Motion Estimation in
In Situ Materials Tests

Dr. Tessa Nogatz
tessa.nogatz.net

Bulk

Fiber

Foam

Fiber Composite

3D Representations of Materials

Foam

Spatially sparse material
Full layers collapse where struts break

Fiber Composites

Matrix fiber bonding crucial
Where many fibers debond, matrix will rupture

Concrete

Very brittle material
Microcracking is forerunner of global failure

Faster Gifs

In Situ Materials Test

A General Framework for Motion Estimation

\( \phi(\ux) = \ux + \ub(\ux), \)
where \(\ub = (u, v, w)\)

\begin{equation} I_0(\ux) = I_1(\phi(\ux)) \end{equation}

\begin{equation} \min_\phi \int_\Omega \left(\class{substep-2-i0}{I_0(\ux)}\class{hidden}{I_0(\ux)}-\class{substep-2-i1}{I_1(\phi(\ux))}\class{hidden}{I_1(\phi(\ux))}\right)^2\,d\ux + \lambda \mathcal{R}(\phi) \end{equation}

\( \mathcal{D} \)

Local Digital Volume Correlation

\begin{equation} \min_{\ub} \class{substep-1}{\sum_{d=1}^{N_c}} \class{substep-2}{\sum_{i=1}^{K^3}} \left(I_1(\class{substep-1}{\ux_d}+\class{substep-2}{m_i}+\class{substep-3-insert}{\ub_d\phi}\class{substep-3-remove}{\ub_h}(\class{substep-1}{\ux_d}))-I_0(\class{substep-1}{\ux_d}+\class{substep-2}{m_i})\right)^2 \end{equation}

Global Digital Volume Correlation

\( \ub_h= \sum_{\alpha,n} a_{\alpha n}\psi_n(\ux)e_\alpha \)

\begin{equation} \min_\ub \int_\Omega \left(I_1\class{substep-1}{(\ux+\ub)}\class{substep-2}{(\ux)+\ub^T\nabla I_0(\ux)} - I_0(\ux)\right)^2\,d\ux \end{equation}

Augmented-Lagrangian DVC

\begin{align} \begin{split} \pazocal{L} = &\sum_i\int_{\Omega_i} \left(I_1(\ux+\ub_i(\ux) + \textbf{F}_i\left(\ux-\ux_i^0\right))-I_0(\ux)\right)^2 \\ &+ \frac{\alpha}{2}\left| \nabla {\hat{\ub}_i} - {\textbf{F}_i} + {\mathbf{W}_i}\right|^2 + \frac{\mu}{2}\left|{\hat{\ub}_i} - {\ub_i} - {\mathbf{v}_i}\right|^2\dx. \end{split} \end{align}

Medical Image Registration

\begin{equation} \min_\phi \int_\Omega \left(I_1(\phi(\ux))-I_0(\ux)\right)^2\dx + \Reg(\phi) \end{equation}

\( \class{hidden}{\phi(\ux)}\class{substep-3-phi}{\phi(\ux)}\class{substep-3-eq}{ = \ux+\ub(\ux)} \)\( \class{hidden}{\Reg}\class{substep-3-reg}{\Reg}^{\class{substep-3-eq}{\text{elas}}}\class{substep-3-eq}{(\ub) = \int_\Omega \frac{\mu}{4} \left(\nabla\ub + \nabla^T \ub\right)^2 + \frac{\lambda}{2}\left(\text{div}\, \ub\right)^2 \dx} \)

Nice physical interpretation:
Regularization forces displacement to follow a materials law

Comparison of State-of-the-Art Methods

\begin{align} u(x,y,z) &= \begin{cases} z\cdot 0.005\cdot K & \text{if } z < D/2,\\ (z-D/2)\cdot 0.005\cdot K & \text{if } z>D/2 \end{cases}\\ v(x,y,z) &= 0\\ w(x,y,z) &= -0.2 - \frac{K-0.2}{\sqrt{0.5\cdot e^{-0.04\cdot(z-D/2)}}} \end{align}

\( w(x,y,z) \)

\( v(x,y,z) \)

\( u(x,y,z) \)

Total Variation Optical Flow

\begin{equation} \min_\ub \int_\Omega |I_0(\ux)-I_1(\ux+\ub)| +|\nabla \ub|\dx \end{equation}

\begin{align} \begin{split} \class{substep-2-t}{T}\class{substep-2-v}{V}\class{hidden}{TV}\class{substep-2-rest}{(f)} &\class{substep-2-rest}{= \int_{\Omega}|D f|} \\ &\class{substep-2-rest}{= \sup\Big\{\int_\Omega f \dvg \g \,dx \;\big|\; \g = (g_1, \ldots, g_n)\in C_0^1(\Omega;\R^n) \;\text{and}\; |\g(\ux)|\leq 1 \;\text{for}\; \ux\in\Omega\Big\}} \end{split} \end{align}

\(T\)otal\(V\)ariation

preserves edges
removes spurios oscillations
functions with finite number of jumps

3DOF

TV Optical Flow Solved by Approximation¹²

\begin{equation} J_{\text{Grad}}(\ub)=\int_\Omega\Psi\left(|\nabla I_0(\ux)-\nabla I_1(\ux+\ub)|\right) \dx \end{equation}

\begin{equation} J_{\text{Data}}(\ub) = \int_\Omega\Psi\left(|I_0(\ux)-I_1(\ux+\ub)|\right) \dx \end{equation}

\begin{equation} J_{\text{Smooth}}(\ub)= \int_\Omega \Psi\left( |\nabla u| + |\nabla v |+|\nabla w|\right) \dx \end{equation}

\( \Psi(s^2) = \sqrt{s^2+\varepsilon^2} \)

\begin{equation} J_{\text{total}}(\ub)=J_{\text{Data}}(\ub)+\lambda J_{\text{Grad}}(\ub)+\mu J_{\text{Smooth}}(\ub) \end{equation}

TVL1

TV Optical Flow Solved by Discretization¹²

\begin{equation} \min_\ub \int_\Omega\ \lambda |\class{substep-0}{I_0(\ux) -I_1(\ux+\ub)}\class{substep-1}{I_0(\ux) -I_1(\ux+\ub_0) - \nabla I_1^T(\ux+\ub_0)(\ub-\ub_0)}\class{substep-2}{\rho(\ub)}| + |\nabla \ub|\dx \end{equation}

Introduce auxiliary variable \(\db\):

\begin{equation} \min_{\ub, \db}\ \int_\Omega |\nabla \ub| + \frac{1}{2\theta}||\ub-\db||_2^2 + \lambda |\rho(\db)|\dx \end{equation}

Solve alternating for \(\ub\) and \(\db\):

\begin{align} \ub^{k+1} &= \min_{\ub} \int_\Omega |\nabla \ub| + \frac{1}{2\theta} ||\ub-\db^k||_2^2 \dx\\ \db^{k+1} &= \min_{\db} \int_\Omega ||\ub^{k+1}-\db||_2^2 + \lambda |\rho(\db)|\dx \end{align}

Comparison of Total Variation Based Methods

\( u(x,y,z) \)

\( v(x,y,z) \)

\( w(x,y,z) \)

In Situ Test of Metal Matrix Composite Foam

Trip Foam

Displacement Magnitude

Displacement 3DOF
Slice View

Displacement ALDVC
Slice View

Fracturing Cell

Full Series of MMC Foam

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Displacement
Field in \(w\)

Displacement for 2% compression

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Displacement for 20% compression (adjusted)

seit 07/2024 Software-Entwicklerin

01/2023 — 06/2024 Postdoktorandin

2022 Dr. rer. nat. Mathematik
(summa cum laude)

03/2021 — 12/2022 Wissenschaftliche Mitarbeiterin

10/2018 — 02/2021 Promotionsstipendiatin

2016 — 2018 M.Sc. Mathematik

2011 — 2015 B.Sc. Mathematik

Szymon Grzesiak, Tessa Nogatz, Christoph de Sousa, Markus Kronenberger, Claudia Redenbach und Matthias Pahn. »In Situ 4D Image Analysis of Crack Width and Strain Changes in GFRP-Reinforced Concrete«. In: Materials and Structures, accepted (2026).
Jelena Zaninović, Tessa Nogatz, Katja Schladitz, Silja Flenner, Imke Greving, Peter Klein, Natalia Konchakova, Claudia Redenbach und Peter Visser. »Leaching in active protective coatings observed in-situ by nano-CT using synchrotron radiation«. In: Corrosion Science (2026).
Sophie Klemm, Tessa Nogatz, Jonathan Perrin, Timm Weitkamp, Claudia Redenbach und Claudia Fleck. »Characterization of Deformation and Failure of Fomes fomentarius Hymenium by Synchrotron Phase-Contrast Enhanced Micro-Computed Tomography, In situ Compression Testing and Three-Dimensional Optical Flow«. In: Small Structures (2025).
Tessa Nogatz, Claudia Redenbach und Katja Schladitz. »MorphFlow: Estimating motion in in-situ tests of concrete«. In: Experimental Mechanics (2025).
Tessa Nogatz, Anne Jung, Tomáš Fíla, Ivana Kumpová und Ondřej Jiroušek. »Analysis of the MesoMechanical Deformation Behavior of Al Foams and Ni/Al Hybrid Foams Using Time-Lapse MicroComputed Tomography Measurements and 3D Optical Flow«. In: Advanced Engineering Materials (2024).
Tessa Nogatz, Anna Nowacka, Claudia Redenbach und Katja Schladitz. »Assessing cracks in 3D images of concrete by a fast and efficient motion estimation algorithm«. In: 20th ICEM. 2023.
Tessa Nogatz, Claudia Redenbach und Katja Schladitz. »3D optical flow for large CT data of materials microstructures«. In: Strain 58.3 (2022), e12412.
Tessa Nogatz, Claudia Redenbach und Katja Schladitz. »Ex and in situ tests of materials: From design to materials parameters via motion estimation«. In: 12th iCT. 2022.
Tessa Kuschnerus, Claudia Redenbach, Katja Schladitz, Thomas Wagner und Friedrich Zerling. »Insitu testing with diffeomorphic mapping«. In: 4th ICTMS. 2019.

Motion Estimation inIn Situ Materials Tests

Dr. Tessa Nogatztessa.nogatz.net

3D Representations of Materials

Foam

Fiber Composites

Concrete

In Situ Materials Test

A General Framework for Motion Estimation

Local Digital Volume Correlation

Global Digital Volume Correlation

Medical Image Registration

Comparison of State-of-the-Art Methods

Total Variation Optical Flow

TV Optical Flow Solved by Approximation12

TV Optical Flow Solved by Discretization12

Comparison of Total Variation Based Methods

In Situ Test of Metal Matrix Composite Foam

Full Series of MMC Foam

DisplacementField in \(w\)

Motion Estimation in
In Situ Materials Tests

Dr. Tessa Nogatz
tessa.nogatz.net

TV Optical Flow Solved by Approximation¹²

TV Optical Flow Solved by Discretization¹²

Displacement
Field in \(w\)