1

Geoffrey Hinton

Boyang Deng

Andrea Tagliasacchi

Kyle Genova

Sofien Bouaziz

Soroosh Yazdani

CvxNet Learnable Convex Decomposition

Family of 3D representations

2

Tatarchenko et al. 2017

Groueix et al. 2018

Mescheder et al. 2019

{voxels}

{patches}

implicits

{gaussians}Genova et al. 2019

Tulsiani et al. 2017

{boxes}

{convexes}Deng et al. 2020

What is the «best» 3D representation?

3

Training Inference

Implicit

Image courtesy of Barill et al. «Fast Winding Numbers for Soups and Clouds» SIGGRAPH 2018

Hybrid

indicator distancevoxel grid

Explicit

triangle souptriangle mesh point cloud

4

Convexes: why are they relevant?

decomposition computed by «v-hacd» – https://github.com/kmammou/v-hacd

K=8

· · ·<latexit sha1_base64="HjmZZteiMwGm516AP0SULoJbluA=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPBi94q2A9oQ9lsNu3azW7YnQil9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YSq4Qc/7dgpr6xubW8Xt0s7u3v5B+fCoZVSmKWtSJZTuhMQwwSVrIkfBOqlmJAkFa4ejm5nffmLacCUfcJyyICEDyWNOCVqp1aORQtMvV7yqN4e7SvycVCBHo1/+6kWKZgmTSAUxput7KQYTopFTwaalXmZYSuiIDFjXUkkSZoLJ/Nqpe2aVyI2VtiXRnau/JyYkMWachLYzITg0y95M/M/rZhhfBxMu0wyZpItFcSZcVO7sdTfimlEUY0sI1dze6tIh0YSiDahkQ/CXX14lrVrVv6jW7i8r9bs8jiKcwCmcgw9XUIdbaEATKDzCM7zCm6OcF+fd+Vi0Fpx85hj+wPn8AbHAjzo=</latexit>

K=32 K=∞universal approximators

rigid body physics Convex Hulls (standard)

Meshes (the «wild west»)

input K=4

Universal approximator of convex domains

5

z }| {Sigmoid(��LogSumExp{�(nh · x+ dh)})

<latexit sha1_base64="+nLrnQlHK1tyF8hch1+eEXU4CRM=">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</latexit>

permutation invariance

controls smoothness

controls indicator dropoff

δ σ

H1

H3

H0

H2

H4

H5

2

664

nx0 ny

0 d0nx1 ny

1 d1· · · · · · · · ·

nx5 ny

5 d5

3

775 ⌘

2

664

H0

H1

· · ·

H5

3

775 ⌘

<latexit sha1_base64="iYH0T2k1b2nTuooGKFC9HE0mbLI=">AAAC0XicbZLPTtwwEMadQFua/tvSYy8Wq1Y9rRLoqj0i9QKXigoWkDbbyHFmFwvHSe0JIoq2qrj27XrrG/Qx6gQLloClaD7/xuNvYjstpTAYhn89f2390eMnG0+DZ89fvHw1eL15bIpKc5jwQhb6NGUGpFAwQYESTksNLE8lnKTnX9r8yQVoIwp1hHUJs5wtlJgLztCiZPAvTmEhVJPmDLW4XAYqCb9f0ve0jbWNWRLSOLY4cjiyuONRy2OeFWjs3Ilb1RWNXdHY7TUOYlDZjRu10x+VuLCx10ZsxRlnstlb2gY6pxWyat1L9RycQRAkg2E4CrtB74vIiSFx4yAZ/Imzglc5KOSSGTONwhJnDdMouAS7cWWgZPycLWBqpWI5mFnT3ciSvrMko/NC208h7ehqRcNyY+o8tSvb3k0/18KHctMK559njVBlhaD4tdG8khQL2l4vzYQGjrK2gnEtbK+UnzHNONpH0B5C1P/l++J4exTtjLa/fRzu7rvj2CBvyRb5QCLyieySPXJAJoR7Xz30lt5P/9Cv/V/+1fVS33M1b8id4f/+DxIQ2c4=</latexit>

E!<latexit sha1_base64="Bj4V2LXC1Z6adls5TZh6YfLS+aM=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRV0GVRBJcV7AOaECbTSTt0JhNmJmIJ+RU3LhRx64+482+ctFlo64GBwzn3cs+cMGFUacf5tipr6xubW9Xt2s7u3v6BfVjvKZFKTLpYMCEHIVKE0Zh0NdWMDBJJEA8Z6YfTm8LvPxKpqIgf9CwhPkfjmEYUI22kwK57HOkJRiy7zQNPcDJGgd1wms4ccJW4JWmAEp3A/vJGAqecxBozpNTQdRLtZ0hqihnJa16qSILwFI3J0NAYcaL8bJ49h6dGGcFISPNiDefq740McaVmPDSTRVK17BXif94w1dGVn9E4STWJ8eJQlDKoBSyKgCMqCdZsZgjCkpqsEE+QRFibumqmBHf5y6uk12q6583W/UWjfV3WUQXH4AScARdcgja4Ax3QBRg8gWfwCt6s3Hqx3q2PxWjFKneOwB9Ynz9LfZSa</latexit>· · ·<latexit sha1_base64="Z9gsgRpMH1WO6HvZkap2BB5vBHE=">AAAB8XicbVBNSwMxEJ31s9avqkcvi0XwVHaroHgqeNFbBfuB7VKy2Wwbmk2WZFYopf/CiwdFvPpvvPlvTNs9aOsLgcd7M8zMC1PBDXret7Oyura+sVnYKm7v7O7tlw4Om0ZlmrIGVULpdkgME1yyBnIUrJ1qRpJQsFY4vJn6rSemDVfyAUcpCxLSlzzmlKCVHrvX9tFIoemVyl7Fm8FdJn5OypCj3it9dSNFs4RJpIIY0/G9FIMx0cipYJNiNzMsJXRI+qxjqSQJM8F4tvHEPbVK5MZK2y/Rnam/O8YkMWaUhLYyITgwi95U/M/rZBhfBWMu0wyZpPNBcSZcVO70fDfimlEUI0sI1dzu6tIB0YSiDaloQ/AXT14mzWrFP69U7y/Ktbs8jgIcwwmcgQ+XUINbqEMDKEh4hld4c4zz4rw7H/PSFSfvOYI/cD5/ABVNkI4=</latexit>

2

664

nx0 ny

0 d0nx1 ny

1 d1· · · · · · · · ·

nx5 ny

5 d5

3

775 ⌘

2

664

H0

H1

· · ·

H5

3

775 ⌘

<latexit sha1_base64="iYH0T2k1b2nTuooGKFC9HE0mbLI=">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</latexit>

input: half-spaces tensor

C(x|�)<latexit sha1_base64="t1nVhbqKg397qakdOpAKAHlqAjA=">AAACGHicbVC7TsMwFHXKq5RXgJHFokIqS0kKEoipogtjkehDaqLKcZzWqvOQ7SCqkM9g4VdYGECItRt/g5NmgJYrWT46517dc48TMSqkYXxrpZXVtfWN8mZla3tnd0/fP+iKMOaYdHDIQt53kCCMBqQjqWSkH3GCfIeRnjNpZXrvgXBBw+BeTiNi+2gUUI9iJBU11M+sa2j5SI4xYkkrreXY8ZLHFD5BywmZK6a++hLLIRKlp0O9atSNvOAyMAtQBUW1h/rMckMc+ySQmCEhBqYRSTtBXFLMSFqxYkEihCdoRAYKBsgnwk7yw1J4ohgXeiFXL5AwZ39PJMgXmT/VmfkWi1pG/qcNYuld2QkNoliSAM8XeTGDMoRZStClnGDJpgogzKnyCvEYcYSlyrKiQjAXT14G3UbdPK837i6qzZsijjI4AsegBkxwCZrgFrRBB2DwDF7BO/jQXrQ37VP7mreWtGLmEPwpbfYDIl6geg==</latexit>

output: function

6

E!<latexit sha1_base64="Bj4V2LXC1Z6adls5TZh6YfLS+aM=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRV0GVRBJcV7AOaECbTSTt0JhNmJmIJ+RU3LhRx64+482+ctFlo64GBwzn3cs+cMGFUacf5tipr6xubW9Xt2s7u3v6BfVjvKZFKTLpYMCEHIVKE0Zh0NdWMDBJJEA8Z6YfTm8LvPxKpqIgf9CwhPkfjmEYUI22kwK57HOkJRiy7zQNPcDJGgd1wms4ccJW4JWmAEp3A/vJGAqecxBozpNTQdRLtZ0hqihnJa16qSILwFI3J0NAYcaL8bJ49h6dGGcFISPNiDefq740McaVmPDSTRVK17BXif94w1dGVn9E4STWJ8eJQlDKoBSyKgCMqCdZsZgjCkpqsEE+QRFibumqmBHf5y6uk12q6583W/UWjfV3WUQXH4AScARdcgja4Ax3QBRg8gWfwCt6s3Hqx3q2PxWjFKneOwB9Ynz9LfZSa</latexit>

… and non-convex domains!

�3,T3

�2,T2

�1,T1

· · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">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</latexit>

�K ,TK<latexit sha1_base64="jJF3wtvmfHhySED0iAqWh9R0aSk=">AAACE3icbVDLSsNAFJ3UV62vqEs3o0UQkZJUQZdFN4KbCn1BE8JkOmmHTiZhZiKUkH9w46+4caGIWzfu/BsnbRbaemGYwzn3cs89fsyoVJb1bZSWlldW18rrlY3Nre0dc3evI6NEYNLGEYtEz0eSMMpJW1HFSC8WBIU+I11/fJPr3QciJI14S01i4oZoyGlAMVKa8sxTx4/YQE5C/aWOTxTKvNQ5vMvOoBMiNfKDtFUwnlm1ata04CKwC1AFRTU988sZRDgJCVeYISn7thUrN0VCUcxIVnESSWKEx2hI+hpyFBLpptObMnismQEMIqEfV3DK/p5IUShz27oz9ynntZz8T+snKrhyU8rjRBGOZ4uChEEVwTwgOKCCYMUmGiAsqPYK8QgJhJWOsaJDsOdPXgSdes0+r9XvL6qN6yKOMjgAR+AE2OASNMAtaII2wOARPINX8GY8GS/Gu/Exay0Zxcw++FPG5w9shJ56</latexit>

input: image, cloud, …

8>>><

>>>:

C(T1(x)| �1)

C(T2(x)| �2)

C(T3(x)| �3)

C(TK(x)| �K)

· · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">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</latexit> · · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">AAAHHnicjVXdbtMwFM4GFCh/Ay65iZgmIYSmZiAN7kDb0CZtrGNbQVqqynFOM4NjR7bTdbLCM3ALVzwNd4hbeBtO2ox0zoSwkvbkfN+xz5/tKONMm07n99z8pctXWlevXW/fuHnr9p2Fu/d6WuaKwiGVXKr3EdHAmYBDwwyH95kCkkYc3kUf10r83QiUZlIcmNMM+ilJBBsySgyqeiGNpdGDhcXOcmcy/KYQVMKiV43u4G5rPowlzVMQhnKi9VHQyUzfEmUY5VC0w1xDRuhHksARioKkoPt24m7hL6Em9odS4SuMP9HOWliSan2aRshMiTnWLlYqL8KOcjN83rdMZLkBQacLDXPuG+mXsfsxU0ANP0WBUMXQV58eE0WowQy126GAEyrTlIjYhvtbrwuLP59CygzYBIQckQGjdBS8KIrz3F63sL1uxTQ514wINqCjTAWrLneX0jdgCjv9r2wkpQLM1KIxe9fIncJ2V3Yq8gkRyQBKR5671FcGizGZ/kyqbEj5+XeJht3+Hsa6V3EzojErnMUyrz1qLy3h45c598vaSK5LVbgOWH8FO6jfzUARI9VjGxKVpEwU2A9J+KSU/kUk4zMiSuf94lLr4ijo27BcmBJut4tBaGBs7GLgBnECLDk2urChTCEhDvq2mE4SRfatawnjDPuiJmy4hOjgDBzagwZIa5A2QFmD0gU5wT4t18V0xtOslso0ikmd8hFRjOB2nuZ71pzhXoaizs2WO7+MPtRxlYxdl5ELPAVmCIcuQcdDhLvHzNFTKUYwnrFccy0zorAW52OLwGBkbopQSUfj4iKqGzAeidijF3o0rhM9dg1FjQkXw1YfajxBZjO52egQQWUMaobS6JFEyTz7X14MLm/d5V1McarDZVb6rRluI7dyqZR4TpYTcdMA0UCyGNHJZrL71Xej/HJoJhu04m3LZD9PN8YZMvHWCNw7oin0VpaDp8sre88WX25V98c174H30HvkBd6q99Lb9LreoUe9D95n74v3tfWt9b31o/VzSp2fq2zue+dG69cfnu+gqg==</latexit>

z }| {O!(x) = max

k{C(Tk(x)|�k)}

<latexit sha1_base64="bbmAhTN/RjD50SZ00diroALNAek=">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</latexit>

output: function

shap

eloca

tion

7

aImplicit functions @ training time

E!<latexit sha1_base64="Bj4V2LXC1Z6adls5TZh6YfLS+aM=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRV0GVRBJcV7AOaECbTSTt0JhNmJmIJ+RU3LhRx64+482+ctFlo64GBwzn3cs+cMGFUacf5tipr6xubW9Xt2s7u3v6BfVjvKZFKTLpYMCEHIVKE0Zh0NdWMDBJJEA8Z6YfTm8LvPxKpqIgf9CwhPkfjmEYUI22kwK57HOkJRiy7zQNPcDJGgd1wms4ccJW4JWmAEp3A/vJGAqecxBozpNTQdRLtZ0hqihnJa16qSILwFI3J0NAYcaL8bJ49h6dGGcFISPNiDefq740McaVmPDSTRVK17BXif94w1dGVn9E4STWJ8eJQlDKoBSyKgCMqCdZsZgjCkpqsEE+QRFibumqmBHf5y6uk12q6583W/UWjfV3WUQXH4AScARdcgja4Ax3QBRg8gWfwCt6s3Hqx3q2PxWjFKneOwB9Ynz9LfZSa</latexit> a a⇡<latexit sha1_base64="jsznNaQENVl2sRLrhSZ5aqy02is=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgh4LXvRWwX5Au5Rsmm1Ds0lIsmJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfjm5nffqTaMCke7ETRMMFDwWJGsHVSu4eV0vKpX674VX8OtEqCnFQgR6Nf/uoNJEkTKizh2Jhu4CsbZlhbRjidlnqpoQqTMR7SrqMCJ9SE2fzcKTpzygDFUrsSFs3V3xMZToyZJJHrTLAdmWVvJv7ndVMbX4cZEyq1VJDFojjlyEo0+x0NmKbE8okjmGjmbkVkhDUm1iVUciEEyy+vklatGlxUa/eXlfpdHkcRTuAUziGAK6jDLTSgCQTG8Ayv8OYp78V79z4WrQUvnzmGP/A+fwCWy4/B</latexit>

O(x)<latexit sha1_base64="A4fI7x5MqCrWFZPPhYZqEW6t1DQ=">AAACAHicbZDLSsNAFIZPvNZ6i7pw4SZYhLopSRV0WXCjKyvYC7ShTKaTduhkEmYmYgnZ+CpuXCji1sdw59s4SbPQ1h8GPv5zDnPO70WMSmXb38bS8srq2nppo7y5tb2za+7tt2UYC0xaOGSh6HpIEkY5aSmqGOlGgqDAY6TjTa6yeueBCElDfq+mEXEDNOLUpxgpbQ3Mw36A1Bgjltym1Zw9P3lMTwdmxa7ZuaxFcAqoQKHmwPzqD0McB4QrzJCUPceOlJsgoShmJC33Y0kihCdoRHoaOQqIdJP8gNQ60c7Q8kOhH1dW7v6eSFAg5TTwdGe2opyvZeZ/tV6s/Es3oTyKFeF49pEfM0uFVpaGNaSCYMWmGhAWVO9q4TESCCudWVmH4MyfvAjtes05q9XvziuNmyKOEhzBMVTBgQtowDU0oQUYUniGV3gznowX4934mLUuGcXMAfyR8fkD9RaWqA==</latexit>

O!(x)<latexit sha1_base64="M7tv25GhbyGTyuHS7eMDWb+5/HE=">AAACB3icbVDLSsNAFJ3UV62vqEtBBotQNyWpgi4LbnRlBfuAJpTJdNIOnWTCzEQsITs3/oobF4q49Rfc+TdO0iy09cCFwzn3cu89XsSoVJb1bZSWlldW18rrlY3Nre0dc3evI3ksMGljzrjoeUgSRkPSVlQx0osEQYHHSNebXGZ+954ISXl4p6YRcQM0CqlPMVJaGpiHToDUGCOW3KQDhwdkhGq55PnJQ3oyMKtW3coBF4ldkCoo0BqYX86Q4zggocIMSdm3rUi5CRKKYkbSihNLEiE8QSPS1zREAZFukv+RwmOtDKHPha5QwVz9PZGgQMpp4OnO7EQ572Xif14/Vv6Fm9AwihUJ8WyRHzOoOMxCgUMqCFZsqgnCgupbIR4jgbDS0VV0CPb8y4uk06jbp/XG7Vm1eV3EUQYH4AjUgA3OQRNcgRZoAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8Ac8iZsg==</latexit>

8>>><

>>>:

C(T1(x)| �1)

C(T2(x)| �2)

C(T3(x)| �3)

C(TK(x)| �K)

· · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">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</latexit> · · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">AAAHHnicjVXdbtMwFM4GFCh/Ay65iZgmIYSmZiAN7kDb0CZtrGNbQVqqynFOM4NjR7bTdbLCM3ALVzwNd4hbeBtO2ox0zoSwkvbkfN+xz5/tKONMm07n99z8pctXWlevXW/fuHnr9p2Fu/d6WuaKwiGVXKr3EdHAmYBDwwyH95kCkkYc3kUf10r83QiUZlIcmNMM+ilJBBsySgyqeiGNpdGDhcXOcmcy/KYQVMKiV43u4G5rPowlzVMQhnKi9VHQyUzfEmUY5VC0w1xDRuhHksARioKkoPt24m7hL6Em9odS4SuMP9HOWliSan2aRshMiTnWLlYqL8KOcjN83rdMZLkBQacLDXPuG+mXsfsxU0ANP0WBUMXQV58eE0WowQy126GAEyrTlIjYhvtbrwuLP59CygzYBIQckQGjdBS8KIrz3F63sL1uxTQ514wINqCjTAWrLneX0jdgCjv9r2wkpQLM1KIxe9fIncJ2V3Yq8gkRyQBKR5671FcGizGZ/kyqbEj5+XeJht3+Hsa6V3EzojErnMUyrz1qLy3h45c598vaSK5LVbgOWH8FO6jfzUARI9VjGxKVpEwU2A9J+KSU/kUk4zMiSuf94lLr4ijo27BcmBJut4tBaGBs7GLgBnECLDk2urChTCEhDvq2mE4SRfatawnjDPuiJmy4hOjgDBzagwZIa5A2QFmD0gU5wT4t18V0xtOslso0ikmd8hFRjOB2nuZ71pzhXoaizs2WO7+MPtRxlYxdl5ELPAVmCIcuQcdDhLvHzNFTKUYwnrFccy0zorAW52OLwGBkbopQSUfj4iKqGzAeidijF3o0rhM9dg1FjQkXw1YfajxBZjO52egQQWUMaobS6JFEyTz7X14MLm/d5V1McarDZVb6rRluI7dyqZR4TpYTcdMA0UCyGNHJZrL71Xej/HJoJhu04m3LZD9PN8YZMvHWCNw7oin0VpaDp8sre88WX25V98c174H30HvkBd6q99Lb9LreoUe9D95n74v3tfWt9b31o/VzSp2fq2zue+dG69cfnu+gqg==</latexit>

Lapprox(!) = Ex⇠R2kO!(x)�O(x)k2<latexit sha1_base64="CrQ9wdD6BqHH3mJ6JQdg/woIk6U=">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</latexit>

prediction ground truth

�3,T3

�2,T2

�1,T1

· · ·<latexit sha1_base64="4hQMhGioUjsffkY19f9I9mmLHY8=">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</latexit>

�K ,TK<latexit sha1_base64="jJF3wtvmfHhySED0iAqWh9R0aSk=">AAACE3icbVDLSsNAFJ3UV62vqEs3o0UQkZJUQZdFN4KbCn1BE8JkOmmHTiZhZiKUkH9w46+4caGIWzfu/BsnbRbaemGYwzn3cs89fsyoVJb1bZSWlldW18rrlY3Nre0dc3evI6NEYNLGEYtEz0eSMMpJW1HFSC8WBIU+I11/fJPr3QciJI14S01i4oZoyGlAMVKa8sxTx4/YQE5C/aWOTxTKvNQ5vMvOoBMiNfKDtFUwnlm1ata04CKwC1AFRTU988sZRDgJCVeYISn7thUrN0VCUcxIVnESSWKEx2hI+hpyFBLpptObMnismQEMIqEfV3DK/p5IUShz27oz9ynntZz8T+snKrhyU8rjRBGOZ4uChEEVwTwgOKCCYMUmGiAsqPYK8QgJhJWOsaJDsOdPXgSdes0+r9XvL6qN6yKOMjgAR+AE2OASNMAtaII2wOARPINX8GY8GS/Gu/Exay0Zxcw++FPG5w9shJ56</latexit>

Ldecomp(!) = Ex⇠R3krelu(sumk

{Ck(x)}� ⌧)k2<latexit sha1_base64="jho5gI/0eLm0W1CzhW4q9/fdwlo=">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</latexit>

Lunique(!) =1H⌃hkdhk2

<latexit sha1_base64="i9urMCJYZj7D7FOIVwPIfHXvGzY=">AAAHVnicjVVtb9xEEHYDzbXHWwof+8UiCioIReeA1CKB1KotSqSGXHpJWik+rPV67Fu63jW768tF2+XH8Gv4Cp/gzyDGdw6+rCOEdXeem+eZ3Z1nxuO04kyb0eivWxvvvHt7c3Dn7vC99z/48KOtex+faVkrCqdUcqlep0QDZwJODTMcXlcKSJlyeJW+edrgr+agNJPixFxWMC1JIVjOKDHoSra+jbnU2taC/VyDexDLEgryefjZd2FsckWojZzdd2E8YUVJEjtD822YJTO8/biXbG2PdkfLK+wbUWtsB+01Tu5tbsSZpHUJwlBOtD6PRpWZWqIMoxzcMK41VIS+IQWcoylICXpql2m6cAc9WZhLhV9hwqV3PcKSUuvLMkVmScxM+1jjvAk7r03+aGqZqGoDgq42ymseGhk2moUZU0ANv0SDUMXwrCGdEVTHoLLDYSzggsqyJCKz8eTge2fx55eYMgO2ACHnJGGUzqNvnLvOPRs7ezZumabmmhHBEjqvVPTQ5x5R+gMYZ1f3NkZSKsCsInqrj408dHa8d9iSL4goEmgO8sinPjFYjOXyV1YbQ5q//27Ri5scY67HLbciGlXhLJN1d6Lhzg5+wkbzsKmN5Lpxxc8A66/gEP1HFShipPrCxkQVJRMO+6GIv2ys/yKSxRURrevnanranUdTGzcbU8LtC5fEBhbGbkd+EhfAipnRzq6a30NfutUiaWpf+pGwqLAvOsJzn5CeXIG5PemBtANpD5QdKH2QE+zTZl+UM1up2jjLNCOd5HOiGMExsNJ7PZzhDADXaXPgry/Tn7q8GsaRz8B5IcUa4dQn6CxHeDxjnp9KMYfFWuRTP7IiCmtxPbcUDGbmS4ROOl+4m6h+wjhKsUdvPNGiE3rhB4oOEz6GrZ5rnCDrSu73OkRQmYFao/R6pFCyrv4vLwOf98zn3UzxqsNl1ZxbN1Pdr1wpJc7JZiFueiAGSJYhunyY7KT93yu/zM3yAW15L2QxqcvniwqZ+NaI/HdE3zjb242+2t07/nr78UH7/rgT3A8+DR4EUfAweBzsB+PgNKDBr8Fvwe/BH5t/bv49uD0YrKgbt9qYT4Jr12DrH76Ss1I=</latexit>

Lmerged(!) =1K⌃k

1N⌃x2NkkReLU(�k(x))k2

<latexit sha1_base64="EbdiLI3gEagUqJ0GcU3q2kY6oXQ=">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</latexit>

convexes should not overlap

only one way to represent a convex

prevents vanishing gradients

H1

H3

H0

H2

H4

H5

primal domain

hullH1

H3

H0

H2

H4H5

dual domain

dual

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dual domain

dualpoint , line

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primal domain

Polygonal meshes @ inference time

CvxN

et

© Luca Prasso

Multi view reconstruction – {Depth}→3D

9

…20x

Ground TruthMesh

CvxNet[Deng et al. 2020]

SIF[Genova et al. 2019]

OccNet SIF CvxNet

OccNet[Mescheder et al. 2019]

10

Input Image Ground Truth Mesh

CvxNet [Deng et al. 2020]

AtlasNet [Groueix et al. 2019]

OccNet [Mescheder et al. 2019]

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

N/A

Single view reconstruction (SVR)N/A

N/A

11

Applications: Retrieval & CorrespondenceCvxNet SelectionInput

Obj

ect A

Obj

ect B

…

…

Retrieved Model

k=27

k=6

k=34

k=25

Ground Truth CvxNet

12

Shape approximation – comparisons

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«structuring»

«union»

Structured Implicits [Genova et al. 2019]

CvxNet [Deng et al. 2019]

13

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

14

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

15

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

16

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

17

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

18

Shape approximation – quality v.s. budget

100506

F-sc

ore%

3

CvxNet

Structured Implicits

# primitives

«union»

CvxNet [Deng et al. 2019]

Ground Truth Mesh

Volumetric Primitives [Tulsiani et al. 2017]

Structured Implicits [Genova et al. 2019]

19

Explore the shape space of primitivesSI

FVP

inter

polat

ion

ex. 1 ex. 2interpolation

Bilinear Interpolation Linear Interpolation Correlation of Hyperplanes

20

OccNet

CvxNet

learnable

Related work @ CVPR 2020

21

Chen et al. «BSP-Net» graph learning, sharp details

Deng et al. «CvxNet» smooth convexes, fixed graph focus

half-spaces

convexes

object

CvxNet Learnable Convex Decomposition

Geoffrey HintonBoyang Deng

Andrea TagliasacchiKyle Genova

Sofien BouazizSoroosh Yazdani

cvxnet.github.io