## Abstract

We propose a generalized framework for quantitatively acquiring multidimensional complex objects based on single-shot phase imaging with a coded aperture (SPICA). In multidimensional SPICA, a propagating field from a multidimensional complex object is sieved by a coded aperture, the sieved field is modulated by an optical element, which is called coding optics, and then the resultant field is captured by a monochrome image sensor. The original complex field is reconstructed from the single captured intensity image by a phase retrieval algorithm with a support constraint of the coded aperture and a sparsity-based reconstruction algorithm based on compressive sensing. We also present theoretical conditions for the proposed method. As a demonstration, we numerically verified an application of this generalized framework for single-shot acquisition of depth-variant multispectral objects.

© 2015 Optical Society of America

## 1. Introduction

Phase imaging is a powerful tool, especially for life science because many biological specimens have low light absorption and are difficult to detect using only the amplitude of propagating light coming from them [1]. A typical approach for phase imaging is an interferometric method such as holography [2, 3]. In general, this approach requires a complex setup for reference light or multiple shots for phase-shifting to quantitatively determine the interference.

Another established approach for phase imaging without using light interference is coherent diffraction imaging (CDI). In CDI, a complex field of an object is reconstructed from a single intensity image of a diffraction pattern from the object with a phase retrieval (PR) algorithm [4]. To solve the reconstruction problem, spatial assumptions about the object, called *support*, are needed to constrain the solution. However, such assumptions severely restrict the size of the object. Ptychography and the transport-of-intensity equation (TIE) are famous multi-shot CDI methods for overcoming the object size limitation [5,6].

We have recently proposed a single-shot CDI method for observing a large complex object. This method, referred to as single-shot phase imaging with a coded aperture (SPICA), has been used to demonstrate acquisition of two-dimensional complex objects [7]. SPICA connects synthetic aperture-based holography and CDI by means of a coded aperture (CA), which is a randomly arranged pinhole array that *sieves* the propagating field from the object, as shown in Fig. 1. The holographic process uses compressive Fresnel holography (CFH). In CFH, a complex object is reconstructed from randomly-sampled sparse holographic data by a numerical algorithm using the sparsity of the object based on compressive sensing (CS) [8]. CS is a recently-emerged sampling technique for observing object data with fewer measurements compared with the conventional sampling theorem [9].

We extend SPICA to single-shot acquisition of multidimensional complex objects in this paper. Several imaging methods for acquiring objects with more than two dimensions via holography or ptychography have been proposed [10–15]. Here we apply them to multidimensional SPICA. We employ an optical element, which is called coding optics in this paper, to modulate the sieved multidimensional complex field after the CA, as shown in Fig. 1. This allows us to realize single-shot multidimensional complex object acquisition with compact hardware that does not need reference light.

A generalized framework for multidimensional SPICA is derived in Section 2, and its reconstruction process is shown in Section 3. Then, as an example, we focus on depth-variant multispectral object acquisition based on the framework in Section 4, where its imaging performance is also demonstrated. A theoretical analysis of the proposed method is provided and numerically verified in Section 5. We summarize this paper in Section 6.

## 2. Generalized system model

We show a generalized imaging process for multidimensional SPICA with matrix operators that are depicted in Fig. 1. First, we treat the matrix operators for describing the imaging process in an abstract form, and then provide their specific forms in the following part of this section and Section 4. In the process description, we refer to *x*-*y* planes in the multidimensional object as a channel. If individual channels along a dimensional axis have coherence between them, the dimension is called a coherent dimension, such as depth, when the depths are included within a coherence length. On the other hand, when individual channels along a dimensional axis do not have coherence between them, the dimension is defined as an incoherent dimension. For example, a spectrum is an incoherent dimension when each spectrum arises from a different illumination source. In the following discussion, we show a system model simply with one coherent dimension and one incoherent dimension. This model can be easily extended to one with multiple coherent and/or multiple incoherent dimensions by rearranging the dimensions. The *y*,*t* and *v*-dimensions that appear in Fig. 1 are omitted for simplicity.

The system model of multidimensional SPICA is composed of two processes, CFH and CDI, which are shown in Fig. 1. The CFH process is written as

First, a multidimensional complex field
$\mathit{f}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{coh}}\times {N}_{c}^{\mathrm{inc}})\times 1}$ of the object illuminated by co- herent light propagates with a Fresnel diffraction operator
$\mathit{P}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{coh}}\times {N}_{c}^{\mathrm{inc}})\times ({N}_{x}\times {N}_{c}^{\mathrm{coh}}\times {N}_{c}^{\mathrm{inc}})}$ to the CA. Next, the propagating field is coherently multiplexed with an integration operator
${\mathit{I}}^{\mathrm{coh}}\in {\mathbb{R}}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times ({N}_{x}\times {N}_{c}^{\mathrm{coh}}\times {N}_{c}^{\mathrm{inc}})}$ along the coherent dimensional axis. Lastly, the multiplexed field is sieved by the CA with a masking operator
$\mathit{M}\in {\mathbb{R}}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times ({N}_{x}\times {N}_{c}^{\mathrm{inc}})}$, as the sieved complex field
$\mathit{a}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times 1}$ just behind the CA. Here, *N _{x}*,
${N}_{c}^{\mathrm{coh}}$, and
${N}_{c}^{\mathrm{inc}}$ are the numbers of elements along the

*x*-axis, and the coherent and incoherent dimensional axes, respectively.

The CDI process is written as

First, channels of the sieved complex field ** a** just behind the CA are modulated with an encoding operator
$\mathit{E}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times ({N}_{x}\times {N}_{c}^{\mathrm{inc}})}$ by the coding optics. The modulated field is then incoherently multiplexed and finally captured by a monochrome image sensor with an integration operator
${\mathit{I}}^{\mathrm{inc}}\in {\mathbb{R}}^{{N}_{x}\times ({N}_{x}\times {N}_{c}^{\mathrm{inc}})}$ along the incoherent dimensional axis, as the single intensity image
$\mathit{g}\in {\mathbb{R}}^{{N}_{x}\times 1}$. The encoding operator

**can be arbitrarily chosen to modulate target dimensions, as shown in Section 4, but it needs to have phase dependencies for the PR process described in Section 3.**

*E*In specific forms, Eq. (1), which shows the CFH process, is rewritten as

The masking matrix ** M** in Eq. (1) is composed of a diagonal sub-matrix
${\mathit{M}}_{0}\in {\mathbb{R}}^{{N}_{x}\times {N}_{x}}$, where the diagonal elements correspond to the mask pattern. The integration matrix

*I*^{coh}for the coherent dimension in Eq. (1) is composed of a sub-matrix ${\mathit{I}}_{0}^{\mathrm{coh}}=[{\mathit{I}}_{0}\cdots {\mathit{I}}_{0}]\in {\mathbb{R}}^{{N}_{x}\times ({N}_{x}\times {N}_{c}^{\mathrm{coh}})}$, where ${\mathit{I}}_{0}\in {\mathbb{R}}^{{N}_{x}\times {N}_{x}}$ is an identity matrix. The propagation matrix

**in Eq. (1) is composed of Toeplitz sub-matrices ${\mathit{P}}_{{C}^{\mathrm{coh}},{C}^{\mathrm{inc}}}\in {\u2102}^{{}^{{N}_{x}\times {N}_{x}}}$, which expresses the convolution of the Fresnel point spread function (PSF) at the**

*P**C*

^{coh}-th coherent and

*C*

^{inc}-th incoherent channel.

In the same manner, Eq. (2), which expresses the CDI process, is rewritten as

The integration matrix *I*^{i}^{nc} for the incoherent dimension in Eq. (2) is composed of the identity matrix *I*_{0}. The encoding matrix ** E** in Eq. (2) consists of sub-matrices
${\mathit{E}}_{{C}^{\mathrm{inc}}}\in {\u2102}^{{N}_{x}\times {N}_{x}}$, which causes each

*C*

^{inc}-th channel of the sieved complex field

**to be differently modulated by the coding optics.**

*a*## 3. Reconstruction process

When the object is two-dimensional
$({N}_{c}^{\mathrm{coh}}=1,{N}_{c}^{\mathrm{inc}}=1)$ the inversion of Eq. (2) is solved by the input-output PR (IOPR) algorithm with a CA-based support constraint [4, 7]. The modified CA-based IOPR for multidimensional SPICA is as follows. In the modified IOPR, the complex field
$\mathit{h}={\left[{({\mathit{h}}_{1})}^{T}\phantom{\rule{0.2em}{0ex}}\cdots \phantom{\rule{0.2em}{0ex}}{\left({\mathit{h}}_{{N}_{c}^{\mathrm{inc}}}\right)}^{T}\right]}^{T}=\mathit{Ea}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times 1}$on the sensor plane before the incoherent integration is also estimated. Here,
${\mathit{h}}_{{C}^{\mathrm{inc}}}\in {\u2102}^{{N}_{x}\times 1}$ is the complex field of the *C*^{inc}-th incoherent channel on the sensor plane, and the superscript *T* is the transpose of a matrix.

- The initial complex field $\stackrel{\u2323}{\mathit{h}}$ on the sensor plane is generated as a random complex field.
- The new estimation
of the complex field on the sensor plane is calculated as*ĥ*$${\widehat{\mathit{h}}}_{{C}^{\mathrm{inc}}}(U)=\left(\frac{|{\stackrel{\u2323}{\mathit{h}}}_{{C}^{\mathrm{inc}}}(U){|}^{2}\mathit{g}(U)}{{\sum}_{{C}^{\mathrm{inc}}}|{\stackrel{\u2323}{\mathit{h}}}_{{C}^{\mathrm{inc}}}(U){|}^{2}}\right)\mathrm{exp}\left(i\mathrm{arg}\left({\stackrel{\u2323}{\mathit{h}}}_{{C}^{\mathrm{inc}}}(U)\right)\right),$$where(*w**J*) shows the*J*-th element of the vector, and arg(*w**w*) is the argument of a complex number*w*. - The complex field
on the sensor plane is back-propagated to the CA plane as $\stackrel{\u2323}{\mathit{a}}={\mathit{E}}^{-1}\widehat{\mathit{h}}\in {\u2102}^{({N}_{x}\times {N}_{c}^{\mathrm{inc}})\times 1}$.*ĥ* - The new estimation
of the sieved complex field just behind the CA is calculated with the CA-based support constraint as*â*=*â*.*Mă* - The sieved complex field
is forward-propagated to the sensor plane as $\stackrel{\u2323}{\mathit{h}}=\mathit{E}\widehat{a}$.*â*

Steps 2 to 5 are iterated until **â** converges.

Since the inversion of Eq. (1) is ill-posed, we use a sparsity-based CS algorithm called two-step iterative shrinkage/thresholding (TwIST) for the inversion [16]. TwIST solves the following optimization problem iteratively:

*ℓ*

_{2}norm, τ is a regularization parameter, and ℛ(•) is a regularizer.

In the system model and the reconstruction of the CFH process, we used the Born approximation (single scattering model) [2, 10]. This approximation degrades the performance of the reconstruction when the object is depth-variant and spatially dense. This issue can be solved by employing some state-of-the-art algorithms used in ptychography [15,17,18].

## 4. Depth-variant multispectral object acquisition

Here, as a special case of Section 2, we show single-shot acquisition of a depth-variant multi-spectral (four-dimensional) complex object ℱ (*x*,*y*,*z*,*λ*), where *x*,*y*, and *z* are the three spatial dimensions, as shown in Fig. 1, and λ represents the spectral dimension. The depth *z* and the spectrum *λ* are the coherent and incoherent dimensions, respectively. To modulate the two dimensions *z* and *λ*, as the coding optics we choose *free space*, described with the encoding operator ** E** in Eq. (2). Free space results in Fresnel propagation, and this propagation process is dependent on the depth

*z*, the wavelength λ, and also the phase. This imaging modality is readily applicable to other dimensions, such as time and polarization. The setup for SPICA-based depth-variant multispectral object acquisition is composed of a CA, a monochrome image sensor, and coherent illumination. The mathematical model of this system is written as

*z*and wavelength λ, $\mathcal{M}$ is the mask pattern of the CA, $\mathcal{A}$ is the sieved complex field just behind the CA, and $\mathcal{G}$ is the single intensity image captured by the monochrome image sensor.

*s*and

*u*are the spatial dimensions on the CA and sensor planes, and

*z*is the distance between the CA and sensor planes, as shown in Fig. 1.

_{A}Equation (7) expresses the CFH process shown in Eq. (1). The depth-variant multispectral complex field
$\mathrm{F}$ of the object illuminated by polychromatic coherent light propagates with the Fresnel diffraction kernel
$\mathcal{P}$ to the CA, the propagating field is coherently multiplexed along the *z*-axis, and the multiplexed field is multiplied with the mask pattern
$\mathcal{M}$ of the CA, resulting in the sieved complex field
$\mathcal{A}$. Equation (8) corresponds to the CDI process shown in Eq. (2). The sieved complex field
$\mathcal{A}$ just behind the CA propagates with the Fresnel diffraction kernel
$\mathcal{P}$ to the monochrome image sensor. The propagating field is incoherently multiplexed along the *λ*-axis onto the monochrome image sensor, generating the captured single intensity image
$\mathcal{G}$

As indicated in the previous paragraph, the imaging process described by Eqs. (7) and (8) is a special case of the generalized system model of multidimensional SPICA described with Eqs. (1) and (2). Here
${N}_{c}^{\mathrm{coh}}={N}_{z}$ is the number of elements along the *z*-axis, and
${N}_{c}^{\mathrm{inc}}={N}_{\lambda}$ is the number of elements along the λ-axis. Also the Toeplitz sub-matrix
${\mathit{P}}_{{C}^{\mathrm{coh}},{C}^{\mathrm{inc}}}={\mathit{P}}_{Z,\Lambda}$ expresses the convolution of the Fresnel PSF at the *Z*-th depth and Λ-th spectral channel, in Eq. (3), and the encoding sub-matrix
${\mathit{E}}_{{C}^{\mathrm{inc}}}={\mathit{P}}_{\Lambda}^{{z}_{A}}$ also expresses the convolution of the Fresnel PSF at distance *z _{A}* and the Λ-th spectral channel, in Eq. (4), respectively.

We verified SPICA-based single-shot depth-variant multispectral object acquisition numerically, as shown in Fig. 2. In the simulation, the depth-variant multispectral complex object ** f** shown in Figs. 2(a)–2(d) was composed of two depths (

*N*= 2) at distances 30 mm (

_{z}*z*

_{1}) and 60 mm (

*z*

_{2}) as the coherent dimension from the CA and three spectrums (

*N*

_{λ}= 3), namely, 460 nm (λ

_{1}), 530 nm (λ

_{2}), and 630 nm (λ

_{3}) as the incoherent dimension. The channels consisted of natural images [19]. The pixel count ${N}_{x}{}^{2}$ was 512

^{2}, and the pixel pitch δ

*was 5*

_{x}*μ*m. A CA whose magnified mask pattern

*M*_{0}is shown in Figs. 2(e) was located 3 mm (

*z*) from the sensor plane. The number

_{A}*N*of the randomly arranged pinholes, shown as white pixels in the figure, was 21845, which was 8.3 % of the total pixel count. The captured intensity image

_{P}**is shown in Fig. 2(f), where Gaussian random noise with a signal-to-noise ratio (SNR) of 30 dB was added. The reconstruction result $\widehat{\mathit{f}}$ after phase bias matching is shown in Figs. 2(g)–2(j). We used the two-dimensional total variation as the regularizer ℛ in Eq. (6) for TwIST [20]. The complex object field was reconstructed successfully, and the peak SNR (PSNR) was 29.5 dB.**

*g*The effects of the measurement noise level and the alignment error of the pinholes in the CA on the reconstruction fidelity are shown in Fig. 3. The former effect is shown in Fig. 3(a), in which the relationship between the measurement SNR and the the reconstruction PSNR are plotted. The latter effect is shown in Fig. 3(b). In this figure, to emulate the alignment error, the locations of multiple pinholes in the CA-based support constraint in the reconstruction process are randomly shifted by one pixel from those on the CA in the simulated imaging process. The ratio of the number of shifted pinholes to the total number of pinholes is defined as the support error. The measurement SNR was fixed to 30 dB in Fig. 3(b). The simulated setup used for obtaining Fig. 3 is the same as that used for the previous simulation in Fig. 2. Those plots show averaged reconstruction PSNRs of ten different mask patterns. The reconstruction PSNRs rise at a measurement SNR of about 20 dB and reach a lower bound at around the support error of 0.25 in this case.

## 5. Theoretical analysis

The theoretical conditions for multidimensional SPICA are derived from the CFH process in Eq. (1) and the CDI process in Eq. (2) [7]. The following discussions assume a depth-variant multispectral complex object as in the previous section. Those discussions are extendable to other dimensions, besides the system model described in Section 2.

When the object distance *z* from the CA satisfies

CFH is approximated as an ideal three-dimensional Fourier encoder [10]. Then the number *N*_{CFH} of the required sampling points in each spectral channel in CFH is calculated, based on a worst-case analysis, to be

*K*is the sparsity in the regularizer domain for all depth channels and spectral channels [21]. In the SPICA case,

*N*

_{CFH}=

*N*, which is the number of the pinholes in the CA. In the previous simulation,

_{P}*z*

^{min}in Eq. (9) with the minimum value of the wavelength λ was 28 mm, and thus the condition was satisfied.

In the CDI process, oversampling is required to find unique autocorrelation of a complex object [4]. In multidimensional SPICA, the oversampling factor may be *N*_{λ}-times larger for PR in each spectral channel compared with that of two-dimensional CDI. The distance *z _{A}* between the CA and sensor planes needs to be

_{2D}is the oversampling factor of PR for a two-dimensional object, because a single pixel on the CA contributes more than ρ

_{2D}

*N*

_{λ}pixels on the image sensor in this condition. The upper bound of the number

*N*of pinholes on the CA is calculated as

_{P}The two-dimensional oversampling factor is known to be ρ_{2D} ≈ 4 [22]. In the previous simulation,
${z}_{A}^{\mathrm{min}}$ in Eq. (11) with the minimum value of the wavelength λ was 0.2 mm and
${N}_{P}^{\mathrm{max}}$ in Eq. (12) was 21845, and thus the conditions were satisfied.

The upper bound of multidimensional SPICA is calculated with Eqs. (10) and (12) to be

*q*is roughly independent for the numbers

*N*and

_{z}*N*

_{λ}of depth channels and spectral channels when the pixel count ${N}_{x}{}^{2}$ of the image sensor is sufficiently large.

The lateral and axial resolutions of multidimensional SPICA are equivalent to those in conventional holographic imaging [10]. The achievable resolutions ${\delta}_{x}^{\mathrm{min}}$ and ${\delta}_{z}^{\mathrm{min}}$ are calculated as

The spectral resolution of multidimensional SPICA is estimated with the Airy disk on the sensor plane using a pinhole of diameter δ* _{x}* on the CA. The diameter

*d*of the first dark annulus of the Airy disk is calculated to be

*d*= 2.44λ

*z*/δ

_{A}*[2], and ∂*

_{x}*d*/∂λ = 2.44

*z*/δ

_{A}*. To change the diameter*

_{x}*d*by more than one pixel pitch δ

*on the image sensor, the achievable spectral resolution ${\delta}_{\lambda}^{\mathrm{min}}$ is calculated to be*

_{x}The theoretical conditions discussed above were verified numerically as shown in Fig. 4. The plot shows the relationship between the required relative sparsity *q* and the numbers *N _{z}* and

*N*

_{λ}of depth channels and spectral channels. The object was composed of point sources randomly distributed four-dimensionally and having the same intensity and different random phases. The sparsity

*K*is the number of sources in this case. The pixel count ${N}_{x}{}^{2}$ was 64

^{2}, the pixel pitch δ

*was 5*

_{x}*μ*m, and the minimum wavelength λ was 530 nm. The nearest plane was located at

*z*

^{min}(= 3 mm) in Eq. (9) with the minimum wavelength. The distance

*z*of the CA was set to ${z}_{A}^{\mathrm{min}}$ (a function of

_{A}*N*

_{λ}) in Eq. (11) with the minimum wavelength. The number

*N*of pinholes on the CA was chosen to be ${N}_{P}^{\mathrm{max}}$ (a function of

_{P}*N*

_{λ}) in Eq (12). The axial pitch was set to be ${\delta}_{z}^{\mathrm{min}}$ in Eq. (15) with the maximum distance (a function of

*N*) and the maximum wavelength (a function of

_{z}*N*

_{λ}). The spectral pitch was set to ${\delta}_{\lambda}^{\mathrm{min}}$ (a function of

*N*

_{λ}) in Eq. (16).

In Fig. 4, each of the numerically calculated relative sparsities *q*, which assume a PSNR of 30 dB as the acceptable reconstruction fidelity, are shown with the blue dots. They are averages obtained using ten combinations of different mask patterns of the CA and distributions of the point sources. The _{1} norm was chosen as the regularizer ℛ in Eq. (6) for TwIST. The analytically calculated relative sparsities *q* shown with the red grid in Fig. 4 were from Eq. (13) with *σ* = 0.6, which was chosen experimentally. The numerical relative sparsity *q* approaches the analytical one, which is approximately independent of the numbers *N _{z}* and

*N*

_{λ}of depth channels and spectral channels in this case, as

*N*and

_{x}*N*

_{λ}increase, although there is a small difference between them based on the worst-case analysis.

## 6. Conclusion

In summary, we extended SPICA to single-shot acquisition of multidimensional complex objects. The generalized system model and its reconstruction process were presented. As an example, assuming a depth-variant multispectral object, the system model and the theoretical conditions were derived and numerically demonstrated. In the multidimensional SPICA method, both the depth and spectrum dimensions are encoded by free space propagation, and the modulated depth and spectral dimensions are coherently and incoherently multiplexed, respectively.

In the numerical demonstrations, we showed the effect of the error between the CA-based support constraint and the actual mask pattern in the reconstruction process. Other modeling errors, such as discretization [23], should also be investigated.

The proposed imaging modality can be readily applied to other optical dimensions, such as time and polarization. Our approach consists of only a CA, a monochrome image sensor, and a coherent light source, and does not require any reference light. Thus, a compact and highly functional imaging system can be realized. In addition, the technique is applicable to incoherent holography and sensing in various spectral regions, such as X-ray and electron imaging.

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