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Performance evaluation of a digital holographic camera under variable source power and exposure time

Gaurav Dwivedi, Lavlesh Pensia, Sanjit K. Debnath, and Raj Kumar

Open Access Open Access

Abstract

In this work, quality evaluation of a reconstructed amplitude image in digital holography is presented. The digital holograms are recorded using three different digital holographic experimental configurations, namely, conventional off-axis digital holography, concave-lens-based digital holography, and the digital holographic camera developed for non-destructive testing applications [Opt. Lasers Eng. 137, 106359 (2021) [CrossRef]  ]. The quality of reconstructed images is measured by calculating the quality evaluation parameters such as speckle index, peak signal-to-noise ratio, and structural similarity index measure for these experimental configurations. Optimization of the power of the light source and exposure time of the recording sensor is performed for the three configurations based on the quality evaluation of reconstructed images. A comparison of the quality of reconstructed images is made for the three experimental configurations to analyze their performance for different source power and exposure time of the recording image sensor.

© 2020 Optical Society of America

1. INTRODUCTION

In many applications of digital holography, such as microscopy, non-destructive testing [1–6], etc., the environmental vibrations can add serious distortions to the information of the object during recording. The effect of vibrations may be reduced by properly optimizing the source power and exposure time of the recording image sensor in digital holography. Both source power and exposure time of the image sensor affect the number of photons interacting with the sensor. This ultimately contributes to the information of the object stored in the digital holograms recorded in the digital holographic technique [7,8]. In digital holography, information of an object is collected by recording the wavefront carrying the information of the object on an image sensor [charge coupled device (CCD)/complementary metal oxide semiconductor (CMOS)]. The recorded wavefront is an interference pattern generated due to the superposition of a reference beam and an object beam formed after scattering from the object’s surface. The information of the object is stored in the form of intensity and shape variations in the recorded interference pattern called a digital hologram. The wavefront recorded in the digital hologram is reconstructed using numerical reconstruction methods [9–12] like angular spectrum, Fresnel diffraction, convolution method, etc. The complex amplitude obtained after numerical reconstruction of the recorded wavefront, contains three-dimensional information of the object’s size, shape, surface texture, surface defects, etc. [13,14]. The intensity and contrast of the object is obtained by calculating the absolute value from the complex amplitude. This absolute value is represented as amplitude image of the object. Similarly, the phase map can be obtained by calculating the argument of complex amplitude. The reconstructed complex amplitude may suffer from structural deformation and speckle noise caused by several factors [15–17] such as un-optimized power of the laser source and exposure time of the recording sensor, environmental vibrations, electronic noise due to the recording sensor, coherent noise of the laser, roughness of the test object’s surface, losses due to optical components, etc. Noise present in complex amplitude affects both the amplitude and phase of the reconstructed image; however, the phase obtained in case of rough surfaces is random, and thus the effect of noise is indiscernible [17]. Therefore, a reconstructed amplitude image from a single digital hologram of rough surface may be more suitable to perceive the effect of noise present in the complex amplitude.

To estimate the attributes of objects such as shape, size, contrast, etc., the properties of source (power, wavelength, etc.) and image sensor (exposure time, resolution, pixel pitch, etc.) play significant roles in recording digital holograms. In this paper, the effects of laser source power and exposure time of digital sensors on the reconstructed images in digital holography are studied by assessing the quality of reconstructed amplitude images. Several image quality parameters are suggested in literature [18–34] for quality estimation in digital holography. Among these, most commonly used parameters, i.e., speckle index, peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM), are employed in this study to evaluate the quality of reconstructed amplitude images. These images are obtained from complex amplitudes computed by applying the Fresnel diffraction method (FDM) on the wavefronts recorded in digital holograms obtained using three different configurations of digital holography, i.e., conventional off-axis digital holography [7], concave-lens-based digital holography [35], and digital holographic camera [36]. The source power and sensor’s exposure time are optimized for the three configurations, and their performances are compared on the basis of estimated quality of reconstructed images.

2. THEORY AND METHODOLOGY

A. Recording and Reconstruction

Three different configurations of digital holography, i.e., conventional off-axis digital holography, concave-lens-based digital holography, and digital holographic camera, are developed to record digital holograms. The digital holograms are recorded at different source powers and sensor exposure times in each configuration. The power of the laser source in different configurations, which are sequentially realized on the same mechanical platform, is controlled by a variable beam splitter. Variation in power of the source is monitored by a power meter placed in the path of the reflected beam from the variable beam splitter. At a certain source power, several digital holograms of the object are recorded with variable exposure times of the recording sensor. These recorded digital holograms are stored in a computer connected to the recording sensor.

The digital holograms are generated due to superimposition of the object beam $O(x,y)$ and reference beam $R(x,y)$, which can be represented as

$$O(x,y) = {A_O}(x,y)\exp (j{\phi _O}(x,y)),$$
$$R(x,y) = {A_R}(x,y)\exp (j{\phi _R}(x,y)),$$
where ${A_O}(x,y)$ and ${\phi _O}(x,y)$ represent amplitude and phase distributions, respectively, of the object beam, and ${A_R}(x,y)$ and ${\phi _R}(x,y)$ represent amplitude and phase distributions, respectively, of the reference beam, and $j = \sqrt{} - {1}$. The intensity distribution of interference fringes $I(x,y)$ obtained after coherent superimposition of object and reference beams is represented as
$$I(x,y) = {\left| {O(x,y) + R(x,y)} \right|^2},$$
$$\begin{split}I(x,y) &= {| {O(x,y)} |^2} + {| {R(x,y)} |^2} + O(x,y){R^*}(x,y) \\&\quad +{O^*}(x,y)R(x,y),\end{split}$$
where the first two terms on right hand side of Eq. (4) are constant terms, and the third and fourth terms represent conjugate orders of the recorded object. The recorded digital holograms can be reconstructed using several numerical reconstruction methods. In present work, the FDM is used in all three configurations, from the commonly used numerical reconstruction methods like the angular spectrum method (ASM), convolution method, etc., as FDM satisfies the distance criterion for propagation [12,37] for a given size of the test object. ASM and convolution-based approaches are exact methods that are restricted to shorter propagation distances and create aliasing in reconstruction for large objects that require longer propagation distances. Zero padding can be applied in both ASM and convolution methods for reconstruction of large objects at longer distances, but it significantly increases the execution time in numerical processing. Therefore, FDM is found to be more suitable in the present case. The complex amplitude of the object recorded in digital holograms can be reconstructed using the Fresnel diffraction formula [38,39], represented as
$$\begin{split}O(\xi ,\eta) &= \frac{{\exp (- jkz)}}{{j\lambda z}}\exp \left({\frac{{- j\pi ({x^2} + {y^2})}}{{\lambda z}}} \right) \\ &\quad*{\rm FT}\left[{O({x,y} )\exp \left({\frac{{- j\pi ({x^2} + {y^2})}}{{\lambda z}}} \right)} \right],\end{split}$$
where $O(\xi ,\eta)$ and $O(x,y)$ are complex amplitude distributions at the object plane and sensor plane, respectively, FT represents Fourier transform, $k = {2}\pi /\lambda$, $\lambda$ is the source wavelength, $z$ is the propagation distance, and $j = \sqrt {} - {1}$. For reconstruction, one of the conjugate orders, mentioned in Eq. (4), is filtered out in the Fourier domain and then propagated using the FDM. This filtering removes the constant DC terms and the other conjugate order.

B. Quality Evaluation Parameters

The quality parameters used in the present work to evaluate the quality of reconstructed images are speckle index, PSNR, and SSIM. Speckle index is an absolute parameter, as it does not require any reference image to evaluate the quality of the test image. Meanwhile, PSNR and SSIM are relative quality parameters, as they use a reference image to compare it with the test image. Speckle index is a ratio of standard deviation and mean of intensity of the test image, which represent local variation of intensity in the image around a mean value. Thus, a low value of the speckle index represents good quality of the test image. The speckle index is represented as [20,40]

$$S = \frac{1}{{({M - 2} )({N - 2} )}}\sum\limits_{m,n = 2}^{M - 1,N - 1} {\frac{{\sigma ({m,n} )}}{{\mu ({m,n} )}}} ,$$
where $S$ is the speckle index, $M {\times} N$ is the resolution of the test image, $m$ and $n$ are integers, and $\sigma$ and ${\mu}$ are standard deviation and mean of intensities of test image, respectively. PSNR is a commonly used parameter in digital holography to estimate the quality of reconstructed images. PSNR is a relative quality parameter that compares the test image with a reference image to estimate its quality. It is a logarithmic ratio of peak intensity value in the test image and mean square error (MSE). The MSE compares pixel-wise intensities of the test image with the reference image. Thus, a low value of MSE and high value of PSNR is desirable for good quality image. PSNR can be represented as [25,41]
$${\rm PSNR} = 10{\log _{10}}\left({\frac{{{P^2}}}{{\rm MSE}}} \right),$$
where $P$ is the peak value of intensity in the test image, and MSE is represented as
$${\rm MSE} = \frac{1}{{{M^\prime} N^{\prime}}}\sum\limits_1^{{N^\prime}} {\sum\limits_1^{{M^\prime}} {{{| {I^\prime - I} |}^2}}} ,$$
where $M^\prime \;{\times}\;N^\prime$ is the resolution of the test image, and $I^\prime$ and $I$ represent intensities of the test image and reference image, respectively. The performance of PSNR is better in estimating the quality of noisy images, but it lacks in discriminating the structural information between test and reference images. Therefore, another quality parameter, i.e., SSIM is also investigated in the present work. It has several advantages over traditionally used image quality parameters such as it does not rely on any threshold value, it evaluates the structural variations between two complex-structured images directly [42], etc. SSIM compares the local variations of pixel intensities between test and reference images and is represented as
$${\rm SSIM}(x,y) = \frac{{(2{\mu _x}{\mu _y} + {C_1})(2{\sigma _{\textit{xy}}} + {C_2})}}{{(\mu _x^2 +\mu_y^2 + {C_1})(\sigma _x^2 + \sigma _y^2 + {C_2})}},$$
where ${{\mu}_x}$ and ${{\mu}_y}$ are mean intensities, ${\sigma _x}$ and ${\sigma _y}$ are standard deviations of the reference image and test image, respectively, ${\sigma _{\textit{xy}}}$ is the covariance between the two images, ${C_1} = {({K_1}L)^2}$ and ${C_2} = {({K_2}L)^2}$ are constants, $L$ is the dynamic range of pixel values of the sensor, and ${K_1}\; \ll{1}$ and ${K_2}\; \ll \;{1}$ are small constants.

C. Reference Image Generation by Simulation

For evaluation of the quality of reconstructed images, a reference image is required, which is obtained by simulating the captured image of the object used in the experiment, to synthesize its digital hologram. For capturing the image of the object, the laser source and recording sensor are same as those used in the experiments to avoid any dissimilarity of the wavelength of the source and pixel pitch of the sensor during quality evaluation. This simulation is performed in the MATLAB environment. To perform simulation [37], first, the object field recorded by the sensor is propagated onto the sensor plane using ASM [43–45]. The object field at the sensor plane can be represented as

$$O(x,y;z) = {F^{- 1}}\left[{F\left\{{{O_0}(x,y)} \right\} \times H(p,q)} \right],$$
where $O(x,y;z)$ and ${O_0}(x,y)$ represent the object field on the sensor plane and the initially recorded object field, respectively, and $H(p,q)$ represents the spatial frequency transfer function for propagation distance “$z$”, expressed as
$$H(p,q) = \exp \left[{- j2\pi z\sqrt {\frac{1}{{{\lambda ^2}}} - {{\left({\frac{p}{{M\Delta}}} \right)}^2} - {{\left({\frac{q}{{N\Delta}}} \right)}^2}}} \right],$$
where $\lambda$ is the source wavelength, $M\;{\times}\;N$ is the resolution of the sensor, $\Delta$ is the pixel pitch of the sensor, and $p$ and $q$ are the integers represented as ${-}M/{2} \le p \le M/{2} - {1}$ and ${-}N/{2} \le q \le N/{2} - {1}$. After propagating the object field onto the sensor, a digital reference beam is added to the object field and squared to create an interference pattern on the sensor plane. The complex amplitude of digital reference beam ${R_D}(x,y)$ can be represented as
$${R_D}(x,y) = {A_{{R_D}}}(x,y)\exp \left({j\frac{{2\pi}}{\lambda}({p + q} )\Delta \sin \theta} \right),$$
where ${A_{{R_D}}}(x,y)$ is the amplitude of the digital reference beam, which can be represented as the average of minimum and maximum values of intensity of the recorded object field, and $\theta$ represents the angle between the object and reference beams for off-axis digital holographic arrangements. The object beam $O(x,y;z)$ and digitally created reference beam ${R_D}(x,y)$ are added and squared to create complex amplitudes of the interference pattern, i.e., the digital hologram. The resultant intensity ${I_D}(x,y)$ of the digital hologram is represented as
$${I_D}(x,y) = {\left| {O(x,y;z) + {R_D}(x,y)} \right|^2},$$
$$\begin{split}{I_D}(x,y) &= {| {O(x,y;z)} |^2} + {| {{R_D}(x,y)} |^2} \\&\quad+ O(x,y;z)R_D^*(x,y)+ {O^*}(x,y;z){R_D}(x,y),\end{split}$$
where ${| {O(x,y;z)} |^2}$ and ${| {{R_D}(x,y)} |^2}$ are the constant terms also known as DC terms, and $O(x,y;z)R_D^*(x,y)$ and ${O^*}(x,y;z){R_D}(x,y)$ are complex conjugate orders representing the real image and complex conjugate image of the object, respectively. After creating the digital hologram in the simulation, it is reconstructed using FDM. For reconstruction, one of the conjugate orders, described in Eq. (14), is filtered out in the Fourier domain and then propagated using FDM using Eq. (5).

3. EXPERIMENTAL CONFIGURATIONS

Three different configurations of the digital holographic setup, as shown in Fig. 1, are developed to record digital holograms. The performance of these three configurations is compared on the basis of quality evaluation of reconstructed amplitude images obtained from the complex amplitude of wavefronts reconstructed from recorded digital holograms. The experimental arrangement of conventional digital holography, as shown in Fig. 1(a), employs a diode laser source of 30 mW power and 660 nm wavelength. The laser beam from source passes through a variable beam splitter that controls the power of the laser source. The laser beam is diverged by a ${45\times}$ microscope objective and divided by a 50:50 cubic beam splitter into two parts. One beam illuminates the object and gets scattered from it to create the object beam. The other beam, called a reference beam, passes through another variable beam splitter, which controls the intensity of reference beam compared to the object beam. The variable beam splitters have a circular shape of 25.4 mm outer diameter, 10 mm inner diameter, and variable optical density (0.4–4). The object beam and reference beam are combined by another 50:50 cubic beam splitter to make an interference pattern on a CMOS image sensor (2.2 µm pixel pitch and 1/2.5 in. sensor size) to serve as digital hologram. This recorded digital hologram is stored in a computer connected to the CMOS sensor for numerical reconstruction.

 figure: Fig. 1.

Fig. 1. Schematic representations of (a) conventional off-axis digital holographic experimental configuration, (b) concave-lens-based digital holographic experimental configuration, and (c) optical configuration of the digital holographic camera. The abbreviations in the figure represent BS, beam splitter; VBS, variable beam splitter; L1, concave lens; L2, convex lens; MO, microscope objective; PM, power meter; CMOS, complementary metal oxide semiconductor.

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The concave-lens-based digital holographic configuration, as represented schematically in Fig. 1(b), is similar to the conventional digital holographic configuration with the addition of a concave lens between the object and beam splitter in the direction of propagation of the object beam. The concave lens has a 25.4 mm diameter and ${-}{50}\;{\rm mm}$ focal length. The distances between the object to the concave lens is 150 mm and the concave lens to the CMOS sensor is 100 mm. The concave lens creates a de-magnified virtual image of the object and thus reduces the spatial frequency content perceived by the recording sensor. Therefore, it can be used to record large size objects at shorter propagation distances in digital holography. Since the concave lens diverges the light beam passing through it, the number of photons arriving at the active medium of the sensor is reduced. This effect causes an increase in exposure time of the sensor, which may contribute to an addition of environmental vibrations during the recording of digital holograms. An arrangement that reduces the exposure time for the same source power and object properties is used in the digital holographic camera [36], the optical configuration of which is represented schematically in Fig. 1(c). The camera employs a combination of concave and convex lenses. The convex lens (25.4 mm diameter and 80 mm focal length) collects the light diverged from the concave lens and converges it onto the CMOS sensor. Therefore, it effectively increases the number of photons arriving at the active medium of the CMOS sensor, and so the exposure time of the sensor is reduced to record the digital hologram. The object used in all three configurations to record digital holograms is an Indian 5 rupee coin, shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Image of an Indian 5 rupee coin used as an object to record digital holograms in all three configurations of digital holographic experiments shown in Fig. 1.

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4. RESULTS AND DISCUSSION

The effect of power of the source and exposure time of the recording sensor in digital holography is estimated by evaluating the quality of amplitude images. These images are obtained from complex amplitudes of reconstructed wavefronts. The numerical reconstruction of wavefronts recorded in digital holograms is performed using FDM. Digital holograms are recorded using three different configurations of digital holography shown in Fig. 1. For comparison of reconstructed images using these configurations, several quality parameters are calculated, e.g., speckle index, PSNR, and SSIM. For calculation of these parameters, a reference image of the coin is generated by simulation of the digital holography process.

A. Reference Image Generation by Simulation

For generation of a reference image of the coin by simulation, an image of the coin is captured using same source and image sensor (i.e., diode laser of 660 nm wavelength and CMOS image sensor with a 2.2 µm pixel pitch and 1/2.5 in. sensor size) as used in the experiments. This captured image of an Indian 5 rupee coin, shown in Fig. 3(a), is propagated to the sensor plane by ASM and then added to the digital reference beam and squared as mentioned in Section 2.B, to create a digital hologram, as shown in Fig. 3(b).

 figure: Fig. 3.

Fig. 3. (a) Captured image of the coin, (b) digital hologram obtained after adding and squaring the digital reference beam with the recorded field of the coin, (c) Fourier spectrum of the digital hologram, and (d) reconstructed image obtained after filtering one of the orders in the Fourier spectrum indicated by rectangle in (c).

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One of the conjugate orders is filtered in the Fourier spectrum, as shown in Fig. 3(c). The filtered order is indicated by a rectangular box. Inverse Fourier transform is applied after filtering, and the resultant hologram is processed using FDM to obtain the reconstructed image shown in Fig. 3(d). The reconstructed image in Fig. 3(d) is used as a reference image for quality evaluation of reconstructed images obtained in the experiments. The propagation distance is maintained in the simulation to equalize the resolution of the reconstructed image with that obtained in the experiments for each of the three configurations. The resolution of the reconstructed image simulated for propagation distances corresponding to all the three configurations are ${539} \times {697}$ pixels for the conventional digital holographic configuration, ${226} \times {302}$ pixels for the concave-lens-based digital holographic configuration, and ${333} \times {431}$ pixels for the digital holographic camera.

B. Recording and Reconstruction of Holograms

Several digital holograms are recorded using the above mentioned configurations of digital holography at different source powers and exposure times of the recording sensor. Figures 4–6 show some of the recorded digital holograms and corresponding reconstructed images for the three configurations.

 figure: Fig. 4.

Fig. 4. (a) Recorded digital holograms using conventional off-axis digital holographic configuration and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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 figure: Fig. 5.

Fig. 5. (a) Recorded digital holograms using concave-lens-based digital holographic configuration and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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 figure: Fig. 6.

Fig. 6. (a) Recorded digital holograms using digital holographic camera and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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The number of photons reaching the recording sensor is controlled by both source power and exposure time of the recording sensor. It can be observed in images of recorded digital holograms in Figs. 4(a), 5(a), and 6(a) that at a particular power an increase in exposure time increases the number of photons and, hence, the intensity of the image of the digital hologram recorded by the sensor. Similarly, at a particular exposure time of the recording sensor, an increase in power of the laser source increases the number of photons reaching the sensor. The variation in mean intensities of recorded digital holograms is plotted in Fig. 7 with the exposure time at different laser powers. It can be observed in Fig. 7 that with an increase in exposure time of the sensor the mean intensity increases rapidly at higher laser power for all configurations, i.e., conventional digital holographic setup [Fig. 7(a)], concave-lens-based digital holographic setup [Fig. 7(b)], and digital holographic camera [Fig. 7(c)]. Also, higher rates of increase in mean intensity of the digital holograms can be observed for the digital holographic camera as compared to the other two configurations. Thus, a larger number of photons can be recorded by the digital holographic camera at lower source power and shorter exposure time of the sensor. The digital holograms recorded in all three configurations are reconstructed numerically using FDM. Similar to the simulation, one of the orders is filtered out in the Fourier domain and further propagated using FDM. The reconstructed images of the coin, corresponding to recorded digital holograms in each configuration, are shown in Figs. 4(b), 5(b), and 6(b).

 figure: Fig. 7.

Fig. 7. Mean values of intensities of recorded images of digital holograms obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera.

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The effect of variation in source power and the sensor’s exposure time is also observable in reconstructed images. On comparing the reconstructed images for all three configurations, it can be observed in Figs. 4(b), 5(b), and 6(b) that, at low source power, i.e., 5 mW, clear reconstructed images are obtained at around 10 ms for a conventional digital holographic configuration and concave-lens-based digital holographic configuration, while the digital holographic camera provides a clear reconstructed image at around 1 ms for 5 mW source power. Similarly, at low exposure time of the sensor, i.e., 0.01 ms, recording of any photon/s is observed (bright circular light) at 20 mW for the conventional digital holographic configuration, 5 mW for the digital holographic camera, while for the concave-lens-based setup, no reconstruction was observed until 28 mW source power. So, it can be inferred from these results that at low power, higher exposure time is required to properly reconstruct the object and vice-versa. Also, the digital holographic camera provides properly reconstructed images at lower source power and shorter exposure time of the sensor compared to the other two configurations. This effect is the result of using a lens system employing concave and convex lenses in the digital holographic camera. This combination has not only increased the imaging area of the system by reducing the spatial frequency using a concave lens, but also reduced the exposure time requirement by converging the light onto the recording sensor using a convex lens.

C. Quality Evaluation of Reconstructed Images

To evaluate the quality of reconstructed images in the presented digital holographic configurations, the reconstructed images obtained at each power and exposure time are compared with simulated reference images in terms of speckle index, PSNR, and SSIM using Eqs. (6), (7), and (9), respectively. The resolution of the simulated reconstructed reference image of the coin is maintained the same as that obtained in the experiments for comparison.

1. Speckle Index

One of the commonly used quality parameters for digital holography is the speckle index, which is the ratio of standard deviation and mean of intensity of the reconstructed image. Figure 8 shows variation of speckle index values of reconstructed images with changes in exposure times at different laser source powers.

 figure: Fig. 8.

Fig. 8. Speckle index values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera.

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These speckle index values are calculated using Eq. (6), which shows that the speckle index represents local variations in intensity of a reconstructed image around a mean value of the intensity. Thus, a low value of the speckle index represents better quality of the reconstructed image. It can be observed in Fig. 8 that a low speckle index is obtained for 5 mW, and, as the source power increases, the value of the speckle index also increases, and its value is maximum for 28 mW source power for each configuration. Although, the reconstructed images obtained by the digital holographic camera provide higher speckle index values compared to the other two configurations, it is more stable in terms of variations in these values, as shown in Fig. 9.

 figure: Fig. 9.

Fig. 9. Speckle index values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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Figure 9 shows plots of comparison of speckle index values for the three configurations at lowest and highest source powers used in the experiments, i.e., 5 mW in Fig. 9(a) and 28 mW in Fig. 9(b). The speckle index surely represents the noise in reconstructed images, but it does not provide any indication about the presence of information in the respective image. For example, as observed in plots in Figs. 8 and 9, for the digital holographic camera, a lower speckle index is obtained at 0.01 ms for all source powers, thus quality of the image should be better. But it can be noticed in Fig. 6(b) that no proper reconstruction was obtained at this exposure time. Thus, other quality parameters, i.e., PSNR and SSIM, are also investigated.

2. Peak Signal-to-noise Ratio

PSNR is another commonly used parameter for quality evaluation of reconstructed images in digital holography. The values of PSNR of reconstructed images, calculated using Eq. (7) for all three configurations, are shown in Fig. 10.

 figure: Fig. 10.

Fig. 10. PSNR values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera after comparing them with the simulated reconstructed image.

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PSNR is a logarithmic ratio of the maximum value of intensity of the image and MSE. Here MSE, as per Eq. (8), compares the pixel-wise intensity of the two reconstructed images, i.e., simulated reference image and experimentally obtained image. For good quality reconstructed images, the value of MSE should be low, and thus the value of PSNR should be high. It can be observed in these plots that with an increase in source power and sensor’s exposure time, the value of PSNR increases. For comparison of the three configurations, variation in PSNR values with exposure time are plotted at lowest and highest source powers used in the experiments, i.e., 5 mW and 28 mW, as shown in Fig. 11. It can be observed in Fig. 11 that for both source powers, i.e., 5 mW [Fig. 11(a)] and 28 mW [Fig. 11(b)], the digital holographic camera gives better PSNR values as compared to the other two configurations, and it is lowest for concave-lens-based setup. PSNR provides the best performance in evaluating the quality of noisy images, but it lacks in distinguishing the structural information in images from the reference image [23]. One such parameter to measure the structural discrimination is SSIM.

 figure: Fig. 11.

Fig. 11. PSNR values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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3. Structural Similarity Index Measure

SSIM is a combination of comparisons of luminance, contrast, and structural correlation between the test image and reference image. The SSIM value of reconstructed images obtained using the three digital holographic configurations is calculated using Eq. (9), and the results are shown in Fig. 12.

 figure: Fig. 12.

Fig. 12. SSIM values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera after comparing them with the simulated reconstructed image.

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It can be observed that a better SSIM value is obtained at higher source powers at particular exposure times for all the three configurations. It has decreased slightly for conventional digital holographic configuration after 30 ms [Fig. 12(a)] and for the digital holographic camera after 10 ms [Fig. 12(c)]. For the concave-lens-based setup, the SSIM values are increasing and may take longer exposure time to saturate/decrease [Fig. 12(b)]. For comparison of the three configurations, again, the variations in SSIM values are plotted with changes in the sensor’s exposure time at the lowest and highest source powers, i.e., 5 mW and 28 mW, used in the experiments, as shown in Fig. 13.

 figure: Fig. 13.

Fig. 13. SSIM values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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Based on SSIM values, the digital holographic camera performs better at 5 mW and 28 mW source powers as compared to the other two configurations shown in Figs. 13(a) and 13(b), respectively. For 5 mW source power, the SSIM values are increasing with exposure time for the conventional digital holographic configuration and the digital holographic camera until 50 ms and may take longer exposure time to saturate. The SSIM values for the concave-lens-based setup deteriorated after 30 ms at 5 mW source power. The maximum value of SSIM achieved is 0.53 with the digital holographic camera at 5 mW source power and 50 ms exposure time of the recording sensor. For 28 mW source power, the SSIM values are slightly deteriorated after 10 ms for the digital holographic camera and 30 ms for the conventional digital holographic configuration. The SSIM values are lowest for the concave-lens-based digital holographic configuration at both powers at a certain exposure time.

D. Optimization of Source Power and Exposure Time of the Sensor

It may be inferred from the reconstructed images and plots of quality parameters that at low source power, long exposure times of the recording sensor are required to obtain good quality of the reconstructed images. Also, with an increase in source power, the requirement of exposure time of the sensor reduces to produce good quality reconstructed images in all three configurations, as shown by PSNR and SSIM values. But, an increase in source power at certain exposure times and vice-versa contributes in increasing the noise in reconstructed images, as shown by speckle index values. Therefore, optimization of source power and the sensor’s exposure time are required to obtain good quality reconstruction in all three configurations. Since environmental vibration plays a critical role in the recording of digital holograms, shorter exposure time of the sensor is required for proper reconstruction, and thus source power needs to be higher. So, on the basis of the first, to the best of our knowledge, occurrence of proper reconstruction in all three configurations, the values of source power, sensor’s exposure time, and quality parameters are mentioned in Table 1 for the conventional digital holographic configuration, Table 2 for the concave-lens-based digital holographic configuration, and Table 3 for the digital holographic camera. For a good quality reconstructed image, low speckle index, high PSNR, and high SSIM values are desired. Also, short exposure time of the recording sensor is required to avoid environmental vibrations during the recording of digital holograms. After analyzing these parameters in Tables 1–3, it can be observed that at 20 mW source power, shorter exposure time of the sensor, low speckle index value, and moderate PSNR and SSIM values are obtained for all three configurations.

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Table 1. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Conventional Digital Holographic Configuration

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Table 2. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Concave-Lens-Based Digital Holographic Configuration

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Table 3. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Digital Holographic Camera

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E. Comparison

After optimizing the parameters, a comparison can be made for all three configurations at 20 mW optimized source power to obtain good quality reconstructed images. Plots of quality parameters at 20 mW source power are shown in Fig. 14 to compare the performance of the three configurations.

 figure: Fig. 14.

Fig. 14. (a) Speckle index, (b) PSNR, and (c) SSIM values for the three configurations at 20 mW source powers.

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It can be observed in Fig. 14(a) that variations in speckle index values for the digital holographic camera are lower as compared to the other two configurations with an increase in exposure time of the sensor. Also, higher PSNR and SSIM values are obtained using the digital holographic camera with increasing the exposure time of the sensor, as shown in Figs 14(b) and 14(c), respectively. Thus, at an optimized 20 mW power of the source, the digital holographic camera performed better compared to the conventional digital holographic and concave-lens-based digital holographic configurations.

5. CONCLUSIONS

The effects of source power and exposure time of the recording sensor on the quality of reconstructed amplitude images obtained by applying FDM on recorded digital holograms are studied. The digital holograms are recorded using three different configurations, namely, conventional digital holography, concave-lens-based digital holography, and the digital holographic camera at different source power and sensor exposure time. To estimate the quality of reconstructed images, several quality parameters are evaluated, such as speckle index, PSNR, and SSIM. A simulated reconstructed image of the object is used as a reference image to calculate PSNR and SSIM values. The simulation was performed for all three configurations of the digital holographic setup to keep the resolution of the reconstructed image in simulation the same as that obtained in the experiments, i.e., ${539} \times {697}$ pixels for the conventional digital holographic configuration, ${226} \times {302}$ pixels for the concave-lens-based digital holographic configuration, and ${333} \times {431}$ pixels for the digital holographic camera. The comparison of quality parameters shows that with an increase in source power at certain exposure times and vice-versa, the values of PSNR and SSIM increases, but at the same time speckle index value also increased. Therefore, an optimum power is determined by comparing the evaluated quality parameters at exposure times where proper reconstruction of the object was observed. In the present case, the optimum power obtained is 20 mW for the recorded surface of the Indian rupee 5 coin. The performance of the three configurations of the digital holographic experiment is also compared on the basis of evaluated quality parameters. The reconstructed images obtained in the experiment performed using the digital holographic camera were found to be of better in quality than that obtained with the other two digital holographic configurations. The relation between source power and exposure time of sensors may be further explored in digital holography on different types of surfaces using quality parameter estimation.

Funding

Department of Science and Technology, Ministry of Science and Technology, India (DST/TSG/NTS/2015/59).

Acknowledgment

Portions of this work were presented at the OSA Imaging and Applied Optics Congress in 2020, paper number—3440878, paper title—“Quality evaluation of reconstructed images in digital holography to analyze the effects of source power and exposure time”.

Disclosures

RK and GD are inventors on a patent describing the method and system for recording digital holograms of larger objects in a non-laboratory environment [U.S. patent application number 16/912604; India patent application number 201911023585].

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Figures (14)
Fig. 1.
Fig. 1. Schematic representations of (a) conventional off-axis digital holographic experimental configuration, (b) concave-lens-based digital holographic experimental configuration, and (c) optical configuration of the digital holographic camera. The abbreviations in the figure represent BS, beam splitter; VBS, variable beam splitter; L1, concave lens; L2, convex lens; MO, microscope objective; PM, power meter; CMOS, complementary metal oxide semiconductor.

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Fig. 2.
Fig. 2. Image of an Indian 5 rupee coin used as an object to record digital holograms in all three configurations of digital holographic experiments shown in Fig. 1.

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Fig. 3.
Fig. 3. (a) Captured image of the coin, (b) digital hologram obtained after adding and squaring the digital reference beam with the recorded field of the coin, (c) Fourier spectrum of the digital hologram, and (d) reconstructed image obtained after filtering one of the orders in the Fourier spectrum indicated by rectangle in (c).

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Fig. 4.
Fig. 4. (a) Recorded digital holograms using conventional off-axis digital holographic configuration and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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Fig. 5.
Fig. 5. (a) Recorded digital holograms using concave-lens-based digital holographic configuration and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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Fig. 6.
Fig. 6. (a) Recorded digital holograms using digital holographic camera and (b) reconstructed amplitude images of the coin. Rows and columns show the variation in power of the laser source and exposure time of the recording sensor, respectively.

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Fig. 7.
Fig. 7. Mean values of intensities of recorded images of digital holograms obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera.

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Fig. 8.
Fig. 8. Speckle index values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera.

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Fig. 9.
Fig. 9. Speckle index values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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Fig. 10.
Fig. 10. PSNR values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera after comparing them with the simulated reconstructed image.

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Fig. 11.
Fig. 11. PSNR values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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Fig. 12.
Fig. 12. SSIM values of reconstructed images obtained using (a) conventional off-axis digital holographic configuration, (b) concave-lens-based digital holographic configuration, and (c) digital holographic camera after comparing them with the simulated reconstructed image.

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Fig. 13.
Fig. 13. SSIM values for the three configurations at (a) 5 mW and (b) 28 mW source powers.

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Fig. 14.
Fig. 14. (a) Speckle index, (b) PSNR, and (c) SSIM values for the three configurations at 20 mW source powers.

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Tables (3)

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Table 1. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Conventional Digital Holographic Configuration

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Table 2. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Concave-Lens-Based Digital Holographic Configuration

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Table 3. Values of Source Power, Sensor’s Exposure Time, and Quality Parameters for the First Occurrence of Properly Reconstructed Image for Digital Holographic Camera

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Equations (14)

Equations on this page are rendered with MathJax. Learn more.

O ( x , y ) = A O ( x , y ) exp ⁡ ( j ϕ O ( x , y ) ) ,
R ( x , y ) = A R ( x , y ) exp ⁡ ( j ϕ R ( x , y ) ) ,
I ( x , y ) = | O ( x , y ) + R ( x , y ) | 2 ,
I ( x , y ) = | O ( x , y ) | 2 + | R ( x , y ) | 2 + O ( x , y ) R ∗ ( x , y ) + O ∗ ( x , y ) R ( x , y ) ,
O ( ξ , η ) = exp ⁡ ( − j k z ) j λ z exp ⁡ ( − j π ( x 2 + y 2 ) λ z ) ∗ F T [ O ( x , y ) exp ⁡ ( − j π ( x 2 + y 2 ) λ z ) ] ,
S = 1 ( M − 2 ) ( N − 2 ) ∑ m , n = 2 M − 1 , N − 1 σ ( m , n ) μ ( m , n ) ,
P S N R = 10 log 10 ( P 2 M S E ) ,
M S E = 1 M ′ N ′ ∑ 1 N ′ ∑ 1 M ′ | I ′ − I | 2 ,
S S I M ( x , y ) = ( 2 μ x μ y + C 1 ) ( 2 σ xy + C 2 ) ( μ x 2 + μ y 2 + C 1 ) ( σ x 2 + σ y 2 + C 2 ) ,
O ( x , y ; z ) = F − 1 [ F { O 0 ( x , y ) } × H ( p , q ) ] ,
H ( p , q ) = exp ⁡ [ − j 2 π z 1 λ 2 − ( p M Δ ) 2 − ( q N Δ ) 2 ] ,
R D ( x , y ) = A R D ( x , y ) exp ⁡ ( j 2 π λ ( p + q ) Δ sin ⁡ θ ) ,
I D ( x , y ) = | O ( x , y ; z ) + R D ( x , y ) | 2 ,
I D ( x , y ) = | O ( x , y ; z ) | 2 + | R D ( x , y ) | 2 + O ( x , y ; z ) R D ∗ ( x , y ) + O ∗ ( x , y ; z ) R D ( x , y ) ,
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