A majorization–minimization-based method for nonconvex inverse rig problems in facial animation: algorithm derivation

Automated methods for facial animation are a necessary tool in the modern industry since the standard blendshape head models consist of hundreds of controllers, and a manual approach is painfully slow. Different solutions have been proposed that produce output in real-time or generalize well for dif...

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Published in:	Optimization letters Vol. 18; no. 2; pp. 545 - 559
Main Authors:	Racković, Stevo, Soares, Cláudia, Jakovetić, Dušan, Desnica, Zoranka
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Subjects:	Computational Intelligence Mathematics Mathematics and Statistics Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation Inverse rig Quadratic blendshape model Majorization–Minimization
ISSN:	1862-4472, 1862-4480
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Abstract	Automated methods for facial animation are a necessary tool in the modern industry since the standard blendshape head models consist of hundreds of controllers, and a manual approach is painfully slow. Different solutions have been proposed that produce output in real-time or generalize well for different face topologies. However, all these prior works consider a linear approximation of the blendshape function and hence do not provide a high-enough level of detail for modern realistic human face reconstruction. A second-order blendshape approximation leads to higher fidelity facial animation but generates a non-linear least squares optimization problem with high dimensionality. We derive a method for solving the inverse rig in blendshape animation using quadratic corrective terms, which increases accuracy. At the same time, due to the proposed construction of the objective function, it yields a sparser estimated weight vector compared to the state-of-the-art methods. The former feature means lower demand for subsequent manual corrections of the solution, while the latter indicates that the manual modifications are also easier to include. Our algorithm is iterative and employs a Majorization–Minimization paradigm to cope with the increased complexity produced by adding corrective terms. The surrogate function is easy to solve and allows for further parallelization on the component level within each iteration. This paper is complementary to an accompanying paper (Racković et al. arxiv preprint. https://arxiv.org/abs/2302.04843 , 2023) where we provide detailed experimental results and discussion, including highly-realistic animation data, and show a clear superiority of the results compared to the state-of-the-art methods.
AbstractList	Automated methods for facial animation are a necessary tool in the modern industry since the standard blendshape head models consist of hundreds of controllers, and a manual approach is painfully slow. Different solutions have been proposed that produce output in real-time or generalize well for different face topologies. However, all these prior works consider a linear approximation of the blendshape function and hence do not provide a high-enough level of detail for modern realistic human face reconstruction. A second-order blendshape approximation leads to higher fidelity facial animation but generates a non-linear least squares optimization problem with high dimensionality. We derive a method for solving the inverse rig in blendshape animation using quadratic corrective terms, which increases accuracy. At the same time, due to the proposed construction of the objective function, it yields a sparser estimated weight vector compared to the state-of-the-art methods. The former feature means lower demand for subsequent manual corrections of the solution, while the latter indicates that the manual modifications are also easier to include. Our algorithm is iterative and employs a Majorization–Minimization paradigm to cope with the increased complexity produced by adding corrective terms. The surrogate function is easy to solve and allows for further parallelization on the component level within each iteration. This paper is complementary to an accompanying paper (Racković et al. arxiv preprint. https://arxiv.org/abs/2302.04843 , 2023) where we provide detailed experimental results and discussion, including highly-realistic animation data, and show a clear superiority of the results compared to the state-of-the-art methods.
Author	Jakovetić, Dušan Racković, Stevo Soares, Cláudia Desnica, Zoranka
Author_xml	– sequence: 1 givenname: Stevo orcidid: 0000-0002-5656-9189 surname: Racković fullname: Racković, Stevo email: stevo.rackovic@tecnico.ulisboa.pt organization: Institute for Systems and Robotics, Instituto Superior Técnico – sequence: 2 givenname: Cláudia orcidid: 0000-0003-3071-6627 surname: Soares fullname: Soares, Cláudia organization: Computer Science Department, NOVA School of Science and Technology – sequence: 3 givenname: Dušan orcidid: 0000-0003-3497-5589 surname: Jakovetić fullname: Jakovetić, Dušan organization: Department of Mathematics, Faculty of Sciences, University of Novi Sad – sequence: 4 givenname: Zoranka surname: Desnica fullname: Desnica, Zoranka organization: 3Lateral Animation Studio, Epic Games Company
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References	WangMBradleyDZafeiriouSBeelerTFacial expression synthesis using a global-local multilinear frameworkComput. Graph. Forum.20203923524510.1111/cgf.13926 NeumannTVaranasiKWengerSWackerMMagnorMTheobaltCSparse localized deformation componentsACM Trans. Graph.201310.1145/2508363.2508417 HoldenDSaitoJKomuraTLearning inverse rig mappings by nonlinear regressionIEEE Trans. Vis. Comput. Graph.201610.1109/TVCG.2016.262803628113940 FengWWKimBUYuYReal-time data driven deformation using kernel canonical correlation analysisACM TOG200810.1145/1360612.1360690 Choe, B., Ko, H.S.: Analysis and synthesis of facial expressions with hand-generated muscle actuation basis. ACM SIGGRAPH 2005 Courses, (2005). https://doi.org/10.1145/1198555.1198595 Racković, S., Soares, C., Jakovetić, D., Desnica, Z: Accurate and Interpretable Solution of the Inverse Rig for Realistic Blendshape Models with Quadratic Corrective Terms. Unpublished manuscript. Retrieved from https://github.com/stevorackovic/manuscripts/blob/main/Rackovic22AcurateInterpretable.pdf TarzanaghDASaeidianZPeyghamiMRMesgaraniHA new trust region method for solving least-square transformation of system of equalities and inequalitiesOptim. Lett.2015330407010.1007/s11590-013-0711-9 ÇetinaslanCOPosition Manipulation Techniques for Facial Animation2020PortugalPorto PorcelliMOn the convergence of an inexact Gauss-Newton trust-region method for nonlinear least-squares problems with simple boundsOptim. Lett.2013302278310.1007/s11590-011-0430-z Deng, Z., Chiang, P.Y., Fox, P., Neumann, U.: Animating Blendshape faces by cross-mapping motion capture data. Proceedings of the 2006 Symposium on Interactive 3D Graphics and Games. (2006) https://doi.org/10.1145/1111411.1111419 BouazizSWangYPaulyMOnline modeling for realtime facial animationACM TOG201310.1145/2461912.2461976 Seol, Y., Seo, J., Kim, P.H., Lewis, J.P., Noh, J.: Artist friendly facial animation retargeting. 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References_xml	– reference: PorcelliMOn the convergence of an inexact Gauss-Newton trust-region method for nonlinear least-squares problems with simple boundsOptim. Lett.2013302278310.1007/s11590-011-0430-z – reference: SongSLShiWReedMAccurate face rig approximation with deep differential subspace reconstructionACM TOG202010.1145/3386569.3392491 – reference: ZhangJChenKZhengJFacial expression retargeting from human to avatar made easyIEEE Trans. Visual Comput. Graph.202210.1109/TVCG.2020.3013876 – reference: Seol, Y., Seo, J., Kim, P.H., Lewis, J.P., Noh, J.: Artist friendly facial animation retargeting. Proceedings of the 2011 SIGGRAPH Asia Conference. (2011) https://doi.org/10.1145/2024156.2024196 – reference: CetinaslanOOrvalhoVSketching manipulators for localized blendshape editingGraph. Models202010.1016/j.gmod.2020.101059 – reference: SongJChoiBSeolYNohJYCharacteristic facial retargetingACM TOG201110.1002/cav.414 – reference: HoldenDSaitoJKomuraTLearning inverse rig mappings by nonlinear regressionIEEE Trans. Vis. Comput. Graph.201610.1109/TVCG.2016.262803628113940 – reference: YuHLiuHRegression-based facial expression optimizationIEEE Trans. Human-Mach. Syst.201410.1109/THMS.2014.2313912 – reference: Wu, C.F.J.: On the convergence properties of the EM algorithm. The Annals of statistics. 95-103 (1938) – reference: Racković, S., Soares, C., Jakovetić, D., Desnica, Z: Accurate and Interpretable Solution of the Inverse Rig for Realistic Blendshape Models with Quadratic Corrective Terms. Unpublished manuscript. Retrieved from https://github.com/stevorackovic/manuscripts/blob/main/Rackovic22AcurateInterpretable.pdf – reference: WangMBradleyDZafeiriouSBeelerTFacial expression synthesis using a global-local multilinear frameworkComput. Graph. Forum.20203923524510.1111/cgf.13926 – reference: BaileySWOmensDDilorenoPO’BrienJFFast and deep facial deformationsACM TOG202010.1145/3386569.3392397 – reference: ÇetinaslanCOPosition Manipulation Techniques for Facial Animation2020PortugalPorto – reference: SongJBlancoRRChoKYouMLewisJPChoiBNohJSparse rig parameter optimization for character animationComput. Graph. Forum.201710.5555/3128975.3128985 – reference: Sifakis, E., Neverov, I., Fedkiw, R.: Automatic determination of facial muscle activations from sparse motion capture marker data. ACM SIGGRAPH 2005 Papers. (2005) https://doi.org/10.1145/1186822.1073208 – reference: LiHYuJYeYBreglerCRealtime facial animation with on-the-fly correctivesACM TOG201310.1145/2461912.2462019 – reference: Seol, Y., Lewis, J.P.: Tuning facial animation in a mocap pipeline. ACM SIGGRAPH 2014 Talks. (2014) https://doi.org/10.1145/2614106.2614108 – reference: LiHWeiseTPaulyMExample-based facial riggingACM TOG201010.1145/1778765.1778769 – reference: BouazizSWangYPaulyMOnline modeling for realtime facial animationACM TOG201310.1145/2461912.2461976 – reference: Holden, D., Saito, J., Komura, T.: Learning an inverse rig mapping for character animation. Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation. (2015) https://doi.org/10.1145/2786784.2786788 – reference: TarzanaghDASaeidianZPeyghamiMRMesgaraniHA new trust region method for solving least-square transformation of system of equalities and inequalitiesOptim. Lett.2015330407010.1007/s11590-013-0711-9 – reference: Choe, B., Ko, H.S.: Analysis and synthesis of facial expressions with hand-generated muscle actuation basis. ACM SIGGRAPH 2005 Courses, (2005). https://doi.org/10.1145/1198555.1198595 – reference: LangeKHunterDRYangIOptimization transfer using surrogate objective functionsJ. Comput. Graph. Stat.2000181986510.1080/10618600.2000.10474858 – reference: NeumannTVaranasiKWengerSWackerMMagnorMTheobaltCSparse localized deformation componentsACM Trans. Graph.201310.1145/2508363.2508417 – reference: FengWWKimBUYuYReal-time data driven deformation using kernel canonical correlation analysisACM TOG200810.1145/1360612.1360690 – reference: RanganathanAThe Levenberg-Marquardt algorithmTutoral LM Algorithm.2004111101110 – reference: SeonghyeonKSunjinJKwanggyoonSRogerBRJunyongNDeep learning-based unsupervised human facial retargetingComput. Graph. Forum.202110.1111/cgf.14400 – reference: Lewis, J.P., Anjyo, K., Rhee, T., Zhang, M., Pighin, F.H., Deng, Z.: Practice and Theory of Blendshape Facial Models. Eurographics 2014–State of the Art Reports. (2014) https://doi.org/10.2312/egst.20141042 – reference: Deng, Z., Chiang, P.Y., Fox, P., Neumann, U.: Animating Blendshape faces by cross-mapping motion capture data. Proceedings of the 2006 Symposium on Interactive 3D Graphics and Games. (2006) https://doi.org/10.1145/1111411.1111419 – reference: ZhangZKwokJTYeungDYSurrogate maximization/minimization algorithms and extensionsMach. Learn.200710.1007/s10994-007-5022-x – ident: 2012_CR19 doi: 10.1145/1186822.1073208 – ident: 2012_CR11 doi: 10.1145/1111411.1111419 – year: 2020 ident: 2012_CR25 publication-title: Graph. Models doi: 10.1016/j.gmod.2020.101059 – ident: 2012_CR18 doi: 10.1145/1198555.1198595 – year: 2017 ident: 2012_CR21 publication-title: Comput. Graph. Forum. doi: 10.5555/3128975.3128985 – year: 2013 ident: 2012_CR3 publication-title: ACM Trans. Graph. doi: 10.1145/2508363.2508417 – ident: 2012_CR13 doi: 10.1145/2614106.2614108 – volume: 39 start-page: 235 year: 2020 ident: 2012_CR4 publication-title: Comput. Graph. Forum. doi: 10.1111/cgf.13926 – ident: 2012_CR26 doi: 10.1145/2024156.2024196 – year: 2013 ident: 2012_CR17 publication-title: ACM TOG doi: 10.1145/2461912.2461976 – volume: 11 start-page: 101 issue: 1 year: 2004 ident: 2012_CR27 publication-title: Tutoral LM Algorithm. – year: 2014 ident: 2012_CR16 publication-title: IEEE Trans. Human-Mach. Syst. doi: 10.1109/THMS.2014.2313912 – ident: 2012_CR24 – year: 2020 ident: 2012_CR8 publication-title: ACM TOG doi: 10.1145/3386569.3392397 – year: 2015 ident: 2012_CR29 publication-title: Optim. Lett. doi: 10.1007/s11590-013-0711-9 – year: 2021 ident: 2012_CR10 publication-title: Comput. Graph. Forum. doi: 10.1111/cgf.14400 – year: 2007 ident: 2012_CR22 publication-title: Mach. Learn. doi: 10.1007/s10994-007-5022-x – year: 2008 ident: 2012_CR15 publication-title: ACM TOG doi: 10.1145/1360612.1360690 – ident: 2012_CR28 – year: 2016 ident: 2012_CR7 publication-title: IEEE Trans. Vis. Comput. Graph. doi: 10.1109/TVCG.2016.2628036 – ident: 2012_CR14 doi: 10.1145/2786784.2786788 – year: 2022 ident: 2012_CR6 publication-title: IEEE Trans. Visual Comput. Graph. doi: 10.1109/TVCG.2020.3013876 – year: 2000 ident: 2012_CR23 publication-title: J. Comput. Graph. Stat. doi: 10.1080/10618600.2000.10474858 – year: 2011 ident: 2012_CR12 publication-title: ACM TOG doi: 10.1002/cav.414 – ident: 2012_CR1 doi: 10.2312/egst.20141042 – volume-title: Position Manipulation Techniques for Facial Animation year: 2020 ident: 2012_CR2 – year: 2013 ident: 2012_CR5 publication-title: ACM TOG doi: 10.1145/2461912.2462019 – year: 2013 ident: 2012_CR30 publication-title: Optim. Lett. doi: 10.1007/s11590-011-0430-z – year: 2020 ident: 2012_CR9 publication-title: ACM TOG doi: 10.1145/3386569.3392491 – year: 2010 ident: 2012_CR20 publication-title: ACM TOG doi: 10.1145/1778765.1778769
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Title	A majorization–minimization-based method for nonconvex inverse rig problems in facial animation: algorithm derivation
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