OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 17 — Aug. 17, 2009
  • pp: 15008–15022
« Show journal navigation

Superresolution imaging method using phase-shifting digital lensless Fourier holography

Luis Granero, Vicente Micó, Zeev Zalevsky, and Javier García  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 15008-15022 (2009)
http://dx.doi.org/10.1364/OE.17.015008


View Full Text Article

Acrobat PDF (1755 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A method which is useful for obtaining superresolved imaging in a digital lensless Fourier holographic configuration is presented. By placing a diffraction grating between the input object and the CCD recording device, additional high-order spatial-frequency content of the object spectrum is directed towards the CCD. Unlike other similar methods, the recovery of the different band pass images is performed by inserting a reference beam in on-axis mode and using phase-shifting method. This strategy provides advantages concerning the usage of the whole frequency plane as imaging plane. Thus, the method is no longer limited by the zero order term and the twin image. Finally, the whole process results in a synthetic aperture generation that expands up the system cutoff frequency and yields a superresolution effect. Experimental results validate our concepts for a resolution improvement factor of 3.

© 2009 Optical Society of America

1. Introduction

More than 40 years ago, Bachl and Lukosz presented a superresolving optical system [1

1. A. Bachl and A. W. Lukosz, “Experiments on superresolution imaging of a reduced object field,” J. Opt. Soc. Am. 57, 163–169 (1967).

] capable to overcome the spatial resolution limit imposed by diffraction [2

2. E. Abbe, “Beitrage zür theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9, 413–468 (1873).

]. Such method took part from a wider collection of techniques where different superresolution strategies were defined as a function of the object classification using a priori information [3

3. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).

,4

4. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit II,” J. Opt. Soc. Am. 57, 932–941 (1967).

]. By stating that the number of degrees of freedom of an imaging system remains constant, Lukosz theorized that any parameter in the system could be extended above the classical limit if any other factor is proportionally reduced. In particular, the spatial bandwidth could be improved by “paying” in others domains in which the object is independent to a given degree of freedom (a priori knowledge). Thus, one can find angular multiplexing for non-extended objects [1

1. A. Bachl and A. W. Lukosz, “Experiments on superresolution imaging of a reduced object field,” J. Opt. Soc. Am. 57, 163–169 (1967).

,3

3. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).

], time multiplexing for temporally restricted objects [4

4. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit II,” J. Opt. Soc. Am. 57, 932–941 (1967).

,5

5. A. Shemer, D. Mendlovic, Z. Zalevsky, J. García, and P. García-Martínez, “Superresolving Optical system with time multiplexing and computer decoding,” Appl. Opt. 38, 7245–7251 (1999).

], spectral encoding for wavelength restricted objects [6

6. A. I. Kartashev, “Optical systems with enhanced resolving power,” Optics Spectrosc. 9, 204–206 (1960).

,7

7. J. D. Armitage, A. W. Lohmann, and D. P. Parish, “Superresolution image forming systems for objects with restricted lambda dependence,” Jpn. J. Appl. Phys. 4, 273–275 (1965).

], spatial multiplexing with one-dimensional objects [3

3. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).

,8

8. M. A. Grim and A. W. Lohmann, “Superresolution image for 1-D objects,” J. Opt. Soc. Am. 56, 1151–1156 (1966).

,9

9. H. Bartelt and A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).

], polarization coding with polarization restricted objects [10

10. A. W. Lohmann and D. P. Paris, “Superresolution for nonbirrefringent objects,” Appl. Opt. 3, 1037–1043 (1964).

,11

11. A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44, 3705–3715 (2005). [PubMed]

], and gray level multiplexing for objects with restricted intensity dynamic range [12

12. Z. Zalevsky, P. García-Martínez, and J. García, “Superresolution using gray level coding,” Opt. Express 14, 5178–5182 (2006). [PubMed]

].

Coming back to the Bachl and Lukosz approach [1

1. A. Bachl and A. W. Lukosz, “Experiments on superresolution imaging of a reduced object field,” J. Opt. Soc. Am. 57, 163–169 (1967).

], the gain in spatial bandwidth is achieved by reducing the object field of view. Two static masks (typically gratings) are inserted into conjugate planes at the object and image space of the experimental setup. The encoding mask (at the object space) allows the transmission of additional diffracted object waves through the limited system aperture in such a way that they will not be transmitted through it in absence of the mask. The role of the decoding mask (at the image space) is to redefine the propagation direction of the new diffracted components as they were generated in the input object. However, a necessary condition must be fulfilled: the object field needs to be limited around the object region of interest in order to avoid image distortion coming from the ghost images produced in the encoding-decoding process.

Some modifications of the Bachl and Lukosz basic setup that also consider static gratings had been proposed along the years [13

13. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Superresolution optical system for objects with finite size,” Opt. Commun. 163, 79–85 (1999).

15

15. J. García, V. Micó, D. Cojoc, and Z. Zalevsky, “Full field of view super-resolution imaging based on two static gratings and white light illumination,” Appl. Opt. 47, 3080–3087 (2008). [PubMed]

]. In any case, a hand-waving explanation of the underlying principle of any superresolving approach starts as follows. For fixed illumination wavelength, the resolving power of an imaging system is limited by diffraction as a function of its numerical aperture (NA) [2

2. E. Abbe, “Beitrage zür theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9, 413–468 (1873).

]. Or in other words, the limited aperture of the imaging system defines a cutoff frequency over the object’s spatial-frequency content. In addition, the aim of the superresolution techniques is to widen such limited aperture allowing the generation of a synthetic aperture which expands up such cutoff frequency limit. This synthetic enlargement in the aperture implies an improvement in the resolution limit without changes in the physical properties of the optical system in comparison with the spatial resolution presented by the same optical system without applying the superresolved approach.

Classically, one of the most appealed payments to allow superresolved imaging is done with the time domain [4

4. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit II,” J. Opt. Soc. Am. 57, 932–941 (1967).

,5

5. A. Shemer, D. Mendlovic, Z. Zalevsky, J. García, and P. García-Martínez, “Superresolving Optical system with time multiplexing and computer decoding,” Appl. Opt. 38, 7245–7251 (1999).

]. Thus, the method for synthetic aperture generation is based on time multiplexing the spatial frequency content diffracted by the input object. Such approaches can be implemented using off-axis illumination in digital holographic microscopy [16

16. Ch. J. Schwarz, Y. Kuznetsova, and S. R. Brueck, “Imaging interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003). [PubMed]

23

23. V. Mico, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express 16, 19260–19270 (2008).

] or by shifting the CCD in digital holography [24

24. F. Le Clerc, M. Gross, and L. Collot, “Synthetic aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett. 26, 1550–1552 (2001).

27

27. J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, “High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning,” Appl. Opt. 47, 5654–5658 (2008). [PubMed]

].

In the last years, the original idea proposed by Bachl and Lukosz in 1967 has been performed in combination with digital holography [28

28. Ch. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81, 3143–3145 (2002).

30

30. M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by two-dimensional dynamic phase grating,” Opt. Express 16, 17107–17118 (2008). [PubMed]

]. The basic idea consists on the recording of a multiplexed hologram composed by the addition of individual ones where each one contains information about different spatial frequency regions of the diffracted object spectrum. Since Fourier-lensless off-axis configuration is implemented in the holographic recording, each band pass image can be recovered by digital fast Fourier transformation of the multiplexed hologram since the hologram diffraction orders do not overlap. And finally a superresolved image is generated by properly managing the different recovered band pass images. However, off-axis holographic configuration suffers from the spatial separation of the different diffraction orders in the hologram’s Fourier domain. This fact means that the whole frequency plane is not accessible under imaging purposes. Thus, strong field of view restrictions must be considered to allow separation in the Fourier plane of the different diffraction orders and band pass images.

The paper is organized as follows. Section 2 provides both qualitative system description and mathematical background of the experiment. Section 3 presents experimental results showing the resolution improvement. Section 4 concludes the paper.

2. Analysis of the proposed method

2.1 System description

The optical assembly used to demonstrate the capabilities of the proposed approach is depicted in Fig. 1. It is basically a Mach-Zehnder interferometric architecture in which a laser beam (incoming from a He-Ne laser source) is used as illumination wavelength. Thus, the object under test is illuminated in transmission mode and a Fresnel diffracted pattern is recorded by the CCD imaging device. Let us first to consider that no other optical elements are placed between the input plane and the CCD. Such diffracted pattern is combined at the CCD with a reference beam incoming from a spatial filter by the action of a beam splitter cube. The reference beam is a spherical divergent beam having the particularity that the distance (z0) between the object and the CCD is equal to the pinhole-CCD distance, configuring a lensless Fourier transform hologram setup [33

33. J. Goodman, Introduction to Fourier Optics2nd ed., (McGraw-Hill, New York, 1996).

].

Fig. 1. Experimental setup used in the validation of the proposed approach.

Under this experimental assembly, the Fourier transform of the recorded interference pattern gives the in-focus on-axis band pass image of the object under test. Such image will have a resolution limit defined by either the NA of the imaging system or the geometrical resolution defined by the pixel size of the CCD detector. Since we are working with low NA values, from now on we assume that the NA which is being the limiting factor. Thus, the CCD full size and the distance between the object and the CCD will define the system NA and thus the cutoff frequency that limits the resolution of the system for a given wavelength. We refer to this configuration as conventional imaging mode along the manuscript.

However, it is possible to overcome the above established resolution limit by placing a diffraction grating between the object and the CCD [28

28. Ch. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81, 3143–3145 (2002).

,30

30. M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by two-dimensional dynamic phase grating,” Opt. Express 16, 17107–17118 (2008). [PubMed]

]. Obviously, the diffraction grating must be properly selected (basic frequency and position). Figure 2 illustrate the operating principle for a 1D case. For sake of simplicity, we consider the axial point of the object. The diffracted spectrum has been divided into rectangular portions according with the CCD size (aperture of the conventional imaging system). Without the grating, only the central portion will reach the CCD area [case (a)]. With the grating, the zero order of the grating does not affect the propagation of the different spectral portions [case (b)] but the grating diffracts additional spatial-frequency portions towards the CCD aperture [case (c)]. Since this new spectral portions reach obliquely the CCD, it will be possible to recover separately each one of them because they will not overlap at the Fourier domain. Once again, this separation depends on the properly selection of the diffraction grating.

Fig. 2. Fig. 2. Schematic figure representative of the proposed approach for a 1D case.

At this point, two methods can be used for recover the complex amplitude distribution of the different band pass images at the Fourier domain. The difference between them is the way that the reference beam is inserted at the CCD plane. On one hand, we can use off-axis holographic recording and windowed Fourier filtering. And on the other hand, we can use on-axis holographic configuration and phase-shift in one of the two interferometric beams (typically the reference beam). The former method needs only one interferogram to recover the band pass images but suffers from the presence of the zero order term and the set of twin images (one to each band pass image) at the Fourier domain. This fact means a high restriction over the accessible area of the frequency plane that is useful for imaging. The way to overcome this limitation is by considering the latter method. The phase-shift in the reference beam modulates those interferometric terms that are multiplied by the reference beam in such a way that it can be recovered separately after applying the phase-shifting algorithm. Since the phase-shifting method does not requires separation of the hologram orders at the Fourier domain, the whole frequency plane can be used for imaging. So, the space-bandwidth product of the system becomes optimized.

In this paper, we propose the use of an on-line configuration with phase-shifting approach to optimize the Fourier domain which is used as imaging space due to the lensless Fourier holographic configuration. Thus, the only restriction is due to the object field of view that can cause overlapping between the different recovered band pass images. Once again, we are in a similar case as in the Bach and Lukosz approach where the object field of view needs to be restricted in order to achieve superresolution effect. However, the restriction imposed by the proposed configuration is not as severe as in the case of the off-axis configuration. In order to control the non-overlapping between the object field of view in the recovered band pass images, we have included an adjustable square diaphragm attached to the input object.

In the phase-shifting procedure, we have applied a saw-tooth displacement in the piezo-mirror of the reference branch to allow continuous shift in the reference beam. Thus, it is simple to calculate the phase-shift cycle by correlation of one captured image with the remaining images that integrate the captured cycle. In our case, 60 subsequent images integrate the full phase-shifting cycle. After applying a conventional phase-shifting algorithm [34

34. T. Kreis, Handbook of Holographic Interferometry, (Wiley-VCH, 2005).

], all the band pass images are recovered in a single image and can be filtered separately in order to assemble the final superresolved image.

But in order to synthesize a high quality superresolved image, two factors must be taken into account. As first one, we find that the different diffracted bands will travel a different optical path before arriving at the CCD. This fact means that each band pass image will present a different global phase after the recovery process and must be compensated in the digital post-processing stage. And as second factor we find the correct repositioning of each recovered spectral band to its original position at the object spectrum. By knowing the distances in the system, the illumination wavelength and the grating period, it is possible to add a linear phase factor to the different band pass images in order to shift the spectral content to a rough position in the spatial-frequency domain. A final fine adjustment is achieved by the addition of smaller linear phase factors in both horizontal and vertical directions. Also, this fine tuning process compensates phase variations incoming from misalignments in the optical setup. This procedure is repeated for every additional band pass image considered in the experiment and the full adjustment can be guided and automated by an image quality criterion.

2.2 Theoretical analysis

In this section we review the mathematical basis of the proposed approach. In our analysis we assume a 1D distribution in order to ease the mathematical treatment. However the expansion for the 2D case is straightforward. We denote by t(x) the amplitude distribution of the input object. To take into account the effect of grating positioned at a given distance from the object, we propagate the field distribution of the input object to the grating location, multiply by the grating and then back propagate the light to the original input plane. Using this strategy, the input field distribution after free space propagation of z1 is proportional to

U(x1)=exp{ik2z1x12}t(x)exp{ik2z1x2}exp{ik2z12x1x}dx
(1)

k being the wave number and x, x1 the spatial coordinates at the input plane and at a plane placed at z1 from the input plane, respectively. Eq. 1 is multiplied by the grating resulting in

U(x1)=exp{ik2z1x12}t(x)exp{ik2z1x2}exp{ik2z12x1x}dxΣnBnexp{i2πnpx1}
(2)

p, n and Bn being the period, the number of diffraction orders, and the coefficients of the different diffraction orders of the grating, respectively. The diffraction grating period will depend on the CCD lateral size, and on the object-CCD and the object-grating distances. The basic condition that must be fulfilled is that the central ray going to the first band pass will be deflected by the grating to the CCD center. Naming α0and α1 the angle of this ray to the optical axis and the angle after the grating deflection, respectively, the grating period must be

p=λsinα0+sinα1
(3)

where the angles are given by

tanα0=Δxz0andtanα1=z1Δxz0(z0z1)
(4)

For the case of paraxial approximation, the grating period has a simple expression

p=λ(z0z1)Δx
(5)

where we can see as the period of the grating decreases as z1 increases from 0 to z0. Thus, when the grating is approaching to the CCD position (z1~z0), we need a small grating period (or high basic frequency) to fulfill our basic condition. And when the grating is placed near to the object (z1~0), we need a large grating period (or low basic frequency). But the first case (z1~z0) will exhibit aliasing problems in the recorded hologram while the second one (z0~0) will produce overlapping of the different band pass images at the Fourier domain. So, the condition defined by z0≅2z1 is the most suitable from an experimental point of view.

Continuing with our propagation procedure, Eq. 2 is now back propagated (-z1) to the input plane. The resulting input’s amplitude distribution including the grating effect is

U(x)=CΣnBnt(xnz1λp)exp{i2πnpx}
(6)

where C is a constant that includes all the constant factors. Note that, if no grating is considered, the amplitude distribution provided by Eq. 6 coincides with that of the input object. Eq. 6 is now propagated in free space a distance of z0 from the input plane to the CCD

U1(x)=C'Σnexp{ik2z0x2}t(xnz1λp)exp{ik2z0x2}exp{i2π(xλz0np)x}dx
(7)

where C’ is a constant. Eq. 7 is gives the amplitude distribution at the CCD plane through the imaging branch. Eq. 7 corresponds with the addition of several Fresnel transformations, each one corresponding with different shifted replicas of the input object function (first term inside the integral). Moreover, such replicas are shifted according to the period and position of the grating in the experimental setup. And those shifts are applied to t(x), that is, to the amplitude distribution of the input object, prior to the propagation. This fact is equivalent to shifting the object at the input plane and it is the source of the vignetting problem in the experimental setup as we will detail in subsection 3.1.

The total amplitude distribution at the CCD plane comes from the addition of Eq. 7 and an on-axis spherical reference beam diverging from the same distance z0:

UR(x,t)=R0exp{ik2z0x2}exp{iϕ(t)}
(8)

where R0 is the amplitude of the reference beam, and ϕ(t) a linear phase variable in time and according to the phase-shifting procedure. Thus, the CCD records the output intensity distribution provided by the addition of Eqs. 7 and 8 and multiplied by the rectangular size of the CCD that trims the recording area at the output plane. For the sake of simplicity, let us assume that the grating has only 3 diffraction orders, that is: n=-1, 0, +1. In this case and leaving aside constant factors, Eq. 7 can be rewritten as

U1(x)=exp{ik2z0x2}t(x+z1λp)exp{ik2z0x2}exp{i2π(xλz0+1p)x}dx
+exp{ik2z0x2}t(x)exp{ik2z0x2}exp{i2πxλz0x}dx
+exp{ik2z0x2}t(xz1λp)exp{ik2z0x2}exp{i2π(xλz01p)x}dx
=O1(x)+O0(x)+O+1(x)=Σn=11On(x)
(9)

where On(x’) represents the different arriving bands at the CCD as consequence of the grating diffraction orders. Hence, the intensity distribution provided by the CCD at a given instant is

ICCD(x)=O1(x)+O0(x)+O+1(x)+UR(x)2
=O1(x)2+O0(x)2+O+1(x)2+UR(x)2
+O1(x)O0*(x)+O0(x)O1*(x)+O0(x)+O+0*(x)+O+1(x)O0*(x)
+O1(x)O+1*(x)+O+1(x)O1*(x)
+[O1(x)+O0(x)+O+1(x)]UR*(x)
+[O1*(x)+O0*(x)+O+1*(x)]UR(x)
(10)

ICCD(x,t)=Σn=11On(x)+UR(x)2
=Σn=11On(x)2+Σn,m=1nm1On(x)Om*(x)+Σn=11On(x)UR*(x)+Σm=11On*(x)UR(x)
=Σn=11On(x)2+Σn,m=1nm1On(x)Om*(x)
+Σn=11On(x)R0exp{ik2z0x2}exp{iϕ(t)}exp{iϕn(x)}
+Σn=11On*(x)R0exp{ik2z0x2}exp{iϕ(t)}exp{iϕn(x)}
(11)

ICCD(x,t)=Σn=11On(x)2+Σn,m=1nm1On(x)Om*(x)+R02
+2R0Re[Σn=11On(x)exp{ik2z0x2}]cos(pϕK+ϕn(x))
(12)

Now, the phase-shift algorithm computes the different intensity distributions stored in time sequence by the CCD and recovers the phase distribution of the different frequency bands of the object [22

22. V. Mico, Z. Zalevsky, and J. García, “Common-path phase-shifting digital holographic microscopy: a way to quantitative imaging and superresolution,” Opt. Commun. 281, 4273–4281 (2008).

,34

34. T. Kreis, Handbook of Holographic Interferometry, (Wiley-VCH, 2005).

]. In particular, we have applied a method that involve m=60 intensity images in one phase-shift period and permits the recovering of the summation of the initial phase distribution according to

Σn=1+1ϕn(x)=arctanΣi=1mIi(x)sin[2πm(i1)]Σi=1mIi(x)cos[2πm(i1)]
(13)

Once the phase-shifting method is applied, the recovered real image term (fifth line in Eq. 10) can be rewritten as

ICCD(x)=[O1(x)+O0(x)+O+1(x)]UR*(x)
=[CΣn=11t(xnz1λp)exp{ik2z0x2}exp{i2π(xλz0np)x}dx]rect(xΔx)
=[CΣn=11FT{t(xnz1λp)}uFT{exp{ik2z0x2}}u]rect(xΔx)
(14)

FT{ICCD(x)}=[Dexp{ik2du2}Σn=11t(u+nz1pz0)exp{i2παu}]FT{rect(xΔx)}
(15)

Conceptually talking, we can extract two conclusions from Eq. 15. From an object field point of view, the object shift at the input plane means a vignetting problem avoidance since the extra-axial points of the object are redirected towards the CCD. Thus, the intensity at the borders of the final reconstructed image will not be distorted. And from a spatial-frequency point of view, the linear exponential means that each shifted image will contain a different spectral range. So a final image having a wider spatial-frequency content could be synthesized in a later stage.

3. Experimental implementation

In this section we present experimental validation of the proposed approach considering two subsections. The first one is aimed to provide a deeper understanding of the approach by simply presenting the method step by step while showing a 1D resolution improvement. And the second one demonstrates a 2D superresolved image incoming from the use of the whole frequency plane as imaging plane. In both cases, we used a He-Ne laser (632nm emitting wavelength) as illumination source, a CCD (Basler A312f, 582×782 pixels, 8.3 µm pixel size, 12 bits/pixel) as imaging device, and a 25 µm pinhole as spherical divergent reference beam.

3.1 Superresolution imaging for a 1D test object case

Fig. 3. (a) and (b) Fourier transformation of the recorded hologram without and with reference beam, respectively. The central spot has been blocked to enhance the contrast of the images.

To recover the complex amplitude distribution of the different band pass images at the Fourier domain, we can use off-axis holographic recording and Fourier filtering [Fig. 4(a)] and on-axis holographic recording and phase-shifting procedure [Fig. 4(b)]. However, the use of on-axis phase-shifting holography provides imaging capabilities in the whole frequency plane, as we can see in both images of Fig. 4. To obtain the image presented in Fig. 4(b), we apply the full phase-shifting cycle which is composed from 60 images. As a result, the virtual image, the autocorrelation terms and the zero order of the recorded hologram are eliminated and the different band pass images can be recovered by simple filtering in the Fourier domain.

Fig. 4. Recovered band pass images when considering (a) off-axis holographic recording and (b) after applying the phase-shifting algorithm. The central spot has been blocked to enhance image contrast.

Finally, a superresolved image is obtained in terms of the generation of a synthetic aperture that expands up the cutoff frequency limit provided by the imaging system. The result is depicted in Figs. 5 and 6. In Fig. 5, the synthetic aperture and the superresolved image is depicted in comparison with the no grating case (conventional imaging mode). Without the grating, the resolution limit is defined by Element 4 of Group 2 (20 lp/mm or 50 µm). When performing the proposed approach, the resolution limit is reduced until Element 9 of Group 2 (60 lp/mm or 16.6 µm), which means a resolution gain factor of 3. For clarity reasons, Fig. 6 depicts the magnified area that is marked with a solid line white rectangle in Fig. 5(d) and plots a section of the last resolved element.

Aside the superresolution effect, we can notice how the vignetting of the conventional image is avoided in the superresolved one. Basically, the vignetting is originated because the aperture of the imaging system, that is, the CCD area, is not placed just at the Fourier plane because there is not a defined Fourier plane in the system. If we look at the borders of the object field of view, the image intensity becomes distorted and shadowed. However, this vignetting in the resulting image is avoided when the grating is inserted in the optical setup. If we pay attention to the left vertical bars of Elements 1 and 2 in the conventional image of the test [dashed white rectangle in Fig. 5(c)], we can see as they appear less defined than the others, that is, they appear a little bit blurred and with less intensity. However, the same vertical lines in the superresolved image appear perfectly defined. So aside a resolution image improvement, the final obtained image is free from vignetting problems.

Fig. 5. (a) The conventional imaging system aperture, (b) the generated synthetic aperture, (c) the conventional image, and (d) the superresolved one.
Fig. 6. (a) Magnified area marked with a white rectangle in Fig. 5(d), and (b) plot along the dashed white line of case (a).

3.2 Superresolution imaging for a 2D test object case

Although the reference beam is not strictly introduced in on-line configuration (we can see as each pair of real and twin band pass images are a little bit shifted), we can see as the different band passes are overlapping [case (b)] and there is no possibility to recover them separately even if we consider off-axis holographic recording. So the only way to do that is by considering phase-shifting process. Figure 8(a) represents the recovery of the different band pass images. We can see as the whole frequency plane becomes optimized since the only restriction to fulfill is the object field of view limitation needed in order that the band pass images will not overlap. In this case, the 3mm diameter of the laser beam is used as object field of view limitation without the need to add an external diaphragm (notice as the intensity of the recovered band pass images decreases from the center to the borders as it corresponds with a Gaussian laser beam profile). Cases (b) to (e) are the central part of the central, right, upper and oblique band pass images, respectively.

Fig. 7. Fourier transformation of the hologram recorded (a) without and (b) with reference beam. The central spot has been blocked to enhance image contrast.
Fig. 8. (a) Whole Fourier domain image with the different band pass images resulting after applying the phase-shifting process and (b) to (e) are the magnified color rectangles of case (a) corresponding with the central region of the different band pass images.

Once the different band pass images are recovered by filtering process, they are used to assemble a synthetic aperture by replacing its spatial-frequency content by its original position in the object spectrum. And finally, a superresolved image is obtained by Fourier transformation of the information contained in the generated synthetic aperture. Figure 9 depicts the aperture of the imaging system in conventional imaging mode [case (a)], its corresponding conventional imaging [case (c)], the synthetic aperture generated as a consequence of the proposed approach [case (b)], and the superresolved image [case (d)].

Fig. 9. (a)–(b) are the conventional imaging aperture and the generated synthetic aperture, respectively, and (c)–(d) are the conventional image and the superresolved one, respectively.

Since the CCD size is rectangular, the conventional imaging defines a different resolution in both main orthogonal directions. Thus, we find a resolution limit of 44 µm (Group 4, Element 4) and 31.25 µm (Group 5, Element 1) for the vertical (horizontal bars) and horizontal (vertical bars), respectively, as we can see in Fig. 9(c). As the position and basic frequency of the gratings are matched for the vertical direction, we should achieve a resolution improvement factor close to 3 in the horizontal bars and 1.4 times lower for the vertical ones. By taking a look at Fig. 9(d), we can see that the resolution limit is improved until 15.6 µm (Group 6, Element 1) for both directions as the inset of Fig. 9(d) enhances. As predicted this new resolution limit means a resolution gain factor of 2.8 and 2 for vertical and horizontal directions, respectively.

4. Conclusions

We have presented a step forward in superresolved digital imaging considering a lensless Fourier holographic configuration. It is based on the insertion of diffraction gratings in the optical assembly in such a way that high order diffracted components are redirected towards the imaging device. The way to recover this additional information inaccessible without using the gratings is by applying on-axis holographic recording and phase-shifting method. This procedure allows the usage of the whole frequency plane as imaging plane and produces an optimization in the information capacity that the imaging system has in comparison with the off-axis holographic recording used in other similar approaches [28

28. Ch. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81, 3143–3145 (2002).

30

30. M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by two-dimensional dynamic phase grating,” Opt. Express 16, 17107–17118 (2008). [PubMed]

]. This fact means that, for a given object field of view, we can improve the gain in resolution by considering a grating having a high number of diffraction orders that will allow a high number of recovered band pass images, or, for a given resolution gain factor, we can get superresolved imaging over a large field of view. Moreover, derived from the fact of inserting a grating in the setup, the final reconstructed image will exhibit reduced vignetting problems.

Acknowledgements

This work was supported by the Spanish Ministerio de Educación y Ciencia and FEDER funds under the project FIS2007-60626. Luis Granero wants to thank AIDO for the time dedicated to this research.

References and links

1.

A. Bachl and A. W. Lukosz, “Experiments on superresolution imaging of a reduced object field,” J. Opt. Soc. Am. 57, 163–169 (1967).

2.

E. Abbe, “Beitrage zür theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9, 413–468 (1873).

3.

W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).

4.

W. Lukosz, “Optical systems with resolving powers exceeding the classical limit II,” J. Opt. Soc. Am. 57, 932–941 (1967).

5.

A. Shemer, D. Mendlovic, Z. Zalevsky, J. García, and P. García-Martínez, “Superresolving Optical system with time multiplexing and computer decoding,” Appl. Opt. 38, 7245–7251 (1999).

6.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Optics Spectrosc. 9, 204–206 (1960).

7.

J. D. Armitage, A. W. Lohmann, and D. P. Parish, “Superresolution image forming systems for objects with restricted lambda dependence,” Jpn. J. Appl. Phys. 4, 273–275 (1965).

8.

M. A. Grim and A. W. Lohmann, “Superresolution image for 1-D objects,” J. Opt. Soc. Am. 56, 1151–1156 (1966).

9.

H. Bartelt and A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).

10.

A. W. Lohmann and D. P. Paris, “Superresolution for nonbirrefringent objects,” Appl. Opt. 3, 1037–1043 (1964).

11.

A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44, 3705–3715 (2005). [PubMed]

12.

Z. Zalevsky, P. García-Martínez, and J. García, “Superresolution using gray level coding,” Opt. Express 14, 5178–5182 (2006). [PubMed]

13.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Superresolution optical system for objects with finite size,” Opt. Commun. 163, 79–85 (1999).

14.

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental results,” J. Opt. Soc. Am. A 18, 514–520 (2001).

15.

J. García, V. Micó, D. Cojoc, and Z. Zalevsky, “Full field of view super-resolution imaging based on two static gratings and white light illumination,” Appl. Opt. 47, 3080–3087 (2008). [PubMed]

16.

Ch. J. Schwarz, Y. Kuznetsova, and S. R. Brueck, “Imaging interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003). [PubMed]

17.

V. Mico, Z. Zalevsky, and J. García, “Superresolution optical system by common-path interferometry,” Opt. Express 14, 5168–5177 (2006). [PubMed]

18.

V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution using multiple off-axis holograms,” J. Opt. Soc. Am. A 23, 3162–3170 (2006).

19.

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46, 993–1000 (2007). [PubMed]

20.

Y. Kuznetsova, A. Neumann, and S. R. J. Brueck “Imaging interferometric microscopy — approaching the linear system limits of optical resolution”, Opt. Express 15, 6651–6663 (2007). [PubMed]

21.

V. Mico, Z. Zalevsky, and J. García, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276, 209–217 (2007).

22.

V. Mico, Z. Zalevsky, and J. García, “Common-path phase-shifting digital holographic microscopy: a way to quantitative imaging and superresolution,” Opt. Commun. 281, 4273–4281 (2008).

23.

V. Mico, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express 16, 19260–19270 (2008).

24.

F. Le Clerc, M. Gross, and L. Collot, “Synthetic aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett. 26, 1550–1552 (2001).

25.

J. H. Massig, “Digital off-axis holography with a synthetic aperture”, Opt. Lett. 27, 2179–2181 (2002).

26.

R. Binet, J. Colineau, and J-C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography”, Appl. Opt. 41, 4775–4782 (2002). [PubMed]

27.

J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, “High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning,” Appl. Opt. 47, 5654–5658 (2008). [PubMed]

28.

Ch. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81, 3143–3145 (2002).

29.

C. Yuan, H. Zhai, and H. Liu, “Angular multiplexing in pulsed digital holography for aperture synthesis,” Opt. Lett. 33, 2356–2358 (2008). [PubMed]

30.

M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by two-dimensional dynamic phase grating,” Opt. Express 16, 17107–17118 (2008). [PubMed]

31.

I. Yamaguchi and T. Zhong, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997). [PubMed]

32.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6185 (2001).

33.

J. Goodman, Introduction to Fourier Optics2nd ed., (McGraw-Hill, New York, 1996).

34.

T. Kreis, Handbook of Holographic Interferometry, (Wiley-VCH, 2005).

OCIS Codes
(050.5080) Diffraction and gratings : Phase shift
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(100.2000) Image processing : Digital image processing
(100.6640) Image processing : Superresolution
(090.1995) Holography : Digital holography

ToC Category:
Image Processing

History
Original Manuscript: May 18, 2009
Revised Manuscript: July 16, 2009
Manuscript Accepted: July 24, 2009
Published: August 10, 2009

Citation
Luis Granero, Vicente Micó, Zeev Zalevsky, and Javier García, "Superresolution imaging method using phase-shifting digital lensless Fourier holography," Opt. Express 17, 15008-15022 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15008


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Bachl and A. W. Lukosz, "Experiments on superresolution imaging of a reduced object field," J. Opt. Soc. Am. 57, 163-169 (1967).
  2. E. Abbe, "Beitrage zür theorie des mikroskops und der mikroskopischen wahrnehmung,"Archiv. Microskopische Anat. 9, 413-468 (1873).
  3. W. Lukosz, "Optical systems with resolving powers exceeding the classical limit," J. Opt. Soc. Am. 56, 1463-1472 (1966).
  4. W. Lukosz, "Optical systems with resolving powers exceeding the classical limit II," J. Opt. Soc. Am. 57, 932-941 (1967).
  5. A. Shemer, D. Mendlovic, Z. Zalevsky, J. García and P. García-Martínez, "Superresolving Optical system with time multiplexing and computer decoding," Appl. Opt. 38, 7245-7251 (1999).
  6. A. I. Kartashev, "Optical systems with enhanced resolving power," Optics Spectrosc. 9, 204-206 (1960).
  7. J. D. Armitage, A. W. Lohmann, and D. P. Parish, "Superresolution image forming systems for objects with restricted lambda dependence," Jpn. J. Appl. Phys. 4, 273-275 (1965).
  8. M. A. Grim and A. W. Lohmann, "Superresolution image for 1-D objects," J. Opt. Soc. Am. 56, 1151-1156 (1966).
  9. H. Bartelt and A. W. Lohmann, "Optical processing of 1-D signals," Opt. Commun. 42, 87-91 (1982).
  10. A. W. Lohmann and D. P. Paris, "Superresolution for nonbirrefringent objects," Appl. Opt. 3, 1037-1043 (1964).
  11. A. Zlotnik, Z. Zalevsky, and E. Marom, "Superresolution with nonorthogonal polarization coding," Appl. Opt. 44, 3705-3715 (2005). [PubMed]
  12. Z. Zalevsky, P. García-Martínez, and J. García, "Superresolution using gray level coding," Opt. Express 14, 5178-5182 (2006). [PubMed]
  13. Z. Zalevsky, D. Mendlovic and A. W. Lohmann, "Superresolution optical system for objects with finite size," Opt. Commun. 163, 79-85 (1999).
  14. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti and I. Kiryuschev, "Superresolution optical system using three fixed generalized gratings: experimental results," J. Opt. Soc. Am. A 18, 514-520 (2001).
  15. J. García, V. Micó, D. Cojoc, and Z. Zalevsky, "Full field of view super-resolution imaging based on two static gratings and white light illumination," Appl. Opt. 47, 3080-3087 (2008). [PubMed]
  16. Ch. J. Schwarz, Y. Kuznetsova and S. R. Brueck, "Imaging interferometric microscopy," Opt. Lett. 28, 1424-1426 (2003). [PubMed]
  17. V. Mico, Z. Zalevsky, and J. García, "Superresolution optical system by common-path interferometry," Opt. Express 14, 5168-5177 (2006). [PubMed]
  18. V. Mico, Z. Zalevsky, P. García-Martínez and J. García, "Synthetic aperture superresolution using multiple off-axis holograms," J. Opt. Soc. Am. A 23, 3162-3170 (2006).
  19. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, "Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms," Appl. Opt. 46, 993-1000 (2007). [PubMed]
  20. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck "Imaging interferometric microscopy - approaching the linear system limits of optical resolution", Opt. Express 15, 6651-6663 (2007). [PubMed]
  21. V. Mico, Z. Zalevsky, and J. García, "Synthetic aperture microscopy using off-axis illumination and polarization coding," Opt. Commun. 276, 209-217 (2007).
  22. V. Mico, Z. Zalevsky, and J. García, "Common-path phase-shifting digital holographic microscopy: a way to quantitative imaging and superresolution," Opt. Commun. 281, 4273-4281 (2008).
  23. V. Mico, Z. Zalevsky, C. Ferreira, and J. García, "Superresolution digital holographic microscopy for three-dimensional samples," Opt. Express 16, 19260-19270 (2008).
  24. F. Le Clerc, M. Gross and L. Collot, "Synthetic aperture experiment in the visible with on-axis digital heterodyne holography," Opt. Lett. 26, 1550-1552 (2001).
  25. J. H. Massig, "Digital off-axis holography with a synthetic aperture," Opt. Lett. 27, 2179-2181 (2002).
  26. R. Binet, J. Colineau, and J-C. Lehureau, "Short-range synthetic aperture imaging at 633 nm by digital holography", Appl. Opt. 41, 4775-4782 (2002). [PubMed]
  27. J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, "High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning," Appl. Opt. 47, 5654-5658 (2008). [PubMed]
  28. Ch. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
  29. C. Yuan, H. Zhai, and H. Liu, "Angular multiplexing in pulsed digital holography for aperture synthesis," Opt. Lett. 33, 2356-2358 (2008). [PubMed]
  30. M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, "Super-resolution in digital holography by two-dimensional dynamic phase grating," Opt. Express 16, 17107-17118 (2008). [PubMed]
  31. I. Yamaguchi and T. Zhong, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997). [PubMed]
  32. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6185 (2001).
  33. J. Goodman, Introduction to Fourier Optics 2nd ed., (McGraw-Hill, New York, 1996).
  34. T. Kreis, Handbook of Holographic Interferometry, (Wiley-VCH, 2005).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited