Comparison and verification of numerical

Optical Engineering 48共10兲, 105802 共October 2009兲

Comparison and verification of numerical reconstruction methods in digital holography Changgeng Liu Dayong Wang Yizhuo Zhang Beijing University of Technology College of Applied Sciences Chao Yang District Beijing 100124 China E-mail: [email protected]

Abstract. The expressions for the reconstructed field from the sample of the diffracted wave, which is produced by illuminating an object, are found by use of different diffraction integrals in the digital holography. The numerical reconstruction methods that truncate and sample this field are compared in overlapping quality, accuracy, pixel resolution, computation window, and speed. The fast Fourier transform 共FFT兲–based direct integration method for the Fresnel integral and the modified FFT-based direct integration method for the Rayleigh-Sommerfeld integral have similar overlapping quality and can flexibly control pixel resolution and computation window size. Meanwhile, the FFT-based angular spectrum method is superior to the FFT-based convolution method in accuracy and speed. The experimental results are presented to verify these consequences. © 2009 Society of Photo-Optical Instrumentation

Engineers. 关DOI: 10.1117/1.3251340兴

Subject terms: digital holography; diffraction integral; reconstruction method. Paper 090388R received May 30, 2009; revised manuscript received Sep. 3, 2009; accepted for publication Sep. 4, 2009; published online Oct. 26, 2009.

1

Introduction

The Rayleigh-Sommerfeld diffraction integral 共RS兲 and the diffraction angular spectrum integral 共AS兲 are found to be equivalent exact solutions to the scalar Helmholtz equation for a linear homogeneous isotropic medium.1,2 The RS can be expressed in the form of a convolution 共CV兲, and it can also evolve into the Fresnel diffraction integral formula 共FR兲 under the paraxial approximation. The numerical reconstruction methods are performed through numerical evaluation of these four integrals and usually based on the fast Fourier transform 共FFT兲. In reality, the applicability of these methods are limited by many factors, such as the pixel resolution, computation window, computation speed, accuracy, etc. Elimination of such limitations has been worked on by many researchers. For instance, the single FFT algorithm 共SFFT兲 is popularly used in digital holography.3–5 However, its pixel resolution will increase in proportion to the reconstruction distance. A zero padding method and multiple-step Fresnel transforms were proposed to control the pixel resolution for the FR.6–10 When the paraxial approximation condition is not satisfied, the exact version of the diffraction integral should be employed to evaluate the diffracted field. For the FFT-based AS method 共FFT-AS兲, the computation window size makes it unsuitable for imaging large objects. Li et al. presented a hybrid method of zero padding strategy and spherical wave illumination to mitigate this difficulty.11 The direct integration for the RS was not applicable due to the large computational load until the FFT-based method for it 共FFT-DIRS兲 was proposed in 2006.12 Later, new versions of the FFTDIRS were presented to control the pixel resolution and improve the speed.13,14 In those papers, the numerical methods are considered as an entirety; thus, some important 0091-3286/2009/$25.00 © 2009 SPIE

Optical Engineering

characteristics, such as the overlapping quality and the relationship between different methods, are not discussed. To explore these characteristics and get more insight into these methods, the reconstruction method is divided into two steps. In the first step, a continuous field is reconstructed from the sample of the complex amplitude of the field at the hologram plane emanating from an object. Because the sensor can only capture the intensity of the field, an off-axis geometry and phase-shifting technique have been proposed to recover the complex amplitude of the field at the hologram plane.15–17 Next, this continuous field is windowed and sampled to generate the desired image. This paper is arranged as follows: In Sec. 2, a brief review of the four integral formulas is given. In Sec. 3, the expressions for the continuous field reconstructed by the four integral formulas are presented. In Sec. 4, the numerical reconstruction methods that window and sample this continuous field are described. In Sec. 5, the overlapping quality, accuracy, pixel resolution, computation window, and speed of these methods are compared and verified by the experimental results. Finally, our conclusions are given. 2 Diffraction Integrals As illustrated in Fig. 1, Uc共␰ , ␩兲 denotes the diffracted field

Fig. 1 Illustration of the coordinate system of the diffraction theory.

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from an object that falls within the sensor aperture. The complex amplitude of the reconstructed field can be given by the RS,

冕 冕 +⬁

Ur共x,y兲 =

−⬁

⫻

1 2␲

d exp关jk冑共x − ␰兲2 + 共y − ␩兲2 + d2兴 U c共 ␰ , ␩ 兲 共x − ␰兲2 + 共y − ␩兲2 + d2

冋冑

1

共x − ␰兲 + 共y − ␩兲2 + d2 2

册

− jk d␰d␩ ,

共1兲

where d is the reconstruction distance and k the wave vector. Ur共x , y兲 can be equally yielded by the AS through the spatial frequency domain as follows:

冕 冕 +⬁

Ur共x,y兲 =

A共␣, ␤兲T共␣, ␤兲exp关j2␲共␣x + ␤y兲兴d␣d␤ ,

−⬁

共2兲 where A共␣ , ␤兲 is the Fourier spectrum of Uc共␰ , ␩兲 and T共␣ , ␤兲 the transfer function of the free-space propagation with the form as follows: T共␣, ␤兲 = exp关jkd冑1 − 共␭␣兲2 − 共␭␤兲2兴,

共3兲

C共x,y兲 =

冋

3

Continuous Reconstructed Field

Once Uc共␰ , ␩兲 is digitized, the diffraction integrals have to be digitized. Note that, at this stage, the coordinates 共␰ , ␩兲 are digitized, while 共x , y兲 remain continuous. Thus, the resulting expression for the reconstructed field is still continuous and varies with the diffraction integral. Assume the sampled version of Uc共␰ , ␩兲 is a M ⫻ N array and ⌬␰ and ⌬␩ are the pixel pitches along horizontal and vertical directions, respectively. Now, let us deduce the expressions of the output fields after propagating a distance d with different diffraction integrals. 3.1 FR For convenience, we define

冋

册

共9兲

f共␰, ␩兲exp关− j2␲共␣␰ + ␤␩兲兴d␰d␩ .

共10兲

f共␰, ␩兲 = Uc共␰, ␩兲exp

Ur共x,y兲 = Uc共x,y兲 丢 g共x,y,d兲,

and

where 丢 denotes the convolution and g共x , y , d兲 is the pulse response, which has the form g共x,y,d兲 =

d exp关jk冑x2 + y 2 + d2兴 2␲共x2 + y 2 + d2兲

冉冑

1 2

x + y 2 + d2

冊

− jk . 共5兲

According to the convolution theorem, Eq. 共4兲 can be calculated as

k共␣, ␤兲 =

冕冕 ⬁

⬁

−⬁

−⬁

jk 2 共 ␰ + ␩ 2兲 2d

The sample of f共␰ , ␩兲 can be denoted by f共n⌬␰ , m⌬␩兲, where n and m are integers. The frequency spectrum of this sample can be expressed as ⬁

Ur共x,y兲 = IFT兵A共␣, ␤兲FT关g共x,y,d兲兴其,

共6兲

共8兲

The numerical reconstruction methods in the digital holography are carried out through numerical evaluation of Eqs. 共1兲, 共2兲, 共6兲, and 共7兲.

where ␭ is the wavelength. Equation 共1兲 can be concisely represented in the form of a two-dimensional linear convolution 共4兲

册

exp共jkd兲 jk 2 2 exp 共x + y 兲 . jkd 2d

⬁

兺 兺

k d共 ␣ , ␤ 兲 = ⌬ ␰ ⌬ ␩

f共n⌬␰,m⌬␩兲

n=−⬁ m=−⬁

where FT and IFT denote Fourier transform and inverse Fourier transform, respectively. Because the Fourier transform of g共x , y , d兲 equals the transfer function T共␣ , ␤兲, the AS and the RS are equivalent.2 Under the paraxial approximation, Eq. 共1兲 can be approximately expressed as the FR exp共jkd兲 Ur共x,y兲 = jkd ⫻exp = C共x,y兲

冕 冕

再

⬁

兺 兺 n=−⬁ m=−⬁

冕冕 ⬁

⬁

−⬁

−⬁

f共␰, ␩兲exp关− j2␲共␣␰

+ ␤␩兲兴␦共␰ − n⌬␰, ␩ − m⌬␩兲d␰d␩ U c共 ␰ , ␩ 兲

冎

−⬁

=

jk 关共x − ␰兲2 + 共y − ␩兲2兴 d␰d␩ 2d

冏冋

Uc共␰, ␩兲exp

jk 2 共 ␰ + ␩ 2兲 2d

⫻exp关− j2␲共␣␰ + ␤␩兲兴d␰d␩

Optical Engineering

⬁

= ⌬␰⌬␩

+⬁

冕冕

with

⫻exp关− j2␲共␣n⌬␰ + ␤m⌬␩兲兴

冏

⬁

⬁

−⬁

−⬁

冉

f共␰, ␩兲comb

冊

␰ ␩ , exp关− j2␲共␣␰ ⌬␰ ⌬␩ ⬁

+ ␤␩兲兴d␰d␩ = k共␣, ␤兲 丢

册

␣=x/␭d,␤=y/␭d

冕冕

冊

⬁

⬁

冉

⬁

兺 兺␦ n=−⬁ m=−⬁

冉

␣−

冊

m n m − = 兺 兺 k ␣− , ␤− . ⌬␩ ⌬␰ ⌬␩ n=−⬁ m=−⬁

, 共7兲

n ,␤ ⌬␰ 共11兲

Comparing Eq. 共7兲 to Eq. 共11兲, the reconstructed field from the sample of Uc共␰ , ␩兲 by the FR can be given as 105802-2

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UFR共x,y兲 = C共x,y兲kd

冉

冊

⬁

⬁

冉

x y , = C共x,y兲 兺 兺 Ur x ␭d ␭d n=−⬁ m=−⬁

冊冒 冉

冋

冉

B共␣, ␤兲 = FT g共x,y,d兲rect

冊

␰ ␩ , N⌬␰ M⌬␩

冊册

共18兲

.

m␭d n␭d , y− . ⌬␰ ⌬␩ 共12兲

The Fourier transform of the sample of the truncated g共x , y , d兲 can be given as

3.2 RS As is described in Sec. 1, the FR is an approximated expression for the RS. It can be anticipated that the reconstructed fields from the same sample for these two formulas should be similar. In this sense, the reconstructed field by the RS can be given as

Similarly, the FFT will be used and the sample of Bd共␣ , ␤兲 becomes

−

m␭d n␭d , y− ⌬␰ ⌬␩

⬁

⬁

URS共x,y兲 ⬇

兺 兺 Ur m=−⬁ n=−⬁

冉

x−

C x−

冊

n␭d m␭d , y− . ⌬␰ ⌬␩

共13兲

In effect, our primary concern is the validity of the period in this expression. If the estimated period works well, then this approximated formula, though imprecise, can tell us whether or not a precise sample of Ur共x , y兲 can be recovered. 3.3 AS From the sampling theorem, it follows that the Fourier transform of the sample of Uc共␰ , ␩兲 can be given as ⬁

⬁

A d共 ␣ , ␤ 兲 =

兺 兺A m=−⬁ n=−⬁

冉

␣−

冊

n m , ␤− . ⌬␰ ⌬␩

共14兲

In practice, the FFT is employed to calculate a sample of Ad共␣ , ␤兲, which can be written as At共␣, ␤兲 = Ad共␣, ␤兲rect共⌬␰␣,⌬␩␤兲,

共15兲

1 , N⌬␰

⌬␤ =

1 . M⌬␩

共16兲

While At共␣ , ␤兲 will not be a precise sample of A共␣ , ␤兲 due to the finiteness of Uc共␰ , ␩兲, it can be seen as a very nice sample of A共␣ , ␤兲, if the Fourier spectrum of Uc共␰ , ␩兲 concentrates in a certain range and the sampling interval is small enough to meet the Nyquist condition. The spectrum analysis can easily identify the suitability of the sampling interval. According to the sampling theorem, an approximated formula of the reconstructed field by the AS can be found ⬁

UAS共x,y兲 ⬇

⬁

兺 兺 Ur共x − nN⌬␰, y − mM⌬␩兲. m=−⬁ n=−⬁

3.4 CV To obtain the reconstructed field for the CV, let Optical Engineering

B d共 ␣ , ␤ 兲 =

兺 兺B m=−⬁ n=−⬁

冉

␣−

冊

n m , ␤− . ⌬␰ ⌬␩

共17兲

共19兲

Bt共␣, ␤兲 = Bd共␣, ␤兲rect共⌬␰␣,⌬␩␤兲,

共20兲

with the sampling intervals given by Eq. 共16兲. If the sampling intervals of Uc共␰ , ␩兲 meet the Nyquist condition and Bt共␣, ␤兲 ⬇ T共␣, ␤兲rect共⌬␰␣,⌬␩␤兲

共21兲

works well, from the sampling theorem, the reconstructed field by the CV can be written as ⬁

⬁

UCV共x,y兲 ⬇

兺 兺

Ur共x − nN⌬␰,y − mM⌬␩兲.

共22兲

m=−⬁ n=−⬁

Noting that expression 共21兲 shows that Bd共␣ , ␤兲 is an approximated truncation of T共␣ , ␤兲, the FFT-AS will be always more precise than the FFT-CV. 4

Windowing and Sampling the Continuous Reconstructed Field In this section, the coordinates 共x , y兲 in Eqs. 共12兲, 共13兲, 共17兲, and 共22兲 will be digitized to recover the desired field and the corresponding numerical reconstruction methods will be described. The pixel resolution and the number of sampling points for these methods will be also presented. The traditional SFFT method can be given as3–5

再

冋

QFR = C共xq,y p兲 ⫻ FFT2 Uc共␰n, ␩m兲 ⫻ exp

with the sampling intervals ⌬␣ =

⬁

⬁

jk 2 2 共␰ + ␩m 兲 d n

册冎

,

共23兲

where C共xq , y p兲 is the phase factor and Uc共␰n , ␩m兲 is the sample of Uc共␰ , ␩兲. QFR is a discrete portion of UFR共x , y兲 with the pixel resolution ⌬x =

␭d , N⌬␰

⌬y =

␭d M⌬␩

共24兲

and the window size Lx =

␭d , ⌬␰

Ly =

␭d . ⌬␩

共25兲

A sample of UFR共x , y兲 can also be obtained by the FFTbased direct integration method for the FR 共FFT-DIFR兲 共see Appendix A兲,13 which is similar to the one described in Ref. 12. The FFT-DIFR can freely control the pixel resolution and the number M ⬘ ⫻ N⬘ of sampling points. It is apparent that the SFFT is a special case of the FFT-DIFR when ⌬x and ⌬y are set as Eq. 共24兲 and M ⬘ ⫻ N⬘ to M

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Fig. 2 The diffracted fields from a die 共a兲 has 512⫻ 512 pixels with the pixel pitch 6.45⫻ 6.45 ␮m2, 共b兲 is obtained by padding 共a兲 with zeros, having 1024⫻ 1024 pixels with the pixel pitch 6.45 ⫻ 6.45 ␮m2, and 共c兲 has 512⫻ 512 pixels with the pixel pitch 12.9 ⫻ 12.9 ␮m2.

⫻ N. The FFT-DIRS12 can get a discrete part of UAS共x , y兲 with the pixel resolution ⌬␰ ⫻ ⌬␩ and the number of sampling points M ⫻ N. A modified FFT method 共MFFT-DIRS兲 共see Appendix B兲 of the FFT-DIRS presented by Ref. 13 can remove the constraints on the number of sampling points and set the pixel resolution as ⌬x =

⌬␰ , w

⌬y =

⌬␩ , h

共26兲

where w and h are positive integers. Obviously, the FFTDIRS is a special case of the MFFT-DIRS. The FFT-AS can be given as QAS = IFFT2兵T共␣n, ␤m兲 ⫻ FFT2关Uc共␰n, ␩m兲兴其,

共27兲

where T共␣n , ␤m兲 is a sample of T共␣ , ␤兲 with the pixel resolution represented by Eq. 共16兲. The FFT-CV can be written as QCV = IFFT2兵FFT2兵g共␰n, ␩m,d兲其 ⫻ FFT2关Uc共␰n, ␩m兲兴其, 共28兲 where g共␰n , ␩m , d兲 a discrete sample of g共x , y , d兲 truncated by the sensor. QAS is produced by windowing and sampling UAS, and QCV by windowing and sampling UCV. Both have the pixel resolution ⌬␰ ⫻ ⌬␩ and the number of sampling points M ⫻ N. In addition, the physical coordinates of all of these discrete reconstructed fields can be freely adjusted. 5 Discussion and Verification The overlapping quality, accuracy, pixel resolution, computation window, and speed of the above-mentioned methods will be discussed in this section. To verify our conclusions, we use the diffracted field from a dice 共about 2 ⫻ 2 ⫻ 2 mm3兲, which is retrieved by the four-step phaseshifting technique. The reconstruction distance is 58.1 mm, the pixel pitch of the sensor 6.45⫻ 6.45 ␮m2, and the wavelength of the light 632.8 nm. As shown, Fig. 2共a兲 has 512⫻ 512 pixels with the pitch 6.45⫻ 6.45 ␮m2; Fig. 2共b兲 is obtained by padding Fig. 2共a兲 with zeros, having 1024 ⫻ 1024 pixels with the pitch 6.45⫻ 6.45 ␮m2; and Fig. 2共c兲 has 512⫻ 512 pixels with the pitch 12.9⫻ 12.9 ␮m2. 5.1 Overlapping Quality From Eqs. 共12兲 and 共13兲, the overlapping periods Px and Py of the SFFT, the FFT-DIFR, and the MFFT-DIRS are Optical Engineering

Fig. 3 Reconstructed images with different methods 共a–c兲 are reconstructed from the field of Fig. 2共a兲 and 共d–f兲 from the field of Fig. 2共c兲. 共a兲 SFFT, 512⫻ 512, 11.13⫻ 11.13 ␮m2, 共b兲 FFT-DIFR, 512 ⫻ 512, 19.35⫻ 19.35 ␮m2, 共c兲 MFFT-DIRS, 1536⫻ 1536, 6.45 ⫻ 6.45 ␮m, 共d兲 SFFT, 512⫻ 512, 5.57⫻ 5.57 ␮m2, 共e兲 FFT-DIFR, 512⫻ 512, 12.9⫻ 12.9 ␮m2, 共f兲 MFFT-DIRS, 512⫻ 512, 12.9 ⫻ 12.9 ␮m2.

Px =

␭d , ⌬␰

Py =

␭d . ⌬␩

共29兲

As shown in Fig. 3, the images in the upper row are reconstructed from the field represented by Fig. 2共a兲 using the SFFT, FFT-DIFR, and MFFT-DIRS, respectively. The period projected by Eq. 共29兲 is 5.7 mm in accordance with the measured value of ⬃5.7 mm. When we enter the field in Fig. 2共c兲, whose pixel pitch is double that in Fig. 2共a兲, the fields reconstructed by the SFFT, FFT-DIFR, and MFFTDIAS, as shown by the images in the lower row of Fig. 3, have a period of ⬃2.85 mm in accordance with the theoretical one, 2.85 mm. The fields yielded by the FFT-DIRS are similar to those by the FFT-DIFR, which verifies the validity of Eq. 共13兲. From Eqs. 共17兲 and 共22兲, when the pixel pitch of the sensor meets the Nyquist condition, the overlapping periods Px and Py of the FFT-AS and FFT-CV are given by Px = N⌬␰,

Py = M⌬␩ ,

共30兲

which equals the computation window size as illustrated by Figs. 4共a兲 and 4共b兲 that are reconstructed from the field of Figure 2共a兲 by the FFT-AS and FFT-CV, respectively. When the pixel pitch does not meet the Nyquist condition, Eqs. 共14兲 and 共19兲 do not hold any more and the noise arises as shown by Figs. 4共c兲 and 4共d兲 that are yielded from the field of Fig. 2共c兲 by the FFT-AS and FFT-CV, respectively. Figures 5共a兲 and 5共c兲 are the spectra of the optical fields represented by Figs. 2共a兲 and 2共c兲, respectively. Figures 5共b兲 and 5共d兲 are the cross sections of Fig. 5共a兲 and 5共c兲, respectively. That the cross section in Figs. 5共b兲 falls to zero on two sides indicates the pixel pitch for the optical field of Fig. 2共a兲 satisfies the Nyquist condition, while the shape of the cross section in Fig. 5共d兲 suggests the pixel pitch for the optical field of Fig. 2共c兲 does not meet the Nyquist condition. It can be seen that the frequency analysis can easily determine the validity of Eqs. 共14兲 and 共19兲. If we simulta-

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Liu, Wang, and Zhang: Comparison and verification of numerical reconstruction methods…

Fig. 6 Demonstration of the segment reconstruction strategy. Images are reconstructed from the optical field shown in Fig. 2共a兲 by FFT-DIFR. 共a–d兲 512⫻ 512, 5.57⫻ 5.57 ␮m2 with the central positions 共−1.43, 1.43兲, 共1.43,1.43兲, 共−1.43, −1.43兲, and 共1.43, −1.43兲 共in millimeters兲, respectively, 共e兲 1024⫻ 1024, 5.57⫻ 5.57 ␮m2.

Fig. 4 Reconstructed images from the field represented by Figs. 2共a兲, 2共c兲, and 2共b兲, from top to bottom using FFT-AS and FFT-CV from left to right. 共a, b兲 512⫻ 512, 6.45⫻ 6.45 ␮m2, 共c, d兲 512 ⫻ 512, 12.9⫻ 12.9 ␮m2, 共e, f兲 1024⫻ 1024, 6.45⫻ 6.45 ␮m2.

neously keep the pixel resolution small enough and increase M and N, the overlapping periods Px and Py will increase proportionally, as indicated by Figs. 4共e兲 and 4共f兲, which are reconstructed from the field of Fig. 2共b兲. 5.2 Accuracy For the FFT-AS and the FFT-CV, the pixel pitch of the field from the object plays an important role in accuracy. When

Fig. 5 Spectra of optical fields of Figs. 2共a兲 and 2共c兲 respectively 共b兲 cross section of 共a兲 and 共d兲 cross section of 共c兲. Optical Engineering

the pixel pitch is too large, Eqs. 共14兲 and 共19兲 cannot be applied. Unexpected noise appears as shown in Figs. 4共c兲 and 4共d兲. Therefore, when the FFT-AS and FFT-CV are applied, the pixel pitch must be small enough to ensure the validity of Eqs. 共14兲 and 共19兲 to avoid such noise. When the pixel pitch reaches such a size, further reduction cannot improve the accuracy. Instead, it is possible to cause the aliasing noise because the overlapping period will decrease. At this point, it is advisable to increase M and N by padding zeros to avoid the aliasing noise as suggested by Figs. 4共e兲 and 4共f兲. In addition, from the quality of Figs. 4共c兲 and 4共d兲, the FFT-AS is more precise than the FFT-CV as predicted in Sec. 3. For the SFFT, the FFT-DIFR and MFFTDIRS, padding zeros is useless in improving the accuracy. Only if the width of Ur共x , y兲 is smaller than the overlapping period given by Eq. 共29兲, the discrete reconstructed image would be a precise sample of UFR共x , y兲 or URS共x , y兲. 5.3 Pixel Resolution and Computation Window The pixel resolution of the SFFT differs with the reconstruction distance; thus, the SFFT is not suitable for the case where pixel resolution is required to be constant when the reconstruction distance changes. A zero padding method and multiple-step Fresnel transform methods have been proposed to control the pixel resolution.6–10 However, it can be seen that the FFT-DIFR is a natural algorithm to control the pixel resolution because it can set the pixel resolution flexibly, as demonstrated by Fig. 3共b兲, where the pixel resolution is set to 19.35⫻ 19.35 ␮m2, and by Fig. 3共e兲, where the pixel resolution is set to 6.45⫻ 6.45 ␮m2. In some cases, we should set the pixel resolution to be small, then M ⬘ and N⬘ have to be set to be large enough to reconstruct the region of interest, thus causing a large computational load. At this time, the segment reconstruction strategy can be employed to improve the speed.12 As shown, Figs. 6共a兲–6共d兲 are reconstructed in sequence with different central positions to form the whole die, and Fig. 6共e兲 is reconstructed at one time with a number of the pixel fourfold that of each of the four segments. The MFFT-DIRS can adjust the pixel resolution as described by Eq. 共23兲. A computational penalty will be generated by the decrease of the pixel size. The segment reconstruction strategy can be also used

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Liu, Wang, and Zhang: Comparison and verification of numerical reconstruction methods… Table 1 Comparison of calculation speeds. Method

Number of FFT Array size of FFT Execution time 共s兲

SFFT

1

512⫻ 512

0.97

FFT-DIFR

3

1023⫻ 1023

15.81

FFT-DIRS

3

1023⫻ 1023

13.34

FFT-AS

2

512⫻ 512

1.22

FFT-CV

3

512⫻ 512

2.36

Acknowledgment We thank Dr. Fucai Zhang for providing us with the experimental data. This research is supported in part by the Science Foundation of Education Commission of Beijing under Grant No KZ200910005001 and by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality 共PHRIHLB兲.

to improve the speed. For the FFT-AS and FFT-CV, the pixel resolution and the computation window size are the same as those of the input field. 5.4 speed Most of the execution time of the reconstruction methods is consumed by the FFT. Therefore, the computational load can be estimated by that of the FFT. If the array the FFT uses has a number of I ⫻ I, then the number of multiplicity can be estimated18 by 2I2 log2I. If the input array has the number of N ⫻ N, for the SFFT, FFT-AS, and FFT-CV, then the size of the array to enter the FFT is N ⫻ N. Assuming the output array of the FFT-DIFR and MFFT-DIRS is set to N⬘ ⫻ N⬘, an array of 共N + N⬘ − 1兲 ⫻ 共N + N⬘ − 1兲 will enter the FFT for the FFT-DIFR, and 共hN + N⬘ − 1兲 ⫻ 共wN + N⬘ − 1兲 for the MFFT-DIRS. For a comparison of the speeds of these methods, we input an array of 512⫻ 512 and set N⬘ ⫻ N⬘ to be 512⫻ 512 and h = w = 1. Table 1 gives the execution time of different methods run by Matlab 7.0 on a computer with CPU 1.79 GHz and memory 256 MB, the number of the FFT involved in different methods, and the array size of the FFT. If the N⬘ ⫻ N⬘ is large, segment reconstruction strategy for the FFT-DIFR and FFT-DIRS can save much execution time, as shown by Fig. 6, where the total execution time of the four segments 关Figs. 6共a兲–6共d兲兴 is 54.4 s, but the time for the whole image 关Fig. 6共e兲兴 is 105.8 s. 6 Conclusions When the pixel pitch of sensor ⌬␰ ⬍ = ␭d / Lim, the SFFT, FFT-DIFR, and MFFT-DIRS can be used to reconstruct the desired image without the aliasing noise, and a further decrease in ⌬x cannot improve the precision. The pixel resolution and the computation window size of the reconstructed image by the SFFT cannot be controlled, while the FFT-DIFR and MFFT-DIRS can decrease the pixel resolution and keep the computation window size, which can be efficiently performed by the segment reconstruction strategy. For the FFT-AS and FFT-CV, ⌬x must meet the Nyquist condition to make Eqs. 共14兲 and 共19兲 work, and the sampling intervals of the Fourier spectrum must satisfy −1 to avoid the aliasing noise. If this precondi共N⌬x兲−1 ⬍ Lim tion is met, Lim can be unlimited in theory because we can always increase N by padding zeros to make the sampling −1 . In intervals of the Fourier spectrum satisfy 共N⌬x兲−1 ⬍ Lim practice, if N is too large, the computational load will exceed the capability of a typical PC. A mixed method of zero Optical Engineering

padding strategy and spherical wave illumination to mitigate this difficulty has been proposed in Ref. 9 Finally, it is worth noting that the FFT-AS is not only more efficient but also more precise than the FFT-CV for the reconstruction in the digital holography.

Appendix A If the center 共M / 2 , N / 2兲 of the sample array Uc共m , n兲 of Uc共␰ , ␩兲 corresponds to the physical position 共0,0兲 and the center 共M ⬘ / 2 , N⬘ / 2兲 of the sample array Ur共p , q兲 of the reconstructed image to 共x0 , y 0兲, then sample sequence number and the physical coordinate satisfy the following relations:

冉 冊

␰= n−

冉

N ⌬␰, 2

␩=− m−

冊

M ⌬␩ 2

共31兲

and

冉 冊

x= q−

冉

N⬘ ⌬x + x0, 2

␩=− p−

冊

M⬘ ⌬y + y 0 . 2

共32兲

Substitute Eqs. 共31兲 and 共32兲 into Eq. 共7兲, yielding M−1 N−1

Ur共p,q兲 = c共p,q兲 兺

兺 a共m,n兲b共p − m,q − n兲p 苸 关0,M ⬘

m=0 n=0

− 1兴,

q 苸 关0,N⬘ − 1兴,

共33兲

where c共p , q兲 is a phase factor and a共m,n兲 = Uc共m,n兲

再

exp共jkd兲 jk 2 exp 关m ⌬y共− ⌬y + ⌬␩兲 jkd 2d

冎 再

+ n2⌬x共− ⌬x + ⌬␰兲兴 exp

jk 关m⌬y共⌬yM ⬘ 2d

冎

− ⌬␩ M + 2y 0兲 − n⌬x共⌬␰N − ⌬xN⬘ + 2x0兲兴 m 苸 关0,M − 1兴, and

再

b共m,n兲 = exp

n 苸 关0,N − 1兴

共34兲

冎

jk 2 关m ⌬y⌬␩ + n2⌬x⌬␰兴 . 2d

共35兲

Equation 共33兲 is a discrete linear convolution, which can be calculated by FFT through being converted into a cyclical convolution, following the steps as 1. if m ⬍ M and n ⬍ N a⬘共m , n兲 = a共m , n兲 else a⬘共m , n兲 = 0. 2. b⬘共m , n兲 = b共m + 1 − M ⬘ , n + 1 − N⬘兲.

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Liu, Wang, and Zhang: Comparison and verification of numerical reconstruction methods…

3. d共m , n兲 = exp兵j2␲关共M − 1兲m / 共M + M ⬘ − 1兲 + 共N − 1兲n / 共N + N⬘ − 1兲兴其. 4. Ur共p , q兲 is the first M ⬘ ⫻ N⬘ elements of the following result: c共p , q兲 ⫻ IFFT2兵FFT2兵a其 ⫻ FFT2兵b其 ⫻ d其. In the first three steps, m 苸 关0 , M + M ⬘ − 2兴 , n 苸 关0 , N + N⬘ − 2兴. Appendix B Substitute Eqs. 共26兲, 共31兲, and 共32兲 into Eq. 共1兲, obtaining M−1 N−1

Ur共p,q兲 = ⌬␰⌬␩ 兺

兺 Uc共m,n兲 ⫻ g兵− 关p − hm + 共hM

m=0 n=0

− M ⬘兲/2兴⌬y + y 0,

关q − wn + 共wN − N⬘兲/2兴⌬x

+ x0,d其p 苸 关0,M ⬘ − 1兴,

q 苸 关0,N⬘ − 1兴.

共36兲

To convert Eq. 共36兲 into a discrete linear convolution, a new array Uc共u , v兲 must be constructed as follows: if u = hm and v = wn Uc共u , v兲 = Uc共m , n兲 else Uc共u , v兲 = 0, where u 苸 关0 , hM − 1兴, v 苸 关0 , wN − 1兴. Take Uc共u , v兲 into Eq. 共32兲, resulting in

9. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. 31共7兲, 897–899 共2006兲. 10. D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47共19兲, D12–D20 共2008兲. 11. J. Li, P. Tankam, Z. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34共5兲, 572–574 共2009兲. 12. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45共6兲, 1102–1110 共2006兲. 13. F. Jia, “Study on the principle and applications of digital holography,” MC’s dissertation, Northwest University, China, Chap. 3 共2008兲. 14. V. Nascov and P. C. Logofatu, “Fast computation algorithm for the Rayleigh—Sommerfeld diffraction formula using a type of scaled convolution,” Appl. Opt. 48共22兲, 4310–4319 共2009兲. 15. U. Schnars and W. Jüptner, “Direct recording of holograms by a sensor target and numerical reconstruction,” Appl. Opt. 33共2兲, 179– 181 共1994兲. 16. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zeroorder and twin-image elimination in digital off-axis holography,” Appl. Opt. 39共28兲, 4070–4075 共2000兲. 17. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22共16兲, 1268–1270 共1997兲. 18. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 共1965兲. Changgeng Liu received his BS in physics from Beijing University of Technology, China, in 2007. Now he is an MS candidate in optics at the College of Applied Sciences, Beijing University of Technology 共BJUT兲. His main research interests include optical information processing and digital holography.

hM−1 wN−1

Ur共p,q兲 = ⌬␰⌬␩

兺 n=0 兺

Uc共u, v兲 ⫻ g兵− 关p − u + 共hM

m=0

− M ⬘兲/2兴⌬y + y 0,

关q − v + 共wN − N⬘兲/2兴⌬x

+ x0,d其p 苸 关0,M ⬘ − 1兴,

q 苸 关0,N⬘ − 1兴.

共37兲

Equation 共37兲 is a discrete linear convolution, which can be calculated by FFT according to the steps given in Appendix A. References 1. J. Goodman, Introduction to Fourier Optics, 3rd ed., Chapter 3, Ben Roberts, Englewood, CO 共2005兲. 2. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formula,” J. Opt. Soc. Am. 57, 546–547 共1967兲. 3. T. M. Kreis and W. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36共8兲, 2357–2360 共1997兲. 4. T. M. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 36, 224–233 共1997兲. 5. U. Schnar and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 共2002兲. 6. P. Ferraro, S. D. Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance Fresneltransform reconstruction of digital holograms,” Opt. Lett. 29共8兲, 854– 856 共2004兲. 7. P. Ferraro, S. D. Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling images parameters in the reconstruction process of digital holograms,” IEEE J. Sel. Top. Quantum Electron. 10共4兲, 829–839 共2004兲. 8. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29共14兲, 1668–1670 共2004兲.

Optical Engineering

Dayong Wang received his BS in optical engineering in 1989 from Huazhong University of Science and Technology, Wuhan, China, and PhD in physics in 1994 from Xi’an Institute of Optics and Fine Mechanics, Chinese Academy of Sciences. From 1994 to 1996, he did his postdoctoral work in Xidian University, China. In 1996, he joined the Department of Applied Physics, BJUT. From 1998 to 2000, he worked at the Weizmann Institute of Science, Israel, as a visiting scientist. Since 2000, he has been a professor in the College of Applied Sciences, BJUT. His research interests include optical information processing, optical storage, holography, and diffractive optical element. Dr. Wang is a Member of the Chinese Optical Society, SPIE, and the Optical Society of America. Yizhuo Zhang received his BS in physics from BJUT in 2005, where currently, he is a PhD candidate in optics. His main research interests include optical information processing and digital holography.

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