## 3 Local Heat Flux Estimation Procedure
 \tag{15}\]  
Nonetheless, the minimization problem of Equation (15)--required to find the optimal regularization parameters for the Gaussian filter--may not exhibit a unique solution, i.e., Equation (15) is inherently ill-posed from a mathematical standpoint. To properly estimate the optimal parameters during space and time regularizations of the data, different approaches or assumptions have been proposed in the literature. [PERSON] et al. [41] employed the function specification method for the space-time heat flux estimation in a cylindrical channel. A bidimensional IHCP problem in transient conditions was considered. The two physics were treated separately: the temporal stabilization parameters were assessed by identifying the time instant at which the width of rate of representation [42] exhibited a slow increase for a single temperature sensor. A physical criterion, specifically designed for the studied application, was thus adopted. On the other hand, the spatial stabilization parameters were estimated in [41] by smoothing the spatial temperature fluctuations of a desired quantity. [PERSON] et al. [43] used the Tikhonov method as a regularization technique for the heat flux estimation in a metallic plate partially heated from one side under transient conditions. Again, space and time were treated separately, while the two optimal regularization parameters were here estimated by minimizing the RMS error between evaluated values and exact values of heat fluxes. Hence, in transient, bidimensional problems, the choice of optimal regularization parameters is strongly dependent on the specific study case. In the present work, another approach was proposed: a relationship between the two space-time regularization parameters was introduced by Equation (16), so that the number of independent variablesin Equation (15) was reduced to one, and the uniqueness of the minimization problem solution was satisfied. \[r_{cut-off}=\frac{u_{c}}{w_{c}} \tag{16}\]  
To determine the value of \(r_{cut-off}\), two auxiliary minimization problems are defined in Equations (17) and (18), varying one parameter at a time, while tending the other to infinity. In other words, the filter is first applied to space only in Equation (17), and therefore to time only in Equation (18), thus treating the two physics separately as in previous works [41, 43]. \[u_{c,opt}= argmin\ fun(u_{c},w_{c}) \tag{17}\] \[u_{c}\ \in\ R^{+}\] \[w_{c}\ \rightarrow\infty\] \[w_{c,opt}= argmin\ fun(u_{c},w_{c})\] (18) \[u_{c}\ \rightarrow\infty\] \[w_{c}\ \in\ R^{+}\]  
Finally, \(r_{cut-off}\) is calculated dividing \(u_{c,opt}\) by \(w_{c,opt}\); the new definition for the transfer function of Equation (12) reads as:  
\[H^{*}(u,v,w)=e^{-(u^{2}+\varphi^{2})/2(w_{c}\tau_{cut-off})^{2}}.e^{-w^{2}/2w_ {c}^{2}} \tag{19}\]  
while Equation (15) reduces to \(\underset{w_{c}\ \in\ R^{+}}{min}fun(w_{c})\), being \(w_{c}\) the only cut-off frequency to be optimized for the final 3D Gaussian filter application. For the considered experimental case, \(\sigma_{T}\) was evaluated in situ through IR acquisitions on the coated device under isothermal conditions (at known temperature of about 30 \({}^{\circ}\)C), and it was found equal to 0.04 K. Specifically, \(\sigma_{T}\) takes into account non-uniformities of the highly emissive coating, as well as imperfect flatness of the device outer surface, which could lead to non-physical, local temperature variations. In addition, \(r_{cut-off}\) was independently evaluated for every test condition and found in between 0.8 and 1.1 for low and high heat loads provided to the evaporator, respectively.