Computational signal-to-noise ratio analysis for optical quadrature microscopy

William C. Warger; Charles A. DiMarzio

doi:10.1364/OE.17.002400

1. Introduction

Optical quadrature microscopy (OQM) was invented in 1997 under the name optical quadrature interferometry to measure the amplitude and phase of an optically transparent object without having to acquire multiple views of the object or introduce multiple light sources [1–3]. The instrument was based on techniques originally developed for radio frequency Doppler radar [4] that were later modified for laser radar [5]. The next generation instrument was implemented within a microscope that was named the quadrature tomographic microscope (QTM) because it could be used to acquire tomographic information and potentially provide viability measurements of live mouse embryos [6–9]. Since work had not been completed to rotate the sample [10–11] or rotate the incident angle of the light source [12], the microscopy technique was named optical quadrature microscopy (OQM) to differentiate between the technique used to acquire a single image of quantitative phase, and an instrument that creates a 3D tomographic reconstruction from the acquisition of multiple views.

OQM is just one of many optical microscopy techniques used to reconstruct a full-field image of quantitative phase. Additional methods include other quadrature detection configurations [13–14], polarization interferometers [15], digital holography [16], Fourier phase microscopy [17], Hilbert phase microscopy [18], diffraction phase microscopy [19], phase-shifting interferometry (PSI) [20–23], quantitative phase microscopy (QPM) [24–25], and reconstruction from DIC images [26–31]. Each of the techniques provides their own advantages for various applications.

OQM has been shown to be non-toxic to live mouse embryos and has been used to count the number of cells accurately beyond the 8-cell stage to extend the use of the cell count for determining embryo viability for in vitro fertilization (IVF) [32–33]. While many of the previous OQM publications have provided a brief mathematical description of the reconstruction technique used to compute the change in phase induced by a sample, a thorough signal-to-noise ratio (SNR) analysis has not been presented. In this paper, we provide an overview of quadrature detection in Section 2 followed by descriptions of the current optical layout and the theory behind the phase reconstructions used for OQM in Sections 3 and 4, respectively. Section 5 provides an overview of a SNR analysis that includes noise terms caused by laser fluctuations, imperfections within the optics, and noise within the CCD cameras to derive an expression for the resultant image of each phase reconstruction, which builds upon previous preliminary results that only included multiplicative noise terms [34]. Section 6 presents a model of the reconstructions using images from the experimental setup to quantify the minimum amount of noise that is inherent in the current phase images.

2. Quadrature detection

A wave is in quadrature with another wave if they are matched in frequency and amplitude, but differ in phase by 90 degrees. Quadrature detection is a technique that has been used in Doppler radar [4] as well as in communications systems [35] to measure the amplitude and phase of an unknown sinusoidal signal by mixing with two reference, or local oscillator, signals that are in quadrature, as shown in Fig. 1. The unknown and reference signals are split into two separate but equal parts, where one part of the reference has a 90-degree phase shift with respect to the second. Two multiplying mixers combine the two parts of the unknown signal with the reference signals that are in quadrature to produce two mixed signals. Low pass filters block the double frequency signal inherent in the mixers and any leakage of the carrier frequencies. The mixed signal with the original reference is the in-phase channel (I channel), and the mixed signal with the 90-degree phase-shifted reference is the quadrature channel (Q channel). The I and Q channels can be interpreted mathematically as the real and imaginary values of a complex number to calculate the amplitude and phase of the unknown signal.

Optical quadrature microscopy utilizes a similar detection technique where the two orthogonal S and Ppolarization states of the laser beam act as the two unknown signal paths in Fig. 1. Polarizing the reference and signal paths at 45 degrees provides equal amounts of S and P polarization. The reference path then transmits through a quarter-wave plate to create circular polarization, where one polarization state has a 90-degree phase shift with respect to the other. The 45-degree polarized signal and circularly polarized reference beams mix within a non-polarizing recombining beamsplitter, and a polarizing beamsplitter separates the two polarization states of the mixed signal into the I and Q channels that are acquired with two CCD cameras. In practice, both outputs of the recombining beamsplitter are separated into two separate I and Q channels in a balanced detection configuration to remove the DC components that result from the additive combination of the fields in the beamsplitters and the square law detection required for optical frequencies.

Fig. 1. Simplified block diagram for quadrature detection in a Doppler radar receiver [4].

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3. Optical layout for OQM

The current optical layout for OQM is shown in Fig. 2. A linearly polarized Helium Neon laser (31-2082-000, Coherent, Santa Clara, California) is fiber coupled with a single-mode optical fiber to a non-polarizing 50/50 beamsplitter that splits the light into the reference and signal paths of a Mach-Zehnder interferometer. A lens at the fiber output of the signal path collimates the beam that travels through a polarizer oriented at 45 degrees relative to the x⃗ and y⃗ basis vectors of the system. The 45-degree linearly polarized light enters the optical path of the Nikon Eclipse TE2000U microscope by reflecting from a narrow bandstop dichroic splitter centered at 633 nm with a 30-nm bandstop defined by the full width at half maximum. The dichroic splitter reflects the 633 nm light from the laser and transmits the white light from the halogen light source of the microscope for brightfield and differential interference contrast (DIC) microscopy [36]. The condenser lens focuses the light that travels through a sample where a phase delay is introduced. An infinity-corrected objective lens collects the light that transmits through the sample and is within the numerical aperture, and a tube lens images the sample at the camera port of the microscope. Because the recombination of the reference and signal paths occurs outside of the microscope body, a lens relays the image of the sample at the camera port to an image plane where the CCD cameras are mounted. A linear polarizer oriented at 45 degrees relative to the x⃗ and y⃗ basis vectors of the system can be positioned before the recombining beamsplitter to ensure the signal path is polarized at 45 degrees before mixing with the reference path.

A lens at the fiber output of the reference path collimates the beam that travels through a linear polarizer oriented at 45 degrees relative to the x⃗ and y⃗ basis vectors of the system, and a quarter-wave plate oriented at 45 degrees to the axis of the polarizer to produce circularly polarized light. A lens matches the wavefront of the beam in the reference path to the wavefront of the beam in the signal path, and the 45-degree linearly polarized signal path mixes with the circularly polarized reference path at the 50/50 non-polarizing recombining beamsplitter. A balanced detection configuration has been implemented such that both outputs of the recombining beamsplitter are acquired instead of just one. The two mixed beams that are output from the recombining beamsplitter travel through polarizing beamsplitters to separate the quadrature components, and four synchronized 8-bit CCD cameras (XC-75, Sony Electronics Inc., Park Ridge, New Jersey) acquire images of the sample from each output of the two polarizing beamsplitters with a framegrabber (Matrox Genesis LC, Dorval, Canada) that has the ability to buffer four simultaneous video channels.

Fig. 2. Optical layout for OQM. The x⃗ and y⃗ basis vectors are labeled along the optical path within the individual arms of the interferometer and after recombination. The unlabeled lenses are single element lenses.

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4. Ideal electric field analysis

To describe the general principles behind the phase reconstructions we provide a mathematical description of OQM that assumes perfect optical elements free from optical aberrations, and CCD cameras with a responsivity of 1 and no dark current noise. Laser light travels through a single-mode fiber to a non-polarizing beamsplitter that separates the light into the reference (E⃗_ref) and signal (E⃗_sig) paths. The 50/50 non-polarizing beamsplitter splits the irradiance of the incident laser light, which is proportional to the intensity or the square of the electric field, thereby providing 1/√2 of the incident field in each path. Each path passes through a linear polarizer oriented at 45 degrees to the basis set to ensure 45-degree polarized light in each path:

{\vec{E}}_{ref} = \frac{1}{\sqrt{2}} E_{R} e^{j (ωt + ϕ)} (\vec{x} + \vec{y})

{\vec{E}}_{sig} = \frac{1}{\sqrt{2}} E_{S} e^{j (ωt + ϕ)} (\vec{x} + \vec{y}),

where E_R/1√2 and E_S/1√2 are the amplitudes of the fields from the laser that pass through the linear polarizers, ϕ is the phase of the laser output, j is √-1, ω is the angular frequency, t is time, and x⃗ and y⃗ are the basis vectors of the system, which are projections of the field in the S and P directions with respect to the polarizing beamsplitters. The reference path travels through a quarter-wave plate with axes oriented at 45 degrees relative to the laser’s polarization axis thereby producing circular polarization:

{\vec{E}}_{ref} = \frac{1}{\sqrt{2}} E_{R} e^{j (ωt + ϕ)} (\vec{x} + j \vec{y}) .

The signal path travels through the sample introducing a change in magnitude (A) and phase (α):

{\vec{E}}_{sig} = \frac{1}{\sqrt{2}} A E_{S} e^{j (ωt + ϕ + α)} (\vec{x} + \vec{y}) .

Assuming a non-scattering sample with a single uniform refractive index, (1-A) is proportional to the absorption of the light within the sample, and α is related to the thickness and refractive index by:

α = \frac{2 π}{λ} (n_{s} - n_{0}) h,

where λ is the wavelength of the laser, n ₀ is the refractive index of the immersion medium, and n_s and h are the refractive index and thickness of the sample, respectively.

The two paths recombine within a second 50/50 non-polarizing beamsplitter. The sum of the input intensity from the square of Eqs. (3) and (4) must be equal to the sum of the output intensity by conservation of energy. Assuming the beamsplitter is lossless the total output is:

{\vec{E}}_{sig}^{2} + {\vec{E}}_{ref}^{2} = {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} + {\vec{E}}_{ref}) ∣}^{2} + {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} - {\vec{E}}_{ref}) ∣}^{2},

where the two terms right of the equals sign correspond to the two outputs of the beamsplitter. Polarizing beamsplitters separate the quadrature components of each output by reflecting the x⃗ component and transmitting the y⃗ component, where the x⃗ and y⃗ components are defined as the S and P polarization states with respect to the polarizing beamsplitters. Four CCD cameras, numbered 0 to 3 and spatially registered with an affine transform [37], capture the interferograms from each output of the polarizing beamsplitters. The two quadrature signals corresponding to the first output of the recombining beamsplitter in Eq. (6) are:

M_{0} = {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} + {\vec{E}}_{ref}) \cdot \vec{x} ∣}^{2}

M_{1} = {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} + {\vec{E}}_{ref}) \cdot \vec{y} ∣}^{2}

and the two quadrature signals corresponding to the second output of the recombining beamsplitter in Eq. (6) are:

M_{2} = {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} + {\vec{E}}_{ref}) \cdot \vec{x} ∣}^{2}

M_{3} = {∣ \frac{1}{\sqrt{2}} ({\vec{E}}_{sig} + {\vec{E}}_{ref}) \cdot \vec{y} ∣}^{2},

where · denotes the dot product and:

M_{0} = \frac{1}{4} (A^{2} E_{S} E_{S}^{*} + E_{R} E_{R}^{*} + A E_{S} E_{R}^{*} e^{jα} + A E_{R} E_{S}^{*} e^{- jα})

M_{1} = \frac{1}{4} (A^{2} E_{S} E_{S}^{*} + E_{R} E_{R}^{*} - j A E_{S} E_{R}^{*} e^{jα} + j A E_{R} E_{S}^{*} e^{- jα})

M_{2} = \frac{1}{4} (A^{2} E_{S} E_{S}^{*} + E_{R} E_{R}^{*} - A E_{S} E_{R}^{*} e^{jα} - A E_{R} E_{S}^{*} e^{- jα})

M_{3} = \frac{1}{4} (A^{2} E_{S} E_{S}^{*} + E_{R} E_{R}^{*} + j A E_{S} E_{R}^{*} e^{jα} - j A E_{R} E_{S}^{*} e^{- jα}) .

Assuming E_S and E_R are real, the complex conjugates of the magnitudes can be removed and Eqs. (11) – (14) reduce to:

M_{0} = \frac{1}{4} (A^{2} E_{S}^{2} + E_{R}^{2} + A E_{S} E_{R} 2 \cos α)

M_{1} = \frac{1}{4} (A^{2} E_{S}^{2} + E_{R}^{2} + A E_{S} E_{R} 2 \sin α)

M_{2} = \frac{1}{4} (A^{2} E_{S}^{2} + E_{R}^{2} + A E_{S} E_{R} 2 \cos α)

M_{3} = \frac{1}{4} (A^{2} E_{S}^{2} + E_{R}^{2} + A E_{S} E_{R} 2 \sin α),

where the squared terms are the DC components from the irradiance of the individual signal and reference paths, and the third terms are the mixing or interference terms that result from the interference between the two paths.

Ideally, balanced mixing via the summation:

E_{BM} = \sum_{k = 0}^{3} j^{k} M_{k} = A e^{jα} E_{S} E_{R},

could remove the DC terms, but in practice variations in irradiance exist along the individual paths from imperfections within the optics and detectors, as shown in the following sections. In Eq. (19) the subscript k defines the signal captured at a specific CCD camera. The subtraction of the pure signal (S_k), reference (R_k), and dark detector voltage (D_k) from the individual signals in Eqs. (11) – (14) removes the aberrations in each path and leads to the reconstruction:

E_{BM, DC} = \sum_{k = 0}^{3} j^{k} (M_{k} - R_{k} - S_{k} + D_{k}) = A e^{jα} E_{S} E_{R},

where the addition of D_k simplifies the notation for the subtraction of the dark detector current from M_k, R_k, and S_k. The images for D_k are subtracted from each image because a value proportional to D_k exists in every image acquired with a CCD camera. Images are acquired for S_k, R_k, and D_k by blocking the signal and reference arms individually and simultaneously. Additional noise results from imperfections in the beamsplitters, such as a 50/50 beamsplitter not splitting the irradiance of the light into equal halves. Thus, the division by the square root of the pure reference images normalizes the camera signals and leads to the current reconstruction that has been used for OQM:

E_{BM, DC, Norm} = \sum_{k = 0}^{3} j^{k} \frac{M_{k} - R_{k} - S_{k} + D_{k}}{\sqrt{R_{k} - D_{k}}} = \frac{A e^{jα} E_{S} E_{R}}{∣ E_{R} ∣} .

Moving the sample out of the field of view and reconstructing a second image with Eq. (21) provides the blank image:

E_{BM, DC, Norm, blank} = \frac{E_{S} E_{R}}{∣ E_{R} ∣} .

Dividing the result of Eq. (21) by the blank image in Eq. (22) provides the complex magnitude and phase induced by the sample:

\frac{E_{BM, DC, Norm}}{E_{BM, DC, Norm, blank}} = A e^{jα} .

The primary difference between the two reconstructions in Eqs. (19) and (21) is the method used to remove the DC components. With balanced mixing in Eq. (19), the DC components are removed mathematically by the summation of the four images. With the current OQM reconstruction in Eq. (21), separate images are acquired for the two DC components and subtracted individually. Thus, the four outputs from the two polarizing beamsplitters are required for balanced mixing, but only two are required for the current OQM reconstruction. However, all four CCD cameras are still used with the current OQM reconstruction to increase the SNR of the final image. The noise that is inherent in each reconstruction is derived in the following sections.

5. SNR analyses for phase reconstructions

The description in the previous section assumes ideal components to explain the concepts behind the phase reconstructions, but additional noise terms exist from temperature fluctuations in the laser, aberrations caused by imperfection in the optics, and noise associated with the CCD cameras. In this section we provide a more complete mathematical description that includes additional noise terms throughout the system for balanced mixing in Eq. (19), balanced mixing and DC term subtraction in Eq. (20), and balanced mixing, DC term subtraction, and camera normalization in Eq. (21). It is important to note that some of the variables within Section 4 are redefined in Section 5 with the inclusion of the noise terms that are incorporated within the model. Section 6 provides the result of each model side-by-side to compare each reconstruction.

5.1. Mathematical description of balanced mixing

The Helium Neon laser outputs an electric field with temperature fluctuations that produce an additive complex noise term. The signal and reference beams are polarized at 45 degrees relative to the x⃗ and y⃗ basis vectors of the system, and the reference path is circularly polarized by the quarter-wave plate oriented at 45 degrees to the laser polarization:

{\vec{E}}_{sig} = \frac{1}{\sqrt{2}} (E_{S} e^{j (ωt + ϕ)} + E_{N} e^{j (ωt + ζ)}) \times (\vec{x} + \vec{y})

{\vec{E}}_{ref} = \frac{1}{\sqrt{2}} (E_{R} e^{j (ωt + ϕ)} + E_{N} e^{j (ωt + ζ)}) \times (\vec{x} + j \vec{y}),

where E_N and Ϛ are the amplitude and phase of the laser fluctuations that pass through the linear polarizers, respectively, and × denotes a multiplication. The signal path transmits through the sample that induces a change in magnitude (A) and phase (α):

{\vec{E}}_{sig} = \frac{1}{\sqrt{2}} (A E_{S} e^{j (ωt + ϕ + α)} + A E_{N} e^{j (ωt + ζ + α)}) \times (\vec{x} + \vec{y}) .

Imperfections within the optics aberrate the field in each path before reaching the recombining beamsplitter:

{\vec{E}}_{ref} = \frac{1}{\sqrt{2}} (E_{R} e^{j (ωt + ϕ)} + E_{N} e^{j (ωt + ζ)}) X_{r} e^{j χ_{r}} E_{C_{k}}^{R} \times (\vec{x} + j \vec{y})

{\vec{E}}_{sig} = \frac{1}{\sqrt{2}} (A E_{S} e^{j (ωt + ϕ + α)} + A E_{N} e^{j (ωt + ζ + α)}) X_{s} e^{j χ_{s}} E_{C_{k}}^{S} \times (\vec{x} + \vec{y}),

where X_r and X_s are the amplitudes and χ_r and χ_s are the phases of the aberrations in the reference and signal paths, respectively. The two paths mix at the recombining beamsplitter and each output travels through a polarizing beamsplitter that separates the quadrature components. Imperfections within the beamsplitters and coherent noise in the CCD cameras provide fixed pattern noise in each path with a magnitude E ^R _{C_k} and E ^S _{C_k} in Eqs. (27) and (28), where the superscript R and S designate the reference and signal paths, respectively. Four CCD cameras convert a general irradiance I_k into a current:

i_{k} = \frac{η_{k} q}{hυ} A_{pixel} I_{k} + i_{D, k},

where $\frac{η_{k} q}{hυ}$ is the responsivity of the camera (A/W), η_k is the quantum efficiency of the silicon chip (electrons/photon), q is the charge of an electron (1.6 × 10^-19 c), h is Planck’s constant (6.6 × 10^-34 Wsec²), ν is the frequency of the laser (sec^-1), A_pixel is the area of a pixel (m²), I_k is the irradiance of the beam (Wm^-2), and i_D,k is the dark current noise. It is important to note that the dark current noise fluctuates randomly from image to image so we have included a different dark current noise term for every image acquired during the reconstruction. The images captured for the mix of the signal and reference paths (M_k) and containing all of the various noise terms can then be expressed:

M_{k} = \frac{1}{4} \frac{η_{k} q}{hυ} A_{pixel} [(A^{2} E_{S} E_{S} + A^{2} E_{S} E_{N} e^{j (ϕ - ζ)}) E_{C_{k}}^{S} E_{C_{k}}^{S} X_{s} X_{s} + {(- j)}^{k} (A E_{S} E_{R} e^{jα} + A E_{S} E_{N} e^{j (ϕ + α - ζ)}) E_{C_{k}}^{S} E_{C_{k}}^{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} +

where i_M,k is the dark current in the mix images and we have assumed the magnitude of each term is real. Balanced mixing of Eq. (30) via the summation in Eq. (19) provides the reconstructed current of the mixed signal:

E_{BM} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} (E_{S} E_{R} + E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

The reconstructed current in Eq. (31) reduces to:

E_{BM} = \frac{A_{pixel} q}{hυ} A e^{jα} E_{S} E_{R},

which is proportional to the result in Eq. (19), assuming the detectors are ideal:

i_{M, k} = 0

η_{k} = 1,

there is no fixed pattern noise within the beamsplitters and CCD cameras:

E_{C_{k}}^{R} = E_{C_{k}}^{S} = 1,

there are no aberrations in the system:

X_{s} e^{j χ_{s}} = X_{r} e^{j χ_{r}} = 1,

and there are no noise fluctuations in the laser:

E_{N} e^{j (ωt + ζ)} = 0 .

Substituting a summation to remove redundant terms, and separating the signal (S) and noise (N_S) terms provides:

S = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} + A e^{- jα} E_{R} E_{S} X_{r} X_{s} e^{- j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} {(- 1)}^{k} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

N_{S} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} (E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N} E_{N}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

where the total image is the sum of Eqs. (38) and (39).

Following the same procedure with laser noise E _B e ^{j(ωt + β)} and dark current in the CCD cameras i_{BM, k}, the reconstruction in Eq. (19) for the blank image with the sample moved out of the field of view provides the signal (B) and noise (N_B) terms:

B = \frac{1}{4} \frac{A_{pixel} q}{hυ} [E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} + E_{R} E_{S} X_{r} X_{s} e^{- j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} {(- 1)}^{k} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

N_{B} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [(E_{S} E_{B} e^{j (ϕ - β)} + E_{B} E_{R} e^{- j (ϕ - β)} + E_{B} E_{B}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +]

Dividing the sum of the signal and noise terms of the sample image by the sum of the signal and noise terms of the blank image, and using a Taylor series expansion for the denominator provides:

\frac{S + N_{S}}{B + N_{B}} \approx \frac{S}{B} + \frac{N_{S}}{B} - \frac{N_{B} S}{B^{2}} \approx \frac{S}{B}

assuming any noise term squared or divided by a signal term is approximately equal to zero. Substituting Eqs. (38)–(41) into Eq. (42) and using Taylor series expansions to remove the denominators provides the reconstructed magnitude and phase induced by the sample:

E_{BM} = A e^{jα} [1 - Φ e^{- j 2 (χ_{s} - χ_{r})} - (Γ_{S} + Γ_{R}) e^{- j (χ_{s} - χ_{r})} - I_{B, M} e^{- j (χ_{s} - χ_{r})}] +

where:

Φ = \frac{\sum_{k = 0}^{3} {(- 1)}^{k} \sqrt{R_{k}} \sqrt{B S_{k}}}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

Γ_{R} = \frac{\sum_{k = 0}^{3} j^{k} R_{k}}{\sum_{k = 0}^{3} \sqrt{B S_{k}} \sqrt{R_{k}}}

Γ_{S} = \frac{\sum_{k = 0}^{3} j^{k} B S_{k}}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

I_{S, M} = \frac{\sum_{k = 0}^{3} j^{k} i_{M, k}}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

I_{B, M} = \frac{\sum_{k = 0}^{3} j^{k} i_{BM, k}}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

assuming the image of the pure signal path with the sample moved out of the field of view and the image of the pure reference path can be approximated by:

B S_{k} \approx \frac{1}{4} \frac{A_{pixel} q}{hυ} η_{k} {∣ E_{C_{k}}^{S} X_{s} E_{S} ∣}^{2}

R_{k} \approx \frac{1}{4} \frac{A_{pixel} q}{hυ} η_{k} {∣ E_{C_{k}}^{R} X_{r} E_{R} ∣}^{2} .

Following the same procedure, a phase image can also be reconstructed without dividing by the blank image, resulting in the reconstructed magnitude and phase:

E_{BM, NB} = [A e^{jα} + A e^{- jα} Φ e^{- j 2 (χ_{s} - χ_{r})} + (A^{2} Γ_{S} + Γ_{R}) e^{- j (χ_{s} - χ_{r})}] e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}} + I_{M}

where:

I_{M} = \sum_{k = 0}^{3} j^{k} i_{M, k} .

However, it is important to emphasize that the resultant phase of Eq. (51) includes the wavefront mismatch between the reference and signal paths in addition to the phase induced by the sample, as will be shown in the reconstructed images in Section 6. This reconstruction is then only useful when a constant wavefront mismatch exists across the field of view, as will be discussed in Sections 6 and 7.

The expressions in Eqs. (43) and (51) for the images reconstructed with balanced mixing remove many of the noise terms in Eq. (30), but noise terms still exist from aberrations and fixed pattern noise in the individual arms of the interferometer (Γ_S and Γ_R). In the following subsection, we subtract images of the individual DC terms to reduce the noise within the resultant image.

5.2. Mathematical description of balanced mixing and DC term subtraction

In addition to the mixed signals in Eq. (30), images can be acquired of the pure signal by blocking the reference path:

S_{k} = \frac{1}{4} \frac{η_{k} q}{hυ} A_{pixel} (A^{2} E_{S}^{2} + A^{2} E_{S} E_{N} (e^{j (ϕ - ζ)} + e^{- j (ϕ - ζ)}) + A^{2} E_{N}^{2}) {(E_{C_{k}}^{S} X_{s})}^{2} + i_{S, k},

the pure reference by blocking the signal path:

R_{k} = \frac{1}{4} \frac{η_{k} q}{hυ} A_{pixel} (E_{R}^{2} + E_{R} E_{N} (e^{j (ϕ - ζ)} + e^{- j (ϕ - ζ)}) + E_{N}^{2}) {(E_{C_{k}}^{R} X_{r})}^{2} + i_{R, k},

and the dark detector current (i_D,k) by blocking both paths, where i_S,k and i_R,k are the dark current terms for the images of the signal and reference images, respectively. Subtracting i_D,k from Eqs. (30), (53), and (54) and substituting into the reconstruction in Eq. (20) provides:

E_{BM, DC} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} (E_{S} E_{R} + E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

which can be separated into the signal (S) and noise (N_S) terms:

S = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

N_{S} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [A e^{jα} (E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

The reconstructed blank image with the sample moved out of the field of view can also be separated into the signal (B) and noise (N_B) terms:

B = \frac{1}{4} \frac{A_{pixel} q}{hυ} [E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} + E_{R} E_{S} X_{r} X_{s} e^{- j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} {(- 1)}^{k} η_{k} E_{C_{k}}^{R} E_{C_{k}}^{S}] +

N_{B} = \frac{1}{4} \frac{A_{pixel} q}{hυ} [(E_{S} E_{B} e^{j (ϕ - β)} + E_{B} E_{R} e^{- j (ϕ - β)} + E_{B}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} η_{k} E_{C_{k}}^{S} E_{C_{k}}^{R} +

where i_BR,k and i_BS,k are the dark current terms for the signal and reference images with the sample moved out of the field of view, respectively. Following the same procedure described in the previous subsection provides the reconstructed magnitude and phase induced by the sample:

E_{BM, DC} = A e^{jα} [1 - Φ e^{- j 2 (χ_{s} - χ_{r})} - I_{B} e^{- j (χ_{s} - χ_{r})}] +

where:

Φ = \frac{\sum_{k = 0}^{3} {(- 1)}^{k} \sqrt{R_{k}} \sqrt{B S_{k}}}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

I_{S} = \frac{\sum_{k = 0}^{3} j^{k} (i_{M, k} - i_{R, k} - i_{S, k} + i_{D, k})}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}}

I_{B} = \frac{\sum_{k = 0}^{3} j^{k} (i_{BM, k} - i_{BR, k} - i_{BS, k} + i_{D, k})}{\sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}}} .

A phase image can also be reconstructed without dividing by the blank image resulting in the reconstructed magnitude and phase:

E_{BM, DC, NB} = [A e^{jα} + A e^{- jα} Φ e^{- j 2 (χ_{s} - χ_{r})}] e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \sqrt{R_{k}} \sqrt{B S_{k}} + I_{BM}

where:

I_{BM} = \sum_{k = 0}^{3} j^{k} (i_{M, k} - i_{R, k} - i_{S, k} + i_{D, k}) .

The expressions in Eqs. (60) and (64) for the images reconstructed with balanced mixing and DC term subtraction removes the fixed pattern noise from the individual arms of the interferometer, and the primary source of noise now derives from the fixed pattern noise in the combination of the signal and reference paths (Φ). If the beamsplitters and cameras had been perfect and the same irradiance was incident on each camera, Φ would go to zero and the reconstruction would be detector noise limited. However, this is not possible with the commercial optics currently available, so we normalize the camera signals by dividing by the square root of the reference images in the following subsection to minimize the fixed pattern noise.

5.3. Mathematical description of balanced mixing, DC term subtraction, and normalization of camera signals

The square root of the difference of the pure reference path in Eq. (54) and the dark current image can be approximated:

\sqrt{R_{k} - i_{D, k}} \approx \frac{1}{2} \sqrt{\frac{η_{k} q}{hυ} A_{pixel}} E_{C_{k}}^{R} X_{r} ∣ E_{R} + E_{N} ∣,

assuming the difference of two dark current terms is much less than the signal terms of the reference image. Substituting Eq. (66) and the subtraction of i_D,k from Eqs. (30), (53), and (54) into the reconstruction in Eq. (21) provides:

E_{BM, DC, Norm} = \frac{1}{2 ∣ E_{R} + E_{N} ∣} \sqrt{\frac{A_{pixel} q}{hυ}} [A e^{jα} (E_{S} E_{R} + E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \frac{\sqrt{η_{k}} E_{C_{k}}^{S} E_{C_{k}}^{R}}{E_{C_{k}}^{R} X_{r}} +

which can be separated into the signal (S) and noise (N_S) terms:

S = \frac{1}{2} \sqrt{\frac{A_{pixel} q}{hυ}} \frac{1}{∣ E_{R} + E_{N} ∣} [A e^{jα} E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \frac{\sqrt{η_{k}} E_{C_{k}}^{S} E_{C_{k}}^{R}}{E_{C_{k}}^{R} X_{r}} +

N_{S} = \frac{1}{2} \sqrt{\frac{A_{pixel} q}{hυ}} \frac{1}{∣ E_{R} + E_{N} ∣} [A e^{jα} (E_{S} E_{N} e^{j (ϕ - ζ)} + E_{N} E_{R} e^{- j (ϕ - ζ)} + E_{N}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \frac{\sqrt{η_{k}} E_{C_{k}}^{S} E_{C_{k}}^{R}}{E_{C_{k}}^{R} X_{r}} +

The blank image with the sample moved out of the field of view can also be separated into the signal (B) and noise (N_B) terms:

B = \frac{1}{2} \sqrt{\frac{A_{pixel} q}{hυ}} \frac{1}{∣ E_{R} + E_{B} ∣} [E_{S} E_{R} X_{s} X_{r} e^{j (χ_{s} - {xχ}_{r})} \sum_{k = 0}^{3} \frac{\sqrt{η_{k}} E_{C_{k}}^{S} E_{C_{k}}^{R}}{E_{C_{k}}^{R} X_{r}} +

N_{B} = \frac{1}{2} \sqrt{\frac{A_{pixel} q}{hυ}} \frac{1}{∣ E_{R} + E_{B} ∣} [(E_{S} E_{B} e^{j (ϕ - β)} + E_{B} E_{R} e^{- j (ϕ - β)} + E_{B}^{2}) X_{s} X_{r} e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \frac{\sqrt{η_{k}} E_{C_{k}}^{S} E_{C_{k}}^{R}}{E_{C_{k}}^{R} X_{r}} +

Following the same procedure described in the previous subsection provides the reconstructed magnitude and phase induced by the sample:

E_{BM, DC, Norm} = A e^{jα} [1 - Ψ e^{- j 2 (χ_{s} - χ_{r})} - I_{B, Norm} e^{- j (χ_{s} - χ_{r})}] +

where:

Ψ = \frac{\sum_{k = 0}^{3} {(- 1)}^{k} \sqrt{B S_{k}}}{\sum_{k = 0}^{3} \sqrt{B S_{k}}}

I_{S, Norm} = \frac{\sum_{k = 0}^{3} j^{k} \frac{i_{M, k} - i_{R, k} - i_{S, k} + i_{D, k}}{\sqrt{R_{k}}}}{\sum_{k = 0}^{3} \sqrt{B S_{k}}}

I_{B, Norm} = \frac{\sum_{k = 0}^{3} j^{k} \frac{i_{BM, k} - i_{BR, k} - i_{BS, k} + i_{D, k}}{\sqrt{R_{k}}}}{\sum_{k = 0}^{3} \sqrt{B S_{k}}} .

A phase image can also be reconstructed without dividing by the blank image resulting in the reconstructed magnitude and phase:

E_{BM, DC, Norm, NB} = [A e^{jα} + A e^{- jα} Ψ e^{- j 2 (χ_{s} - χ_{r})}] e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \sqrt{B S_{k}} + I_{BM, Norm}

where:

I_{BM, Norm} = \sum_{k = 0}^{3} j^{k} \frac{i_{M, k} - i_{R, k} - i_{S, k} + i_{D, k}}{\sqrt{R_{k}}} .

The expressions in Eqs. (72) and (76) for the images reconstructed with balanced mixing, DC term subtraction, and camera normalization removes the noise from the individual arms of the interferometer and reduces the noise from the fixed pattern noise in the reference path. The sources of noise in these images are dependent on the fixed pattern noise in the signal path (Ψ) and the dark current noise in the detectors. The following section describes the images that were created for the noise terms in the SNR analysis to create a model for the resultant images that are acquired with the microscope.

6. Modeling of the phase reconstructions

Images from a data collection of a PolyMethylMethAcrylate (PMMA) bead immersed in oil were used to model the variables within the resultant expressions for each phase reconstruction described in Section 5. The difference of the phase aberration terms (χ_s - χ_r) corresponds to the wavefront mismatch between the reference and signal paths when the sample is not within the field of view. Using the same approximations described in Section 5.3, the reconstruction in Eq. (21) for the image of the blank provides the expression:

E_{BM, DC, Norm, blank} = [1 + Ψ e^{- j 2 (χ_{s} - χ_{r})}] e^{j (χ_{s} - χ_{r})} \sum_{k = 0}^{3} \sqrt{B S_{k}} + I_{BM, Norm} .

Assuming the unwrapped phase of an experimental reconstruction using Eq. (21) with the bead moved out of the field of view is approximately equal to the wavefront mismatch between the reference and signal paths, the difference of the phase terms can be modeled by fitting the Zernike polynomials for bias, tilt, and focus [38] in Fig. 3.

Fig. 3. (a) Experimental image of the wavefront mismatch between reference and signal paths. (b) Model of the wavefront mismatch created from a Zernike polynomial fit that included bias, tilt, and focus.

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The images acquired with the reference path blocked and the sample moved out of the field of view provided the images for B_Sk in Eq. (49), and the images acquired with the signal path blocked provided the images for R_k in Eq. (50). Seven sets of images with the same field of view and containing random numbers between 0 and 2 were created for each camera to model the dark current noise terms i_M,k, i_BM,k, i_S,k, i_BS,k, i_R,k, i_BR,k, and i_D,k. The images for B_Sk, R_k, and the dark current noise were then substituted into Eqs. (44)–(48), (52), (61)–(63), (65), (73)–(75), and (77) to create the images shown in Fig. 4 that model Ψ, Φ, Γ_S, Γ_R, I_S,M, I_B,M, I_M, I_S, I_B, I_BM, I_S,Norm, I_B,Norm, and I _BM,Norm.

Fig. 4. Images for the noise terms that are associated with the phase reconstructions. The real (Re) and imaginary (Im) parts of the complex noise terms are shown in separate images. All of the dark current noise terms have similar distributions to those shown in (g) and (h).

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An image of an ideal PMMA bead immersed in oil was created to validate the model. A brightfield image of the PMMA bead was used to determine an accurate measure of the bead diameter. The maximum ideal phase of the bead was calculated to be -23.0 radians using Eq. (5), where the maximum thickness of the bead (h), the refractive index of the bead (n_s), and the refractive index of the immersion oil (n ₀) were set to 99.12 μm, 1.489, and 1.5124, respectively. An image of the ideal phase for the bead (α) was created with the bead centered on the same pixel in the field of view as the experimental image. Fig. 5(a), 5(d), and 5(g) show the resultant wrapped phase from the reconstructions in Eqs. (43), (60), and (72), and Fig. 5(b), 5(e), and 5(h) show the wrapped experimental results from the reconstructions in Eqs. (19), (20), and (21) that were also divided by a blank image created from the same reconstructions with the sample moved out of the field of view, respectively. Fig. 5(c), 5(f), and 5(i) show plots of the unwrapped phase values through the center of both the model and experimental phase reconstructions to show the accuracy of the phase measurement of a 99 μm PMMA bead immersed in oil, and the correlation of the noise inherent in each phase reconstruction.

Fig. 5. Comparison of the SNR model to experimental images of wrapped phase from balanced mixing in Eq. (19), balanced mixing and DC term subtraction in Eq. (20), and balanced mixing, DC term subtraction, and camera normalization in Eq. (21) when the sample images were divided by a blank image created from the same reconstruction. The plot for each phase reconstruction shows the unwrapped phase values through the center of both the model and experimental beads.

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To visualize the effect of dividing the sample image by the blank image, we also created images for the reconstructions in Eqs. (51), (64), and (76) that do not divide the image of the sample by the blank image. Fig. 6(a), 6(d), and 6(g) show the resultant wrapped phase from reconstructions in Eqs. (51), (64), and (76), and Fig. 6(b), 6(e), and 6(h) show the wrapped experimental results from the reconstructions in Eqs. (19), (20), and (21), respectively. Fig. 6(c), 6(f), and 6(i) show plots of the unwrapped phase values through the center of both the model and experimental phase reconstructions to show the correlation of the noise inherent in each phase reconstruction. The wavefront mismatch is clearly visible in these images and must be accounted for to determine the phase induced by the sample alone. Thus, a constant wavefront mismatch must be incorporated into the system in order to remove the need to divide by the blank image.

Fig. 6. Comparison of the SNR model to experimental images of wrapped phase from balanced mixing in Eq. (19), balanced mixing and DC term subtraction in Eq. (20), and balanced mixing, DC term subtraction, and camera normalization in Eq. (21) when the sample images were not divided by a blank image. The wavefront mismatch between the reference and signal paths is clearly visible in the resultant images. The plot for each phase reconstruction shows the unwrapped phase values through the center of both the model and experimental beads.

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6.1. Inherent noise in the phase reconstructions

The noise in the phase reconstructions was approximated by substituting a model of constant phase across the field of view for the phase induced by the sample, and the models of the noise terms discussed in the previous subsection into Eqs. (43), (60), and (72) for the sample images divided by the blank images, and into Eqs. (51), (64), and (76) for the reconstructions that include the wavefront mismatch between the reference and signal paths. It is important to note that a constant wavefront mismatch was assumed for Eqs. (51), (64), and (76) to provide a constant ideal phase across the field of view. Incorporating the wavefront mismatch shown in Fig. 4 would produce a range of phase values for the ideal phase image, and would not provide an accurate representation of the reconstruction error for the reconstructions that do not divide by a blank. The wavefront mismatch was included for the reconstructions that do divide by a blank.

The plots in Fig. 7 show the mean phase error of each reconstruction, calculated by subtracting the expected phase from the result of each reconstruction, versus the constant phase values between -4π and 4π. The error bars show the standard deviation of the phase error across the field of view. The maximum RMS error across the 8π measurements was 0.31 radians, 0.14 radians, and 0.08 radians for the reconstructions in Eqs. (43), (60), and (72) that divide the sample image by the blank image, respectively, and 0.20 radians, 0.10 radians, and 0.05 radians for the reconstructions in Eqs. (51), (64), and (76) that assume a constant wavefront mismatch across the field of view and do not divide by a blank image, respectively. The periodicity of the error can be explained by the strong contribution of multiplicative noise in the reconstructions from the fixed pattern noise. DC term subtraction and camera normalization reduces the amplitude of the periodicity, but the fixed pattern noise must be reduced in the experimental system for the noise to approach a constant value over the range -4π to 4π. It is important to note that the amplitude of the periodic error is 0.06 radians for the current OQM reconstruction, which is less than 0.3% of the maximum phase that we are measuring from 100 μm diameter mouse embryos [33].

Fig. 7. Phase reconstruction error for balanced mixing from (a) Eq. (43) and (b) Eq. (51), balanced mixing and DC term subtraction from (c) Eq. (60) and (d) Eq. (64), and balanced mixing, DC term subtraction, and camera normalization from (e) Eq. (72) and (f) Eq. (76). The plots in the left column show the result of dividing the sample images by a blank image using the same reconstruction, while the plots in the right column do not divide the sample image by a blank. The plots in the right column also assume a constant wavefront mismatch across the field of view to calculate the error with a constant ideal phase.

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7. Conclusion

The balanced mixing reconstructions in Eqs. (43) and (51) are the same mathematically as a phase-shifting system that acquires four sequential images with a 90-degree phase shift in between each image [39]. However, the imperfections within the optics and CCD cameras contribute to a maximum RMS error of 0.31 radians in the phase reconstruction when the sample image is divided by a blank image to remove the wavefront mismatch between the signal and reference paths. The maximum RMS error can be reduced from 0.31 radians to 0.14 radians by subtracting separate images of the reference and signal paths, and to 0.08 radians by also dividing by the square root of the reference image to normalize the camera signals. However, the fixed pattern noise within the system produces a periodic error over the range of phase values that is negligible when imaging large phase objects such as 100 μm PMMA beads or live mouse embryos that induce a change in phase on the order of 8π radians, but must be reduced to image samples that induce a change in phase on the order of 1 cycle.

A telecentric imaging system would provide a constant wavefront mismatch across the field of view and remove the need to divide by the blank. Such a system would allow the use of the balanced mixing phase reconstruction without dividing by a blank image and provide a maximum RMS error of 0.20 radians. The maximum RMS error can be further reduced to 0.10 radians by subtracting separate images of the reference and signal paths, and to 0.05 radians by also normalizing the camera signals, assuming the noise terms are comparable to the current experimental setup. Images must be collected with a telecentric system to recreate the models of the noise terms and calculate the noise in such a setup accurately.

While DC term subtraction and camera normalization provides the most accurate measurement of quantitative phase, this technique complicates the acquisition of real-time images because multiple images are required to reconstruct a single phase image. Phase unwrapping further complicates real-time imaging of samples that produce a change in phase greater than 2π. We have used the L^P-norm algorithm to unwrap our phase images [40] because it seems to provide the most repeatable and accurate results for circular objects, such as PMMA beads and mouse embryos [33]. However, this algorithm requires time on the order of minutes to unwrap a single 640 × 480 image of quantitative phase on a computer using a 1.8 GHz processor and 1.5 GB of RAM. Additional work is required to create the hardware necessary to unwrap the images in near-real-time [41], and to develop approximations to the noise terms to remove the need for multiple image acquisition.

The SNR model presented within this paper incorporates noise terms from the experimental setup, but it does not include sample dependent effects, such as refraction and diffraction. We have ignored these effects and assumed a projection model in our previous work because the change in refractive index between the sample and the immersion medium is approximately 0.02 [33]. However, these effects will increase as the refractive index mismatch increases and as the location of the sample changes with respect to the focal plane of the microscope.

It is important to note that the phase reconstruction errors reported in this paper describe the noise inherent in the current system and are not the limiting noise of the techniques. Further optimization of the components and of the optical layout will reduce the error in the resultant phase images. Until this optimization has been completed and the SNR model presented within this paper has been used to recalculate the error, the RMS error of the OQM phase images can be recorded as 0.08 radians.

Acknowledgments

This work was supported in part by the Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-CenSSIS), under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821).

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Computational signal-to-noise ratio analysis for optical quadrature microscopy

Abstract

1. Introduction

2. Quadrature detection

3. Optical layout for OQM

4. Ideal electric field analysis

5. SNR analyses for phase reconstructions

5.1. Mathematical description of balanced mixing

5.2. Mathematical description of balanced mixing and DC term subtraction

5.3. Mathematical description of balanced mixing, DC term subtraction, and normalization of camera signals

6. Modeling of the phase reconstructions

6.1. Inherent noise in the phase reconstructions

7. Conclusion

Acknowledgments

References and links

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Figures (7)

Equations (108)

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