This study proposes the following two contributions:

1.

A retroreflective model that considers diffraction and ray shifts

2.

A method for estimating the model parameters using differentiable rendering

3.1 Retroreflective model

As shown in Fig. 2, a retroreflector consists of two layers: a surface coating layer (surface) and corner-cube arrays (retroreflective layer). The proposed model f is expressed as the sum of a surface specular reflection \(f_R\) and an internal retroreflection \(f_\) (Eq. (1)). The symbols and mathematical expressions are shown in Table 1.

$$\begin f = f_R + f_ \end$$

(1)

3.1.1 Surface reflection

The surface reflection \(f_R\) was represented using Microfacet BRDF by Walter et al. [12], following Saito et al [5]. The BRDF \(f_r\) of surface reflection and the BTDF \(f_t\) of transmitted light are given by Eq. (2) and (3), respectively.

$$\begin \begin f_R(}, }_, })&= f_r(}, }_, }) \\ &= \frac}, }_r)G(}, }_, }_r)D(}_r)}}\cdot }||}_\cdot }|} \end \end$$

(2)

$$\begin \begin f_t(}, }_i, }) =&\frac}\cdot }_t||}_i\cdot }_t|}}\cdot ||}_i\cdot }|} \\&\frac^2(1-F(}, }_t))G(}, }_i, }_t)D(}_t)}(}\cdot }_t)+\eta _(}_i\cdot }_t))^2} \end \end$$

(3)

3.1.2 Retroreflection

The retroreflection \(f_\) represents light that transmits through the surface (\(f_t\)), is retroreflected at the corner surface (\(f_\)), and then transmits again(\(f_t\)), as expressed by Eq. (4). Diffraction is expressed as \(f_\), and ray shift is reproduced by the distribution obtained by ray simulation.

$$\begin f_(}, }_) = f_(}, }_i, })f_(}_i, }_o)f_(}_o, }_, }) \end$$

(4)

In this study, the intensity distribution of diffracted light in the retroreflector is approximated by a sinc-squared function. When viewed from above, as shown on the right side of Fig. 3, light entering the central hexagonal region of the corner surface is retroreflected [16]. Because incident light follows six different paths depending on the incident positions on the corner cube, diffraction occurs at each sub-aperture formed by dividing the hexagonal region into six rectangles. Because the intensity distribution of diffracted light from a rectangular aperture follows a sinc-squared function, the same distribution is applied to the diffracted light in the corner cube.

During rendering, rays are sampled based on the sinc-squared function as a probability distribution of outgoing directions. As shown on the left side of Fig. 3, when light enters at an angle \(\theta _i\), it exits at a deviated angle \(\theta _o\) toward the mid-air image located a distance R from the corner surface. The arrival position shifts by \(x = R\tan \). Since the positional deviation x follows the sinc-squared function, the retroreflective model is expressed by Eq. (5). Here, \(\mu\) denotes the reflectance of retroreflection, and a represents the spread of sinc function.

$$\begin f_(\theta _i, \theta _o) = \frac)})^2} \end$$

(5)

Fig. 3

Diffraction and symbols in Eq. (5) (left), and top view of the corner cube and diffraction aperture (right). Diffraction is represented by a sinc-squared function, which is treated as a probability distribution of the outgoing direction. The incident rays follow six distinct paths depending on the incident position on the corner cube (one path is shown in the figure). Diffraction then occurs through a square aperture formed by isolating the central hexagonal region, as indicated by the gray area

For reproducing ray shifts, the distribution of shift amount is obtained through ray simulations in the renderer. As illustrated in Fig. 4, rays are emitted from a generation plane toward the 3D model of the corner cube. The displacement between incident and outgoing ray positions after three reflections is calculated as a relative value with the corner cube edge length normalized to 1.0. Rays are generated from varying positions on the plane, and a histogram of shift amount is constructed. This process is repeated while changing the azimuth angle \(\phi\) and incident angle \(\theta\), generating distributions for each angle. Figure 5 shows an example of the ray-shift distribution for \(\phi = 0^\) and \(\theta = 0^\).

During rendering, the shift amount is calculated based on the distribution obtained from ray simulation, and the outgoing position is changed. When light enters at an incident angle \(\theta\), the corresponding angular distribution is used to compute the shift amount. Since the corner surface size varies with the retroreflector, the corner size c (defined as the edge length of the corner cube in Fig. 4) is used to scale the displacement. The direction of the ray shift is determined randomly under the assumption of isotropy.

Fig. 4

Ray shift is reproduced by ray simulation within the renderer. Rays are emitted toward a 3D model of the corner cube, and the shift amount between the incident and outgoing positions is calculated. A histogram of the shift amount is then generated for each incident angle

Fig. 5

Example of ray-shift distribution at \(\phi = 0^\) and \(\theta = 0^\)

3.1.3 Model parameters

The proposed model includes three parameters: a, c, and \(\mu\). The diffraction parameter a represents the degree of diffraction in the retroreflected light. Smaller values of a result in a wider sinc function and therefore stronger diffraction. The corner size c varies depending on the retroreflectors, and an appropriate value must be determined individually for each retroreflector. The reflectance \(\mu\) represents the proportion of incident light that is retroreflected by the retroreflector.

Fig. 6

End-to-end parameter estimation pipeline based on differentiable rendering. The parameters a and c are estimated by minimizing loss functions defined using MTF-based blur metrics, including the MSE of the MTF curves (Eq. (6)) and the difference in slopes of the average MTF plotted as a function of floating distance (Eq.(7)). The reflectance parameter \(\mu\) is estimated by minimizing the loss based on the luminance ratio difference between the light source and the mid-air image under real and simulated conditions (Eq. (8))

3.2 Differentiable rendering pipeline

First, a and c are jointly estimated using the Modulation Transfer Function (MTF) as a blur metric to reproduce blur variation with floating distance. The MTF of the mid-air image viewed from the front is measured, and the mean value below 1 lp/mm is plotted as a function of the floating distance, showing a linear relationship (top right of Fig. 6). At the intercept of the fitted lines of the real and CG data (at the shortest floating distance), the Mean Squared Error (MSE) \(\mathcal (a)\) of the MTF curve is used as a loss function (Eq. (6)) to estimate a. Here, \(y(\nu _i)\) denotes the estimated MTF curve, and \(t(\nu _i)\) denotes the measured MTF curve of real mid-air image. The slope difference of fitted lines (\(d_\), \(d_\)) is used as a loss function to estimate c (Eq. (7)). This enables reproduction of blur variations with respect to floating distance.

Furthermore, at the shortest floating distance, a is estimated for each observation angle using the MSE of the MTF curve as a loss function (Eq. (6)). This enables the reproduction of blur variations depending on observation angle. Parameter c is estimated only at an observation angle of \(0^\) because ray shifts caused by oblique incident angle are reproduced using the shift distribution.

$$\begin \mathcal (a) = \frac\sum _^N(y(\nu _i) - t(\nu _i))^2 \end$$

(6)

$$\begin \mathcal (c) = (d_ - d_)^2 \end$$

(7)

Next, the reflectance \(\mu\) is estimated using the luminance ratio between the light source and the mid-air image. First, the luminance ratio \(r_\) of the real mid-air image is calculated. Then, \(\mu\) is estimated for each observation angle by minimizing the loss function defined in Eq. (8), so that the CG luminance ratio \(r_\) approaches \(r_\).

$$\begin \mathcal (\mu ) = (r_ - r_)^2 \end$$

(8)

In our pipeline, differentiable rendering is used as a core component of an end-to-end optimization framework for estimating AIRR parameters. Specifically, the differentiable computation graph spans from the AIRR model parameters to the rendered mid-air images through differentiable optical image formation implemented in Mitsuba 3. The MTF is computed from the rendered images using the slanted-edge method and is treated as a post-processing analysis step; therefore, it is not included in the differentiable computation graph. Despite this indirect formulation, the optimization was empirically stable in our experiments. This stability is attributed to the low-dimensional parameter space, the use of aggregated MTF-based metrics, and the controlled experimental setup with fixed imaging conditions.

Table 2 Results of parameter estimation