Calibration of a phase-shifting DIC microscope for ... - CiteSeerX

Calibration of a phase-shifting DIC microscope for quantitative phase imaging Sharon V. King1, Ariel R. Libertun1, Chrysanthe Preza2, Carol J. Cogswell1, 3 1

Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO USA Department of Electrical and Systems Engineering, University of Memphis, Memphis, TN USA 3 CDM Optics, Inc. Boulder, CO USA

2

ABSTRACT Phase-shifting differential interference contrast (DIC) provides images in which the intensity of DIC is transformed into values linearly proportional to differential phase delay. Linear regression analysis of the Fourier space, spiral phase, integration technique shows these values can be integrated and calibrated to provide accurate phase measurements of objects embedded in optically transparent media regardless of symmetry or absorption properties. This approach has the potential to overcome the limitations of profilometery, which cannot access embedded objects, and extend the capabilities of the traditional DIC microscope, which images embedded phase objects, but does not provide quantitative information. Keywords: DIC (Nomarski) microscopy, quantitative phase imaging, phase shifting, spiral phase

1. INTRODUCTION Differential interference contrast (DIC) microscopy is well known for its ability to image transparent phase objects that otherwise produce very little contrast in bright-field microscopy. Its particular advantages over other phase imaging techniques include: applicability at high numerical apertures (NA’s), lack of aberrations and its ability to image phase objects embedded within a transparent material. Additionally, the differential delay of its interferometer makes DIC very sensitive to small phase gradients over a large (cm) transverse scale. Although DIC’s advantages are useful in a large range of applications, it is hampered by a number of limitations, which prevent it from becoming a useful quantitative imaging tool. One such limitation is that there is no linear correlation between image intensity and the true slope of the specimen phase gradient. This is a fundamental limitation of the DIC system due to the fact that the complex transmission function of the specimen is encoded in an intensity measurement in the DIC image. This non-linearity is demonstrated by the asymmetric transfer function of the DIC system (see fig. 1), which is described in detail in an earlier paper by Cogswell1. 0.6

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Fig. 1. The phase gradient transfer function for a DIC system with equal condenser and objective apertures and a bias phase of 90 degrees, showing signal strength as a function of normalized phase gradient m/mo. Reprinted with permission from Cogswell and Sheppard1.

Several research groups have addressed various aspects of DIC’s inherent difficulties in an attempt to overcome them. A summary of previous related work was presented in the 2006 proceedings of this conference2. Recent additions include; (1) an iterative phase estimation method developed for reflection DIC which incorporates the use of an atomic force microscope, (2) a method applying non-iterative de-convolution, with an approximate MTF, to phase modulated DIC images in the weak phase regime developed for generally shaped phase objects in reflection and (3) results from a quantitative method, employing phase-shifting techniques similar to those used in the method discussed here, showing

Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIV, edited by Jose-Angel Conchello, Carol J. Cogswell, Tony Wilson, Proc. of SPIE Vol. 6443, 64430M, (2007) 1605-7422/07/$18 · doi: 10.1117/12.725061 Proc. of SPIE Vol. 6443 64430M-1

that the Abel transform can be used to numerically integrate linear phase gradients of rotationally symmetric objects with high accuracy3-6. A truly quantitative phase imaging microscope, based on DIC, would allow quantification of objects; such as transparent and partially absorptive objects embedded in transparent media, and phase changes within a transparent media, currently inaccessible by reflective interferometric or stylus based profilometers. The discussion in this paper focuses on the calibration of a method of deriving quantitative phase images from traditional DIC intensity through non-iterative integration of phase-shifted DIC (PS-DIC) images in Fourier-space. This method, previously presented in these proceedings, has been shown to produce quality linear phase images provided that the phase gradient of the object does not vary more than either modulus π or the value of the numerical aperture over the shear distance, which ever is greater7, 8. Based on a Fourier filtering technique; this method has been dubbed spiral phase integration due to the spiral nature of the filtering function’s phase angle. Derived from a geometric optical model of the DIC microscope, the symmetry of this function allows integration of phase gradients regardless of symmetry without the directional artifacts seen in numerical integration9, 10. This technique is applicable at the same high NA as is traditional DIC. However, it has also been shown that the spiral phase function can be implemented optically and used to imitate a low resolution DIC system whose images become quantitative with similar digital post processing11. The application of Fourier-space, spiral phase integration to PS-DIC images produces new images whose pixel values are linearly proportional to the true phase of the imaged objects. Their potential to provide accurate and repeatable phase measurements can be realized through careful calibration of the proportionality relationship. Comparison of the accuracy and sensitivity of SPI phase imaging to other quantitative phase imaging methods will depend on the accuracy of this calibration. In this paper we show results of an initial calibration of results obtained with the SPI method. Section 2 briefly describes the acquisition of data for calibration. Three types of data are defined according to the origin of the values input into the SPI algorithm. Details of the application of phase-shifting to DIC images and the application of SPI to PS-DIC can be found in previous proceedings2, 7-10. Sections 3 and 4 present the method used for calibration of SPI images and results.

2. IMAGE ACQUISITION The calibration method described here is applied to three sets of SPI results; a numerical gradient set, a simulated DIC set and an empirical DIC set. These three sets are the result of different types of input into the SPI algorithm, defined in detail below. In both the numerical gradient case and the simulated DIC case, the object was a 2D computer generated test object or phantom. This phantom’s phase properties were designed to match those of a 2µm polystyrene sphere embedded in optical cement. Preparations of 2µm polystyrene beads embedded in multiple optical cements of varying index were used to capture empirical DIC SPI results. Numerical gradient SPI results were produced by numerically differentiating the phase of the phantom using the standard Matlab routine, gradient.m, which performs a differencing in real space. An example phantom is shown in figure 2a. Two orthogonal directions of the numerical differential were input into the SPI algorithm in place of the PSDIC image data (fig. 2b & 2c). These data are the ideal input into the SPI algorithm, which is in essence performing a numerical integration; therefore, results from these data represent the best possible SPI result (fig. 2d).

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c) d) Fig. 2. a) A 64x64 pixel computer generated blob phantom with phase values 0-0.4 radians. b) The numerical gradient in the horizontal direction of the bead phantom phase. c) The numerical gradient in the vertical direction of the bead phantom phase. d) The SPI result of the complex vector combination of b) and c) in which pixel values are proportional to phase in radians.

Simulated DIC SPI results were produced from simulated DIC images of the phantom. A second example phantom is shown in figure 3a. This simulation was implemented using Matlab in the Fourier domain with a 2D DIC transfer function for coherent illumination1, 12. The optical properties of the simulated imaging system were chosen to match those of the experimental microscope as closely as possible, with an NA = 0.5, objective power = 20x, camera pixel size = 7.4µm. The wavelength was approximated to be 0.55µm. Four simulated images with phase bias =0, π/2, π and 3π/2 were created and used to calculate a simulated PS-DIC image, shown in figure 3b, which, based on the geometric optical model, is given by ∆θ = tan−1

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This process creates an image from which the amplitude information has been removed. As a result, the pixel values of the PS-DIC image are linearly proportional to the phase gradient of the object10. A second PS-DIC image, shown in figure 3c, was created exactly the same way, but with the object rotated ninety degrees to capture the orthogonal phase gradient. These two images combined in the form of a complex vector ⎛ ∆θ g ( x, y ) = ⎜⎜ x ⎝ ∆x

⎞ ⎛ ∆θ y ⎟⎟ + i⎜ ⎜ ⎠ ⎝ ∆y

⎞ ⎟ = ∆θ ( x , y ) ⎟ ⎠

were input to the SPI algorithm to produce simulated DIC SPI results (fig.3d).

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Fig. 3. a) A 64x64 pixel computer generated 2µm bead phantom, n=1.59, embedded in a medium, n=1.504. b) The simulated PS-DIC image of the bead phantom with shear in the horizontal direction. c) The simulated PS-DIC image of the bead phantom with shear in the vertical direction. d) The SPI result of the complex vector combination of b) and c) in which pixels are proportional to phase in radians.

Empirical DIC SPI results were produced from experimental DIC images. Images were captured with a Photometrics Cascade camera with 7.4µm square pixels mounted on a Zeiss Axioplan DIC upright microscope using a 20x, 0.5 NA objective, and a halogen lamp source filtered with a 515-565nm bandpass optical filter. Four images with phase bias =0, π/2, π and 3π/2 were recorded and used to calculate the empirical PS-DIC images. The phase bias was controlled using a Sernamont compensator rather than by laterally shifting the Nomarski prism. Figure 4 shows this configuration of the DIC microscope. Rotating analyzer Quarter wave Beam combining Nomarski Prism Objective back focal plane Object Shear Condenser front focal plane Beam splitting Nomarski Prism Polariser (45 Degrees) Fig. 4. The setup of the PS-DIC microscope is essentially a traditional DIC microscope with a rotating upper analyzer and an additional quarter wave plate as shown in this schematic. The Senarmont compensator configuration allows precise control of the amount of added phase shift.

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A rotating stage allowed the preparations to be rotated ninety degrees. Details on the registration of images for SPI and experimental considerations in the application of SPI are found in King et al8. Two empirical PS-DIC images containing orthogonal linear phase gradient values, with amplitude information removed, were combined according to equation (2) and were input to the SPI algorithm to produce empirical DIC SPI results (fig.5a-d). iu4 3.2 2.8 3.1 2.4

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c) d) Fig. 5. a) Amplitude removed, using phase shifting, from experimental DIC images of 2µm polystyrene bead, n=1.59, embedded in a medium, n=1.504. b) The empirical PS-DIC image of the polystyrene bead with shear in the horizontal direction. c) The empirical PS-DIC image of the polystyrene bead with shear in the vertical direction. d) The SPI result of the complex vector combination of b) and c) in which pixels are proportional to phase in radians.

The goal of calibration is to determine the proportionality factor between the SPI result and the true phase of the object. This requires a calibration object whose phase properties are well understood and well suited for the subsequent planned measurements. For comparison of numerical and simulated data, we chose a 64x64 pixel computer generated object or phantom of an asymmetric blob with phase values from 0 to 0.4 radians (fig. 2). For comparison of simulated and empirical data we chose 2µm polystyrene beads (n=1.59) embedded in two different optical cements (n = 1.504 and 1.52) (fig. 3). We chose an experimental object small enough to be considered within the depth of field of the objective and whose maximum absolute phase was less than π, for the sake of simplicity. Since the two-micron bead size falls within the depth of field of our optical system, the simulated DIC phantom used in this comparison is a 64x64 computer generated 2D projection of the 3-D phase of the polystyrene bead. In the case of SPI, calibrating the SPI result against simply the true phase is not enough. Due to the assumptions of the SPI algorithm derivation, any phase modulation due to diffraction at the aperture of the objective is also integrated2. Therefore, it is important to calibrate the SPI result against a computed phase referred to from here on as the true phase. In the numerical gradient case, the true phase is equal to the computer-generated object or phantom. In the case of simulated data and empirical data, the true phase was calculated by convolving the phantom with the bright-field PSF for an equivalent, non-DIC, optical system. In the empirical case, since the two-micron bead size falls within the depth of field of our optical system, we again approximate the 3-D phase of the polystyrene bead with the phantom, which is a 2D projection of the bead phase. The true phase is then compared to the SPI result to establish the proportionality factor.

3. CALIBRATION METHOD

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Arnison et al. have shown that, with linear phase gradient values as its input, the SPI result is linearly proportional to the true phase7. The relationship between the SPI result and the true phase is therefore assumed to be of the form:

Y = aΘ + C

(3)

where Y is the SPI result, a the proportionality factor, Θ, true object phase and C is a constant of integration, which follows from the integration of the linear phase gradient values, Y’

Y ' = a∆θ + b ∫ Y ' = ∫ [a∆θ + b] + C '

Y = aΘ + b'+C ' = aΘ + C

(4).

The problem of calibration then becomes the problem of finding a and C. Linear regression statistics allow both the determination of these calibration parameters as well as an analysis of the validity of the linear calibration model. For the application of linear regression to the calibration of the SPI method, the SPI result is considered the response or dependent variable, while the true phase is considered the explanatory or independent variable. These data are plotted against each other pixel by pixel and an unconstrained nonlinear optimization of the least squared error between the fit and the SPI result, also called residual, is used to fit a line to the plot (fig. 6). Several statistical metrics; residual error, correlation coefficient R2, and prediction and confidence intervals, are used to analyze the strength of the fit of the calibration model to the data. A plot of the residual versus the true phase can be used to easily identify any slight curvature in the data with respect to the fit as well as to observe any patterns in the variance of the data (fig. 6). The correlation coefficient is a measure of the difference between the sum of the residual error and the standard deviation of the dependant variable,

R2 = n

(SSY − SSE ) SSE

where

2

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^ ^ ⎞ ⎛ ⎜⎜ Yi − Yi ⎟⎟ Yi = SPI result, Y i = linear fit, ∑ i =1 ⎝ ⎠ n

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which varies between 0 and 1, with 1 indicating the strongest possible fit13. The confidence interval indicates the uncertainty in the regression line based on the variance of the independent parameter. − 2 ⎛ ⎛ ⎞ ⎞⎟ ⎜ − X X ⎜ i ⎟ ⎛ n −1 ⎞ ⎜1 ⎝ ⎠ ⎟ where t is the 95% point of 2 Confidence interval = t × ⎜ ⎟ var(Y ) − a × var( X ) ⎜ + n (n − 1) × var( X ) ⎟ ⎝n−2⎠ ⎜⎜ ⎟⎟ ⎝ ⎠

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the t distribution with n-2 degrees of freedom 13. The prediction interval indicates the uncertainty in the prediction of a true phase value based on the variance in the SPI result.

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− 2 ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ Yi − Y ⎟ 1 ⎛ n −1 ⎞ ⎜ ⎟ ⎝ ⎠ 2 Prediction interval = t × ⎜ ⎟ var(Y ) − a × var( X ) ⎜1 + + n (n − 1) × var( X ) ⎟ ⎝n−2⎠ ⎜⎜ ⎟⎟ ⎝ ⎠

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b) Fig. 6. a) A linear regression analysis on numerical gradient SPI results compared with b), a linear regression analysis on simulated DIC SPI results, no noise added, of a 64x64 pixel computer generated asymmetric blob phantom with 00.4 radians of phase. The upper plot of the regression analysis shows the pixel values of the SPI result on the y axis plotted against the pixel values of the true phase on the x axis along with the least squares fit of the calibration model to the data and the 95% confidence and prediction intervals for that fit. The inset box shows the correlation coefficient, R2. The bottom plot of the regression analysis shows the residual error between the fit and the SPI result on the y axis plotted against the pixel values of the true phase on the x axis.

4. CALIBRATION RESULTS Figure 6 shows a linear regression analysis on numerical gradient SPI results compared with a linear regression analysis on simulated DIC SPI results of the asymmetric blob phantom described in section 3. The upper plot of the regression

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analysis shows the pixel values of the SPI result on the y axis plotted against the pixel values of the true phase on the x axis along with the least squares fit of the calibration model to the data and the 95% confidence and prediction intervals for that fit. The bottom plot of the regression analysis shows the residual error between the fit and the SPI result on the y axis plotted against the pixel values of the true phase on the x axis. Visual inspection of both plots, along with high R2 values for both sets of data, indicates the linear calibration model may be a good fit. However, both plots show a slight curvature in the data, best demonstrated by the residuals, which is not reflected in the calibration model. The appearance of this slight curvature in the analysis of the numerical gradient SPI results indicates that whatever deviation from the linear model is present in the SPI result is due to the SPI algorithm itself rather than the effects of simulated DIC imaging. On the other hand, no noise was added to the simulated DIC images of the blob from which the SPI result was calculated. Therefore, any increase in the non-linearity of the SPI result/true phase relationship seen in the regression analysis on the simulated DIC SPI result of the blob phantom can only be attributed to the effects of simulated DIC imaging and not to the effects of additional noise. Application of the calibration parameters from the above linear regression on simulated DIC SPI results of the blob phantom demonstrates the strengths and weakness of the calibration. Figure 7a shows a horizontal profile through the simulated DIC SPI result of a 64x64 pixel computer generated 2µm bead phantom, n=1.59, embedded in a medium, n=1.57, shown in fig. 7b, calibrated with those parameters. Again, no noise has been added to the simulated DIC images of the 2µm bead phantom. The calibrated SPI result is plotted with error bars given by the prediction interval of the regression line, which are equivalent to a standard deviation of approximately 0.04 radians or λ/157. The true phase is also plotted for comparison. In general, the values of true phase at each location fall with in the prediction interval of the calibration. Not surprisingly, the weakest agreement between the true phase and the calibrated SPI result is found at the highest values of phase for which the fit of the calibration model, from which the calibration parameters were taken, is also weakest. The strongest agreement between the true phase and the calibrated SPI result is found at the weakest values of phase where the fit of the calibration model is very good. Blmul.t.d SPI r..ult 2um bd 1 .e o.llbr.t.d wfth Blmul.t.d blob; m.x ph... 0.4 r.dl.n. —+—True Phase —e— Calibrated SP Iresu It

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It is also worth noting that the calibration object, the blob phantom, contains significantly more values representing the low end of its dynamic range than the high end. As a result the fit of the calibration model, using this calibration object, is inevitably stronger for weaker phase values. Figure 8 shows a linear regression analysis on simulated DIC SPI results of the 2µm bead phantom, n=1.59, described in section 3, embedded in a medium n=1.504, compared with a linear regression analysis on empirical DIC SPI results of a 2µm polystyrene bead, n=1.59, embedded in optical cement, n=1.504. In this case 20dB Poisson noise was added to the simulated DIC images used for calculation of the simulated DIC SPI result for more realistic comparison with empirical results. Plots of the analysis are shown in exactly the same style as in fig. 6, where the upper plot of the regression analysis shows the pixel values of the SPI result plotted against the pixel values of the true phase along with the least

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squares fit of the calibration model to the data and the 95% confidence and prediction intervals for that fit, while the lower plot shows the residual error between the fit and the SPI result plotted against the true phase. While, the slight deviation of the data from a perfect linear calibration model can still be seen in both residual plots, again, visual inspection of both plots, along with high R2 values for both sets of data, indicates that the linear calibration model is a very good fit. + data

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True P haue (radianu) b) Fig. 8. a) A linear regression analysis on simulated DIC SPI results, 20dB Poisson noise added, of a 64x64 pixel computer generated 2µm bead phantom, n=1.59, embedded in a medium, n=1.504, compared with b) a linear regression analysis on empirical DIC SPI results, of a 2µm polystyrene bead, n=1.59, embedded in a medium, n=1.504. The upper plot of the regression analysis shows the pixel values of the SPI result on the y axis plotted against the pixel values of the true phase on the x axis along with the least squares fit of the calibration model to the data and the 95% confidence and prediction intervals for that fit. The inset box shows the correlation coefficient, R2. The bottom plot of the regression analysis shows the residual error between the fit and the SPI result on the y axis plotted against the pixel values of the true phase on the x axis.

However, in this case, because the calibration object is small laterally, the number of pixels representing significant phase strength compared to the dynamic range is small. To avoid including too many pixels representing weak phase values and thus influencing the calibration parameters significantly toward these values as discussed with regard to fig. 7 above, the linear regression shown involves only the center 8x8 pixels of the 64x64 SPI result. The majority of this subsection of pixel values contains pixels with values greater than 20% of the dynamic range of the SPI result. Consequently the regression analysis of these results is much coarser than that shown in fig. 6 with 64 times fewer pixels

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sampling a range of phase three and a half times larger. The smaller number of values included in the regression cause the widening of the confidence and prediction intervals and make small non-linearities harder to detect. Based on the regression statistics, the fit of the linear calibration model to the empirical DIC SPI result does not appear degraded compared to the models fit to the simulated DIC SPI result. The R2 value is three percent lower, but the residuals are slightly less. The prediction interval for the simulated DIC SPI result is 0.2 radians, while the prediction interval for the empirical DIC SPI result is 0.25 radians. The main disparity between the regression analyses on the two sets of data is the difference in slope and offset parameters given by the fit of the calibration model to the data. Application of the calibration parameters from the above linear regression on empirical DIC SPI results of the 2µm polystyrene bead demonstrates the applicability of the empirical calibration to an empirical measurement. Figure 9a shows a horizontal profile through the 8x8 empirical DIC SPI result of a 2µm polystyrene bead, n=1.59, embedded in a medium, n=1.52, shown in fig. 9b, calibrated with parameters given by the empirical linear regression shown in fig. 8b. For comparison, fig. 9c shows a horizontal profile through the 8x8 simulated DIC SPI result of a 2µm bead phantom, n=1.59, embedded in a medium, n=1.52, shown in fig. 9d, calibrated with parameters given by the simulated linear regression shown in fig. 8a. Again, 20dB Poisson noise was added to the simulated DIC images of the 2µm bead phantom for a more realistic comparison. The calibrated SPI result is plotted with error bars given by the prediction interval of the regression line along with the true phase. The values of true phase at each location fall with in the prediction interval of the calibration. For this calibration, having artificially removed values representing very weak phase from the linear regression as described above, the weakest agreement between the true phase and the calibrated SPI result is found at the weakest values of phase.

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regression line, which are equivalent to a standard deviation of approximately 0.25 radians. c) A horizontal profile through the 8x8 simulated DIC SPI result of a 64x64 pixel computer generated 2µm bead phantom, n=1.59, embedded in a medium, n=1.52, shown in b), calibrated with parameters a = 1.1115 and C = 0.1012 obtained from the linear regression on the 8x8 simulated DIC SPI result of a 64x64 pixel computer generated 2µm bead phantom, n=1.59, embedded in a medium, n=1.504. The calibrated SPI result is plotted with error bars given by the prediction interval of the regression line, which are equivalent to a standard deviation of approximately 0.2 radians. 20dB Poisson noise was added to the simulated DIC SPI result.

5. CONCLUSION Linear regression analysis on SPI results from three data sets with varying types of input into the SPI algorithm show the relationship between the SPI result and true object phase is approximately linear. While slight deviations of the data from the linear calibration model are seen, they are seen in all three data sets, and are shown to originate from the SPI algorithm itself. The implication of this result is that a more sophisticated implementation of the SPI filter might increase the strength of agreement between the SPI result and a linear calibration model. Additionally, no increase in nonlinearity of the SPI result/true phase relationship is observed in phase gradient values affected by an imaging point spread function. This observation extends to empirical DIC SPI results, which agree well with the simulated DIC SPI results in all respects other than the exact values of the calibration parameters. One reason for this discrepancy, not yet thoroughly investigated, may be the difference in illumination exposure between the simulated and empirical DIC images used to calculate the corresponding SPI results. An initial empirical calibration, supported by comparison with a simulated calibration, demonstrates the potential of PSDIC with SPI to quantitatively image phase as small as 0.25 radians. This was demonstrated with a partially absorptive object embedded in a medium and therefore not accessible to a traditional profilometer. Simulation shows that this technique is equally applicable to non-symmetric objects. It is clear that the accuracy of this quantitative phase imaging method will always be limited by the quality of its calibration, which is shown to depend strongly on the suitability of the calibration object. Linear regression is a well developed statistical method applicable to an enormous range of applications including the analysis of instrumentation response.13, 14 It is used extensively in this paper for calibration and analysis of the response of the SPI algorithm to linear phase gradient input. Such analysis is not only useful for determination of calibration parameters and validation of the calibration model but is also a valuable aid in the understanding of calibrated results.

6. ACKNOWLEDGMENTS This research is supported by the National Science Foundation under the collaborative research grants DBI-0455365 awarded to C. Preza and DBI-0455408 awarded to C. Cogswell. Additional thanks to Rafael Piestun, Assistant Professor, University of Colorado, Boulder, CO for useful discussions and guidance.

REFERENCES 1.

Carol J. Cogswell and C.J.R. Sheppard, “Confocal differential interference contrast (DIC) microscopy”, Journal of Microscopy, 165, 81-101(1992).

2.

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