In general, a UV printer prints the bottom layer with a matte white ink to remove the translucency of base materials. On the other hand, we rather utilize the translucency without printing the white layer. The translucency of the printed object depends on both of the translucency of the base materials and inks and can be changed by a combination of the factors. If the translucency of the printed objects in all combinations can be measured, it is easy to control the translucency in an example-based manner. However, it is almost impossible to print and measure in all the combinations. Therefore, we fuse such a manner with a different manner based on physics model.

The proposed method consists of measurement and fabrication steps, as shown in Fig. 2. In the measurement step, the individual translucencies of the base materials and inks are measured. The translucency of the printed object in a combination is rendered based on a physics model. The translucency in all the combinations can be rendered by simulation with few measurements. Therefore, it enables to build a lookup table between the combinations and translucency, like the example-based manner. In the fabrication step, given a query about translucency, either measured or manually designed, a combination can be found in the lookup table so that the translucency of the printed object is the most similar to the query. The rest of this section explains the rendering method, measuring the translucency, and building the lookup table and finding the best combination.

### 2.1 Rendering the translucency of a layered object

The translucency of the printed object depends on both of the translucency of the base material and ink, as mentioned above. A key feature is that the printed object has a layered structure, as shown in Fig. 1b. Thus, we apply Kubelka’s layer model [6] to render the translucency. The original model formulates scalar reflectance *r* and transmittance *t* of a two-layered object, as follows:

$$\left\{ \begin{array}{l} r = r_1 + t_{1}^2 r_2 (1 + r_{1} r_{2} + \cdots) = r_{1} + \frac{t_{1}^{2} r_{2}}{1 - r_{1} r_{2}}, \,\,\,\,\,\,\,\,\,(1)\\ t = t_{1} t_{2} (1 + r_{1} r_{2} + \cdots) = \frac{t_{1} t_{2}}{1 - r_{1} r_{2}}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2) \end{array} \right. $$

where *r*_{1} and *t*_{1} are the reflectance and transmittance of the top layer and *r*_{2} and *t*_{2} those of the bottom one, respectively. Now, let us take subsurface scattering into consideration because it is a main cause why an object looks translucent. Since subsurface scattering diffusely spreads light, it is modeled by point spread function (PSF). The PSF can also be separated into reflective and transmissive PSFs. The reflective and transmissive PSFs of the top layer are defined as *R*_{1}(*x*),*T*_{1}(*x*), respectively. As well, those of the bottom one are defined as *R*_{2}(*x*),*T*_{2}(*x*). The reflective PSF of the two-layered object is, therefore, written as

$$\begin{array}{*{20}l} R(x) =& R_{1}(x) + ((T_{1} * R_{2}) * T_{1})(x) \\ &+ ((((T_{1} * R_{2}) * R_{1}) * R_{2}) * T_{1})(x) + \cdots, \end{array} $$

(3)

where “ ∗” is the convolution operator. The transmissive PSF can also be written as well but it is omitted here for saving the space. By Fourier transform, Eq. (3) is transformed into

$$\begin{array}{*{20}l} {\mathcal F}[\!R] &= {\mathcal F}[R_{1}] + {\mathcal F}[\!T_{1}] {\mathcal F}[\!R_{2}] {\mathcal F}[\!T_{1}] + \cdots \\ &= {\mathcal F}[\!R_{1}] + \frac{{{\mathcal F}[\!T_{1}]}^{2} {\mathcal F}[\!R_{2}]}{1 - {\mathcal F}[\!R_{1}] {\mathcal F}[\!R_{2}]}, \end{array} $$

(4)

where \({\mathcal F}[\!\cdot ]\) means Fourier transform and the argument *x* is omitted. If the PSFs are isotropic, the imaginary parts in the frequency domain become zero. Therefore, \({\mathcal F}[\!R]\) can be regarded as the reflective modulation transfer function (MTF), which is defined as \(\hat {R}(f_{x})\), where *f*_{
x
} is the spatial frequency. Finally, the reflective and transmissive MTFs of the two-layered object are written, as follows:

$$ \left\{ \begin{array}{l} \hat{R}(f_{x}) \,= \hat{R_{1}}(f_{x}) + \frac{\hat{T_{1}}^{2}(f_{x}) \hat{R_{2}}(f_{x})}{1 - \hat{R_{1}}(f_{x}) \hat{R_{2}}(f_{x})}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(5) \\ \hat{T}(f_{x}) = \frac{\hat{T_{1}}(f_{x}) \hat{T_{2}}(f_{x})}{1 - \hat{R_{1}}(f_{x}) \hat{R_{2}}(f_{x})}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6) \end{array} \right. $$

where \(\hat {A}(f_{x})\) means the corresponding MTF to a PSF *A*(*x*).

Once the individual reflective and transmissive MTFs of the base materials and inks are given, those of the printed object can be rendered by using Eqs. (5) and (6). Recursively applying Eqs. (5) and (6), it is also possible to render the MTFs of a multi-layered object.

### 2.2 Measuring the modulation transfer functions

The MTFs are employed for explaining the translucency, as mentioned above. In this paper, to measure the MTFs, we apply the modulated imaging, proposed by Cuccia et al. [7] for measuring the quantitative scattering and absorption coefficients of a medium. The modulated imaging is based on measuring MTFs in a Pro-Cam system. Projecting a sinusoidal pattern onto the medium, light scattering in the medium blurs the pattern, and hence, the amplitude of the measured sinusoidal pattern is attenuated in comparison with that of the projected one. The MTF consists of the rates of attenuation at different frequencies. We refer the readers to [7] for the details.

We can directly measure the MTFs of the base materials in the Pro-Cam system. However, it is impossible to directly measure the MTFs of the inks because the ink layer has to be printed on an object. Therefore, we estimate those by utilizing Eq. (5). As printing an ink whose reflective and transmissive MTFs are \(\hat {R_{1}}(f_{x})\) and \(\hat {T_{1}}(f_{x})\), respectively, on a mirror, the reflective MTF of the printed object \(\hat {M_{1}}(f_{x})\) is, as below:

$$\begin{array}{*{20}l} \hat{M_{1}}(f_{x}) = \hat{R_{1}}(f_{x}) + \frac{\hat{T_{1}}^{2}(f_{x})}{1 - \hat{R_{1}}(f_{x})}, \end{array} $$

(7)

where we assume \(\hat {R_{2}}(f_{x}) = 1\) in Eq. (5) because of only specular reflection on a mirror. Also, as printing the same ink twice, the reflective MTF \(\hat {M_{2}}(f_{x})\) is, as below:

$$\begin{array}{*{20}l} \hat{M_{2}}(f_{x}) = \hat{R_{1}}(f_{x}) + \frac{\hat{T_{1}}^{2}(f_{x}) \hat{M_{1}}(f_{X})}{1 - \hat{R_{1}}(f_{x}) \hat{M_{1}}(f_{X})}. \end{array} $$

(8)

Now, the reflective and transmissive MTFs of the ink, \(\hat {R_{1}}(f_{x})\) and \(\hat {T_{1}}(f_{x})\), can be estimated from Eqs. (7) and (8) at least because both of \(\hat {M_{1}}\) and \(\hat {M_{2}}\) can be measured. It is also possible to increase the number of the same ink layers and then use it for a stable estimation in a least-squares method.

### 2.3 The lookup table

Finally, it is possible to build the lookup table by using the rendering method with the measurements. The base materials should be translucent, such as rubber and wax, not transparent and opaque.

The lookup table is built by rendering in all the combinations. Building the lookup table is a time-consuming process but it is required only once.

In the fabrication step, given a query about translucency, a combination is searched in the lookup table so that the translucency of the printed object is the most similar to the query. Here, it is required to define distance for representing how much a MTF is close to another. Thus, we employ a root-mean-square error (RMSE) for that. The distance *E*_{
A
B
} between MTFs \(\hat {A}(f_{x})\) and \(\hat {B}(f_{x})\) is defined as

$$ E_{AB} = \sqrt{\frac{1}{|\mathbb{F}|} \sum_{f_{x} \in \mathbb{F}}\left\{ \hat{A}(f_{x}) - \hat{B}(f_{x}) \right\}^{2}}, $$

(9)

where \(\mathbb {F}\) is a set of discrete frequencies to be used for calculating the distance.