9NiN9U0wWSEB7RjqcSC6iCb3z5FZ1bpoX6sbEmD5urviQwLrScisp0Cm6Fe9mbif14nNf3rIGMySQ2VZLGon3Jkf5wFgXpMUWL4xBJMFLO3IjLEChNj4yraEPzll1dJ86LiexX/4bJcvc3jKMAJnMI5+HAFVbiHGjSAwBie4RXenKnz4rw7H4vWNSefOYY/cD5/AGtrk2M=</latexit>
Ul
mn(g)<latexit sha1_base64="woHgSUEeoUwd+EkfFXpVHDVuWKs=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LXjxWMG0hjWWz3bRLN7thdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHbS0zRahPJJeqG2FNORPUN8xw2k0VxUnEaSca3878zhNVmknxYCYpDRM8FCxmBBsrBf4j7+eJmNaH5/1qzW24c6BV4hWkBgVa/epXbyBJllBhCMdaB56bmjDHyjDC6bTSyzRNMRnjIQ0sFTihOsznJ0/RmVUGKJbKljBorv6eyHGi9SSJbGeCzUgvezPxPy/ITHwd5kykmaGCLBbFGUdGotn/aMAUJYZPLMFEMXsrIiOsMDE2pYoNwVt+eZW0Lxqe2/DuL2vNmyKOMpzAKdTBgytowh20wAcCEp7hFd4c47w4787HorXkFDPH8AfO5w/VKpDx</latexit><latexit sha1_base64="woHgSUEeoUwd+EkfFXpVHDVuWKs=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LXjxWMG0hjWWz3bRLN7thdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHbS0zRahPJJeqG2FNORPUN8xw2k0VxUnEaSca3878zhNVmknxYCYpDRM8FCxmBBsrBf4j7+eJmNaH5/1qzW24c6BV4hWkBgVa/epXbyBJllBhCMdaB56bmjDHyjDC6bTSyzRNMRnjIQ0sFTihOsznJ0/RmVUGKJbKljBorv6eyHGi9SSJbGeCzUgvezPxPy/ITHwd5kykmaGCLBbFGUdGotn/aMAUJYZPLMFEMXsrIiOsMDE2pYoNwVt+eZW0Lxqe2/DuL2vNmyKOMpzAKdTBgytowh20wAcCEp7hFd4c47w4787HorXkFDPH8AfO5w/VKpDx</latexit><latexit sha1_base64="woHgSUEeoUwd+EkfFXpVHDVuWKs=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LXjxWMG0hjWWz3bRLN7thdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHbS0zRahPJJeqG2FNORPUN8xw2k0VxUnEaSca3878zhNVmknxYCYpDRM8FCxmBBsrBf4j7+eJmNaH5/1qzW24c6BV4hWkBgVa/epXbyBJllBhCMdaB56bmjDHyjDC6bTSyzRNMRnjIQ0sFTihOsznJ0/RmVUGKJbKljBorv6eyHGi9SSJbGeCzUgvezPxPy/ITHwd5kykmaGCLBbFGUdGotn/aMAUJYZPLMFEMXsrIiOsMDE2pYoNwVt+eZW0Lxqe2/DuL2vNmyKOMpzAKdTBgytowh20wAcCEp7hFd4c47w4787HorXkFDPH8AfO5w/VKpDx</latexit><latexit sha1_base64="woHgSUEeoUwd+EkfFXpVHDVuWKs=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LXjxWMG0hjWWz3bRLN7thdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHbS0zRahPJJeqG2FNORPUN8xw2k0VxUnEaSca3878zhNVmknxYCYpDRM8FCxmBBsrBf4j7+eJmNaH5/1qzW24c6BV4hWkBgVa/epXbyBJllBhCMdaB56bmjDHyjDC6bTSyzRNMRnjIQ0sFTihOsznJ0/RmVUGKJbKljBorv6eyHGi9SSJbGeCzUgvezPxPy/ITHwd5kykmaGCLBbFGUdGotn/aMAUJYZPLMFEMXsrIiOsMDE2pYoNwVt+eZW0Lxqe2/DuL2vNmyKOMpzAKdTBgytowh20wAcCEp7hFd4c47w4787HorXkFDPH8AfO5w/VKpDx</latexit>
eika
<latexit sha1_base64="CWIz0scLdnN3TIkTSf25Gm5QM84=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaWDbbSbtkswm7G6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvSAXXxnW/ndLa+sbmVnm7srO7t39QPTxq6yRTDFssEYnqBlSj4BJbhhuB3VQhjQOBnSC6nfmdJ1SaJ/LBTFL0YzqSPOSMGit18DHnEZ0OqjW37s5BVolXkBoUaA6qX/1hwrIYpWGCat3z3NT4OVWGM4HTSj/TmFIW0RH2LJU0Ru3n83On5MwqQxImypY0ZK7+nshprPUkDmxnTM1YL3sz8T+vl5nw2s+5TDODki0WhZkgJiGz38mQK2RGTCyhTHF7K2FjqigzNqGKDcFbfnmVtC/qnlv37i9rjZsijjKcwCmcgwdX0IA7aEILGETwDK/w5qTOi/PufCxaS04xcwx/4Hz+AIpsj7A=</latexit><latexit sha1_base64="CWIz0scLdnN3TIkTSf25Gm5QM84=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaWDbbSbtkswm7G6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvSAXXxnW/ndLa+sbmVnm7srO7t39QPTxq6yRTDFssEYnqBlSj4BJbhhuB3VQhjQOBnSC6nfmdJ1SaJ/LBTFL0YzqSPOSMGit18DHnEZ0OqjW37s5BVolXkBoUaA6qX/1hwrIYpWGCat3z3NT4OVWGM4HTSj/TmFIW0RH2LJU0Ru3n83On5MwqQxImypY0ZK7+nshprPUkDmxnTM1YL3sz8T+vl5nw2s+5TDODki0WhZkgJiGz38mQK2RGTCyhTHF7K2FjqigzNqGKDcFbfnmVtC/qnlv37i9rjZsijjKcwCmcgwdX0IA7aEILGETwDK/w5qTOi/PufCxaS04xcwx/4Hz+AIpsj7A=</latexit><latexit sha1_base64="CWIz0scLdnN3TIkTSf25Gm5QM84=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaWDbbSbtkswm7G6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvSAXXxnW/ndLa+sbmVnm7srO7t39QPTxq6yRTDFssEYnqBlSj4BJbhhuB3VQhjQOBnSC6nfmdJ1SaJ/LBTFL0YzqSPOSMGit18DHnEZ0OqjW37s5BVolXkBoUaA6qX/1hwrIYpWGCat3z3NT4OVWGM4HTSj/TmFIW0RH2LJU0Ru3n83On5MwqQxImypY0ZK7+nshprPUkDmxnTM1YL3sz8T+vl5nw2s+5TDODki0WhZkgJiGz38mQK2RGTCyhTHF7K2FjqigzNqGKDcFbfnmVtC/qnlv37i9rjZsijjKcwCmcgwdX0IA7aEILGETwDK/w5qTOi/PufCxaS04xcwx/4Hz+AIpsj7A=</latexit><latexit sha1_base64="CWIz0scLdnN3TIkTSf25Gm5QM84=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaWDbbSbtkswm7G6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvSAXXxnW/ndLa+sbmVnm7srO7t39QPTxq6yRTDFssEYnqBlSj4BJbhhuB3VQhjQOBnSC6nfmdJ1SaJ/LBTFL0YzqSPOSMGit18DHnEZ0OqjW37s5BVolXkBoUaA6qX/1hwrIYpWGCat3z3NT4OVWGM4HTSj/TmFIW0RH2LJU0Ru3n83On5MwqQxImypY0ZK7+nshprPUkDmxnTM1YL3sz8T+vl5nw2s+5TDODki0WhZkgJiGz38mQK2RGTCyhTHF7K2FjqigzNqGKDcFbfnmVtC/qnlv37i9rjZsijjKcwCmcgwdX0IA7aEILGETwDK/w5qTOi/PufCxaS04xcwx/4Hz+AIpsj7A=</latexit>
ˆf(k) =
1
2⇡
Z
R
f(x)e ikx
dx
f(x) =
Z
R
ˆf(k)eikx
dk
<latexit sha1_base64="U8J5Wfcb/kA2DWD4Bxikmj2UaHQ=">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</latexit><latexit sha1_base64="U8J5Wfcb/kA2DWD4Bxikmj2UaHQ=">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</latexit><latexit sha1_base64="U8J5Wfcb/kA2DWD4Bxikmj2UaHQ=">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</latexit><latexit sha1_base64="U8J5Wfcb/kA2DWD4Bxikmj2UaHQ=">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</latexit>
ˆfl
mn =
Z
G
f(g)Ul
mn(g) dg
f(g) =
X
l,m,n
dl
ˆfl
mnUl
mn(g)
<latexit sha1_base64="dcdTyJ8XNME+w6KZIVnfMkZXfoU=">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</latexit><latexit sha1_base64="dcdTyJ8XNME+w6KZIVnfMkZXfoU=">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</latexit><latexit sha1_base64="dcdTyJ8XNME+w6KZIVnfMkZXfoU=">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</latexit><latexit sha1_base64="dcdTyJ8XNME+w6KZIVnfMkZXfoU=">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</latexit>
Fourier Correlation
• S2 Fourier 

•
!10
(f ? )(R) =
Z
S2
(R 1
x)f(x) dx
=
Z
S2
X
l,m
ˆl
mY l
m(R 1
x)
X
l0,m0
ˆfl0
m0 Y l0
m0 (x) dx
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Y l
m(R 1
x) =
X
n
Dl
mn(R 1
)Y l
n(x)
<latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit>
SO(3) Wigner-D function
S2 

→ S2 (?)

S2 SO(3) : S2 = SO(3)/SO(2)
Fourier Correlation
• S2 Fourier
(f ? )(R) =
Z
S2
(R 1
x)f(x) dx
=
Z
S2
X
l,m
ˆl
mY l
m(R 1
x)
X
l0,m0
ˆfl0
m0 Y l0
m0 (x) dx
=
X
l,m
X
l0,m0
X
n
ˆl
m
ˆfl0
m0 Dl
mn(R 1
)
Z
S2
Y l
n(x)Y l0
m0 (x) dx
<latexit sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit><latexit sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit><latexit sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit><latexit sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit>
=
SO(3) Fourier
ll0 nm0<latexit sha1_base64="VjJmjm+yNwOlDddWGs0qp4b/MWY=">AAACA3icbVDLSgMxFM3UV62vUXe6CRapG8uMCLosunFZwT6gHYZMJtOGJpkhyQhlGHDjr7hxoYhbf8Kdf2PajqCtBy6cnHMvufcECaNKO86XVVpaXlldK69XNja3tnfs3b22ilOJSQvHLJbdACnCqCAtTTUj3UQSxANGOsHoeuJ37olUNBZ3epwQj6OBoBHFSBvJtw/6IWEa+RljtRz+PE4Fr+W+XXXqzhRwkbgFqYICTd/+7IcxTjkRGjOkVM91Eu1lSGqKGckr/VSRBOERGpCeoQJxorxsekMOj40SwiiWpoSGU/X3RIa4UmMemE6O9FDNexPxP6+X6ujSy6hIUk0Enn0UpQzqGE4CgSGVBGs2NgRhSc2uEA+RRFib2ComBHf+5EXSPqu7Tt29Pa82roo4yuAQHIET4IIL0AA3oAlaAIMH8ARewKv1aD1bb9b7rLVkFTP74A+sj2+Wqpd3</latexit><latexit sha1_base64="VjJmjm+yNwOlDddWGs0qp4b/MWY=">AAACA3icbVDLSgMxFM3UV62vUXe6CRapG8uMCLosunFZwT6gHYZMJtOGJpkhyQhlGHDjr7hxoYhbf8Kdf2PajqCtBy6cnHMvufcECaNKO86XVVpaXlldK69XNja3tnfs3b22ilOJSQvHLJbdACnCqCAtTTUj3UQSxANGOsHoeuJ37olUNBZ3epwQj6OBoBHFSBvJtw/6IWEa+RljtRz+PE4Fr+W+XXXqzhRwkbgFqYICTd/+7IcxTjkRGjOkVM91Eu1lSGqKGckr/VSRBOERGpCeoQJxorxsekMOj40SwiiWpoSGU/X3RIa4UmMemE6O9FDNexPxP6+X6ujSy6hIUk0Enn0UpQzqGE4CgSGVBGs2NgRhSc2uEA+RRFib2ComBHf+5EXSPqu7Tt29Pa82roo4yuAQHIET4IIL0AA3oAlaAIMH8ARewKv1aD1bb9b7rLVkFTP74A+sj2+Wqpd3</latexit><latexit sha1_base64="VjJmjm+yNwOlDddWGs0qp4b/MWY=">AAACA3icbVDLSgMxFM3UV62vUXe6CRapG8uMCLosunFZwT6gHYZMJtOGJpkhyQhlGHDjr7hxoYhbf8Kdf2PajqCtBy6cnHMvufcECaNKO86XVVpaXlldK69XNja3tnfs3b22ilOJSQvHLJbdACnCqCAtTTUj3UQSxANGOsHoeuJ37olUNBZ3epwQj6OBoBHFSBvJtw/6IWEa+RljtRz+PE4Fr+W+XXXqzhRwkbgFqYICTd/+7IcxTjkRGjOkVM91Eu1lSGqKGckr/VSRBOERGpCeoQJxorxsekMOj40SwiiWpoSGU/X3RIa4UmMemE6O9FDNexPxP6+X6ujSy6hIUk0Enn0UpQzqGE4CgSGVBGs2NgRhSc2uEA+RRFib2ComBHf+5EXSPqu7Tt29Pa82roo4yuAQHIET4IIL0AA3oAlaAIMH8ARewKv1aD1bb9b7rLVkFTP74A+sj2+Wqpd3</latexit><latexit sha1_base64="VjJmjm+yNwOlDddWGs0qp4b/MWY=">AAACA3icbVDLSgMxFM3UV62vUXe6CRapG8uMCLosunFZwT6gHYZMJtOGJpkhyQhlGHDjr7hxoYhbf8Kdf2PajqCtBy6cnHMvufcECaNKO86XVVpaXlldK69XNja3tnfs3b22ilOJSQvHLJbdACnCqCAtTTUj3UQSxANGOsHoeuJ37olUNBZ3epwQj6OBoBHFSBvJtw/6IWEa+RljtRz+PE4Fr+W+XXXqzhRwkbgFqYICTd/+7IcxTjkRGjOkVM91Eu1lSGqKGckr/VSRBOERGpCeoQJxorxsekMOj40SwiiWpoSGU/X3RIa4UmMemE6O9FDNexPxP6+X6ujSy6hIUk0Enn0UpQzqGE4CgSGVBGs2NgRhSc2uEA+RRFib2ComBHf+5EXSPqu7Tt29Pa82roo4yuAQHIET4IIL0AA3oAlaAIMH8ARewKv1aD1bb9b7rLVkFTP74A+sj2+Wqpd3</latexit>
X
l,m,n
ˆl†
m
ˆfl
nDl
mn(R)
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Fourier Correlation
• Fourier Correlation 

• Fourier G-FFT Correlation
(f ? )(R) = F 1
SO(3) FS2 (f)FS2 ( )†
(R)
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• S2-Correlation Feature map SO(3) 

• 2 SO(3)-Correlation 

• Correlation R2 Feature map R2
R2-Correlation CNN 

• SO(3) Translation SO(3) SO(3)
Fourier
!13
(f ? )(R) :=
Z
SO(3)
(R 1
Q)f(Q)dQ
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SO(3) Translation
SO(3) Haar measure
S2-Conv
SO(3)-Conv
SO(3)-Conv
…
• Stereographic projection of rotated MNIST

• 

• S2 SO(3)-Translation NR
R
!14
S2-Conv → ReLU → SO(3)-Conv → ReLU → FC → Softmax
Conv → ReLU → Conv → ReLU → FC → Softmax
• 3D shape recognition

• 3DCAD Ray casting 

• S2-CNN
!15
S2-Conv → BN → ReLU → SO(3)-Conv → BN → ReLU
→ SO(3)-Conv → BN → ReLU → MaxPool → BN → Linear
• CNN 

• Correlation Convolution Fourier


•
Fourier 

• 3DCAD
S2-CNN 

• 3
SE(3)
!16

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Spherical CNNs paper reading

  • 1. Spherical CNNs 2018/5/26 ICLR @PFN Kohta Ishikawa [@_kohta] !1
  • 2. • S2 CNN • CNN building block Cross-correlation (Convolution) • Fourier !2
  • 3. Fourier Correlation • Fourier (f ? )(x) := Z R (x0 x)f(x0 ) dx0 = Z R dx0 Z R ˆ(k0 )eik0 (x0 x) dk0 Z R ˆf(k)eikx0 dk = Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) Z R dx0 ei(k+k0 )x0 = (2⇡) Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) (k + k0 ) = (2⇡) Z R dk ˆ( k) ˆf(k)eikx = (2⇡) Z R dk ˆ† (k) ˆf(k)eikx <latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit> Filter Signal Translation !3
  • 4. Fourier Correlation • Fourier (f ? )(x) := Z R (x0 x)f(x0 ) dx0 = Z R dx0 Z R ˆ(k0 )eik0 (x0 x) dk0 Z R ˆf(k)eikx0 dk = Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) Z R dx0 ei(k+k0 )x0 = (2⇡) Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) (k + k0 ) = (2⇡) Z R dk ˆ( k) ˆf(k)eikx = (2⇡) Z R dk ˆ† (k) ˆf(k)eikx <latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit> !4
  • 5. S2-Correlation? • • Translation? • (f ? )(x) := Z R (x0 x)f(x0 ) dx0 = Z R dx0 Z R ˆ(k0 )eik0 (x0 x) dk0 Z R ˆf(k)eikx0 dk = Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) Z R dx0 ei(k+k0 )x0 = (2⇡) Z R dk0 ˆ(k0 )e ik0 x Z R dk ˆf(k) (k + k0 ) = (2⇡) Z R dk ˆ( k) ˆf(k)eikx = (2⇡) Z R dk ˆ† (k) ˆf(k)eikx <latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">AAAEF3icpVPNi9NAFJ9N/FjrV1ePXgaL2xRpSRZBEYRFLx5XsbsLnW6ZTCbtkMkHMxNpCfkvvPivePGgiFe9+d84maRgmy578EHg8d7v471Hxs84k8p1/+xZ9rXrN27u3+rcvnP33v3uwYNTmeaC0DFJeSrOfSwpZwkdK6Y4Pc8ExbHP6Zkfvan6Zx+pkCxNPqhVRqcxnicsZAQrXZodWEMnhEgqLCDKJBs4ywE8fPkKIpaoGYqxWvh+8b40TWfZh0OoAaHOBhDBQBcQ6hzugNe97eoCq6JSKp1IC9CLgkX9taohRZeQQs1oCMt+jbzE2UjscBpqq2XZRq/BtUOrX21hfJ0IPoVay9gbZ+cIZWwH5f8GCChXeG12ldOm0bDib1/rymk3NC4KFOD5nIp6mJbYrNtzR64J2E68JumBJk5m3d8oSEke00QRjqWceG6mpgUWihFOyw7KJc0wifCcTnSa4JjKaWH+6xI+0ZUAhqnQX6Kgqf7LKHAs5Sr2NbLaSG73quKu3iRX4YtpwZIsVzQhtVGYc6hSWD0SGDBBieIrnWAimJ4VkgUWmCj9lDr6CN72yu3k9GjkuSPv3bPe8evmHPvgEXgMHOCB5+AYvAUnYAyI9cn6Yn2zvtuf7a/2D/tnDbX2Gs5DsBH2r79N6EoP</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit> ? !5
  • 6. SO(3) S2-Translation • = R , = 3 = SO(3) !6
  • 7. S2 • • Haar Z S2 f(x)dx = 1 4⇡ Z 2⇡ 0 d↵ Z ⇡ 0 sin d f(↵, ) <latexit sha1_base64="7tdbAsxkeD8/MCIPf6wyQN6q7Dw=">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</latexit><latexit sha1_base64="7tdbAsxkeD8/MCIPf6wyQN6q7Dw=">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</latexit><latexit sha1_base64="7tdbAsxkeD8/MCIPf6wyQN6q7Dw=">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</latexit><latexit sha1_base64="7tdbAsxkeD8/MCIPf6wyQN6q7Dw=">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</latexit> !7 Z S2 f(Rx)dx = Z S2 f(x)dx, R 2 SO(3) <latexit sha1_base64="ZAaVkLcdfIm/x6XmS0Ch7l0cOo0=">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</latexit><latexit sha1_base64="ZAaVkLcdfIm/x6XmS0Ch7l0cOo0=">AAACJHicbZDLSgMxFIYz9VbrbdSlm2ARWpAyUwUFEYpu3Flbe4G2lkwm04ZmMmOSkZahD+PGV3Hjwgsu3PgsppeFrR4I/Hz/OZyc3wkZlcqyvozEwuLS8kpyNbW2vrG5ZW7vVGUQCUwqOGCBqDtIEkY5qSiqGKmHgiDfYaTm9C5Hfu2BCEkDfqsGIWn5qMOpRzFSGrXNsyblqh2X7/JD6GVK/Sx0+/AcztAxPITN+wi5sDTyYPk6c5Rtm2krZ40L/hX2VKTBtIpt873pBjjyCVeYISkbthWqVoyEopiRYaoZSRIi3EMd0tCSI5/IVjw+cggPNHGhFwj9uIJj+nsiRr6UA9/RnT5SXTnvjeB/XiNS3mkrpjyMFOF4ssiLGFQBHCUGXSoIVmygBcKC6r9C3EUCYaVzTekQ7PmT/4pqPmdbOfvmOF24mMaRBHtgH2SADU5AAVyBIqgADB7BM3gFb8aT8WJ8GJ+T1oQxndkFM2V8/wBed6F+</latexit><latexit sha1_base64="ZAaVkLcdfIm/x6XmS0Ch7l0cOo0=">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</latexit><latexit sha1_base64="ZAaVkLcdfIm/x6XmS0Ch7l0cOo0=">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</latexit> parameterization ZYZ-Euler angle SO(3) parameterization S2 parameterization α β
  • 8. S2-Correlation • Correlation dx S2 • !8 (f ? )(R) := Z S2 (R 1 x)f(x)dx <latexit sha1_base64="r7g0bV6C+3hSdApedhDMsTASdxs=">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</latexit><latexit sha1_base64="r7g0bV6C+3hSdApedhDMsTASdxs=">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</latexit><latexit sha1_base64="r7g0bV6C+3hSdApedhDMsTASdxs=">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</latexit><latexit sha1_base64="r7g0bV6C+3hSdApedhDMsTASdxs=">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</latexit> (f ? )(R) := Z S2 X k k(R 1 x)f(x)dx <latexit sha1_base64="CbfLlgXmACTgIAiGdTB2Ddf4dLg=">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</latexit><latexit sha1_base64="CbfLlgXmACTgIAiGdTB2Ddf4dLg=">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</latexit><latexit sha1_base64="CbfLlgXmACTgIAiGdTB2Ddf4dLg=">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</latexit><latexit sha1_base64="CbfLlgXmACTgIAiGdTB2Ddf4dLg=">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</latexit> CNN
  • 9. Fourier • Fourier = !9 Fourier Fourier Fourier ga : x 7! x + a<latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit><latexit sha1_base64="dTfD7IDp4J8cJwrOGTMfOiBhDVE=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWQRDKjAiKq6IblxXsBdphOJOmbWiSGZKMWIaCr+LGhSJufQ53vo1pOwtt/SHw8Z9zyDl/lHCmjed9O4Wl5ZXVteJ6aWNza3vH3d1r6DhVhNZJzGPVikBTziStG2Y4bSWKgog4bUbDm0m9+UCVZrG8N6OEBgL6kvUYAWOt0D3oh4Cv8CPuCEi0iS2dYgjdslfxpsKL4OdQRrlqofvV6cYkFVQawkHrtu8lJshAGUY4HZc6qaYJkCH0aduiBEF1kE3XH+Nj63RxL1b2SYOn7u+JDITWIxHZTgFmoOdrE/O/Wjs1vcsgYzJJDZVk9lEv5dieOckCd5mixPCRBSCK2V0xGYACYmxiJRuCP3/yIjTOKr5X8e/Oy9XrPI4iOkRH6AT56AJV0S2qoToiKEPP6BW9OU/Oi/PufMxaC04+s4/+yPn8Ab2zlAo=</latexit> gR : x 7! 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  • 10. Fourier Correlation • S2 Fourier • !10 (f ? )(R) = Z S2 (R 1 x)f(x) dx = Z S2 X l,m ˆl mY l m(R 1 x) X l0,m0 ˆfl0 m0 Y l0 m0 (x) dx <latexit sha1_base64="1VAoOk6wySR0KgL+snPOYqfgqvQ=">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</latexit><latexit sha1_base64="1VAoOk6wySR0KgL+snPOYqfgqvQ=">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</latexit><latexit sha1_base64="1VAoOk6wySR0KgL+snPOYqfgqvQ=">AAACi3icbVFdT9swFHWyr64bWzce92Kt2kglViWABJpAQqBJe2Rs5UNNGzmuQy1sJ7JvUCsrf2Y/aW/8G5wSYCu7kq3jc8/xte9NC8ENhOG15z95+uz5i9bL9qvXK2/edt69PzF5qSkb0Fzk+iwlhgmu2AA4CHZWaEZkKthpenlY50+vmDY8V79gXrCRJBeKZ5wScFTS+R1ksQGicVwY3guOe/jzHo65gsT+HG9UCzo4HtsvUTXrZcGsh2M8meE4bi/pTCkTK9alg1MCtvZVY5FIfF7v91fcCdfW5VojzaqxO1eJrZnzB3xXLOl0w364CPwYRA3ooiaOks6feJLTUjIFVBBjhlFYwMgSDZwKVrXj0rCC0EtywYYOKiKZGdlFLyv8yTETnOXaLQV4wf7tsEQaM5epU0oCU7Ocq8n/5YYlZDsjy1VRAlP0tlBWCgw5rgeDJ1wzCmLuAKGau7diOiWaUHDja7smRMtffgxONvpR2I9+bHX3D5p2tNAH9BEFKELbaB99R0dogKjX8vretrfjr/ib/ld/91bqe41nFf0T/rcbtA/BNg==</latexit><latexit sha1_base64="1VAoOk6wySR0KgL+snPOYqfgqvQ=">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</latexit> Y l m(R 1 x) = X n Dl mn(R 1 )Y l n(x) <latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit><latexit sha1_base64="YE1O/ZAplnGnU4lSB3uVBFXbUXk=">AAACGnicbVDLSgMxFM3UV62vqks3wSK0C8uMCLoRirpwWcU+pJ0OmTRtQ5PMkGSkZeh3uPFX3LhQxJ248W/MtLPQ6oHA4ZxzubnHDxlV2ra/rMzC4tLySnY1t7a+sbmV396pqyCSmNRwwALZ9JEijApS01Qz0gwlQdxnpOEPLxK/cU+kooG41eOQuBz1Be1RjLSRvLxz12EeL9504kNnMirBM9hWEfcEvDR6zMUktUowCYriqOTlC3bZngL+JU5KCiBF1ct/tLsBjjgRGjOkVMuxQ+3GSGqKGZnk2pEiIcJD1CctQwXiRLnx9LQJPDBKF/YCaZ7QcKr+nIgRV2rMfZPkSA/UvJeI/3mtSPdO3ZiKMNJE4NmiXsSgDmDSE+xSSbBmY0MQltT8FeIBkghr02bOlODMn/yX1I/Kjl12ro8LlfO0jizYA/ugCBxwAirgClRBDWDwAJ7AC3i1Hq1n6816n0UzVjqzC37B+vwG2pme3g==</latexit> SO(3) Wigner-D function S2 → S2 (?) S2 SO(3) : S2 = SO(3)/SO(2)
  • 11. Fourier Correlation • S2 Fourier (f ? )(R) = Z S2 (R 1 x)f(x) dx = Z S2 X l,m ˆl mY l m(R 1 x) X l0,m0 ˆfl0 m0 Y l0 m0 (x) dx = X l,m X l0,m0 X n ˆl m ˆfl0 m0 Dl mn(R 1 ) Z S2 Y l n(x)Y l0 m0 (x) dx <latexit sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit><latexit 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sha1_base64="+alGSqHLAgEFxrPTySWE/DkpTNg=">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</latexit><latexit 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  • 12. Fourier Correlation • Fourier Correlation • Fourier G-FFT Correlation (f ? )(R) = F 1 SO(3) FS2 (f)FS2 ( )† (R) <latexit sha1_base64="1tsKI/tx1zCxZJZQv9CRkMqtBIU=">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</latexit><latexit sha1_base64="1tsKI/tx1zCxZJZQv9CRkMqtBIU=">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</latexit><latexit sha1_base64="1tsKI/tx1zCxZJZQv9CRkMqtBIU=">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</latexit><latexit sha1_base64="1tsKI/tx1zCxZJZQv9CRkMqtBIU=">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</latexit>
  • 13. • S2-Correlation Feature map SO(3) • 2 SO(3)-Correlation • Correlation R2 Feature map R2 R2-Correlation CNN • SO(3) Translation SO(3) SO(3) Fourier !13 (f ? )(R) := Z SO(3) (R 1 Q)f(Q)dQ <latexit sha1_base64="Xp0H3ixXX1RSZQy33e2LsM5UDv8=">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</latexit><latexit sha1_base64="Xp0H3ixXX1RSZQy33e2LsM5UDv8=">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</latexit><latexit sha1_base64="Xp0H3ixXX1RSZQy33e2LsM5UDv8=">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</latexit><latexit sha1_base64="Xp0H3ixXX1RSZQy33e2LsM5UDv8=">AAACH3icbVDLSsNAFJ34rPVVdelmsAjJwpKoqAiC6MadtloVmlgm04kOnUzCzI1QQv/Ejb/ixoUi4q5/4/SxUOuBezmccy8z94Sp4Bpct2dNTE5Nz8wW5orzC4tLy6WV1WudZIqyOk1Eom5DopngktWBg2C3qWIkDgW7Cdunff/mkSnNE3kFnZQFMbmXPOKUgJGapT078jUQhf1Uc8euOfjwCPtcQjO/PLd3nO7AsGt3+ZbXrTo4sk1rVZulsltxB8DjxBuRMhrholn68lsJzWImgQqidcNzUwhyooBTwbpFP9MsJbRN7lnDUElipoN8cF8XbxqlhaNEmZKAB+rPjZzEWnfi0EzGBB70X68v/uc1MogOgpzLNAMm6fChKBMYEtwPC7e4YhRExxBCFTd/xfSBKELBRFo0IXh/Tx4n19sVz6141d3y8ckojgJaRxvIRh7aR8foDF2gOqLoCb2gN/RuPVuv1of1ORydsEY7a+gXrN43I+2fRw==</latexit> SO(3) Translation SO(3) Haar measure S2-Conv SO(3)-Conv SO(3)-Conv …
  • 14. • Stereographic projection of rotated MNIST • • S2 SO(3)-Translation NR R !14 S2-Conv → ReLU → SO(3)-Conv → ReLU → FC → Softmax Conv → ReLU → Conv → ReLU → FC → Softmax
  • 15. • 3D shape recognition • 3DCAD Ray casting • S2-CNN !15 S2-Conv → BN → ReLU → SO(3)-Conv → BN → ReLU → SO(3)-Conv → BN → ReLU → MaxPool → BN → Linear
  • 16. • CNN • Correlation Convolution Fourier • Fourier • 3DCAD S2-CNN • 3 SE(3) !16