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Abstract
Mask-based lensless cameras break the constraints of traditional lens-based cameras, introducing highly flexible imaging systems. However, the inherent restrictions of imaging devices lead to low reconstruction quality. To overcome this challenge, we propose an explicit-restriction convolutional framework for lensless imaging, whose forward model effectively incorporates multiple restrictions by introducing the linear and noise-like nonlinear terms. As examples, numerical and experimental reconstructions based on the limitation of sensor size, pixel pitch, and bit depth are analyzed. By tailoring our framework for specific factors, better perceptual image quality or reconstructions with 4× pixel density can be achieved. This proposed framework can be extended to lensless imaging systems with different masks or structures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Imaging systems have followed a point-to-point imaging structure to record the scene since the invention of the earliest pinhole camera. Traditional cameras remain in this scheme and use lenses to amplify luminous flux at the cost of expense and complexity. Lensless imaging circumvents this trade-off by breaking the one-to-one mapping relation. By replacing lenses with the mask near to a sensor, one can get a cheap, thin, and flexible [1,2] lensless camera which is widely used in short-wavelength imaging [3,4], wide-field on-chip microscopy [5–7], fluorescence microscopy [8], in vivo imaging [9], etc.
In the process of imaging with a lensless camera, the mask acts as a coded aperture, and the scene is reconstructed by applying algorithms to the measurement. The mask can be an amplitude or phase modulator [10–12], a diffuser [13], or even a programmable device [14,15]. A wisely chosen mask can enhance reconstruction performance at robustness [16,17], super-resolution [14,18,19], and depth acquisition [20]. As to the forward imaging models and backward reconstruction algorithms, a universal imaging forward model for various masks is the convolutional model [10,11,13] which expressed the imaging process compactly and effectively. To solve the inverse problem, optimization-based algorithms is commonly used for high quality reconstruction. Those algorithms can utilize the properties of compressed sensing, which expand the capability boundaries of imaging system [10,21–25]. For example, one can achieve hyperspectral image [26,27], 3D image [11,13], or video [28] from a single shot. Deep learning-based algorithms had also be proposed to reconstruct the encoded measurement [29–35]. Those data-driven methods usually have better performance on reconstruction quality and speed at a cost of high training cost and low interpretability. Besides, special masks can also introduce new models from the aspect of separability [36] or moire fringe [37]. In conclusion, masks and algorithms offer lensless imaging systems a very high degree of flexibility.
Previous works have got remarkable progress by employing the convolutional model and compressed sensing algorithms [10,11,13,26–28] or taking the sensor size into consideration [13]. However, the success of the traditional convolutional model still highly relies on sufficient information acquisition. A practical coded aperture lensless camera has inherent distorted property caused by multifaceted restrictions, for example, limited sensor size and limited sensor bit depth. Traditional convolutional model fails when the measurement is under-sampled or affected by serious noise due to the restrictions.
In this work, we aim at presenting an explicit-restriction convolutional framework whose forward model takes the limited factors into account. The restrictions are categorized into the linear part (for example, limited sensor size and large pixel pitch) and the noise-like nonlinear part (for example, limited sensor bit depth) according to how they behave in the imaging process. The alternating direction method of multipliers (ADMM) algorithm [38] is employed as the reconstruction method for the forward model. We will show that the proposed explicit-restriction convolutional framework is versatile to both hardware restrictions and introduced factors, which means it has great potential to achieve superior image quality for lensless imaging systems.
2. Model and algorithm
2.1 Traditional convolutional forward model
The basic structure of the convolutional forward model for lensless cameras is shown in Figs. 1(a) and 1(b). The object of size $l_i$ is illuminated by incoherent light which cast the scene on the sensor after being modulated by the mask of size $l_m$. Without loss of generality, the scene, mask, and sensor are considered to be squares.
To analyze this process, a simple case of pinhole imaging is considered. As illustrated in the red beam in Fig. 1(b), a pinhole camera zooms the scene at a factor of ${d_2}/{d_1}$, where $d_1$ is the distance between scene and mask, $d_2$ is the distance between mask and sensor. The scaled scene is denoted as $a$ with size $L_i=\left ({d_2}/{d_1}\right )l_i$.
Fig. 1. Overview of the explicit-restriction convolutional model. (a) Configuration of the convolutional lensless imaging system. (b) Side view of (a). (c) reconstruction by ADMM. Prior knowledge of image sparsity and intensity boundaries are introduced to solve the ill-conditioned problem.
Then we consider the case that the object is a single point. As shown in the blue beam in Fig. 1(b), the response on the plane $z=d_2$ of a point source at $\left (x_0,y_0,-d_1\right )$ is represented by the point spread function $PSF_{mask}\left (x,y,d_2\right )$, which is calculated by the operator $\mathscr {P}$. $\mathscr {P}$ contains a free space propagation of distance $d_1$, a modulation by the mask, and a free space propagation of distance $d_2$
Notice that $\mathscr {P}$ is a linear operator, which means
This so-called memory effect can also be obtained by analogy with disordered masks [40,41]. Therefore, the lensless imaging process can be expressed as a convolution between zoomed scene $a$ and $PSF_{mask}$
Here we use bold capital letters for matrices and bold lowercase letters for vectors. $\boldsymbol {\bar {F}}$ is the matrix representation of the FFT operator. Notably, Eq. (5) is just a compact representation of convolution. Operators in Eq. (4) is more feasible for calculation because they can be implemented in computational complexity of $\mathscr {O}(n$log$n)$ [42].
2.2 Explicit-restriction convolutional forward model
As analyzed above, the traditional convolutional forward model in Eq. (5) has two key features.
Feature 1 Linear. The captured image $\boldsymbol {b}$ is a linear combination of scene $\boldsymbol {a}$ so the transformation can be represented by a compact matrix $\boldsymbol {M}$.
Feature 2 Solvable. The transform matrix $\boldsymbol {M}$ has a special structure, making efficient solving feasible.
Any forward model that meets the two features can be considered as a generalization of the traditional convolution model. This fact inspires us to eliminate restrictions by generalizing the forward models. Previous works have given an excellent example by introducing the cropping operator. We would like to present how an extension can deliver a vast generalization of the convolutional model in the next argument and examples.
To effectively study the restrict effect, we divide those restrictions into two categories. The first type, named linear restrictions, can be expressed as a linear transformation just like convolution operator $\boldsymbol {M}$. The situation is more complicated for another type which is called nonlinear restrictions because there is lack of concise expression. Fortunately, a particular kind of nonlinear restriction that behaves like noise can be solved by regularization.
Therefore, the proposed explicit-restriction convolutional forward model for lensless imaging systems is
2.3 ADMM for explicit-restriction convolutional forward model
Simple backpropagation algorithm hardly works for an explicit-restriction forward model. First, linear restrictions $\boldsymbol {L}$ is not a full rank matrix in many cases (see examples in Section 3), making Eq. (6) an ill-posed problem. Second, the noise-like nonlinear restrictions $\boldsymbol {\epsilon }$ is amplified during the backpropagation process. Instead, the compress sensing method is applicable in this situation due to the redundancy property of lensless imaging. Based on the gradient domain sparsity of natural images, total variation (TV) regularization is widely used in imaging reconstruction algorithms. By solving the $l_1$ norm minimization problem Eq. (7), we can pursue a sparse image in the gradient domain and therefore get the reconstruction $\hat {v}$ [24].
where $\mathcal {T}(f,x)={\rm sgn}(f) {\rm max}\left (|f|-x,0\right )$ is the soft threshold function. "${\rm sgn}(\cdot )$" is the signum function that values $+1$ for a positive number and $-1$ for a negative number. $\boldsymbol {I}$ is the identity matrix, and $(\cdot )^{T}$ is the conjugate transpose operator. $\boldsymbol {r}_k=\boldsymbol {M^{T}}(\mathrm{\mu} _1\boldsymbol {w}_{k+1}-\boldsymbol {\xi }_k)+ \boldsymbol {\Psi ^{T}}(\mathrm{\mu} _2\boldsymbol {u}_{k+1}-\boldsymbol {\eta }_k)+(\mathrm{\mu} _3\boldsymbol {w}_{k+1}-\boldsymbol {\rho }_k)$.
3. Examples
In this section, three numerical simulated examples with different kinds of restrictions are given, the first two of which demonstrate two linear effects and the third of which shows a kind of noise-like nonlinear restriction. Then, the proposed framework is validated by optical experiments. As an overall demonstration presented in Fig. 2, the measurements are distorted due to the restrictions of devices (the second column), causing the failure or low quality of reconstruction by the traditional forward model (the third column). In contrast, the explicit-restriction convolutional framework out-performs.
Fig. 2. Overall demonstration of the three practical examples. (a) Imaging system with a full sampling sensor (b) Limited sensor size. (c) Limited pixel pitch. (d) Limited sensor bit depth.
Some assumption for simplicity should be clarified first. For an amplitude modulated mask, $d_2$ is so small that the diffraction effect can be neglected. In this case, the $PSF_{mask}$ of size $L_P$ is $\left (1+{d_2}/{d_1}\right )$ times magnification of the mask pattern. That is to say, $L_P=\left (1+{d_2}/{d_1}\right )l_m$.
Fresnel zone aperture (FZA), which is inspired by inline holography [10,43] is used as the mask. The transmission function of the FZA mask is
The system sizes are chosen to be $d_1=30$ cm, $d_2=3$ mm, and $l_m=5.07$ mm. For scene with size $l_i=25.6$ cm, we have $L_P=5.12$ mm and $L_i=2.56$ mm. By assuming the sensor pixel pitch $L_{pix}=10$ ${\mathrm{\mu}}$m, the dimension of the reconstruction is 256 by 256 pixels. On the algorithmic side, the penalty parameters $\mathrm{\mu} _1$, $\mathrm{\mu} _2$, and $\mathrm{\mu} _3$ in Algorithm 1 are chosen to keep the primal and dual residual norms converging at close speeds [38]. We launch 300 iterations for each reconstruction.
3.1 Limited sensor size
As shown in Fig. 2(b), there exists a trade-off between light flux and information acquisition in lensless cameras as to the fact that the mask should be designed bigger than the sensor to maximize light flux and field of view (FoV). Otherwise, light utilization will be limited by the mask. Consequently, edges of the scene information is not captured when the system gets high light efficiency.
As shown in Fig. 3, to model the restriction of limited sensor size, $\boldsymbol {L}$ in Eq. (6) can be chosen as a cropping matrix $\boldsymbol {L_1}$ representing cropping operator from size $\left (L_P+L_i\right )$ to size $L_s$, where $L_s=R l_m$ is the sensor size. $R$ is defined as the size ratio of mask to sensor. Assuming $n=\left (L_P+L_i\right )/L_{pix}$ and $m=L_s/L_{pix}$, the cropping matrix $\boldsymbol {L}_1$ is a $m^{2}$ by $n^{2}$ non-full rank matrix. For each row of $\boldsymbol {L_1}$, only one element is $1$ and the rest elements are $0$.
The reconstructions of "cameraman" are shown in Fig. 4. Peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM) are employed as evaluation indexes. The final reconstruction quality versus size ratio of mask to sensor $R$ is shown in Fig. 4(c). As $R$ decreases, the non-convexity property of the minimize problem Eq. (8) becomes increasingly prominent, causing the poorer quality of reconstruction. Figures 4(a) and 4(b) show the reconstructed images and iteration process when $R$ equals to $0.4$, $0.6$, $0.8$, and $1.0$, respectively. The degradation of reconstruction quality is mainly caused by the artifacts.
Fig. 4. Reconstruction for different size ratio of mask to sensor $R$. (a) Reconstructed images. (b) PSNR and SSIM for different iteration numbers. (c) PSNR and SSIM after 300 iterations for different $R$.
By reconstructing the 1951 USAF Test Target, we studied the resolution ability of the proposed framework with the size limitation of the sensor. As Fig. 5 shows, the artifact effect becomes more serious when $R$ goes down while the maximum resolution ability does not change significantly.
Fig. 5. Resolution ability of the proposed model for the $R$ of $0.4$, $0.6$, $0.8$, and $1.0$, respectively.
3.2 Limited pixel pitch (super-resolution)
For a lensless imaging system, the low resolution of the reconstructed image caused by the pixelated sensor is also a noteworthy effect. As shown in Fig. 2(c), if the sensor pixel pitch is $L_p$, the reconstructed image has $\left (L_i/L_p\right )$ by $\left (L_i/L_p\right )$ pixels. However, $L_i$ is small due to the tiny distance $d_2$ between the mask and the sensor, which means the sensor’s resolution ability is not utilized sufficiently.
The explicit-restriction convolutional model can be used to circumvent the above limit. As shown in Fig. 6, assuming that one replaces the actual sensor with an imaginary sensor whose sensor pixel pitch is $1/R_{SR}$ times as much as the actual one. Consequently, the actual sensor records a subsample of the imaginary sensor’s response. In a similar way with section 3.1, $\boldsymbol {L_2}$ is the matrix representation of a uniformly sub-sampling operator at a sampling rate of $R_{SR}$. Particularly, this example is considered to be a super-resolution technique rather than a hardware restriction [44].
As shown in Fig. 7(a), the reconstruction resolution with the proposed framework is enhanced to a large degree compared to the traditional method. The obscureness of reconstruction is caused mainly by the undersampled measurements. The regularization term also leads to the loss of details, but it is necessary to eliminate latticed artifacts. Whereas, one can still get a high-quality image with four times pixel density according to Figs. 7(b) and 7(c).
Fig. 6. $\boldsymbol {L_2}$ is the subsampling operator when the proposed model is used to achieve a high-resolution image.
Fig. 7. Reconstruction for different super-resolution rate $R_{SR}$. (a) Reconstructed images. (b) PSNR and SSIM for different iteration numbers. (c) PSNR and SSIM after 300 iterations for different super-resolution rate $R_{SR}$.
A resolution test is also conducted and Fig. 8 presents the main results. As a significant example, when sensor with big pixel pitch whose $L_{pix}=40$ ${\mathrm{\mu}}$m is used for measurement. The traditional method can only present a low-resolution reconstructed test target with $64\times 64$ pixels, of which No. 4 to 6 elements in group -1 (zoomed part) is totally blurred (the third column). As a contrast, the proposed framework presents a reconstructed image with higher resolution ($256\times 256$) as the last column shows.
Fig. 8. Comparison of reconstructed targets before and after super-resolution for the super-resolution rate $R_{SR}$ of 2 and 4, respectively.
3.3 Limited sensor bit depth
Practical sensors have limited bit depth $B$, as Fig. 2 (d) shows, meaning that each pixel of the measurement is only allowed to take $2^{B}$ different discrete values. The limitation of sensor dynamic range is not separately considered because it can be absorbed into the bit depth problem by properly setting the exposure time.
When considering the effect of limited sensor bit depth, a pixel’s groundtruth value is dragged to its nearest step-wise value as illustrated in Fig. 9. Thus it can be approximately treated as adding random noise to the captured image, which can be attenuated by the regularization term. Assume each pixel of the measurement has intensity uniformly distributed in $[0,1]$, the intensity range is evenly divided into $2^{B}$ subintervals of length $2^{-B}$. Noticing the fact that for intensity in each subinterval, the sensor can only record a specific value. Thus, it is easy to calculate that the equivalent noise satisfies a uniform distribution in the interval $\left [-2^{-(B+1)},2^{-(B+1)}\right ]$.
Fig. 9. Limited sensor bit depth effect is equivalent to adding a random noise $\boldsymbol {\epsilon }$
As illustrated in Fig. 10(a), when the available sensor bit depth goes down, the weight of the regularization term is increased to against stronger equivalent noise. As a side effect, the reconstruction loses more details. Figure 10(c) shows the quantitative results of different sensor bit depths. The image quality goes down as $B$ decreases but is still acceptable when the available sensor bit depth is not less than eight. The simulation results provide guidance for the selection of masks. The mask $C_{mask}$ should be designed to minimize the DC term of the point spread function $PSF_{mask}$ so as to achieve the maximum utilization rate of the sensor bit depth.
Fig. 10. Reconstruction for different bit depth $B$. (a) Reconstructed images. (b) PSNR and SSIM for different iteration numbers. (c) PSNR and SSIM after 300 iterations for different $B$.
3.4 Experimental validation
The explicit-restriction convolutional framework is validated by optical experiments. As shown in Fig. 11(a), the FZA mask used in the experiments has a radius of 4.56 mm, and $r_0$ is chosen to be 0.325 mm. The mask is placed 3 mm away from the sensor. Grayscale images of size 20 cm$\times$20 cm are displayed on an LCD screen that is 30 cm away from the mask. With the above parameters, $L_P=9.21$ mm and $L_i=2$ mm, which means the encoded pattern on the sensor plane has size of 11.21 cm$\times$11.21 cm.
Fig. 11. Experimental validations. (a) FZA mask based lensless camera. (b) Measurements and reconstructions for sensor A and sensor B, respectively.
We test two measurements in the experiments, named "sensor A" and "sensor B", which correspond to the sensor size restriction and the pixel area restriction, respectively. For sensor A, the sensor size $L_s$ is 6.45 cm and the sensor pixel pitch $L_{pix}$ is 3.8 ${\mathrm{\mu}}$m. For sensor B, $L_s$=11.21 cm and $L_{pix}$=15.2 ${\mathrm{\mu}}$m. Both measurements have bit depth $B$=16. The measurements called sensor A and sensor B are respectively cropped and sub-sampled from a raw measurement by a QHY163M CMOS image sensor. Fifty iterations are launched for each reconstruction. It takes about 40 minutes to get a reconstruction of 526 by 526 pixels with a laptop (1.6GHz CPU and 8GB RAM). The experimental results are shown in Fig. 11(b).
The scenes are successfully reconstructed with incomplete measurements. Compared with the traditional method, the reconstruction of sensor A by the proposed approach has repressed artifacts. For sensor B, reconstruction with four times pixel density than the sensor is achieved. Edges of both reconstructed "THU" patterns are blurred because the diffraction effect is not taken into account. Notably, the proposed framework is valid for multi-wavelength scene by using an RGB sensor. The process is reconstructing RGB channels independently and combining then into final color image.
4. Conclusion
To conclude, we proposed an explicit-restriction convolutional framework for the lensless imaging system. By introducing the linear and noise-like nonlinear parts to the forward model, the proposed framework can effectively take physical restrictions into account. Notably, artificially introduced factors like super-resolution can also be incorporated. Both numerical and experimental tests indicate that our framework presents reconstructions with higher resolution and fewer artifacts than the traditional convolutional model. This versatile, extensible, and flexible framework has promising application scenarios for lensless imaging.
Funding
National Natural Science Foundation of China (61827825).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Ref. [45].
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