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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 16 — Aug. 1, 2011
  • pp: 14779–14793
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Focus detection criterion for refocusing in multi-wavelength digital holography

Li Xu, Mike Mater, and Jun Ni  »View Author Affiliations


Optics Express, Vol. 19, Issue 16, pp. 14779-14793 (2011)
http://dx.doi.org/10.1364/OE.19.014779


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Abstract

The majority of focus detection criteria reported is based on amplitude contrast. Due to phase wrapping, phase contrast was previously reported unsuitable for focus finding tasks. By taking the advantage of multi-wavelength digital holography, we propose a new focus detection criterion based on phase contrast. Experimental results are presented to prove the feasibility of the developed criterion. Possible applications of the developed technology include inspecting machined surfaces in the auto industry.

© 2011 OSA

1. Introduction

Applications of focus detection in holographic imaging have been reported in quite a few articles [1

1. J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recognit. Lett. 9(1), 19–25 (1989). [CrossRef]

6

6. J. Liu, X. Song, R. Han, and H. Wang, “Autofocus method in digital holographic microscopy,” Proc. SPIE 7283, 72833Q, 72833Q-6 (2009). [CrossRef]

]. Typical focus detection criteria are based on a serial of reconstructed frames along the axial direction, and determine the best focused frame via a certain kind of “sharpness indicator”. The sharper the image is, the closer its position to the best focus. Commonly adopted indicators include: entropy indicator [1

1. J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recognit. Lett. 9(1), 19–25 (1989). [CrossRef]

], variance indicator [2

2. L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. Soc. Am. A 6, 396–400 (2004).

], spectral based indicator [3

3. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008). [CrossRef] [PubMed]

] and correlation coefficient indicator [4

4. Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17(4), 761–770 (2006). [CrossRef]

]. The majority of focus-finding applications is based on amplitude analysis, even though in many cases it is phase contrast that is actually of interest. As reported in [5

5. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14(13), 5895–5908 (2006). [CrossRef] [PubMed]

], best focus with sharpest phase contrast can be determined, when it corresponds to the lowest amplitude contrast. Such objects are referred to as “pure phase” objects. In more general cases, however, when phase contrast is not strongly dependent on amplitude contrast, the task of determine “the sharpest phase contrast plane” is difficult. Examples of using phase information for refocusing purpose are limited to a few simple cases, such as MEMS structures reported in [7

7. P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, “Extended focused image in microscopy by digital Holography,” Opt. Express 13(18), 6738–6749 (2005). [CrossRef] [PubMed]

,8

8. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, C. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35(11), 1840–1842 (2010). [CrossRef] [PubMed]

], where refocusing distances are determined by counting phase wrapping stripes. This method can be performed only when the object slope is continuous and gentle, so that phase stripes are continuous and clear enough to be counted. It is even believed that phase contrast is not usable for more complicated focus-finding purposes, such as step like height structures, since jumps at 2π caused by phase wrapping would be misunderstood as real sharp features [3

3. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008). [CrossRef] [PubMed]

]. Phase unwrapping algorithms are available, yet they are not satisfactory solutions, since unwrap errors would significantly reduce the accuracy of the detection.

Digital holographic technology applying dual / multiple wavelengths has experienced substantial development in recent years. M. K. Kim reported an evident improvement of axial resolution by conducting multi-wavelength interference in tomography [9

9. M. K. Kim, “Wavelength-scanning digital interference holography for optical section imaging,” Opt. Lett. 24(23), 1693–1695 (1999). [CrossRef] [PubMed]

]. Frederic Montfort et al. experimentally demonstrated the constructive effect of multiple complex waves at a selected plane, while destructive effect could be clearly observed at other planes within the axial extent determined by wavelength interval [10

10. F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, “Submicrometer optical tomography by multiple-wavelength digital holographic microscopy,” Appl. Opt. 45(32), 8209–8217 (2006). [CrossRef] [PubMed]

]. J. Gass et al. has proposed a novel noise reduction algorithm for reconstructing multi-wavelength topography [11

11. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003). [CrossRef] [PubMed]

]. A variety of methods have been developed in order to reduce recording time, whose ultimate goal is to achieve real-time multi-wavelength holography [12

12. Y. Zou, G. Pedrini, and H. Tiziani, “Surface contouring in a video frame by changing the wavelength of a diode laser,” Opt. Eng. 35(4), 1074–1079 (1996). [CrossRef]

14

14. M. T. Rinehart, N. T. Shaked, N. J. Jenness, R. L. Clark, and A. Wax, “Simultaneous two-wavelength transmission quantitative phase microscopy with a color camera,” Opt. Lett. 35(15), 2612–2614 (2010). [CrossRef] [PubMed]

]. Demonstrated applications of multi-wavelength digital holography involves, for example, observing bio samples and bio structures [15

15. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express 7(9), 305–310 (2000). [CrossRef] [PubMed]

17

17. J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]

], contouring mechanical objects [18

18. D. Carl, M. Fratz, M. Pfeifer, D. M. Giel, and H. Höfler, “Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths,” Appl. Opt. 48(34), H1–H8 (2009). [CrossRef] [PubMed]

] and developing endoscopic equipment [19

19. J. Kandulla, B. Kemper, S. Knoche, and G. von Bally, “Two-wavelength method for endoscopic shape measurement by spatial phase-shifting speckle-interferometry,” Appl. Opt. 43(29), 5429–5437 (2004). [CrossRef] [PubMed]

]. Among all these reported development, a core advantage of multi-wavelength digital holography is that, by greatly expanding the unambiguous range, phase wrapping problems can be overcome in practice.

Multi-wavelength interferometry (MWI), which has recently been implemented as a powerful surface quality monitoring tool in the auto industry [20

20. C. C. Aleksoff, “Multi-wavelength digital holographic metrology,” Proc. SPIE 6311, 63111D, 63111D-7 (2006). [CrossRef]

,21], presents a realistic need of phase contrast based focus detection: Over one machined part, there can be several separated surfaces of inspection interest. Due to either measurement time restrictions or geometric structure limitations, not every surface can be measured in-focus. Out-of-focus imaging impedes inspection quality by reducing measurement resolution. Holographic refocusing with correct refocusing distance helps evidently on resolution enhancement of out-of-focus objects [22

22. Li Xu and J. Ni, “3D resolution enhancement in multi-wavelength interferometric imaging by digital refocusing,” (Submitted manuscript to Appl. Opt. ). [PubMed]

]. Nevertheless, dependence between amplitude contrast and phase contrast can be fairly loose due to flatness and roughness variation. Thus indirect focus detection criteria are not always a viable choice. In the same time, stripe-counting strategy is also made impossible due to discontinuity and shape-complexity of surfaces.

2. Multi-wavelength interferometric (MWI) refocusing

MWI imaging generates a set of holograms each of a different wavelength for the same object. A typical system arrangement is shown in Fig. 1
Fig. 1 MWI imaging system.
. A wavelength tunable laser (New Focus, TLB-6316, 838nm~853nm wavelength range) serves as the coherent light source. Its output is separated into a reference beam and an object beam. The reference beam contains a controllable phase shifter to generate phase shifted interferograms. In the object beam, part to be measured (object), parabolic mirror (for off-axis aberration correction purpose) and lens, and the camera (Kodak, KAI-4021, 2048 by 2048 pixels) form a reflective imaging system. For each wavelength λ, its hologram can be generated by applying phase shifted interferometry (PSI): Interferograms captured by CCD can be expressed as [20

20. C. C. Aleksoff, “Multi-wavelength digital holographic metrology,” Proc. SPIE 6311, 63111D, 63111D-7 (2006). [CrossRef]

]:
I(φ)=Iobj+Iref+2IobjIrefcos[(γobjγref)φ]
(1)
where Iobj / γobj and Iref / γref stands for the intensity / phase of the object and reference beam, respectively. φ is a phase shift term generated by the phase shifter. The phase difference term (γobj - γref) physically results from the optical path difference H between the object and reference beam:
γobjγref=2πHλ=2nπ+θ
(2)
where n is an integer, and θ is the residual phase angle ranging from –π to π. θ can be mathematically derived from interferograms of designed φ values. For example, if φ is set to be 0, π/2, π and 3π/2, then [23

23. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38(34), 7085–7094 (1999). [CrossRef] [PubMed]

]:
θ=atan2(I(3π/2)I(π/2)I(0)I(π))
(3)
where atan2 is a four-quadrant inverse tangent function in accordance with the definition in MATLAB [24

24. MATLAB, by Mathworks, http://www.Mathworks.com.

]. The calculated hologram of wavelength λ can then be represented by a complex field:
E(x,y)=2Iobj(x,y)Iref·eiθ(x,y)
(4)
where 2Iobj(x,y)Iref is the amplitude term and eiθ(x,y) is the phase term for each pixel of the camera.

In a single wavelength case, due to the periodicity of the cosine function in Eq. (1), H remains undefined to the additive integer n multiplies 2π as shown in Eq. (2). In other words, θ alone is insufficient to fully represent H, unless n is predetermined. This requirement restricts the varying range of γobj - γref to be within –π to π. The unambiguous measurement range of object height h, which equals to half of H in the shown system, is limited to λ/2. For a larger measurement range, at least one more wavelength should be added in order to deduce a correct object height. By generating a synthetic wavelength λsyn=λ1λ2/(λ1λ2), the unambiguous measurement range will be expanded to λsyn/2. Further introduction of extra wavelengths would make the deduction process more robust, as a Fourier transform based height scan envelope [20

20. C. C. Aleksoff, “Multi-wavelength digital holographic metrology,” Proc. SPIE 6311, 63111D, 63111D-7 (2006). [CrossRef]

]
S(h)=|1Pp=1Pexp[j(4πhλpθ(λp))]|
(5)
can be depicted for each and every point over the whole field of view, in order to deduce the correct value of h, which is consistent with the highest peak position of S(h). Experimentally, applying 14 selected wavelengths ranging from 838 to 853nm with pm level accuracy would result in over ± 150mm unambiguous measurement range.

Similar to single wavelength holographic imaging, MWI imaging of an out-of-focus object would result in blurred details. To deblur the image, the captured holograms of each wavelength should be digitally refocused to the focus plane respectively, before the height scanning step of Eq. (5). A practical refocusing equation for MWI holograms is [22

22. Li Xu and J. Ni, “3D resolution enhancement in multi-wavelength interferometric imaging by digital refocusing,” (Submitted manuscript to Appl. Opt. ). [PubMed]

]:
E2(x',y')=Fx',y'1{exp[jkdλ22(u2+v2)]×[Fu,v+1E1(x,y)]}
(6)
where E2(x',y') / E1(x,y) stands for the out-of-focus / in-focus complex hologram respectively (Fig. 2
Fig. 2 Coordinate system for digital refocusing.
). d is the distance between the planes. k is wave number, which equals to 2π/λ. u and v are spatial frequencies along x and y axis, respectively. Fη,ξ±1f(α,β) denotes 2D Fourier / inverse Fourier transform:
Fη,ξ±1f(α,β)=exp[j2π(αη+βξ)]f(α,β)dαdβ
(7)
Comparing with commonly applied single wavelength Fresnel diffraction formula [25

25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed (McGraw-Hill, 1996).

], Eq. (6) get the common phase term exp(jkd) eliminated from the right hand side: Due to enormous magnitude difference between refocusing distance d and wavelength λ, this term would introduce evident phase error even for pm level wavelength inaccuracy. Such phase errors among applied wavelengths would result in phase mismatching in the height scanning step of Eq. (5), making refocused observation no longer valid for metrology purpose. After digitalization, Eq. (6) turns into:
E2(X'ΔM,Y'ΔM)=X,Y,U,V=0N11N2E1(XΔM,YΔM)×
×exp[jπλdM2N2Δ2(U2+V2)]×exp{j2πN[(XX')U+(YY')V]}
(8)
where the whole field captured is of N by N pixels. Δ represents pixel interval. M is the magnification of the imaging system. X’, Y’, X and Y all range from 0 to N-1. Discrete spatial frequencies U and V also range from 0 to N-1.

Maximum refocusing range is limited by aliasing error in Eq. (8) [23

23. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38(34), 7085–7094 (1999). [CrossRef] [PubMed]

]. For a small magnification factor of M = 1/20, if N = 2048, Δ = 7.5μm, λ≈850nm, a theoretical maximum refocusing distance dmax can be as large as 27.1m.

A typical MWI refocusing process can be summarized as follows:
  • Capture multiple interferograms for each wavelength λp
  • Generate the individual out-of-focus complex fields, based on Eq. (4)
  • Calculate refocused complex fields, using Eq. (8)
  • Height scanning (Eq. (5) to reconstruct an in-focus height map h(x,y) from the refocused fields.
As an example, height maps of before and after refocusing a one cent penny is shown in Fig. 3
Fig. 3 Experimental measurement of a 200mm out-of-focus penny.
. The penny is placed 200mm out of focus. The height maps are threshold filtered to emphasize features within the range from 1.250mm to 1.650mm above its background.

3. Focus detection for MWI imaging

Object height h derived in Eq. (5) originates from the phase term cos[(γobjγref)φ] of the captured interferograms, rather than the amplitude term 2Iobj·Iref in Eq. (1). The focus detection criterion derived below for h(x,y) is thereby based on phase contrast, which is a significant difference of the proposed work from its predecessors.

A major difficulty of setting up a practical focus-finding criterion on height maps is that the useful height contrast information is relatively weak, comparing with its background noise. Take Fig. 3 as an example, the reconstructed features over the penny are typically around 100μm in height, 1/40 of the experimental height scanning range (−1mm to 3mm). Their occupied area is less than 5% of the whole field of view (200 by 200 pixels). Therefore, an effective feature extraction and enhancement process should be the core part of the criterion.

After generating refocused height maps h(x,y){d} with a serial of refocusing distance d along z axis, the proposed focus detection criterion can be conducted in the following steps:

  • Impulse detector based noise filtering
  • Finding the difference between two reconstructed height maps of certain interval
  • Eliminate measurement uncertainty
  • Feature enhancement
  • Implement differential indicator η{d}

Step 1: Impulse detector based noise filtering

Figure 3 demonstrates obvious salt-and-pepper noise. This is introduced by practical small errors of λps and θ(λp)s. As we are taking a real-time measurement of wavelengths experimentally, the main contributor of errors in λps is considered to be rounding errors. On the other hand, the causes of errors of θ(λp) are manifold, including vibration, temporal air turbulence along optical path and speckle induced degradation, etc. Laser speckles due to the roughness of the target object surface randomly degrade intensity modulation, on which we depend to derive θ(λp) in Eq. (3). Since the height scanning process (Eq. (5) is ill-posed, small differences of λp and θ(λp) in (4πh/λpθ(λp)) would result in unpredictable peak position shift. When errors of λp and θ(λp) are random small values, such shift should be point-by-point: one outlier point will not affect its neighbors. Under such conditions, impulse detector based noise filtering should be effective. The filtered height maps are noted as h¯(x,y){d}.

Step 2: Finding the difference between two reconstructed height maps of certain interval

Other than focusing on the information of a single frame, the developed criterion here uses two reconstructed frames: the subtraction of two height maps with a certain interval, named “height difference map”, is employed. Assuming the frame interval of the reconstruction serial is δd, then the difference interval between two frames is usually selected to be l times of δd for convenience, and the resulting height difference map:
h¯_diff(x,y){d,dlδd}=h¯(x,y){d}h¯(x,y){dlδd}
(9)
This operation is more or less similar to the correlation coefficient (CC) indicator [4

4. Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17(4), 761–770 (2006). [CrossRef]

], as it can be viewed as a kind of “robustness enhancement” procedure: larger the interval is, more evident the difference between two frames should be. The idea of using difference instead of the frames themselves has a physical meaning in MWI imaging: An out-of-focus height map provides a “blurred version” of a correct height distribution. Subtraction between reconstructed frames is then just a height difference calculation. The majority points in the two height maps should be similar. After subtraction, they should thereby be around zero. Only those points along the edges of features remain obviously different from zero in a height difference map.

Step 3: Eliminate measurement uncertainty

In the last step, although the majority points are of similar heights in the two frames, their difference may not be exactly zero. This is because of measurement uncertainty introduced by either digitalization error or microscopic surface fluctuation. Although they may take up a considerable percentage of the field of view, such non-zero values do not reflect feature sharpness. They must be eliminated in order not to be mistakenly amplified by the following feature enhancement step. Threshold filtering provides a possible solution: any value between ± ε should be set to zero in the resultant height difference map h¯_diff¯(x,y){d,dlδd}.

Step 4: Feature enhancement

After step 3, the remaining non-zero values in height difference maps stand for feature sharpness variation. However, from an image processing point of view, their distribution in gray level (the height scanning range) needs to be further adjusted by a histogram regularization step. The purposes of this step are: 1) enhance weak features / small height differences and 2) reshape the histogram, to make it suitable for a selected sharpness calculation.

In this work, feature enhancement of a height difference map after step 3 is accomplished via a revised histogram equalization process: First, pick out all non-zero values; next, do histogram equalization form these values; then calculate the mean of the equalized values; finally, set all zero points in the original height difference map to be this mean value. The feature-enhanced height difference maps are noted as h¯_diff¯eq(x,y){d,dlδd}

The above described process is developed especially for variance calculation: on one hand, the majority zero valued points contain no feature information, thus should be designed to have no contribution to the variance; on the other hand, variance based indicator would be most efficient when the histogram is reshaped closer to a Gaussian distribution. If another sharpness calculation is to be adopted in step 5, the process of feature enhancement may be adjusted accordingly.

Step 5: Implement differential indicator η{d}

The differential indicator described below is based on variance calculation. The major difference is, instead of the maximum of variance, the symmetry around the focus is emphasized: at each position d, two height difference maps, namely, h¯_diff¯eq(x,y){d,dlδd} and h¯_diff¯eq(x,y){d+lδd,d} are considered. If their variances, Var_diffeq{d,dlδd} and Var_diffeq{d+lδd,d}, are equal, d will be recognized as the location of best focus plane. The regularized indicator to estimate how close a selected plane is to the best focus is thereby:
η{d}=Var_diffeq{d+lδd,d}Var_diffeq{d,dlδd}Var_diffeq{d,dlδd}+Var_diffeq{d+lδd,d}
(10)
Figure 4
Fig. 4 Sequential height maps and focus detection curves.
illustrates the idea of η{d}. Symmetry detection implements a further subtraction upon height difference maps, which can be viewed as a kind of differential operation to a maximum finding curve. By combining information from three frames: h(x,y){dlδd}, h(x,y){d} and h(x,y){d+lδd}, each point on η{d} curve contains more information than the existing indicators, and should therefore be more robust: a sharper indication of best focus with less “false alarms” would be expected. As an extra benefit, η{d} directly tells the relative location of the best focus by its sign. This would be especially helpful when either the best focus position or the capture position is unknown.

The proposed focus detection criterion is summarized in flow chart in Fig. 5
Fig. 5 Flow chart of the differential focus detection criterion.
.

4. Experimental results

Parameter selection

For comparison purpose, performances of a few other indicators are also displayed:Ent: entropy indicator [1

1. J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recognit. Lett. 9(1), 19–25 (1989). [CrossRef]

]
Ent=xy{p(x,y)×log[p(x,y)]}
(11)
Var: variance indicator [2

2. L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. Soc. Am. A 6, 396–400 (2004).

]
Var=xy[p(x,y)pmean]2
(12)
Spec: spectral indicator [3

3. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008). [CrossRef] [PubMed]

]
Spec=uvlog{1+|[Fx,y+1p(x,y)](u,v)|}
(13)
Fx,y+1: two dimensional Fourier transform CC: correlation coefficient indicator [4

4. Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17(4), 761–770 (2006). [CrossRef]

]
CC=xy[p1(x,y)p1mean][p2(x,y)p2mean]{xy[p1(x,y)p1mean]2}{xy[p2(x,y)p2mean]2}
(14)
p(x,y) stands for the height map to be processed, and pmean stands for its mean value. For variance, spectral and correlation coefficient indicators, p(x,y) are median filtered height maps, i.e., h¯(x,y){d}; For entropy, since available features differences are too weak, h¯(x,y){d} is pre-processed by a direct histogram equalization operation.

In correlation coefficient indicator, the special interval between p1(x,y) and p2(x,y) is selected to be 6 × 25mm, in accordance with lδd for the developed indicator.

Results

Experimental result of the 200mm out-of-focused penny in Fig. 3 is shown. Its height scanning range is from −1mm to 3mm, with a 1μm step size. Features over the penny (typically around 100μm in height) are almost invisible in the original height map h(x,y){d} in Fig. 6(a)
Fig. 6 height map before and after median filtering.
); on the contrary, salt-and-pepper noises are much more evident. After median filtering, noises are removed in Fig. 6(b)). Difference height map h¯_diff(x,y){0mm,150mm} is shown in Fig. 7(a)
Fig. 7 Feature enhancement: (a) h¯_diff(x,y){0mm,150mm}; (b) histogram of (a); (c) histogram equalization without uncertainty elimination; (d) histogram of (c); (e) h¯_diff¯eq(x,y){0mm,150mm} feature enhancement after uncertainty elimination; (f) histogram of (e).
), and its corresponding histogram is demonstrated in Fig. 7(b)). It can be observed that over 91% of the points are within ± 5μm. In spite of their large percentage, these points are irrelevant towards feature sharpness. Histogram equalization is directly implemented on h¯_diff(x,y){0mm,150mm}, and the resultant gray image and corresponding histogram are shown in Fig. 7(c)) and Fig. 7(d)). The tiny difference with measurement uncertainty is amplified because of their percentage superiority, while the true feature sharpness information are still crowded in two narrow bands. Figure 7(e)) and Fig. 7(f)) display feature enhanced image h¯_diff¯eq(x,y){0mm,150mm} and its corresponding histogram resulting from step 3 and step 4. It is clear that the grey bars other than zeros are widely spread in the whole range, indicating a more effective feature sharpness enhancement operation.

The differential curve η{d} is shown in Fig. 8
Fig. 8 Focus detection curve η{d} for the out-of-focus penny in Fig. 3.
. Its zero-passing is at 210.3 mm. Comparing with system’s depth of view, this result is of good accuracy.

Figure 9(a)
Fig. 9 Result of other indicators: (a) entropy indicator; (b) variance indicator; (c) spectral indicator; (d) correlation coefficient indicator.
9(d) demonstrate results by entropy (Ent) indicator, variance (Var) indicator, spectral (Spec) indicator and correlation coefficient (CC) indicator. In Fig. 9(a)), entropy curve does not tell the correct best focus position in this case; Although with limited accuracy, variance curve and spectral curve do show some kind of local maxima in Fig. 9(b) and 9(c). Nevertheless, local maxima are insufficient for automatic focus detection purpose; The CC indicator successfully overcomes the problem of local maxima and finds the correct best focus position. However, it displays a wide top range from approximately 125mm to 300mm in Fig. 9(d)). Such flat top phenomenon indicates that the result is likely to be corrupted by residual noise, and the method is in general less accurate.

Two more examples are provided in Fig. 10
Fig. 10 result of imaging a key 200mm out of focus.
and Fig. 11
Fig. 11 result of “coherix” image. For clearance, in (d) the refocused height map is displayed within −60μm to + 35μm range after 3-by-3 median filtering.
. Figure 10 demonstrates a result of “concave” features by measuring a key 200mm out of focus. Different from the penny example, main features over the key (the characters) are below their background. Surface quality of the key is also worse, as more salt-and-pepper noise points can be observed. The detected best focus position is 210.1mm from its capturing origin.

Figure 11 shows a possible solution of “weak feature” recognition. An artificial part with characters “coherix” has been captured 300mm out of focus. The characters are approximately 1mm above their background. The detected refocusing distance is 311.5mm. Over the machined background, there are tooling marks typically of 10μm to 30μm in height. These tooling marks are barely above the uncertainty filtering threshold and containing very limited refocusing information. Tooling marks are of observation interest in machining process analysis. Due to measurement time or structure limitations, some of the machined surfaces have to be observed out-of-focus. Details of tooling marks can therefore be covered by blurring. Since they contribute little in focus detection, a direct focus detection based on tooling marks can be severely deviated. The refocusing result shown in Fig. 11(d)) demonstrates a solution, which relates weak features (tooling marks) to strong features (characters), in order to achieve satisfactory observation quality for the former.

5. Conclusion

By implementing multi-wavelength interferometry, holographic focus detection criterion can be based on phase contrast distribution. A specific criterion has been developed in this article with experimental results on step like height structures proving its feasibility. The developed technology can be especially helpful in practical applications such as surface quality inspection in the auto industry, for it provides a possibility of recognizing micron level tooling marks over surfaces far out of focus. Future studies involve questions such as how the criterion can be improved for a “boundary” condition, with surface features between gentle and step like. Another question of interest is how much the selected parameters in section 4 depend on a specific system and the certain object investigated. Furthermore, studies on how to reduce phase error further in a practical system would also be of remarkable value.

Acknowledgments

The authors are greatly thankful to Prof. Jeffrey A. Fessler and Dr. Carl C. Aleksoff for the fruitful discussions and helpful comments on the manuscript. The presented work builds on the foundation established from NSF as well as NIST funded research projects, including NSF grant (#9453491) and a NIST ATP project: “High definition metrology and processes: 2µm manufacturing.”

References and links

1.

J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recognit. Lett. 9(1), 19–25 (1989). [CrossRef]

2.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. Soc. Am. A 6, 396–400 (2004).

3.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008). [CrossRef] [PubMed]

4.

Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17(4), 761–770 (2006). [CrossRef]

5.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14(13), 5895–5908 (2006). [CrossRef] [PubMed]

6.

J. Liu, X. Song, R. Han, and H. Wang, “Autofocus method in digital holographic microscopy,” Proc. SPIE 7283, 72833Q, 72833Q-6 (2009). [CrossRef]

7.

P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, “Extended focused image in microscopy by digital Holography,” Opt. Express 13(18), 6738–6749 (2005). [CrossRef] [PubMed]

8.

T. Colomb, N. Pavillon, J. Kühn, E. Cuche, C. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35(11), 1840–1842 (2010). [CrossRef] [PubMed]

9.

M. K. Kim, “Wavelength-scanning digital interference holography for optical section imaging,” Opt. Lett. 24(23), 1693–1695 (1999). [CrossRef] [PubMed]

10.

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, “Submicrometer optical tomography by multiple-wavelength digital holographic microscopy,” Appl. Opt. 45(32), 8209–8217 (2006). [CrossRef] [PubMed]

11.

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003). [CrossRef] [PubMed]

12.

Y. Zou, G. Pedrini, and H. Tiziani, “Surface contouring in a video frame by changing the wavelength of a diode laser,” Opt. Eng. 35(4), 1074–1079 (1996). [CrossRef]

13.

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007). [CrossRef] [PubMed]

14.

M. T. Rinehart, N. T. Shaked, N. J. Jenness, R. L. Clark, and A. Wax, “Simultaneous two-wavelength transmission quantitative phase microscopy with a color camera,” Opt. Lett. 35(15), 2612–2614 (2010). [CrossRef] [PubMed]

15.

M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express 7(9), 305–310 (2000). [CrossRef] [PubMed]

16.

A. Khmaladze, M. Kim, and C.-M. Lo, “Phase imaging of cells by simultaneous dual-wavelength reflection digital holography,” Opt. Express 16(15), 10900–10911 (2008). [CrossRef] [PubMed]

17.

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]

18.

D. Carl, M. Fratz, M. Pfeifer, D. M. Giel, and H. Höfler, “Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths,” Appl. Opt. 48(34), H1–H8 (2009). [CrossRef] [PubMed]

19.

J. Kandulla, B. Kemper, S. Knoche, and G. von Bally, “Two-wavelength method for endoscopic shape measurement by spatial phase-shifting speckle-interferometry,” Appl. Opt. 43(29), 5429–5437 (2004). [CrossRef] [PubMed]

20.

C. C. Aleksoff, “Multi-wavelength digital holographic metrology,” Proc. SPIE 6311, 63111D, 63111D-7 (2006). [CrossRef]

21.

http://www.coherix.com/automotive.

22.

Li Xu and J. Ni, “3D resolution enhancement in multi-wavelength interferometric imaging by digital refocusing,” (Submitted manuscript to Appl. Opt. ). [PubMed]

23.

F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38(34), 7085–7094 (1999). [CrossRef] [PubMed]

24.

MATLAB, by Mathworks, http://www.Mathworks.com.

25.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed (McGraw-Hill, 1996).

OCIS Codes
(090.2880) Holography : Holographic interferometry
(090.1995) Holography : Digital holography
(100.3175) Image processing : Interferometric imaging

ToC Category:
Holography

History
Original Manuscript: April 6, 2011
Revised Manuscript: May 27, 2011
Manuscript Accepted: June 12, 2011
Published: July 18, 2011

Citation
Li Xu, Mike Mater, and Jun Ni, "Focus detection criterion for refocusing in multi-wavelength digital holography," Opt. Express 19, 14779-14793 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-14779


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References

  1. J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recognit. Lett. 9(1), 19–25 (1989). [CrossRef]
  2. L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. Soc. Am. A 6, 396–400 (2004).
  3. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008). [CrossRef] [PubMed]
  4. Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17(4), 761–770 (2006). [CrossRef]
  5. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14(13), 5895–5908 (2006). [CrossRef] [PubMed]
  6. J. Liu, X. Song, R. Han, and H. Wang, “Autofocus method in digital holographic microscopy,” Proc. SPIE 7283, 72833Q, 72833Q-6 (2009). [CrossRef]
  7. P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, “Extended focused image in microscopy by digital Holography,” Opt. Express 13(18), 6738–6749 (2005). [CrossRef] [PubMed]
  8. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, C. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35(11), 1840–1842 (2010). [CrossRef] [PubMed]
  9. M. K. Kim, “Wavelength-scanning digital interference holography for optical section imaging,” Opt. Lett. 24(23), 1693–1695 (1999). [CrossRef] [PubMed]
  10. F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, “Submicrometer optical tomography by multiple-wavelength digital holographic microscopy,” Appl. Opt. 45(32), 8209–8217 (2006). [CrossRef] [PubMed]
  11. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003). [CrossRef] [PubMed]
  12. Y. Zou, G. Pedrini, and H. Tiziani, “Surface contouring in a video frame by changing the wavelength of a diode laser,” Opt. Eng. 35(4), 1074–1079 (1996). [CrossRef]
  13. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007). [CrossRef] [PubMed]
  14. M. T. Rinehart, N. T. Shaked, N. J. Jenness, R. L. Clark, and A. Wax, “Simultaneous two-wavelength transmission quantitative phase microscopy with a color camera,” Opt. Lett. 35(15), 2612–2614 (2010). [CrossRef] [PubMed]
  15. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express 7(9), 305–310 (2000). [CrossRef] [PubMed]
  16. A. Khmaladze, M. Kim, and C.-M. Lo, “Phase imaging of cells by simultaneous dual-wavelength reflection digital holography,” Opt. Express 16(15), 10900–10911 (2008). [CrossRef] [PubMed]
  17. J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]
  18. D. Carl, M. Fratz, M. Pfeifer, D. M. Giel, and H. Höfler, “Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths,” Appl. Opt. 48(34), H1–H8 (2009). [CrossRef] [PubMed]
  19. J. Kandulla, B. Kemper, S. Knoche, and G. von Bally, “Two-wavelength method for endoscopic shape measurement by spatial phase-shifting speckle-interferometry,” Appl. Opt. 43(29), 5429–5437 (2004). [CrossRef] [PubMed]
  20. C. C. Aleksoff, “Multi-wavelength digital holographic metrology,” Proc. SPIE 6311, 63111D, 63111D-7 (2006). [CrossRef]
  21. http://www.coherix.com/automotive .
  22. Li Xu and J. Ni, “3D resolution enhancement in multi-wavelength interferometric imaging by digital refocusing,” (Submitted manuscript to Appl. Opt. ). [PubMed]
  23. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38(34), 7085–7094 (1999). [CrossRef] [PubMed]
  24. MATLAB, by Mathworks, http://www.Mathworks.com .
  25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed (McGraw-Hill, 1996).

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