Depth from defocus measurement method based on liquid crystal lens

DFD optic model

Suppose the focal length of the main lens is fg, the optical power is Pg = 1/fg. In this study, the aperture of the main lens is 2.0 mm and fg = 8.0 mm. The power of the compound lens is then P = Pg + PLC-dPgPLC. So, the power range of the compound lens is [Pmin, Pmax] = [Pg+Pmax −−dPgPmax −, Pg+Pmax+−dPgPmax+].

We define ∇P as:Let the image on the image sensor of one object O of object distance u is I when the optical power of the system is P, we introduce a parameter ∂∈(−1,1) to describe P, that is:P = (Pmax + Pmin)/2 when ∂ = 0, P = Pmax when ∂ = + 1, and P = Pmin when ∂ = −1. The images I, I1, and I2 of the object O correspond, respectively, the optical power of P, Pmax, and Pmin, as shown in Fig.

Efficient DFD computation methods have been proposed in previous works [25–27]. In this paper, we just select an approach based on rational filters [14] which is better for real time computation. Rational filters are broadband linear filters. We can implement them with small convolution kernels, low computational cost and high spatial resolution. In our system, we also adopt the simple model: MP=M(u,v,∂)P(u,v,∂)=GP1(u,v)GM1(u,v)β+GP2(u,v)GM1(u,v)β3. The theory of the rational filters generation, implementation has been clarified and detailed by Watanabe, et al [16].

In fact, the final design issue pertains to maximum frequency frmax. Since the discrete Fourier transform of a kernel of size ks has the minimum discrete frequency period of 1/ks. The maximum frequency shall be above 1/ks. According to Nyquist theorem, we can specify the condition as: frmax≥2/ks where frmax = 0.73(u0-fg)/(2afgu0∇P). This condition can be interpreted as follows: The max blur circle diameter 2afgu0∇P/(u0-fg) must be smaller than 73% of the kernel size ks.

For our imaging system with LC lens, if the radius of the aperture is 2.0 mm, the focal length fg = 8 mm, the focal power of the LC lens is P∈[-2, 2] m−1, one can easily get the maximum radius of defocused spot is rmax = 32 μm. Provided the pixel size is 2.2 μm, rmax = 14.544 pixels. From this computation we can clear acknowledge how large the rational filter we need and how to setup the experiment.