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Augmented Lagrangian Digital Volume Correlation (ALDVC)

Abstract

Digital volume correlation (DVC), the volumetric extension of the popular digital image correlation (DIC) technique, is a powerful experimental tool for measuring 3D volumetric full-field displacements and strains. Most current DVC algorithms can be categorized into either local or finite-element-based global methods. As with most experimental approaches, there are drawbacks with each of these methods. In the local method the subvolume deformations are estimated independently and the computed displacement field may not necessarily be kinematically compatible. Thus, the deformation gradients can be noisy, especially when using small volumetric subsets. Although the global method often enforces kinematic compatibility, it generally incurs substantially greater computational costs than its local counterpart, which is especially significant for large volumetric data sets. To address these shortcomings, we present a new hybrid DVC algorithm, called augmented Lagrangian digital volume correlation (ALDVC), which combines the advantages of both the local (fast computation time) and global (compatible displacement field) methods. This new algorithm builds on our recent work on the augmented Lagrangian digital image correlation (2D-ALDIC) technique and solves the general motion optimization problem by using the alternating direction method of multipliers (ADMM). We demonstrate that our ALDVC algorithm has high accuracy and precision while maintaining low computational cost, and is a significant improvement compared to current local and global DVC methods. ALDVC is a computationally efficient algorithm to measure 3D volumetric displacements and strains. An open-source Matlab implementation is freely available.

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Notes

  1. 1.

    Both tri-cubic and tri-cubic spline interpolations are commonly used and interpolation bias errors are O(10− 3) voxels, cf. [27].

  2. 2.

    Practically, ALDVC ADMM can be stopped after \(3 \sim 5\) iterations.

  3. 3.

    To probe baseline local FFT performance, rather than that of a specific algorithm, we reduce the algorithm of [8] to a relatively generic “local FFT” method by removing the in-built IDM subset refinement and filtering steps.

  4. 4.

    This includes the iterative deformation method (IDM), but at a strictly enforced subset size, which will produce suboptimal results when compared to the self-refining FIDVC algorithm

  5. 5.

    Besides using a constant regularization coefficient, there are also methods to optimize a spatially variable, dependent regularization coefficient α, to achieve better performance [17]. However, these methods usually are extremely expensive.

  6. 6.

    See https://github.com/FranckLab

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Acknowledgment

We gratefully acknowledge funding support from the Office of Naval Research (Dr. Timothy Bentley; grant N000141712058) and the National Institutes of Health (grant R01 AI116629). The authors thank Prof. Kaushik Bhattacharya, Dr. Mohak Patel, and Dr. Orion Kafka for helpful discussions.

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Appendices

Appendix A: Local FFT-based and Iterative Image Deformation Method

In the local FFT-based DVC method, the volume of interest (VOI) is divided into local subvolumes (subsets) and the degrees of freedom (DoFs) governing the local deformation of each subvolume are assumed to be represented by a piecewise constant translation

$$ \mathbf{y}(\mathbf{X}) = \mathbf{X}+ \mathbf{u}(\mathbf{X}) = \mathbf{X} + \sum\limits_{i} \left( \mathbf{u}_{i} \right) \chi_{i} (\mathbf{X}), $$
(14)

where ui is the translation vector of the center of each local subset Ωi, and χi is the characteristic or index function

$$ \chi_{i} = \left\lbrace \begin{array}{ll} &1, \quad \mathbf{X} \in {\Omega}_{i}, \\ &0, \quad \mathbf{X} \notin {\Omega}_{i}. \end{array} \right. $$
(15)

Using this piecewise translation formulation (14), the optimization problem (2) decomposes into a number of independent optimization problems over translation vector variables, where the objective function can be computed very efficiently using the fast Fourier transform (FFT) method [4]

$$ C_{\text{CC}}(\mathbf{u}) = \mathcal{F}^{-1} \left[ \overline{\mathcal{F}(f)} \odot \mathcal{F}(g) \right], $$
(16)

where “\(\overline {\ \cdot \ }\)” denotes the complex conjugate, and “⊙” is the Hadamard product where multiplication is conducted element-wise. The displacement vector u can be calculated with sub-voxel resolution by fitting the 33 voxel cross correlation peak to a Gaussian polynomial or a quadratic polynomial [4].

To account for large material deformations including large stretches, rotations, and shear, Bar-Kochba et al. [8] significantly improved on prior local FFT methods by iteratively warping the reference and deformed images using a linearized local displacement field that is interpolated from the current displacement field until the reference and deformed images converge to the same final configuration, while introducing several filtering steps to improve accuracy and convergence. This can be further sped up by using an initial guess transfer scheme [44] and improved by the introduction of quality factors of the cross-correlation space to detect and remove poor FFT results among subvolumes [11].

Appendix B: Non-FFT-based Local IC-GN DVC Method

Similarly to the FFT-based methods, for non-FFT-based DVC methods each subvolume is assumed to be independent (although initial guess propagation is often used, e.g, [28]) with regard to its neighboring subvolumes and the deformation field has the general piecewise affine deformation formulation

$$ \mathbf{y}(\mathbf{X}) = \mathbf{X}+ \mathbf{u}(\mathbf{X}) = \mathbf{X} + \sum\limits_{i} \left( \mathbf{u}_{i} + \mathbf{F}_{i} (\mathbf{X}- \mathbf{X}_{i0} ) \right) \chi_{i} (\mathbf{X}), $$
(17)

where Xi0 is the center point of local subvolume Ωi, ui is the displacement of Xi0 and Fi is the affine deformation gradient tensor of Ωi minus identity.

The optimization problem (3) decomposes into a number of decoupled problems with, typically, twelve degrees of freedom {ux,uy,uz,Fxx,Fxy,Fxz,Fyx,Fyy,Fyz,Fzx, Fzy,Fzz} for the subvolume’s first order shape function and can be solved in parallel. This optimization problem is as follows:

$$ \begin{array}{@{}rcl@{}} C_{\text{SSD} i} &=& {\int}_{{\Omega}_{i}} \left| f(\mathbf{X}) - g(\mathbf{X}+ \mathbf{u}_{i} + \mathbf{F}_{i} (\mathbf{X}- \mathbf{X}_{i0} ) ) \right|^{2} d \mathbf{X}\\ &\rightarrow& \text{minimize over } \lbrace \mathbf{F}_{i}, \mathbf{u}_{i} \rbrace \end{array} $$
(18)

and can be solved efficiently using an inverse compositional Gauss-Newton (IC-GN) scheme. Given the current iteration of the deformation map yk, we seek the updated deformation map yk+ 1. It is convenient to define the inverse map φk such that φk(yk(X)) = X. We also define the increment ψk through yk+ 1 = ψkyk as shown in Fig. 1(a). At each IC-GN iteration, we make the approximation \(\boldsymbol {\psi ^{k}} \approx \mathbf {z} + \mathbf {v} + \mathbf {H}(\mathbf {z}-\mathbf {z}_{0})\), and use a change of variables to minimize the SSD correlation function in the current iteration configuration

$$ \begin{array}{@{}rcl@{}} C_{\text{SSD} i} &=& {\int}_{{{\Omega}_{i}^{k}}} \left| f(\boldsymbol{\varphi}^{k}(\mathbf{z})) - g(\mathbf{z}) - \nabla g(\mathbf{z})\right.\\ &&\left. \cdot \left( \mathbf{v}+ \mathbf{H}(\mathbf{z}-\mathbf{z}_{0}) \vphantom{\boldsymbol{\varphi}^{k}}\right) \right|^{2} d \mathbf{z}. \end{array} $$
(19)

Minimizing over {v,H}, we obtain

$$ \left( \begin{array}{cc} a_{lp} & b_{lqr} \\ b_{mnp} & c_{mnqr} \end{array}\right) \left( \begin{array}{cc} w_{p} \\ H_{qr} \end{array}\right) = \left( \begin{array}{cc} d_{l} \\ e_{mn} \end{array}\right) $$
(20)

where

$$ \begin{array}{@{}rcl@{}} a_{lp} &=& 2 {\int}_{{{\Omega}_{i}^{k}}} g_{,l} g_{,p} d \mathbf{z}, \end{array} $$
(21)
$$ \begin{array}{@{}rcl@{}} b_{lqr} &=& {\int}_{{{\Omega}_{i}^{k}}} g_{,l}g_{,q}(z_{r} - z_{0r}) d \mathbf{z}, \end{array} $$
(22)
$$ \begin{array}{@{}rcl@{}} c_{mnqr} &=& 2 {\int}_{{{\Omega}_{i}^{k}}} g_{,m} (z_{n} - z_{0n}) g_{,q} (z_{r} - z_{0r}) d \mathbf{z}, \end{array} $$
(23)
$$ \begin{array}{@{}rcl@{}} d_{l} &=& {\int}_{{{\Omega}_{i}^{k}}} (f-g) g_{,l} d \mathbf{z}, \end{array} $$
(24)
$$ \begin{array}{@{}rcl@{}} e_{mn} &=& {\int}_{{{\Omega}_{i}^{k}}} (f-g) g_{,m} (z_{n} - z_{0n}) d\mathbf{z} \end{array} $$
(25)

and g,l = g/zl, etc. We solve (20) for {v,H} to obtain ψk. We then obtain the new (inverse) deformation φk+ 1 = φk ∘ (ψk)− 1. In practice, we compute the integrals (or voxel-wise sums) over the final deformed configuration instead of the intermediate iterating configurations. This significantly decreases computational cost because all the gradients ∇g only need to be computed once and remain unchanged during each IC-GN iteration as summarized in Algorithm 2.

figureb

Appendix C: Global DVC Method

For the global DVC method, we represent the global deformation using a global basis set, often based on a finite element formulation, such that the compatibility or continuity of the displacement field is automatically guaranteed (see Fig. 1(b)), i.e.,

$$ \mathbf{y}(\mathbf{X}) = \mathbf{X} + \mathbf{u}(\mathbf{X})=\mathbf{X} + \sum\limits_{p} {u}_{p} \boldsymbol{\psi}_{p} (\mathbf{X}) $$
(26)

where ψp(X) are chosen global basis functions and up are the unknown degrees of freedom. Thus, equation (3) becomes

$$ \begin{array}{@{}rcl@{}} C_{g} &=& {\int}_{\Omega} \left| f(\mathbf{X}) - g(\mathbf{X} + \sum\limits_{p} {u}_{p} \boldsymbol{\psi}_{p} (\mathbf{X}) ) \right|^{2} d \mathbf{X}\\ &\to& \text{minimize over } \{u_{p}\} . \end{array} $$
(27)

We can solve this problem iteratively by setting uk+ 1 = uk + δu and using the first order approximation

$$ g(\mathbf{y}(\mathbf{X})) = g(\mathbf{X}+\mathbf{u}_{k}(\mathbf{X}) + \delta \mathbf{u}) \approx g(\mathbf{X} + \mathbf{u}_{k}(\mathbf{X}) )+ \nabla g \cdot \delta \mathbf{u}(\mathbf{X}) $$
(28)

such that

$$ \begin{array}{@{}rcl@{}} C_{g} &\approx& {\int}_{\Omega} \left|\vphantom{\sum\limits_{p}~} f(\mathbf{X}) - g(\mathbf{X}+\mathbf{u}_{k}(\mathbf{X}))\right.\\ && \left. - \left( \sum\limits_{p} \delta u_{p} \boldsymbol{\psi}_{p} (\mathbf{X}) \right) \cdot \nabla g(\mathbf{X}) \right|^{2} d \mathbf{X}. \end{array} $$
(29)

This leads to a linear equation in δu

$$ M_{pq} \delta u_{q} = b_{p} $$
(30)

where

$$ \begin{array}{@{}rcl@{}} M_{pq} & = & \displaystyle{ {\int}_{\Omega} \boldsymbol{\psi}_{p}^{T} (\mathbf{X}) \left( \nabla g \right) \left( \nabla g \right)^{T} \boldsymbol{\psi}_{q}(\mathbf{X}) d \mathbf{X} }, \end{array} $$
(31)
$$ \begin{array}{@{}rcl@{}} b_{p} & = & \displaystyle{{\int}_{\Omega} \left( f(\mathbf{X}) - g(\mathbf{X} + \mathbf{u}_{k}(\mathbf{X})) \right) \boldsymbol{\psi}_{p}^{T}(\mathbf{X}) \nabla g (\mathbf{X}) d \mathbf{X} } . \end{array} $$
(32)

In this paper, we use an 8-node hexahedron (HEX8) finite element mesh in our global DVC method, and the algorithm is summarized in Algorithm 3. Alternately, if the displacements are small, we can treat (30) as a linear problem with δu as the incremental displacement.

Global DVC is usually computationally expensive since the size of the linear problem (30) is equal to the number of basis functions or the size of the finite element discretization. While parallel implementation strategies exist, they can be cumbersome to utilize in practice. The problem is exacerbated when analyzing volumetric time-lapse data with multiple image pairs.

figurec

Appendix D: Synthetic 3D Volume Images

The synthetic digital volume images in “Assessing the Accuracy and Precision of the ALDVC Algorithm” are generated to mimic actual volumetric experimental images. In each reference volume, isolated spherical beads are randomly seeded using a 3D Gaussian intensity profile as an approximation of a random, isotropic image pattern (e.g., mimicking the point spread function (PSF) of a laser scanning confocal microscope [8, 43]). A typical Gaussian PSF with amplitude A and spread (i.e., standard deviation) σ is expressed as

$$ \text{PSF}(x) = A \exp \left( - \sum\limits_{i=1}^{3} \frac{{x_{i}^{2}}}{2 \sigma^{2}} \right). $$
(33)

A PSF with a spread σ= 1 approximates a spherical particle in the volume image with a diameter of approximately 5 voxels. All the beads are sampled randomly with seeding density 0.006 beads per voxel. To avoid beads overlapping in the synthetic images, a Poisson disc sampling algorithm is used to seed center-point locations in the volume images with a minimum separation distance between particles equal to the particle diameter (see [43]), see Fig. 9. The particle positions in the deformed image are calculated via the imposed displacement field and all the deformed volume images are warped from the reference to deformed configuration using tri-cubic interpolation [27].

Fig. 9
figure9

(a) Representative, synthetically generated DVC volume using a typical Gaussian-like point spread function (PSF) mimicking typical diffraction-limited optical systems. (b) Inset from (a)

Appendix E: Indentation Experiment Preparation

In our experiment, polyacrylamide (PA) hydrogels of approximately 400 μm in thickness were polymerized in the well of a glass-bottomed 24-well plate, pre-treated with 0.5% 3-aminopropyl-trimethoxysilane (Sigma-Aldrich, MO) and 0.5% glutaraldehyde (Polysciences, Inc., PA) as described previously [45,46,47]. The hydrogels were fabricated using 3% acrylamide (Bio-Rad, CA) and 0.06% bis-acrylamide (Bio-Rad, CA), following a previously described protocol [45, 46, 48] with an approximate final elastic modulus of 480 Pa. Cross-linking of the PA hydrogels was achieved with the addition of ammonium persulfate (Sigma-Aldrich, MO) and N,N,N,N-tetramethylethylenediamine (ThermoFisher Scientific, MA). Hydrogels were doped with 10% (w/v) 1 μm diameter carboxylate-modified fluorescent microspheres (ThermoFisher Scientific, MA) as fiducial markers. Hydrogels were left to fully swell in deionized water overnight. All the related parameters are summarized in Table 5.

Table 5 Details of hydrogel indentation experiment parameters

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Yang, J., Hazlett, L., Landauer, A. et al. Augmented Lagrangian Digital Volume Correlation (ALDVC). Exp Mech 60, 1205–1223 (2020). https://doi.org/10.1007/s11340-020-00607-3

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Keywords

  • Digital volume correlation (DVC)
  • Augmented Lagrangian
  • Alternating direction method of multipliers (ADMM)