## Motion compensation and noise tolerance in phase-shifting digital in-line holography

Optics Express, Vol. 14, Issue 10, pp. 4286-4299 (2006)

http://dx.doi.org/10.1364/OE.14.004286

Acrobat PDF (1398 KB)

### Abstract

We present a technique for phase-shifting digital in-line holography which compensates for lateral object motion. By collecting two frames of interference between object and reference fields with identical reference phase, one can estimate the lateral motion that occurred between frames using the cross-correlation. We also describe a very general linear framework for phase-shifting holographic reconstruction which minimizes additive white Gaussian noise (AWGN) for an arbitrary set of reference field amplitudes and phases. We analyze the technique’s sensitivity to noise (AWGN, quantization, and shot), errors in the reference fields, errors in motion estimation, resolution, and depth of field. We also present experimental motion-compensated images achieving the expected resolution.

© 2006 Optical Society of America

## 1. Introduction

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

4. S. Lai, B. King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital in-line holography,” Opt. Comm. **173**, 155–160 (2000). [CrossRef]

## 2. Technique

### 2.1. Physical apparatus and data acquisition

*M*+1 raw sensor frames, where

*M*≥3. For the

*j*-th frame, the reference field has complex amplitude

*R*

_{j}. The first and last frames use the same reference field,

*R*

_{1}=

*R*

_{M}

_{+1}.

### 2.2. Motion compensation

*j*=1) and last (

*j*=

*M*+1) frames can be estimated using the cross correlation of those frames. Because those two frames use the same reference field, they differ only due to the motion of the object. That is,

*I*

_{1}(

*x*,

*y*)=

*I*

_{M}

_{+1}(

*x*+Δ

*x*,

*y*+Δ

*y*) where Δ

*x*and Δ

*y*are the distances by which the object has moved in the

*x*and

*y*directions, respectively. In this ideal limit, the cross-correlation

*x′*=Δ

*x,y′*=Δ

*y*, where ℱ and ℱ

^{-1}indicate Fourier transform and inverse Fourier transform, respectively.

*x*and Δ

*y*have been estimated, the frames can be un-shifted according to

*Ī*

_{j}(

*x*,

*y*) represent the intensity patterns that would have been recorded if the object had remained at the same position (it’s position for frame

*j*=1) for all frames.

*λ*, such phase shifts can become nearly random, making reconstruction difficult. There are, however, techniques for estimating the phase shifts from the image data itself [1, 5

5. K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express **9**, 236–253 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-5-236. [CrossRef] [PubMed]

### 2.3. Reconstruction

*f*(

*x*,

*y*) from the intensity measurements

*Ī*

_{j}. Many mathematical formalisms for such reconstruction have been developed [1], but we present here a very general framework allowing for linear reconstruction of the electric field using

*M*≥3 frames, each with a different reference field. Having served its purpose in the motion compensation, frame

*M*+1 is no longer needed. For each frame

*j*, the recorded intensity can be written as

*R*

_{j}(

*f*) is the complex amplitude of the reference (object) field at the sensor. A single prime indicates the real part of a quantity, and a double prime indicates the imaginary part. Here, the spatial dependence of

*Ī*

_{j}and

*f*

_{j}have been suppressed for clarity. The reference field

*R*

_{j}is a plane wave and so has no spatial dependence. This equation holds for all pixels independently. If the values of

*R*

_{j}are known, then we can define the quantity

*M*exposures, this can be written in matrix form as

^{T}indicates transpose and

**P**can be found such that both

**P**exist as long an

*M*≥3 and

**R**is rank-2. As discussed in Subsection 3.2, a good choice is

**P**=

^{+}indicates the Moore-Penrose pseudo-inverse. Once

**P**is found, the field at the sensor plane can be calculated as

**f**=

**PD**/

**2**.

**D**becomes an

*M*×

*N*matrix and

**f**becomes 2×

*N*, where each column represents one of

*N*pixels.

### 2.4. Back-propagation

*f*(

*x*,

*y*) calculated, estimating the field at the object plane

*g*(

*x*,

*y*) is simply a matter of Fresnel-Kirchoff back-propagation according to

*K*(

*x*,

*y*) is the Fresnel-Kirchoff diffraction kernel

*ĝ*(

*x*,

*y*) is an estimate of

*g*(

*x*,

*y*) because the finite-aperture diffraction process acts as a filter, as discussed below.

*M*=4 and the reference fields are

*R*,

*iR*,-

*R*,-

*iR*,

*R*. The illumination is provided by a He:Ne laser operating at 632.8 nm and the raw images are captured on a CCD with 765×511 pixels, each 9

*µ*m×9

*µ*m. Each image is 500×500 pixels and represents a 4.5 mm×4.5 mm area. Figure 2(a) shows a traditional (non-holographic) image of the object acquired using a 75 mm focal-length lens in a one-to-one imaging configuration. For the holographic imaging, the object was moved horizontally by approximately 100

*µ*m between each frame. The intensity at the object plane (

*z*=107 mm), reconstructed

*without*motion compensation is shown in Fig. 2(b). Reconstruction from the same data with motion compensation yields Fig. 2(c). For comparison, a single raw captured frame of the holographic data is shown in Fig. 2(d).

*z*in the back-propagation calculation, it is possible to “focus” on different object depths computationally from a single acquisition, as shown in Fig. 3. Both Figs. 3(a) and 3(b) are reconstructions of the same holographic data. The top and bottom targets were placed at different distances from the sensor.

## 3. System design and performance limitations

### 3.1. Optical design

*p*, where

*p*is the pixel separation. In the lensless case, such a fringe pattern on the sensor is generated by a grating of the same period at the object.

_{m}=sin

^{-1}(

*mλ*/Δ). If we insist that at least two scattering orders (including

*m*=0) fall on the sensor regardless of the position of the localized grating, then we require

*N*

_{p}≥2

*λd*/Δ, where

*N*is the linear number of pixels and

*d*is the distance between the object and sensor. The diffraction-limited object resolution is then

*r*

_{R}≈1.22λ

*d*/(

*N*

_{p}).

*d*

_{opt}≈

*d*

_{opt}=511×(9×10

^{-6})

^{2}/(632.8×10

^{-9})=0.065 m, yielding an achievable diffraction-limited resolution of

*r*

_{d}=9

*µ*m, the pixel size. A larger distance (0.107 m) was used above due to mechanical constraints, resulting in a predicted resolution of

*r*

_{d}=15

*µ*m. The observed resolution is approximately 17

*µ*m in the vertical direction and 25

*µ*m in the horizontal direction, suggesting that the horizontal motion led to a reduction in resolution by approximately one pixel, as described in Subsection 3.5.

*M*

_{i}=

*s*

_{i}/

*s*

_{o}and the resolution becomes

*D*

_{l}must satisfy

*D*

_{l}≥λ

*s*

_{i}/

*p*. Also, the plane reference field must either be mixed with the scattered object light after the lens (as shown in Fig. 5) or the reference field must be prepared such that it has planar wavefronts as it strikes the sensor.

### 3.2. Noise and dynamic range

*f̂*at the sensor is given by

*I*

_{max}is the maximum measurable intensity,

*b*is the number of bits, and the quantity 2√3 comes from the assumption of a uniform intensity distribution. This assumption need only be valid on the scale of a quantization level

*I*

_{max}/2

^{b}. Calculating the quantization noise variance on

*f̂*is very similar to the AWGN case, leading to

*α*which depends on the wavelength and sensor properties. The resulting noise variance is given by

*E*

_{f}=

*M*|

*f*|

^{2}and from the reference

*E*

_{R}=

*MR*

^{2}is held fixed, then the noise variance becomes

_{E}contribution from shot noise does not depend on

*M*, whereas the contribution from detector AWGN and quantization noise decreases with increasing

*M*. Therefore, the best noise performance is achieved with small

*M*.

_{E}vs. sensor SNR=

*I*

_{max}/σ

_{Ig}for two values of

*M*using both the linear inversion technique described above and also a nonlinear technique. The nonlinear approach chooses

**f**to minimize the magnitude of (

**2R**+

**1f**

^{T})

**f**-

**D**at each pixel. The simulation includes both AWGN and shot noise. In the high-sensor-noise region on the left, the performance is dominated by sensor noise. In that limit, the linear dependence of SNR

_{E}on the sensor SNR dominates and the 1/√

*M*dependence is clearly visible. Also, we see that there is a slight improvement (less than 1 dB) in SNR

_{E}achieved by using nonlinear reconstruction. At high sensor SNR, performance is dominated by shot noise. In that regime, nonlinear reconstruction continues to provide improved SNR

_{E}, but the 1/√

*M*dependence is lost.

*f*

_{max}

*R*

_{max}, where

*f*

_{max}and

*R*

_{max}are the maximum amplitudes of the object and reference fields, respectively. However, the sensor places a practical upper limit on the field strengths because the total intensity may not exceed

*I*

_{max}. Therefore, one must impose the constraint

*I*

_{max}/4. If both object and reference field strengths can be adjusted, then one should choose

*H*(

*k*

_{x},

*k*

_{y})|=|ℱ[

*K*(

*x*,

*y*)]|. This filtering process can also be thought of as introducing spatial correlations. Fig. 8 shows plots of the ATF as a function of

*k*

_{x}for a 500×500 sensor with 9

*µ*m pixels for several reconstruction distances

*z*. In all cases, the high frequencies are attenuated, but for larger

*z*the coloring effect is more extreme. The vertical lines mark the spatial frequencies

*k*

_{x}=1/(2

*r*

_{d}), where

*r*

_{d}is the resolution derived in Eq. (11). Figure 9 shows several simulated images corresponding to the same conditions. For larger

*z*, the noise is less grainy, suggesting weaker high-frequency components. Obviously, the object field also experiences the same low-pass filtering, leading to worse resolution and a blurry image.

*f*(

*x*,

*y*)|

^{2}). Therefore, calculating its final spatial correlations is difficult, although like the AWGN, it will be (additionally) colored by the ATF.

### 3.3. Errors in Rj

**D**due to noise. It is also likely that the actual values of the reference fields will differ from the desired values. The impact of such error is easily seen by considering an estimated reference matrix

**R̂**, the corresponding inversion matrix

**P̂**, and the resulting estimated field

**f̂**. Equation (5) can be solved using the same measurements

**D**with either the ideal or estimated versions of

**R**,

**P**, and

**f**. By equating both versions, one finds

**E**is a 2×2 error matrix. This matrix is constant for all pixels. In general, it is difficult to predict the value of this matrix given arbitrary

**R**and arbitrary error. However, we present a few common cases here.

**R̂**=(1+

*ε*)

**R**, then

**E**=-

*ε*

**I**/(1+

*ε*) and the resulting field is also scaled in amplitude. If all reference fields have a constant phase error

*R̂*

_{j}=

*e*

^{iϕ}

*R*

_{j}, then the resulting field measurement is

*f̂*=

*e*-

^{iϕ}

*f*.

*R̂*

_{j}=(1+

*ε*

_{j})

*R*

_{j}with small Gaussian-distributed

*ε*

_{j}, then the RMS of the error matrix is given approximately by

*ε*

_{j}. We see that the variance of the error in

*f*depends only on the variance of the error in

*R*

_{j}. Also, amplitude error in

*R*

_{j}tends to create amplitude error in

*f*, and phase error creates phase error. However, these hold only in the limit that σ

_{ε}≪1; for larger σ

_{ε}, phase and amplitude errors begin to mix.

### 3.4. Depth of field and reconstruction

*f*(

*x*,

*y*) at the sensor and then reconstructs the field

*g*(

*x*,

*y*) at any position

*z*, it is useful to define two quantities: the depth of field (DOF) and depth of reconstruction (DOR). The DOF has a similar meaning to its use in traditional imaging; it is the range over which an object remains (approximately) in focus for a single reconstruction. In contrast, the DOR is the range of

*z*over which a given

*f*(

*x*,

*y*) can be used to reconstruct

*g*(

*x*,

*y*) with good resolution.

*z*/(

*N*

_{p}) for the lensless case. Therefore, a factor of two loss in resolution occurs at

*z*=2

*z*<

*z*+Δ

*z*, where

*z*is the reconstruction depth, is identical to a traditional lens with focal length

*F*=

*z*/2 and diameter

*D*=

*N*

_{p}, as illustrated in Fig. 10. This is because the lens and propagation through distance

*z*

_{0}introduce the same quadratic phase term as back-propagation by a distance -z0 [2]. Therefore, the DOF is roughly given by twice the Rayleigh range of the equivalent lens system,

*rd*. For the experimental results presented in Fig. 2, we find DOF=0.56 mm.

### 3.5. Motion-related error

*t*

_{e}

*v*

_{o}long compared to the pixel size

*p*, where

*t*

_{e}is the exposure time and

*v*

_{o}is the object velocity. However, this form of blur affects each frame identically and results in final-image blur that is very similar to the blur in a conventional image. However, in a traditional image, the intensity

*I*(

*x*,

*y*) is blurred, whereas in this case, the field

*f̂*(

*x*,

*y*) (and

*ĝ*(

*x*,

*y*)) is blurred. That is, a traditional images is blurred according to

*Ī*(

*x*,

*y*)=

*I*(

*x*,

*y*)∗

*B*(

*x*,

*y*), where

*B*(

*x*,

*y*) is a blur function. Because the blur is applied to each

*I*

_{j}(

*x*,

*y*) identically, the result is

6. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. **37**, 247–260 (1998). [CrossRef]

*M*are shown and because total shift increases with

*M*, the MSE is also larger for larger

*M*. This figure demonstrates that in order to avoid losing resolution as a result of axial motion, one must keep the axial translation small compared to the DOF.

## 4. Conclusions

## Acknowledgments

## References and links

1. | T. Kreis, |

2. | J. W. Goodman, |

3. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

4. | S. Lai, B. King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital in-line holography,” Opt. Comm. |

5. | K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express |

6. | R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(110.6880) Imaging systems : Three-dimensional image acquisition

**ToC Category:**

Holography

**History**

Original Manuscript: February 27, 2006

Revised Manuscript: April 27, 2006

Manuscript Accepted: April 27, 2006

Published: May 15, 2006

**Citation**

Michael D. Stenner and Mark A. Neifeld, "Motion compensation and noise tolerance in phase-shifting digital in-line holography," Opt. Express **14**, 4286-4299 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4286

Sort: Year | Journal | Reset

### References

- T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, Weinheim, 2005).
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, Greenwood Village, 2004).
- I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]
- S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000). [CrossRef]
- K. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9, 236-253 (2001). [CrossRef] [PubMed]
- R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998). [CrossRef]

## 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.