ROBUST OFF-AXIS DIGITAL HOLOGRAPHIC IMAGING SYSTEM IN COMPRESSIVE SENSING FRAMEWORK

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ISSN Print: 0976-6480 and ISSN Online: 0976-6499 DOI: 10.34218/IJARET.12.1.2021.072

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ROBUST OFF-AXIS DIGITAL HOLOGRAPHIC

IMAGING SYSTEM IN COMPRESSIVE SENSING

FRAMEWORK

School of Electronics Engineering, Vellore Institute of Technology (VIT), Chennai, Tamil Nadu, India

*Corresponding Author

ABSTRACT

Digital holographic imaging technique based on digital sensor arrays such as CCD/CMOS requires a larger number of pixels to capture the interference pattern into holograms with high resolution. These holographic images hence require large memory for data storage and transmission of 3D information. This paper demonstrates a compressive sensing (CS) method merged with off-axis digital Fresnel holography to reconstruct object wavefront from incomplete hologram pixels detection. Two different sampling masks, i.e. random sampling mask and uniform subsampling mask are considered to reduce the digital hologram samples to reconstruct the object wavefront from fewer hologram measurements. The robustness of the proposed method is validated by reconstructing intensity and phase images of good quality using Haar wavelet sparsified CS approach. The quantitative analysis shows that the proposed compressive off-axis digital holographic system is capable of reconstructing superior quality intensity and phase images of the object wavefront with only 25% hologram pixels. The quantitative comparisons of the results are presented from a numerical experiment to show the proof of the concept.

Key words: Compressive sensing, Off-axis digital holography, Wavelet sparsification,

Digital hologram.

Cite this Article: B. Lokesh Reddy and Anith Nelleri, Robust Off-Axis Digital

Holographic Imaging System in Compressive Sensing Framework, International

Journal of Advanced Research in Engineering and Technology, 12(1), 2021,

pp. 792-801.

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=12&IType=1

imaging techniques used in various applications [5-8], such as quantitative phase imaging, microscopic and macroscopic imaging, measurements and characterization. Compressive Sensing (CS) [9-14] is a signal processing technique used for efficiently recovering the original sparse signal of spatial dimension 𝑀 × 𝑀 from a finite number of measurements 𝐾 exploiting the sparsity of the signal. In compressive digital holography [15-24], the original sparse object wave is reconstructed from a lesser number of hologram samples. The signal is said to be sparse if it has a very less number of samples. The signal shall be sparse either in the spatial domain or in the transform domain like FFT, wavelet, DCT etc. The CS has been applied in many inline digital holographic schemes [15-23] like CS based phase shift digital holography (CS-PSDH) [22] and CS based parallel phase shift digital holography (CS-PPSDH) [23] etc. The recording procedure of PSDH requires multiple holograms sensing with different reference wave phase shifts, whereas the PPSDH scheme requires only a single shot hologram with reference wave phase shifts addressed on pixels in an interleaved manner. The advantage of CS-PPSDH [23] over CS-PSDH [22] is its suitability for imaging the moving or dynamic specimens. Recently, the compressive off-axis digital holography has attracted many researchers as it involves only the recording of single shot hologram and it is very useful for quantitative phase imaging of dynamic specimens.

Li et. al. [25] and Clemente et. al. have [26] demonstrated CS based inline four-step phase

shifting digital holography (PSDH) for hologram acquisition process using single-pixel detector. The object wavefront from the hologram is reconstructed using signal recovery algorithm in CS framework. Leportier et. al. [27] has demonstrated the principle of single-pixel imaging by CS theory and random linear measurements were obtained using binary mask technique. In another study reported by Leportier et. al. [28], the DCT and FFT transforms were utilized as an efficient way to reconstruct hologram with better sparsity by using CS framework. The experimental and simulation results show that the DCT as sparsifying operator gives better sparsification compared to FFT sparsification. The literature shown in Ref. [15-28] involve CS-based inline digital holographic methods with improved reconstruction accuracy for both intensity and phase information of object wavefront. Recently, Lokesh et. al [24] has demonstrated a compressive complex wave retrieval method from single off-axis digital Fresnel holographic system with Haar wavelet sparsification for the micro-lens characterization. In the above study, it was demonstrated by authors that CS based reconstruction method has given an accurate reconstruction of both intensity and quantitative phase of the micro-lens sample as compared to that of the conventional reconstruction method.

The strength of digital holography lies in its ability to image and computationally reconstruct both intensity and phase of the object. The linearity in the measurement model is an important requirement for CS implementation. The complex wave retrieval method from an off-axis digital Fresnel hologram is a linear model [29] developed using non-linear change of variables under appropriate assumptions and it is found suitable for CS application. If the object wave is not sparse in spatial domain, a separate sparsification of the object wavefront is required. The present work demonstrates the compressive complex wave retrieval method by considering only a fewer hologram pixels detection from a single off-axis digital Fresnel hologram and its robustness in whole object wave field reconstruction. We have evaluated the intensity and phase reconstruction quality of the object wavefront from a randomly sampled and subsampled hologram to study the robustness of the proposed method. Study on the efficacy of object wave sparsification is also carried out for above cases using the Haar wavelet. The necessary mathematical model of the CS implementation in off-axis holography is discussed, and the results of the numerical simulations are reported.

2. MATHEMATICAL MODEL FOR COMPRESSIVE OFF-AXIS

DIGITAL HOLOGRAPHY WITH SPARSIFICATION

The mathematical framework for the complex object wave retrieval using CS approach applied to off-axis digital Fresnel holography is presented in this section. In off-axis digital holography, a single-shot hologram is obtained by the interference between the object wave and plane reference wave. Let us consider the complex object wave 𝑂_𝑜 of size 𝑀 × 𝑀pixels. Then Fresnel transform is performed on 𝑂_𝑜 with propagation distance 𝑑 to the detector plane and the complex Fresnel field 𝑓 can be obtained as

𝑓 = ℱ𝜆,𝑑{𝑂𝑜} (1)

where 𝑓 denotes complex Fresnel field, ℱ denotes the Fresnel transform and 𝜆 represents the wavelength of the light source. Interference between the Fresnel field 𝑓 at a distance 𝑑 from the object plane and the off-axis plane reference wave 𝑅, is detected using a spatial electronic sensor array. The resulting off-axis Fresnel hologram obtained at the detector plane is given by

𝐻 = |𝑓 + 𝑅|2 (2)

In order to demonstrate the quantitative object information retrieval from a fewer samples or incompletely measured data 𝐼 of the detected hologram 𝐻, an under sampling is performed using mask 𝐴 of dimension 𝑀 × 𝑀on the originally detected hologram 𝐻. Now the hologram is multiplied with 𝐴 to obtain 𝐼 measurements as described in Eq. (3).

𝐼 = 𝐴. 𝐻 (3)

In the present paper, the measurement matrix 𝐴 is modeled with two different sampling scenarios: (i) random sampling mask and (ii) uniform subsampling mask. The random sampling mask is produced numerically using random sequences of 0 and 1. A uniform subsampling matrix is produced using a uniform 2×2 pixel kernel with considering only one pixel out of 4 pixels is one and the remaining three pixels are zero. The samples ratio of the hologram on the recording plane is calculated using the formula, (𝐾/𝐿) × 100 where 𝐾the sample is size of compression (𝐾 ≪ 𝐿) and 𝐿 is the total size of the hologram pixels 𝑀 × 𝑀. In the off-axis hologram reconstruction process, only 𝐾 << 𝐿 incomplete measurements are considered.

In the numerical reconstruction process, the complex wave retrieval algorithm [29] from a single digital Fresnel hologram is used to retrieve the Fresnel field in the recording plane. By implementing a non-linear change of variables, the non-linear holography is approximated to a linear process that can be solved by a linear algorithm to determine the Fresnel field 𝑓̃. The

inverse Fresnel propagation of the Fresnel field 𝑓̃ reconstructs the object wave 𝑂_𝑜. Thus the linearity property of the complex wave retrieval algorithm makes it suitable to be applied as a sensing matrix ℱ̃_𝑑 in the compressive complex wave retrieval method (CS method). This improves the quality of the reconstructed intensity and phase of the object wave from an incomplete digital Fresnel hologram measurement. The CS framework is used to reconstruct the object information from the incomplete single-shot off-axis digital hologram measurements 𝐼 by solving the following optimization problem [11].

𝑚𝑖𝑛 ⏟ 𝑂𝑜 1 2‖𝑓̃ − ℱ̃𝑑𝑂𝑜‖2 2 + 𝜏‖𝑂_𝑜‖₁ (4)

Finally, the reconstructed Fresnel field using compressive complex wave retrieval method is inverse Fresnel transformed to reconstruct original object wavefield.

𝑂_𝑜= ℱ_𝜆,𝑑−1{𝑓} (5)

In this method, the intensity and phase reconstruction of the sparse object wave 𝑂_𝑜 is implemented from the incomplete measurements 𝑓̃ by solving CS unconstrained optimization

sparse in the spatial domain but sparse in transform domain such that 𝑂_𝑜= 𝜙𝒔, 𝜙 is an 𝑀 × 𝑀 basis matrix and 𝒔 is sparse in nature, Eqn. (4) can be expressed as in Eqn. (6)

𝑚𝑖𝑛 ⏟ 𝑂𝑜 1 2‖𝑓̃ − ℱ̃𝑑𝜙𝑠‖2 2 + 𝜏‖𝑠‖₁ (6)

Here ℱ̃_𝑑and 𝜙 are termed as sensing and sparsifying matrices respectively. The properties of ℱ̃_𝑑 and 𝜙 are to be mutually non-coherent for the good quality reconstruction of the signal 𝑂𝑜 [22, 24]. The Haar wavelet [24] is used for sparsification in the place of the operator 𝜙.In

this method, the off-axis digital Fresnel holography and CS reconstruction framework are combined to retrieve both the intensity and phase information of a complex object wavefield from fewer samples of the hologram.

3. SIMULATION RESULTS AND DISCUSSION

In order to verify the reconstruction procedure of compressive off-axis digital Fresnel hologram, numerical simulations were conducted using MATLAB software. In this study, a gray scale USAF resolution chart was used as the intensity of the complex object image of dimension 1024 × 1024 pixels. This intensity varies between (0, 1) and phase part is also distributed in the same interval. Fig.1 shows the simulated intensity and phase of the input complex USAF resolution image. The parameters used in the simulation were 𝜆 = 632.8𝑛𝑚, pixel pitch of the recording sensor ∆= 6𝜇𝑚 and recording distance 𝑑 = 90𝑚𝑚. The Fresnel field of the complex USAF image was obtained at the detector plane over a propagation distance 𝑑 = 90𝑚𝑚 using the Fresnel transform.

Figure 1 Input complex USAF image (a) intensity and (b) phase.

The Fresnel field 𝑓 and the reference plane wave 𝑅 interfered at an angle 𝜃 = 1.95° _to

obtain an off-axis hologram on the recording plane using Eqn. (2). The mask matrix A of size 1024 × 1024 was used in the simulation with random sequences of 0 and 1. Fig. 2(a-c) show the simulated random sampling masks used for discarding 25%, 50% and 75% of samples respectively of the hologram H that was originally detected. A uniform subsampling mask of size 1024 × 1024 was generated with a kernel of size 2 × 2 = [1 0

0 0] is shown in Fig. 2(d).

Figure 2. Simulated random sampling mask with various random sample ratios (a) 75% measurements (786432pixels retained), (b) 50% measurements (524288 pixels retained) and (c) 25% measurements

(262144 pixels retained), (d) uniform subsampling mask (˷262144 pixels).

Out of 1024 × 1024 pixels (𝐿) of the originally detected hologram H, only 𝐾 pixels were considered such that 𝐾 << 𝐿 using the mask matrix A. Here, 𝐿 = 1024 × 1024 = 1048576 pixels. Fig. 3(a-c) show the randomly sampled digital holograms using the masks specified in Fig 2(a-c) respectively with various sampling ratios. Fig. 3(d) shows the computed digital Fresnel hologram using uniformly spaced subsampling mask.

Figure 3.Computed off-axis digital Fresnel hologram using various sample ratios (a) 75% measurements (786432 pixels retained), (b) 50% measurements (524288 pixels retained) and (c) 25%

measurements (262144 pixels retained), (d) uniform subsampling mask (˷262144 pixels).

The approximated Fresnel field was numerically reconstructed at recoding plane using complex wave retrieval algorithm and it is inverse Fresnel propagated using Eqn. (5). The Fig.4

a _b

c d

a b

complex wave retrieval method (without CS). The compressive complex wave retrieval method (CS method) was implemented by solving unconstrained optimization problem given in Eqn. (6) with Haar wavelet sparsifier.

Figure 4. Intensity and phase of the reconstructed complex USAF resolution chart using direct complex wave retrieval method (without CS) retaining various percentages of hologram pixels (a-b)

75% (c-d) 50% (e-f) 25%.

Figure 5. Intensity and phase of the reconstructed complex USAF resolution chart using CS algorithm without sparsification and retaining various percentages of hologram pixels (a-b) 75% (c-d) 50%

(e-f) 25%. a b C d e f CS CS CS a b C _d e _f

The intensity and phase of the reconstructed complex USAF resolution chart obtained by using CS method without any sparsification and using CS method with Haar wavelet sparsification are shown Fig.5 and Fig. 6 respectively. From the results shown in Fig (4-6), it can be subjectively observed that the robustness of CS method in reconstructing the complex object wave (both intensity and phase) was superior than conventional direct method even after retaining just 25% hologram pixels. From Fig 6, it can also be observed that the Haar wavelet sparsification improves the results further. It can be inferred that the reconstructed results of CS method with Haar wavelet sparsifier has effectively minimized the noise of both intensity and phase when compared to other methods.

Figure 6. Intensity and phase of the reconstructed complex USAF resolution chart using CS algorithm with Haar wavelet sparsification and retaining various percentages of hologram pixels (a-b) 75% (c-d)

50% (e-f) 25%.

As a special case, a uniform subsampling mask was applied by evenly selecting one pixel out of every four pixels in the mask matrix 𝐴 as described in Sec 2. Applying a subsampling mask in CS has lot of practical importance [30-31]. The results presented in Fig.7(a-b) show the reconstructed intensity and phase of the object wavefront obtained using conventional direct method. The reconstruction results obtained by the CS method using a subsampling mask, without sparsification and with Haar wavelet sparsification are shown in Fig.7(c-d) and Fig.7(e-f) respectively.

a b

C d

Figure 7. Intensity and phase of the reconstructed complex USAF resolution chart with subsampling mask (a-b) Conventional direct method (c-d) CS method without sparsification (e-f) CS method with

Haar wavelet sparsification.

Table. 1 MSE of the reconstructed object wavefront for all the methods with just 25% of hologram pixels were retained.

Reconstruction methods Mean Square Error (MSE) of reconstructed object

wave

Intensity Phase Complex wave retrieval method (without CS) 153.50 203.84 CS method without sparsification using random mask 125.64 33.05 CS method with Haar wavelet sparsification using

random mask

101.31 17.87 CS method without sparsification using uniform

subsampling mask

168.57 46.02 CS method with Haar wavelet sparsification using

uniform subsampling mask

141.84 28.66

To quantitatively appraise the quality of the intensity and phase images of the object wave reconstructed, the MSE was calculated between the input complex object field and reconstructed complex object field. Table. 1 shows the MSE of reconstructed object wave for various reconstructions. In order to highlight the degree of robustness of the method, we have presented the results of the reconstruction with just 25% of hologram pixels used. It was observed from Table. 1 that the MSE for the CS method with Haar wavelet sparsification has least error for both the cases, i.e., for random and subsampling masks and thus found to be a robust technique.

CS

c _d

e f

4. CONCLUSIONS

A robust compressive complex wave retrieval method is demonstrated in this study. This has enabled a good quality of object wavefront reconstruction by using a fewer hologram pixels measurements. The CS reconstruction algorithm is realized by solving an l1-norm optimization

problem. The linearity requirement of the CS algorithm has been satisfied using the complex wave retrieval method from single off-axis digital Fresnel hologram as the sensing matrix. The proposed CS method is proved to be very effective in improving the reconstruction quality of the of the complex object wave using minimal number of hologram pixels detection. The results showed that the CS method with Haar wavelet sparsification performs better in minimizing the noise present in both intensity and phase of the reconstructed object wave. The robustness of the method is analyzed by computing the MSE of the retrieved USAF resolution chart for the cases of random sampling mask and uniform subsampling masks applied for all reconstruction cases. The presented CS method is feasible and a straightforward approach to reconstruct the whole object wavefield from a fewer number of hologram samples and extends its application for imaging dynamic specimens.

ACKNOWLEDGEMENT

This work was supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under the grant no. CRG/2018/003906.

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