Detail displaying difference of the digital holographic reconstructed image between the convolution algorithm and Fresnel algorithm

doi:10.1364/OE.19.023621

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Detail displaying difference of the digital holographic reconstructed image between the convolution algorithm and Fresnel algorithm

Liyun Zhong, Hongyan Li, Tao Tao, Zhun Zhang, and Xiaoxu Lu »View Author Affiliations

Optics Express, Vol. 19, Issue 23, pp. 23621-23630 (2011)
http://dx.doi.org/10.1364/OE.19.023621

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Abstract

To reach the limiting resolution of a digital holographic system and improve the displaying quality of the reconstructed image, the subdivision convolution algorithm and the subdivision Fresnel algorithm are presented, respectively. The obtained results show that the lateral size of the reconstructed image obtained by two kinds of subdivision algorithms is the same in the central region of the reconstructed image-plane; moreover, the size of the central region is in proportional to the recording distance. Importantly, in the central region of the reconstructed image-plane, the reconstruction can be performed by the subdivision Fresnel algorithm instead of the subdivision convolution algorithm effectively, and, based on these subdivision approaches, both the displaying quality and the resolution of the reconstructed image can be improved significantly. Furthermore, in the reconstruction of the digital hologram with the large numerical aperture, the computer's memory consumed and the calculating time resulting from the subdivision Fresnel algorithm is significantly less than those from the subdivision convolution algorithm.

1. Introduction

Since the complex amplitude distribution of the measured object can be conveniently obtained by digital holography(DH), in which CCD or CMOS is employed to record hologram and the reconstruction is performed by the numerical calculation [1

J. W. Goodman and P. W. Lawrence, “Digital image formulation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967). [CrossRef]

], DH has been a powerful tool in inspecting the micro-deformation and micro-vibration microscopic observation, information encryption, particle measurement, three dimensional recognition [2

D. D. Aguayo, F. Mendoza Santoyo, M. H. De la Torre-I, M. D. Salas-Araiza, C. Caloca-Mendez, and D. A. Gutierrez Hernandez, “Insect wing deformation measurements using high speed digital holographic interferometry,” Opt. Express 18(6), 5661–5667 (2010). [CrossRef] [PubMed]

–10

L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. 31(7), 897–899 (2006). [CrossRef] [PubMed]

]. However, due to both the resolution and the size of CCD or CMOS are greatly less than those of the traditional silver-salt recording material, to reach the limiting resolution of DH system and improve the displaying quality of the reconstructed image, the reconstructed algorithm is a important research content for DH's development and application in quantitative measurement.

In general, the displaying quality of DH's reconstructed image is determined by the lateral resolution of DH's recording system, meanwhile the power of DH's resolution is related to the detail-displaying quality of the reconstructed image. For the digital hologram with the large numerical aperture, the reconstruction is performed by the complex Rayleigh-Sommerfeld diffraction integral and named as the convolution algorithm. In contrast, for the digital hologram with the small numerical aperture, assume Fresnel diffraction approximate can be fulfilled, the reconstruction can be performed by Fresnel diffraction integral and named as Fresnel algorithm [11

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

,12

U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).

], obviously, Fourier transform can be directly used in the reconstruction of Fresnel algorithm. In addition, based on the convolution theorem, the reconstruction of the convolution algorithm also can be performed by fast Fourier transform(FFT), thus the pixels spacing of the reconstructed image-plane (PSRI) is the same with that of CCD [12

U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).

]. However, while the minimum resolving spacing of DH system is less than the pixels spacing of CCD, the displaying quality of the reconstructed image obtained by the convolution algorithm cannot represent the resolution of DH system. To improve the displaying quality of the reconstructed image, a kind of hologram's frequency-domain zero-padding approach, in which the digital hologram is transformed to its frequency-domain and then implemented by the zero-padding operation, is proposed to shorten the PSRI [13

L. P. Chen and X. X. Lu, “The recording of digital hologram at short distance and reconstruction using convolution approach,” Chin. Phys. B 18(1), 189–194 (2009).

]. However, while the resolution of DH system is very high, the size of data matrix come from the convolution algorithm will be larger than the limiting of the computer's memory, thus DH's reconstruction cannot be completed effectively, in contrast, due to the PSRI obtained by Fresnel algorithm is only half of the minimum resolving spacing of DH system [12

U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).

], the corresponding reconstructed calculating time and the size of data matrix are less than those of the convolution algorithm significantly, therefore Fresnel algorithm should be a better candidate used as the complementary and the alternative of the convolution algorithm. In addition, due to there are some differences of the approximation between the convolution algorithm and Fresnel algorithm, and it is difficult to evaluate the difference of the reconstructed result between two kinds of algorithms by the analytical method, in general, the numerical calculation method supply a better solution for this problem. Specially, while the detail size of the recorded sample reaches the minimum resolving spacing of DH system, the displaying quality of the reconstructed image obtained by Fresnel algorithm is also greatly lower, to solve this problem, the subdivision Fresnel algorithm in which the hologram is directly implemented by the zero-padding operation can be employed to perform the reconstruction [14

A. V. Oppenheim, Signals & Systems (Pearson Education, Singapore, 1997).

,15

Q. Fan, J. L. Zhao, and S. Li, “Detail displaying and vision aberration rectifying of reconstructed image in digital holography,” Chin. J. Lasers 32, 1401–1405 (2005).

In this paper, for both the convolution algorithm and Fresnel algorithm, the relationship between the PSRI and the resolution of DH system are carefully discussed, respectively. Moreover, to improve the displaying quality of the reconstructed image, the reconstructed results obtained by the subdivision convolution algorithm and the subdivision Fresnel algorithm are also respectively presented. Specially, while Fresnel approximation is not fulfilled, the detail-displaying difference of the digital holographic reconstructed image between the subdivision convolution algorithm and the subdivision Fresnel algorithm is also shown carefully. Importantly, these research results will supply a powerful approach for DH's application in quantitative measurement.

2. Resolution of DH recording system

Resolution of DH recording system is determined by the maximum spatial frequency

f_{\max}

of the object wave which can be recorded by DH system. Usually

f_{\max}

can be written as

f_{\max} = \sin u / λ

, and the corresponding limiting resolution

ξ

of DH system can be expressed as [13

L. P. Chen and X. X. Lu, “The recording of digital hologram at short distance and reconstruction using convolution approach,” Chin. Phys. B 18(1), 189–194 (2009).

]

ξ = \frac{1}{ε} = \frac{\sin u}{λ} = \frac{1}{λ} \frac{L_{C C D} / 2}{\sqrt{Z_{O H}^{2} + {(L_{C C D} / 2)}^{2}}}

(1)

where λ is the laser wavelength, and ε is the limiting resolving spacing of DH recording system, sinu is the numerical aperture (NA) of DH system, Z_OH is the distance between the object-plane and CCD target-plane, and L_CCD is the size of CCD target-plane. In general, assume the recording wavelength

λ

of DH system is unchanged, the limiting resolution

ξ

is determined by Z_OH and L_CCD, the shorter of the recording distance Z_OH or the larger of CCD size L_CCD, the higher of the limiting resolution

ξ

. While the center of the object is located in the central normal line of CCD, and then

Z_{O H} > > L_{C C D}

, that is

{(Z_{O H} / L_{C C D})}^{2} > > 1

, Eq. (1) can be simplified as

ξ = \frac{1}{ε} = \frac{N A}{λ} \approx \frac{L_{C C D}}{2 λ Z_{O H}}

(2)

From Eq. (1) and Eq. (2), it is clearly shown that resolution of DH system is in proportional to the NA and in inverse proportional to the wavelength λ. Obviously, three kinds of approaches can be employed to improve the resolution of DH system as following: (1)To record the hologram by a shorter wavelength laser of λ; (2) To shorten the recording distance Z_OH; (3) To enlarge the effective size L_CCD of CCD target.

3. PSRI obtained by different algorithms

3.1 PSRI obtained by the convolution algorithm

In the reconstruction of the digital hologram with the large NA, due to Fresnel diffraction condition is not fulfilled, the complex Rayleigh-Sommerfeld diffraction integral is employed to perform the diffraction calculation. Assume the coordinates of the object-plane, the hologram-plane (CCD target plane) and the reconstructed image-plane are respectively expressed as

(x_{O}, y_{O})

(x_{H}, y_{H})

and

(x_{I}, y_{I})

, and the distance between the hologram-plane and the object-plane is Z_OH, the distance between the reconstructed image-plane and the hologram-plane is Z_HI, as shown in Fig. 1 , thus the digital hologram and the reconstructed wave can be written as

I (x_{H}, y_{H})

and

C (x_{H}, y_{H})

, respectively.

Fig. 1 Coordinates relationship between the object-plane (x_O,y_O), the recorded hologram-plane(x_H,y_H) and the reconstructed image-plane(x_I,y_I).

Based on Rayleigh-Sommerfeld diffraction formula, the object's image can be expressed as [11

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

]

O (x_{I}, y_{I}) = \iint I (x_{H}, y_{H}) C (x_{H}, y_{H}) g (x_{I} - x_{H}, x_{I} - y_{H}) d x_{H} d y_{H}

(3)

where

g (x_{I} - x_{H}, x_{I} - y_{H})

is the point spread function of DH system

g (x_{I} - x_{H}, y_{I} - y_{H}) = - \frac{Z_{H I}}{j λ} \frac{\exp [- j k \sqrt{Z_{H I}^{2} + {(x_{I} - x_{H})}^{2} + {(y_{I} - y_{H})}^{2}}]}{Z_{H I}^{2} + {(x_{I} - x_{H})}^{2} + {(y_{I} - y_{H})}^{2}}

(4)

Based on the convolutional characteristics, Eq. (3) can be expressed by Fourier transform of the frequency domain

O (x_{I}, y_{I}) = F^{- 1} {F [I (x_{H}, y_{H}) C (x_{H}, y_{H})] F [g (x_{I} - x_{H}, y_{I} - y_{H})]}

(5)

where

F

and

F^{- 1}

respectively denote the fast Fourier transform(FFT) and its inverse transform, and

F [g (x_{I} - x_{H}, y_{I} - y_{H})]

is the Fourier transform of the point spread function, in general, it is also named as the transfer function of DH system.

From Eq. (5), it can be seen that the reconstruction of the convolution algorithm can be performed by two times of Fourier transform and one time of inverse Fourier transform, and in this case, the corresponding PSRI can be expressed as¹²

Δ x_{I} = Δ x_{H} Δ y_{I} = Δ y_{H}

(6)

where

Δ x_{I}

and

Δ y_{I}

are the PSRI in x and y direction,

Δ x_{H}

and

Δ y_{H}

are the pixel spacing of CCD in the x and y directions, respectively, it is clearly shown that the this PSRI is equal to the pixels spacing of the digital hologram, that to say, this PSRI is independent of the resolution of DH recording system. However, while the pixels spacing of CCD is larger than the minimum resolving distance of DH system, the PSRI obtained by the convolution algorithm is enlarged, thus the resolving power of DH system cannot be represented effectively, therefore it is needed to shorten the PSRI to half of the limiting resolution of DH system, that is

Δ x_{I} \leq ε_{x} / 2, Δ y_{I} \leq ε_{y} / 2

, thus the minimum detail displaying which can be identified by DH system will be presented in the reconstructed image, that to say, the smaller of the PSRI, the better of the displaying quality of the reconstructed image.

3.2 PSRI obtained by Fresnel algorithm

Assume Fresnel approximation condition is fulfilled, the object's image can be described by Fresnel diffraction formula [12

U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).

]

\begin{matrix} O (x_{I}, y_{I}) = A \iint I (x_{H}, y_{H}) C (x_{H}, y_{H}) \exp [- j \frac{π}{λ Z_{H I}} (x_{H}^{2} + y_{H}^{2})] \exp [j \frac{2 π}{λ Z_{H I}} (x_{H} x_{I} + y_{H} y_{I})] d x_{H} d y_{H} \\ = A F^{- 1} {I (x_{H}, y_{H}) C (x_{H}, y_{H}) \exp [- j \frac{π}{λ Z_{H I}} (x_{H}^{2} + y_{H}^{2})]} \end{matrix}

(7)

where A is the phase factor which is independent of the integral

A = \frac{1}{j λ Z_{H I}} \exp (- j k Z_{H I}) \exp [- \frac{j k}{2 Z_{H I}} (x_{I}^{2} + y_{I}^{2})]

(8)

Assume the hologram is sampled as M × N pixels,

Δ x_{H}

and

Δ y_{H}

are the pixels spacing in the x and y direction, respectively, then the discrete form of Eq. (7) can be written as

O (m Δ x_{I}, n Δ y_{I}) = A F^{- 1} {I (k Δ x_{H}, l Δ y_{H}) C (k Δ x_{H}, l Δ y_{H}) \exp [- j \frac{π}{λ Z_{H I}} (k^{2} Δ x_{H}^{2} + l^{2} Δ y_{H}^{2})]}

(9)

where m, n are integers,

0 \leq m \leq M - 1, 0 \leq n \leq N - 1

Δ x_{I}

and

Δ y_{I}

are the PSRI in x and y directions, respectively. According to the relationship of the discrete Fourier transform between N sampled points and the corresponding N discrete frequency [12

U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).

Δ x_{I}

and

Δ y_{I}

can be expressed as

Δ x_{I} = \frac{λ Z_{H I}}{M Δ x_{H}}, Δ y_{I} = \frac{λ Z_{H I}}{N Δ y_{H}}

(10)

From Eq. (10), it can be clearly seen that the PSRI obtained by Fresnel algorithm is in proportional to the wavelength λ and the reconstructed distance

Z_{H I}

, and in inverse proportional to the pixels number of M or N, as well as the pixels spacing of

Δ x_{H}

Δ y_{H}

of the hologram. In general, the reconstructed wave is set up the same with the reference wave, that is

Z_{O H} = Z_{H I}

, moreover, due to

M Δ x_{H} = L_{x C C D}, N Δ y_{H} = L_{y C C D}

, From Eq. (2) and Eq. (10), it can be founded that

Δ x_{I} = ε_{x} / 2, Δ y_{I} = ε_{y} / 2

, that to say, the PSRI is only half of the limiting resolving spacing of DH system. From the above analysis, it is clearly shown that while Fresnel algorithm is employed to perform DH's reconstruction, the PSRI can reach the limiting resolution of DH system. However, to improve the detail displaying quality of the reconstructed image, it is still needed to further shorten the PSRI.

4. Detail-displaying difference of the reconstructed image between the convolution algorithm and Fresnel algorithm

As described above, for both the convolution algorithm and Fresnel algorithm, the subdivision approach can be used to shorten the PSRI and improve the displaying quality of reconstructed image. In general, due to the PSRI obtained by the convolution algorithm is equal to the pixels spacing of the hologram, and the PSRI obtained by Fresnel algorithm is in inverse proportional to the total pixels number of the hologram, and in proportional to the recording distance, therefore there are some differences between the subdivision convolution algorithm and the subdivision Fresnel algorithm. In the former, the subdivision is performed in the frequency domain of the original hologram by the zero-padding operation, and in the latter, the subdivision is performed by the zero-padding operation in the original hologram directly.

Usually, in the subdivision convolution algorithm, based on the frequency domain zero-padding operation, the pixels spacing of the hologram can be shortened effectively. Figure 2 is ascheme for the process of the hologram's frequency domain zero-padding operation. Firstly, as shown in Figs. 2a, 2b, the original hologram with the pixels number

M_{1} \times N_{1}

is performed the FFT, and then the corresponding frequency spectrum of the hologram is obtained. Secondly, based on the zero-padding operation in the frequency spectrum, an enlarged frequency spectrum with the pixels number

M_{2} \times N_{2}

is obtained, as shown in Fig. 2c. Thirdly, by performing the IFFT to Fig. 2c, as shown in Fig. 2d, a new hologram with the pixels number

M_{2} \times N_{2}

is obtained, in which both the size and the distribution of the interference fringe are still the same with the original hologram, due to the pixels number of the hologram is enlarged from

M_{1} \times N_{1}

M_{2} \times N_{2}

, thus pixels spacing of the new hologram is shortened to only

M_{1} / M 2

times or

N_{1} / N 2

times of the original hologram. Finally, by using Eq. (5), the subdivision hologram is multiplied with a digital reconstructed wave, and then performed the FFT and multiplied with the transfer function, finally performed the IFFT, thus the reconstructed image obtained by the above subdivision convolution algorithm can be presented. In contrast, in Fresnel algorithm, due to the PSRI is in proportional to the recording distance and in inverse proportional to the total pixels number of the hologram, the total pixels number of the hologram can be increased by the zero-padding operation in the original hologram directly [14

A. V. Oppenheim, Signals & Systems (Pearson Education, Singapore, 1997).

,15

Q. Fan, J. L. Zhao, and S. Li, “Detail displaying and vision aberration rectifying of reconstructed image in digital holography,” Chin. J. Lasers 32, 1401–1405 (2005).

], thus the PSRI also can be shortened conveniently. For example, assume the pixels number of the hologram is increased from

M_{1} \times N_{1}

M_{3} \times N_{3}

by the zero-padding operation in the original hologram, by using of Eq. (9), the size of the reconstructed image obtained from the new hologram with the pixels number of

M_{3} \times N_{3}

is the same with that obtained by the original hologram with the pixels number of

M_{1} \times N_{1}

, however, the pixels spacing of the new hologram is shorten to M₁/M₃ times and N₁/N₃ times of the original hologram.

Fig. 2 Scheme for the process of the hologram's frequency domain zero-padding operation.

5. Experiments and results

A phase-shifting in-line lensless Fourier transform experimental setup is employed to record the digital hologram [16

H. Li, L. Zhong, Z. Ma, and X. Lu, “Joint approach of the sub-holograms in on-axis lensless Fourier phase-shifting synthetic aperture digital holography,” Opt. Commun. 284(9), 2268–2272 (2011). [CrossRef]

], in which the hologram with the large NA can be obtained and the reconstructed zero-order image and its conjugate image can be eliminated easily, thus the ratio of signal to noise of the reconstructed image can be improve significantly. Parameters of the system are as following: Laser wavelength: 632.8 nm, Computer's CPU: Intel Core2 E7400, Clock: 2.80GHz, Memory: 2GB, Pixels number of CCD: 768H × 576V, Pixels spacing of CCD: 10μm × 10μm. In addition, the minimum 25th unit of No.3 resolution test plate, in which each line-group has 17 line-pairs and the corresponding line spacing is 20 μm (50 lp/mm), is used as the recorded sample. In our experiment, two digital holograms in the different recording distance Z_OH = 65.7mm and 27.64 mm are recorded, as shown in Fig. 3 , in which Fresnel approximation condition are respectively not fulfilled and completely not fulfilled, and the corresponding minimum resolving spacing are respectively 14.44μm and 6.07μm.

Fig. 3 Recorded phase-shifting digital holograms (a) one of four-step phase-shifting digital holograms in the recording distance Z_OH = 65.7mm, (b) one of four-step phase-shifting digital holograms in the recording distance Z_OH = 27.64mm.

Firstly, to present the displaying difference between the convolution algorithm and Fresnel algorithm, Fig. 4 and Fig. 5 respectively give the reconstructed results of the original hologram obtained by the above two kinds of algorithms in which the recording distance are respectively Z_OH = 65.7 mm and Z_OH = 27.64 mm, and the pixels number of the reconstructed image obtained by two kinds of algorithms are both 576H × 576V, which is the same with that of the original hologram. Figures 4a, 4b (or Figs. 5a, 5b) are the reconstructed image respectively obtained by the convolution algorithm and Fresnel algorithm, Figs. 4c, 4d (or Figs. 5c, 5d) are respectively the magnified images marked with the white real line in Figs. 4a, 4b (or Figs. 5a, 5b), and Figs. 4e, 4f (or Figs. 5e, 5f) are respectively the intensity distribution of the same column marked with the black real arrow line in Figs. 4c, 4d (or Figs. 5c, 5d). In Fig. 4, in which Z_OH = 65.71mm and Fresnel approximation condition is a little not fulfilled, it can be clearly seen that the PSRI obtained by the convolution algorithm is larger than that obtained by Fresnel algorithm, therefore the displaying quality of the reconstructed image obtained by the convolution algorithm(Fig. 4a) is lower than that obtained by Fresnel algorithm (Fig. 4b), and from Figs. 4e,4f, it can be calculated that the PSRI are respectively 10.0 μm for the convolution algorithm and 7.27 μm for Fresnel algorithm. Importantly, these results are the same with the results obtained by the theoretical analysis of Eq. (6) and Eq. (10). In contrast, in Fig. 5, in which Z_OH = 27.64 mm and Fresnel approximation condition is not fulfilled completely, as shown in Fig. 5a, in the convolution algorithm, it is observed that the displaying quality of the reconstructed image is almost the same with that in Fig. 4a, in which Z_OH = 65.71 mm and Fresnel approximation condition is a little not fulfilled, and from Fig. 5e, it can be calculated that the PSRI obtained is still 10μm, however, this image's displaying quality (Fig. 5a) is lower than that obtained by Fresnel algorithm (Fig. 5b) significantly, and from Fig. 5f, it can be calculated that the PSRI is only 3.05μm.

Fig. 4 Reconstructed results obtained from the original hologram directly in the recording distance Z_OH = 65.71mm: (a) the reconstructed image obtained by the convolution algorithm; (b) the reconstructed image obtained by Fresnel algorithm; (c)the magnification marked with the white real line in (a); (d)the magnification marked with the white real line in (b); (e)the intensity distribution of one column marked with the black real arrow line in (c); (f)the intensity distribution of one column marked with the black real arrow line in (d).

Fig. 5 Reconstructed results obtained from the original hologram directly in the recording distance Z_OH = 27.64 mm: (a) the reconstructed obtained by the convolution algorithm; (b) the reconstructed obtained by Fresnel algorithm; (c)the magnification marked the white real line in (a); (d)the magnification marked the white real line in (b); (e)the intensity distribution of one column marked with the black real arrow line in (c); (f)the intensity distribution of one column marked with the black real arrow line in (d).

Figures 6 and 7 give the reconstructed results obtained by different subdivision approaches in which the recording distance are respectively Z_OH = 65.7 mm and Z_OH = 27.64 mm. To compare the detail displaying difference of the reconstructed image between the subdivision convolution algorithm and the subdivision Fresnel algorithm conveniently, assume the PSRI obtained by two kinds of subdivision algorithms are set up as 2μm. In the subdivision convolution algorithm, the pixels number of the hologram's frequency domain is enlarged from 576H × 576V to 2880H × 2880V by the zero-padding operation in both the recording distance Z_OH = 65.7 mm and 27.64 mm. And in the subdivision Fresnel algorithm, the pixels number of the hologram is enlarged from 576H × 576V to 2079H × 2079V by the zero-padding operation in the recording distance Z_OH = 27.64 mm, as well as 576H × 576V to 874H × 874V in the recording distance Z_OH = 27.64 mm.

Fig. 6 Reconstructed results obtained from different subdivision approaches in the recording distance Z_OH = 65.71mm (a) the reconstructed image obtained by the subdivision convolution algorithm in which the hologram is implemented by the zero-padding operation in frequency domain (b) the reconstructed image obtained by the subdivision Fresnel algorithm in which the hologram is directly implemented by the zero-padding operation; (c)the magnification marked the white real line in area in (a); (d) the Magnification marked the white real line in (b); (e)Intensity distribution of the same column marked with the black real arrow line in (c) and (d), respectively.

Fig. 7 Reconstructed results obtained from different subdivision approaches in the recording distance Z_OH = 27.64 mm (a) the reconstructed image obtained by the subdivision convolution algorithm in which the hologram is implemented by the zero-padding operation in frequency domain (b) the reconstructed image obtained by the subdivision Fresnel algorithm in which the hologram is directly implemented by the zero-padding operation; (c)the magnification marked the white real line in area in (a); (d) the Magnification marked the white real line in (b); (e)Intensity distribution of the same column marked with the black real arrow line in (c) and (d), respectively.

Figures 6a, 6b (or Figs. 7a, 7b) are the reconstructed image respectively obtained by the subdivision convolution algorithm and the subdivision Fresnel algorithm, Figs. 6c, 6d (or Figs. 7c, 7d) are the magnified images respectively obtained from the white square in Figs. 6a, 6b (or Figs. 7a, 7b), and Fig. 6e (or Fig. 7e) are the intensity distribution respectively obtained from the same column in Figs. 6c, 6d (or Figs. 7c, 7d), in which the solid line and the circle respectively denote the result obtained by the above two kinds of subdivision algorithms.

Comparing Fig. 4 with Fig. 6, as well as Fig. 5 with Fig. 7, it is clearly showed that the displaying quality of the reconstructed image obtained by the subdivision convolution algorithm is better than that obtained by the direct convolution algorithm, and in the subdivision Fresnel algorithm, the displaying quality of the reconstructed image is related to the recording distance, assume the PSRI is unchanged, the larger the recording distance, the better the displaying quality of the reconstructed image.

In addition, there is a greatly difference of the reconstructed calculating time between the subdivision convolution algorithm and the subdivision Fresnel algorithm, for example, in the reconstruction of the same hologram, assume the PSRI obtained by two kinds of subdivision algorithms is the same, in general, due to the subdivision convolution algorithm need four times Fourier transform, moreover, the higher of the resolution of DH system, the greater size of the hologram's frequency domain matrix implemented by the zero-padding operation, in contrast, the subdivision Fresnel algorithm need only one time Fourier transform, and the higher of the resolution of DH system, the smaller size of the hologram's matrix implemented by the zero-padding operation, therefore, it can be concluded that both the calculating time and computer memory consuming in the subdivision convolution algorithm are more than that in the subdivision Fresnel algorithm significantly, that to say, while DH's reconstruction performed by the subdivision Fresnel algorithm takes less than 1 second, and the reconstruction performed by the convolution algorithm requires 22 seconds.

Specially, assume the PSRI obtained by two kinds of algorithms is setup the same, the obtained results show that the size of the reconstructed image obtained by two kinds of algorithms is exactly the same in the central region, however, in the non-central region, there is a little lateral size difference of the reconstructed image between two kinds of algorithms, moreover, the size of the central region is in proportional to the recording distance. In our experiment, while the recording distance is 65.7mm, the size of the central region of the reconstructed image is larger than 1.5mm, moreover, all parts of the reconstructed image are located in the central of the region, while the recording distance is 27.64 mm, the size of the central region of the reconstructed image is larger than 0.5mm, the difference of the reconstructed image between the subdivision convolution algorithm and the subdivision Fresnel algorithm is increased from the central region to the edge, however, even in the edge of the reconstructed image, the difference between two kinds of algorithms is less than 1μm, specially, it is far less than the minimum resolving spacing 6.07μm of DH system.

6. Conclusion and discussion

For the digital hologram with the large NA, to improve the displaying quality of the reconstructed image and reach the limiting resolution of DH system, it is a better solution to perform the subdivision for the hologram, thus the detail displaying quality of the reconstructed image can be improved significantly. In the subdivision convolution algorithm, due to the reconstruction is performed by Rayleigh-Sommerfeld diffraction formula, based on the convolutional characteristics, this kind of subdivision algorithm can be implemented by the zero-padding operation in hologram's frequency domain, however, for the reconstruction of the hologram with the large NA, due to the data matrix come from this kind of subdivision convolution algorithm is greatly large, and leads to the increase of the computer's memory consuming and the reconstructed calculating time, thus the reconstruction cannot be completed effectively, in contrast, in the subdivision Fresnel algorithm, the memory consuming and the calculating time is less than that of the former significantly, that to say, the subdivision Fresnel algorithm should be a better solution for the reconstruction of the digital hologram with the large NA. Specially, in both the subdivision convolution algorithm and the subdivision Fresnel algorithm, the lateral size of the reconstructed image is the same in the central region, moreover, these results are consistent with Goodman's derivation in which the lateral size of the reconstructed image is not less than

4 {(λ Z_{H I})}^{1 / 2}

, in addition, our research results also show that even if the reconstructed image is located in two times size of the central region, the difference between the above two kinds of algorithms is less than a pixel spacing. Importantly, in the reconstruction of the digital hologram with the large numerical aperture, these research results supply a powerful approach for improvement the displaying quality of the reconstructed image and shortening the reconstructed calculating time, thus promote DH's development in measuring accuracy and application in quantitative measurement.

Acknowledgments

This work is supported by National Nature Science Foundation of China grants (60877070, 60978065, 61078064).

References and links

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3.	F. Joud, F. Laloë, M. Atlan, J. Hare, and M. Gross, “Imaging a vibrating object by Sideband Digital Holography,” Opt. Express 17(4), 2774–2779 (2009). [CrossRef] [PubMed]
4.	C. Trillo, A. F. Doval, F. Mendoza-Santoyo, C. Pérez-López, M. de la Torre-Ibarra, and J. L. Deán, “Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method,” Opt. Express 17(20), 18014–18025 (2009). [CrossRef] [PubMed]
5.	C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008). [CrossRef] [PubMed]
6.	F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17(15), 13071–13079 (2009). [CrossRef] [PubMed]
7.	M. Salvador, J. Prauzner, S. Köber, K. Meerholz, J. J. Turek, K. Jeong, and D. D. Nolte, “Three-dimensional holographic imaging of living tissue using a highly sensitive photorefractive polymer device,” Opt. Express 17(14), 11834–11849 (2009). [CrossRef] [PubMed]
8.	W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17(22), 20342–20348 (2009). [CrossRef] [PubMed]
9.	X. Wu, G. Gréhan, S. Meunier-Guttin-Cluzel, L. Chen, and K. Cen, “Sizing of particles smaller than 5 microm in digital holographic microscopy,” Opt. Lett. 34(6), 857–859 (2009). [CrossRef] [PubMed]
10.	L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. 31(7), 897–899 (2006). [CrossRef] [PubMed]
11.	J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).
12.	U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).
13.	L. P. Chen and X. X. Lu, “The recording of digital hologram at short distance and reconstruction using convolution approach,” Chin. Phys. B 18(1), 189–194 (2009).
14.	A. V. Oppenheim, Signals & Systems (Pearson Education, Singapore, 1997).
15.	Q. Fan, J. L. Zhao, and S. Li, “Detail displaying and vision aberration rectifying of reconstructed image in digital holography,” Chin. J. Lasers 32, 1401–1405 (2005).
16.	H. Li, L. Zhong, Z. Ma, and X. Lu, “Joint approach of the sub-holograms in on-axis lensless Fourier phase-shifting synthetic aperture digital holography,” Opt. Commun. 284(9), 2268–2272 (2011). [CrossRef]

OCIS Codes
(090.0090) Holography : Holography
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: August 3, 2011
Revised Manuscript: October 18, 2011
Manuscript Accepted: October 21, 2011
Published: November 4, 2011

Citation
Liyun Zhong, Hongyan Li, Tao Tao, Zhun Zhang, and Xiaoxu Lu, "Detail displaying difference of the digital holographic reconstructed image between the convolution algorithm and Fresnel algorithm," Opt. Express 19, 23621-23630 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23621

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References

J. W. Goodman and P. W. Lawrence, “Digital image formulation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967). [CrossRef]
D. D. Aguayo, F. Mendoza Santoyo, M. H. De la Torre-I, M. D. Salas-Araiza, C. Caloca-Mendez, and D. A. Gutierrez Hernandez, “Insect wing deformation measurements using high speed digital holographic interferometry,” Opt. Express18(6), 5661–5667 (2010). [CrossRef] [PubMed]
F. Joud, F. Laloë, M. Atlan, J. Hare, and M. Gross, “Imaging a vibrating object by Sideband Digital Holography,” Opt. Express17(4), 2774–2779 (2009). [CrossRef] [PubMed]
C. Trillo, A. F. Doval, F. Mendoza-Santoyo, C. Pérez-López, M. de la Torre-Ibarra, and J. L. Deán, “Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method,” Opt. Express17(20), 18014–18025 (2009). [CrossRef] [PubMed]
C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express16(13), 9753–9764 (2008). [CrossRef] [PubMed]
F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express17(15), 13071–13079 (2009). [CrossRef] [PubMed]
M. Salvador, J. Prauzner, S. Köber, K. Meerholz, J. J. Turek, K. Jeong, and D. D. Nolte, “Three-dimensional holographic imaging of living tissue using a highly sensitive photorefractive polymer device,” Opt. Express17(14), 11834–11849 (2009). [CrossRef] [PubMed]
W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express17(22), 20342–20348 (2009). [CrossRef] [PubMed]
X. Wu, G. Gréhan, S. Meunier-Guttin-Cluzel, L. Chen, and K. Cen, “Sizing of particles smaller than 5 microm in digital holographic microscopy,” Opt. Lett.34(6), 857–859 (2009). [CrossRef] [PubMed]
L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett.31(7), 897–899 (2006). [CrossRef] [PubMed]
J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).
U. Schnars and W. Jueptner, Digital Holography (Springer, Berlin, 2005).
L. P. Chen and X. X. Lu, “The recording of digital hologram at short distance and reconstruction using convolution approach,” Chin. Phys. B18(1), 189–194 (2009).
A. V. Oppenheim, Signals & Systems (Pearson Education, Singapore, 1997).
Q. Fan, J. L. Zhao, and S. Li, “Detail displaying and vision aberration rectifying of reconstructed image in digital holography,” Chin. J. Lasers32, 1401–1405 (2005).
H. Li, L. Zhong, Z. Ma, and X. Lu, “Joint approach of the sub-holograms in on-axis lensless Fourier phase-shifting synthetic aperture digital holography,” Opt. Commun.284(9), 2268–2272 (2011). [CrossRef]

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