## 3 Background  
For brevity, we will give a brief overview of fluid flow, and then introduce the previous method of optical flow solve [5]. Throughout this paper we will use symbol (e.g., \(u\)) to denote continuous value, and bold symbol (e.g., \(\mathbf{u}\)) to denote the discrete counterpart. The symbol with subscript (e.g., \(\phi_{r}\)) is usually used to represent the derivative of the corresponding variable. Thus, a subscript will not denote the component of a vector except for the special statement. **Fluid Flow**. The fluid flow is governed by incompressible Navier-Stokes equations [24], and a set of partial different equations that are supposed to hold throughout the fluid  
\[\frac{\partial v}{\partial t}+v\cdot\
abla v=f-\frac{\
abla P}{\rho}+\gamma \
abla^{2}v\ \ \text{and}\ \ \
abla\cdot v=0 \tag{1}\]  
where \(v\) is used in fluid for the velocity of the fluid, \(\frac{\partial v}{\partial t}\) is the commonly used material derivative w.r.t. time \(t\) for advection, \(f\) is the external body forces(to make the fluid behave in some desired way), the letter \(\rho\) stands for the fluid density, \(P\) is the force due to pressure that the fluid exerts on anything, and \(\gamma\) is the \"kinematic viscosity\". Following the description of [19], the observable concentration of a marker fluid is denoted with \(\phi\) and the advection of a scalar field \(\phi\) based on a velocity field \(v\) can be expressed as \(\phi_{t}+\
abla\cdot(v\phi)-\gamma\
abla^{2}\phi=s\), where \(s\) is a source term, including internal and external sources. Assume that the marker fluid and the ambient fluid have approximately the same mass density, the flow will perform a pure advection when the negligible diffusivity \((\gamma=0)\), incompressibility \((\
abla\cdot v=0)\), and \(s=0\), the standard form of advection is given by  
\[\phi_{t}+v\cdot\
abla\phi=0. \tag{2}\]  
Equation (2) is equivalent to the brightness-constancy constraint of optical flow as introduced by Horn and Schunck algorithm [17]. **Optical Flow solve**. The work of [5] uses the \(L_{2}\) norm based optical flow solve to perform a registration step. In this setting, two animations from different simulation setups are taken as the inputs: a source and a target. Each animation is a sequence of surfaces over time, where the surfaces can be concatenated as time slices of a 4D volume. The registration mainly transforms source to further fit it into target through deformations \(u\). A flow simulations can be represented by 4D signed-distance values \(\phi\), and the parameter \(r\) is an external control knob to modify the simulation from source to target. The regular optical flow usually recovers the motion of dynamic object from two images taken at different times. [PERSON] [5] instead uses optical flow to recover the motion among surfaces of different simulations. The algorithm employs the brightness-constancy assumption to recover the flow with motion discontinuities. Please refer to [5] for further details. Applying the optical flow alone is underdetermined because of the non-linearity, so that various additional constraints or priors are imposed to regularize the problem. The Tikhonov regularizer term restricts the deformation to be as small as possible. \(E_{smooth}\) defines a \(L_{2}\) regularizer which smoothes the consistent motions along spatial as well as temporal dimensions. Given the optical flow constraint and some common regularizers, the estimation of deformation field \(u\) is achieved by minimizing the following energy  
\[\mathop{\mathbf{min}}_{u}E_{data}(u)+\beta_{T}E_{Tikhonov}(u)+\beta_{S}E_{ smooth}(u) \tag{3}\]  
where \(\beta_{S}\) and \(\beta_{T}\) are the weights directly controlling the importance of each energy term.