Theoretical Development and Experimental Evaluation of Imaging ...

Published in the J. Opt. Soc. Am. A/Vol. 16, No. 9, 1999.

Theoretical Development and Experimental Evaluation of Imaging Models for Dierential Interference Contrast Microscopy Chrysanthe Preza, Donald L. Snyder, Institute for Biomedical Computing, Washington University 700 South Euclid Avenue, St. Louis, Missouri 63110 and Department of Electrical Engineering Washington University, 1 Brookings Drive, St. Louis, Missouri 63130

Jose-Angel Conchello Institute for Biomedical Computing, Washington University 700 South Euclid Avenue, St. Louis, Missouri 63110

Abstract In this paper imaging models for Dierential-Interference-Contrast (DIC) microscopy are presented. Two- and three-dimensional models for DIC imaging under partially-coherent illumination were derived and tested using phantom specimens viewed with several conventional DIC microscopes and quasi-monochromatic light. DIC images recorded using a CCD camera were compared to model predictions which were generated using theoretical point-spread functions, computer-generated phantoms, and estimated imaging parameters such as bias and shear. Results show quantitative and qualitative agreement between model and data for several imaging conditions. Keywords (OCIS code): Nomarski DIC microscopy (999.9999), image formation theory (110.2990), 3-D microscopy (180.6900), 3-D image acquisition (110.6880).

1 Introduction Transmitted-light Nomarski Dierential-Interference-Contrast (DIC) microscopy [1, 2] is a widely used microscopy modality that was developed over forty years ago for the study of unstained transparent biological specimens [3]. Such specimens, known as phase objects, cannot be seen when in-focus under an ordinary transmitted-light microscope. Phase objects retard or advance light that passes through them due to spatial variations in their refractive index and/or thickness, and thus, they can be examined with special microscopes that allow the visualization of phase variations. Alternatively, phase variations can be converted to intensity variations by the use of interference methods. A widely used interference method is DIC microscopy in which a two-dimensional (2-D) image is formed from the interference of two mutually coherent waves that have a lateral dierential displacement of a few tenths of a micrometer (called the shear) and are phase-shifted relative to each other [4]. This is accomplished by illuminating a specimen with a plane-polarized beam that has been split into two orthogonally polarized, mutually coherent components by a Wollaston prism and afterwards recombined into a single beam by another prism and analyzer before being detected (see Figure 1). The amplitude, and thus the intensity of the resulting beam is a function of the phase dierence between the two waves. Therefore, the intensity distribution in measured DIC images is given by a nonlinear function of approximately the spatial gradient of a specimen's 1

Emerging Light Analyzer α

Cx

⊗ Cy

Sliding Wollaston Prism

d z y ⊗

Objective Shear Specimen Plane

x

Condenser

Cx

α

⊗ Cy

Compensator Wollaston Prism

Polarizer Entering Light

Figure 1: Schematic of a DIC microscope with two typical Wollaston prisms (the Nomarski DIC system uses modi ed Wollaston prisms referred to as Nomarski prisms [28]). Note that the angular wave splitting and the shear distance are exaggerated for clarity and they should be very small relative to the size of the eld of view.

2

optical-path-length distribution (integral of refractive index over length) along the direction of shear (perpendicular to the optical axis). Conventional DIC microscopy has been used to study both thin and thick living specimens. A three-dimensional (3-D) image of a thick specimen is obtained via the method of optical sectioning by collecting a set of 2-D images while the specimen is moved through focus. DIC microscopy has been known to have good optical-sectioning capability (i.e. it has a shallow depth of eld), which is a consequence of the combination of a high numerical aperture (NA) used for the illumination and the shearing of the wavefronts [1]. However, comparisons of confocal and non-confocal 3-D DIC images have shown that a confocal DIC microscope rejects more out-of-focus information than a conventional DIC microscope [5, 6]. As in other microscopy modalities, improvements in DIC images can be achieved by computational methods designed to undo degradations introduced by the optical system. To be able to undo these degradations, it is necessary to have a mathematical description of the image formation process. Therefore, the development of model-based computational methods for DIC microscopy is essential for quantitative interpretation of DIC images. A rst step towards improving DIC images is the derivation of a reasonable imaging model that can predict these degradations. In this paper, a general model for DIC image formation is derived, and results from testing dierent cases of the model are presented. The paper is organized as follows. Section 2 summarizes related work. In Section 3, the model is derived for 2-D imaging and then extended to three dimensions. Section 4 describes the data collection and the calculation of the model predictions. Section 5 is a comparison study between measured DIC images and the model predictions. The last section summarizes our results. For completeness, we include in this paper portions of our work that were presented elsewhere [7, 8].

2 Related work Although a detailed description of DIC optics can be found in the literature [1, 2, 4, 9] and in references therein, very little has been published until recently concerning the mathematical theory of DIC image formation. In the last decade, there have been a few isolated studies based on dierent approaches for the development of a DIC imaging model [5, 9, 10]. The results of these eorts provide a useful starting point for the development of more general models, and they also oer simplifying cases to which more accurate models should reduce under appropriate conditions. We brie y review related work below in a chronological order. The rst eort to predict images obtained with a DIC microscope was by Galbraith [11] who, predicted the DIC image of a point using computer simulations and compared it to measured DIC images of a pinhole for various imaging parameters. Because the image of a pinhole corresponds to the squared magnitude of the DIC point-spread function (PSF), we have used Galbraith's results as a reference for testing our software for the computation of the DIC PSF. Five years later, Holmes and Levy [10] attempted to predict mathematically the DIC image of an arbitrary 2-D phase object. Based on the physical description of the Nomarski optics, and on the theory of Fourier optics, they derived an expression for the intensity of a 2-D DIC image. Although it was not stated, this expression is based on coherent illumination (as we show later in Section 3.3.1) and coincides with a limiting case of our more general model. Using this expression, and a computer-generated phantom, Holmes and Levy generated synthetic DIC images which were judged to have a DIC appearance. However, their model was not evaluated with comparisons of model predictions to measured DIC images from an actual phantom specimen. The second eort to predict DIC images with a model was in 1992 by Dana [9], who developed a 3-D linear model using several simplifying assumptions: 1, very small phase variations due to the specimen; 2, bias equal to 90 degrees; 3, incoherent illumination; and 4, that the blurring due to the objective lens can be incorporated in the model by convolving the intensity PSF with the intensity of the \ideal" DIC image derived without diraction eects. In a DIC microscope the objective lens blurs the two wavefronts before they interfere to form an image. Thus, DIC imaging is better described when diraction eects are incorporated by blurring wave amplitudes with the amplitude PSF before the image intensity is computed. This leads to a nonlinear model as we show in Section 3. 3

In 1992, Cogswell and Sheppard [5], who had developed a confocal re ected-light DIC microscope, derived a 2-D spatial-frequency transfer theory of image formation for re ected-light DIC microscopy as part of a comparison study between the frequency transfer properties of a conventional DIC microscope and a confocal one for re ection optics. In their study, the eect of the shear and the bias on the performance of conventional and confocal DIC microscopy was demonstrated for two special cases: 1, imaging a weak phase specimen; and 2, imaging a specimen with a constant phase gradient. This is the rst study that quanti es the eect of these system parameters on the DIC image. Their theoretical comparison of frequency transfer as well as their experimental comparison of DIC images showed that conventional DIC microscopes do not have the same optical-sectioning capabilities as confocal DIC microscopes because they cannot exclude out-of-focus, low-spatial-frequency components from an optical-section image. We have extended their frequency transfer theory to incorporate partially coherent illumination.

3 Imaging Model Prior to the work described in this paper, there had been only a few studies modeling DIC image formation, as reviewed in the previous section. Although informative, these studies did not provide a general model for two reasons. First, previous models for DIC imaging were developed based on coherent illumination. However, in practice, ideal coherent illumination cannot be achieved in microscopes that use Kohler illumination (see Figure 7) because the condenser aperture cannot be stopped down to a single point. Even if this were possible, it would not be desirable because it would degrade the resolution and reduce the illuminating light considerably. In practice, the condenser aperture is at least partially open for DIC imaging, and thus, the illumination is partially coherent. Second, previous models describe only 2-D image formation with the exception of Dana's [9] simpli ed model. Furthermore, none of these models were validated by comparing measured DIC images of phantom specimens to model predictions. We have derived and evaluated a general image-formation model for DIC microscopy under partially coherent illumination. This was accomplished by rst deriving a 2-D model and then extending it to three dimensions, assuming weak optical interactions within the specimen. The derivation of the 2-D model was based on two dierent approaches, each of which led to the same result. First we extended the frequency theory of Cogswell and Sheppard [5] from coherent to partially coherent illumination, by propagating the mutual intensity of the light through a thin specimen and the optical system based on the theory of image formation in partially coherent light for transmitted-light optics described by Born and Wolf [12] (pp. 526-532). We arrived at the same model by propagating the complex amplitude of the illuminating wave eld under Kohler illumination through a thin specimen and the optical system. In what follows, we rst present the DIC PSF and then derive the general imaging model. Special cases of this model are shown to be equivalent to models developed by others. Evaluation of the model under certain imaging conditions is presented in Section 5.

3.1 DIC point-spread function

In a DIC microscope, the image is formed from the dierence between the amplitudes of two waves that are phase-shifted relative to each other and are separated by a lateral shear, 2
x, (expressed in length units) introduced by the Wollaston prism (see Figure 1). Without loss of generality, we assume that the shear is along the x direction. The phase dierence between the two waves is due to the fact that the waves travel through dierent regions of the specimen. Furthermore, a uniform phase dierence between the two waves, the bias retardation, 2
 (expressed in radians), can be introduced by translating the sliding prism along the shear direction. Assuming coherent illumination, perfect polarizing components, and that there is no instrumental stray light, then, when the phase dierence due to the specimen is zero, we can write the amplitude dierence between the two waves as: h(x; y) = (1 ? R) exp(?j
) k(x ?
x; y) ? R exp(j
) k(x +
x; y); (1) 4

(a)

(b)

Figure 2: A calculated 2-D DIC PSF (of a 10/0.3-NA lens) with bias equal to 1.57 radians, shear distance equal to 1:1m along the horizontal axis, and illumination light wavelength equal to 550 nm: (a) real part of the complex amplitude; and (b) imaginary part of the complex amplitude. The direction of shear is along the horizontal axis. The scale bar is approximately 1.64 m. The black regions in the images represent negative values. Pro les through the center of the images are shown in Figure 3. where R is the amplitude ratio - the amplitude of one wave eld divided by the sum of amplitudes of the two wave elds, and k(x; y) is the amplitude PSF for transmitted-light optics under coherent illumination [13]. The complex-valued function h(x; y) is the amplitude PSF for DIC optics; a calculated h(x; y) is shown in Figure 2. It should be noted that the square magnitude of the DIC PSF, jh(x; y)j2 , predicts the DIC image of a pinhole [11], i.e. an intensity point, rather than the DIC image of a delta phase function (see derivation in the Appendix). As is evident from Equation (1), in addition to the numerical aperture of the objective lens, several other physical parameters determine the DIC PSF. First, the shear which is xed in conventional DIC microscopes, is determined by the deviation angle of the Nomarski prism and the focal length of the objective lens, and it is usually of the order of the resolving power of the microscope objective (see Pluta [4], p. 158). Second, the amplitude ratio, R, is determined by the orientation of the polarizer and the analyzer, and for crossed polars R is equal to 0.5 (yielding equal-strength illuminating beams) which is the optimum setting for imaging purely phase objects. Furthermore, the bias retardation is adjusted by sliding the second Wollaston prism in order to optimize contrast (see Figure 3 in [5]) in the DIC image. The eect of the bias on the DIC PSF is shown in Figure 3. The 3-D PSF of a DIC system, h(x; y; z ), can be obtained from Equation (1) and the defocused PSF for transmitted-light optics, k(x; y;
z ), which can be modeled as the 2-D Fourier transform of the generalized pupil function p(x; y;
z) = jp(x; y;
z)j exp(j 2 w(x; y;
z)=); where w(x; y;
z ) is an eective path-length error (see Goodman [14], p. 121) and
z is the amount of defect in the focus due to a displacement of the object plane along the z (optical) axis. Detailed derivations of k(x; y;
z ) can be found in [12, 15, 16, 17]). A review and comparison of various models for the defocused PSF and the order of approximations made in each model can be found in [15]y. In our work, computation of k(x; y;
z ) is based on the PSF model by Gibson and Lanni [17] which was shown to be accurate enough for 3-D serial-sectioning microscopy. Section images through the volume of a calculated h(x; y; z ) are shown in Figure 4. y A model introduced in [15] to describe the 3-D diraction eld for o-axis points was shown to be more accurate for larger eld angles than models derived for on-axis points.

5

1 0.00 0.79 1.57 2.36 3.14

Re{ h(x,0) }

0.5

0

-0.5

-1 -6

-4

-2

0 2 x [microns]

4

6

4

6

0.2 0.00

Im{ h(x,0) }

0 -0.2

0.79

-0.4

1.57

-0.6 2.36

-0.8

3.14

-1 -6

-4

-2

0 2 x [microns]

Figure 3: Eect of the bias value on the DIC PSF. Horizontal pro les from the DIC PSF of a 10/0.3-NA dry lens. The pro les plot the real part (top panel) and the imaginary part (bottom panel) of h(x; 0) along the direction of shear (x-axis). The shear distance is 1.1m. The bias values are shown by the curves and they are in radians. The PSF was normalized so that the real part has a peak equal to 1.0 for bias equal to 0.0 radians, and the imaginary part has a peak equal to -1.0 for bias equal to 3.14 radians.

6

x y

z x

(a)

(b)

(c)

Figure 4: xy- and xz -section images from a 3-D calculated DIC PSF of a 10/0.3 NA lens with bias equal to 0 radians: (a) real part of the complex amplitude; (b) imaginary part of the complex amplitude; and (c) squared magnitude. In the images, black represents the minimum negative value in (a) and (b), and a zero value in (c), while white represents the maximum positive value of each image. For the xy images, z = 1:7m away from focus. The shear distance is equal to 0.68m and the direction of shear is along the x-axis. The scale bar is approximately 3m.

3.2 Frequency support of the PSF

The Fourier transform of the 2-D DIC PSF, h(x; y), assuming R = 0:5 in Equation (1), can be written as H (f; g) = ?j sin(2f
x +
)K (f; g); (2) where K (f; g) is the coherent transfer function, de ned as the Fourier transform of the coherent PSF, k(x; y). For a circular aperture, K (f; g) is circularly symmetric, and it is equal to 1.0 inside a circle with radius equal to fc = NA= and zero outside the circle, where NA is the numerical aperture of the objective lens, and  is the wavelength of the illumination light. Thus, the cuto frequencies of H (f; g) are fc = gc = NA=. Because K (f; g) is a real function, the real part of H (f; g) is zero. Pro les through calculated H (f; g) functions for dierent bias values are shown in Figure 5. As evident from Figure 5(b) and Equation (2), H (0; 0) equals zero for bias (2
) equal to zero while for a non-zero bias, H (0; 0) = ?j sin
. Furthermore, for a non-zero bias H (0; g) is not zero along the g-axis (i.e. perpendicular to the direction of shear) as it is for a zero bias as evident by the vertical pro les in Figure 5(a). This is important because it shows that some frequencies of an object with a phase function that does not change along the shear direction are passed through the objective lens. Such an object has frequency components only in the direction perpendicular to the shear direction, and thus, with a non-zero bias some of these frequency components pass through the DIC microscope. In fact, some information about such a specimen is actually imaged by the microscope (see [18], p. 173). The Fourier transform, H (f; g; j ), of the 3-D DIC PSF, h(x; y; z ), is shown in Figure 6. H (f; g; j ) is the DIC 3-D amplitude transfer function for a closed condenser aperture. The spatial-frequency cutos of H (f; g; j ) are fc = gc = NA= in the fg plane, and jc = NA2 =(n) along the j -axis, where n is the refractive index of the objective len's immersion medium. Thus, based on the Nyquist frequency required by the sampling theorem, an adequate discrete representation of the PSF can be obtained with samples dx = dy  =(2NA) in the xy plane, and dz  n=(2NA2 ) along the z axis. 7

0.00

0

Im{ H(0,g) }

−0.2

0.79

−0.4 −0.6

1.57

−0.8

2.36

−1

3.14

−1.2 −1.5

−1

−0.5

0

0.5

g [cycles/microns]

1

1.5

1 0.00 0.79 1.57 2.36 3.14

0.8 0.6 Im{ H(f,0) }

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 f [cycles/micrometers]

1

Figure 5: Eect of the bias on the Fourier transform of the PSF of a 10/0.3-NA dry lens. Vertical and horizontal pro les through the center of the imaginary part of H (f; g) are plotted in the top and bottom panels respectively. The bias values are shown by the curves and they are in radians. We note that the edges of the curves are sloped because the sampling is coarse.

8

f

g

fc jc j

f

(a)

(b)

(c)

Figure 6: Fourier transform, H (f; g; j ), of the 3-D DIC PSF of a 10/0.3-NA lens with bias = 0.0 radians, shear= 1.0 m, and 550 nm illumination light wavelength: (a) real part of the complex amplitude; (b) imaginary part of the complex amplitude; and (c) squared magnitude. The fgsection images (top row) are cuts through the 3-D image at the lines shown in the fj -section images, while the fj -section images (bottom row) are through the center of the volume as shown by the lines in the fg-section images. The direction of shear is along the f -axis. The spatialfrequency cutos are fc = gc = 0:545m?1 , and jc = 0:083m?1 . Black regions in (a) and (b) represent negative values while in (c) they represent values equal to zero.

9

k z-axis

Source

Objective lens

Auxilary lens Condenser lens

Field stop Illuminator

Aperture stop Dc Front-focal plane [ ξ, η ]

specimen plane [ xo, yo ]

image plane [ x, y ]

Figure 7: Kohler illumination. An incoherent light source is focused in the front-focal plane of the condenser lens, and thus, the specimen is illuminated by plane waves with normal vectors [kx ; ky ; kz ], kx = 2=fcon, ky = 2=fcon , kz = (2=)2 ? kx 2 ? ky 2 , which make angles x = sin?1 (=fc ) and y = sin?1 (=fc ) with the yz - and xz -plane respectively (see p. 49 in Goodman [14]). The direction of each plane wave depends on the illuminating point [; ] in the front-focal plane of the condenser lens. The DIC components are not shown here for simplicity.

3.3 Derivation of the 2-D imaging model

The derivation given in this section is based on propagating the complex amplitude of the illuminating wave eld under Kohler illumination through a thin specimen and the optical system. In Kohler illumination an extended light source emitting incoherent light is focused by a collector lens in the front-focal plane of the condenser lens [12] as shown in Figure 7. The condenser then illuminates the specimen which is speci ed by a complex amplitude transmission function f (x; y), and is assumed to be planar with a small thickness and transparent with spatially varying refractive index. In the following analysis we consider paraxial light waves only, and assume that f (x; y) is independent of the angle of illumination (i.e. the angle between the normal to the illuminating wave and the optical axis). We also assume that the light source is quasi-monochromatic with a mean wavelength . Let  = [; ], xo = [xo ; yo ], and x = [x; y] denote points in the front-focal plane of the condenser lens, in the object plane, and in the image plane respectively. Let the complex amplitude of the wave eld in the front-focal plane of the condenser lens, Us (), be a white random process with zero mean and covariance E fUs ()Us ( )g = () ( ?  ), where ( ) is the intensity of the light source and equals zero at points that lie outside the circular aperture, Dc , of the condenser lens, ( ) is a 2-D Dirac delta impulse function, and denotes complex conjugate. Then the complex amplitude, Uc (), of the wave eld in the back-focal plane of the condenser lens and right before the specimen can be obtained by the Fresnel superposition integral 0

Uc(xo ) =

+1 Z

?1

Us() hc(; xo ) d;

0

(3)

where hc ( ; xo ) = 1=jfcon exp(j 2(xo t )=fc ) speci es the complex amplitude of the illuminating plane waves (see Figure 7), and fcon is the focal length of the condenser lens. Under the paraxial approximation the complex amplitude of the wave transmitted by the specimen is simply Uo(xo ) = f (xo) Uc(xo ). Assuming that the DIC microscope (excluding the condenser lens) is a linear shift-invariant system characterized by the PSF h(x) (Equation (1)), we can express the

10

complex amplitude of the wave eld in the image (detector) plane as

Ui(x) =

+1 Z ?1

Uo (xo ) h(x ? xo) dxo:

(4)

The intensity in the image is then obtained from i(x) = E fUi (x)Ui (x)g, which yields the expression 2 +1 Z

+1 Z i(x) = () f (xo ) h(x ?1 ?1



? xo) hc(; xo) dxo d:

(5)

Equation (5) is a general formulation of the intensity in the image plane of a DIC microscope and describes partially coherent imaging. Limiting cases of the model can be obtained by specifying the intensity, ( ), of the source. The coherent-illumination limit of the model, obtained by restricting ( ) to be zero except at a single point, predicts Holmes and Levy's [10] intensity expression, and yields the same frequency transfer function reported by Cogswell and Sheppard [5] (see Sections 3.3.1 and 3.3.2 below).

3.3.1 Coherent illumination

To describe coherent illumination we let ( ) = a ( ) which corresponds to closing the condenser aperture down to a single point. In this case Equation (5) becomes +1 Z i(x) = a1 f (xo ) h(x ?1

?

2 xo ) dxo ;

(6)

where a1 = a=(fcon )2 . We refer to Equation (6) as the point-aperture model. It is easy to show that Equation (6) predicts the expression derived by Holmes and Levy [10]. First we substitute Equation (1) (with R = 0:5) in Equation (6) and let +1 Z p c(x) = 0:5 a1 f (xo) k(x ? xo) dxo ?1

= jc(x)j exp(?jc (x)): Then, Equation (6) may be rewritten as:

(7)

i(x; y) = j exp(?j
)c(x ?
x; y) ? exp(j
)c(x +
x; y)j2 = jc(x1 ; y)j2 + jc(x1 + 2
x; y)j2 ? 2 jc(x1 ; y)j jc(x1 + 2
x; y)j cos[c (x1 ; y) ? c (x1 + 2
x; y) + 2
]; (8) where x1 = x ?
x. Equation (8) is the same as Equation (28) in [10] with the speci ed shear and bias parameters. It can also be shown that Equation (6) predicts the expression given by Pluta [4] which assumes geometric optics. Assuming an ideal psf (i.e. ignoring the blurring eects of the PSF), h(x; y) = 0:5 exp(?j
)(x ?
x; y) ? 0:5 exp(j
)(x +
x; y); (9) and that the specimen's transmission function is f (x) = exp(?j(x)), Equation (6) reduces to:

i(x; y) = a1 sin2 [0:5 f(x ?
x; y) ? (x +
x; y)g +
] : 11

(10)

Equation (10) is a geometric-optics model for DIC imaging which is usually reduced by assuming that the wavefront shear, 2
x, is very small (dierential) relative to the size of the specimen, and thus, the phase dierence in Equation (10) can be written approximately as a phase gradient: 



(x; y) +
 : i(x; y) = a1 sin2
x @@x

(11)

Equation (11) agrees with Equation (7.27) in [4]. Thus, for ideal imaging under coherent illumination, the intensity in the DIC image is related to the gradient of the specimen's phase function along the direction of shear, and it is independent of the specimen if the specimen's phase function is constant in that direction. We emphasize that the latter is true only for ideal imaging, and it is due to the geometric optics approximation for the PSF (Equation (9)). In general, some information about a specimen's phase function that is constant along the direction of shear is expected to be imaged by the DIC microscope (see [18], p. 173).

3.3.2 Frequency transfer theory

The purpose of this section is to compare our image-formation model with the frequency transfer theory derived by Cogswell and Sheppard [5] and not to present a detailed analysis of frequency transfer through the DIC microscope as in [5]. In order to do that, we rst rewrite the detected intensity (Equation (5)) as follows,

i(x) =

+1 Z +1 Z

f (xo) f  (xo ) js (xo ? xo)h(x ? xo) h (x ? xo ) dxo dxo ;

?1 ?1

where

js (xo ? xo ) = E 0



Uc(xo )U (x0 ) = 1=(fcon )2

c

o

0

0

0

0

(12)

+1 Z

( ) exp(j 2([xo ? xo ]t )=fcon) d 0

?1

is the mutual intensity of the wave eld illuminating the specimen y. Equation (12) can be derived directly by propagating the mutual intensity, js (xo ? xo), through the system in the same way that it was derived in [12] for transmitted-light optics. Equation (12) can then be rewritten (see Born and Wolf [12], p. 529) as 0

i(x) =

+1 Z +1 Z

TDIC (f ; f ) F (f ) F  (f ) exp(j 2(f ? f )t x) df df ; 0

?1 ?1

0

0

0

(13)

where F (f ) is the 2-D Fourier transform of f (xo ),

TDIC (f ; f ) = 0

+1 Z

00

H (f + f ) H  (f + f )Js (f ) df ; 0

?1

00

(14)

is the transmission cross-coecient of the system, H (f ) and Js (f ) are the Fourier transforms of h(0x) 0 and js(x) respectively, and all bold letters denote vectors of spatial frequencies (i.e., f = [f ; g ]). TDIC (f ; f ) expresses the combined eect of the illumination and the optics on the object, 0

0

Note that for Kohler illumination, the mutual intensity depends on the dierence of the coordinates, i.e., (xo ; xo ) = js (xo ? xo ). y

js

0

0

12

and characterizes the frequency transfer properties of an optical system under partially coherent illumination as can be shown by the Fourier transform of the image intensity (Equation (13)),

I (f ) =

+1 Z

TDIC (f + f ; f ) F (f + f ) F  (f ) df : 0

?1

0

0

0

(15)

0

Another function that characterizes frequency transfer in an optical system under partially coherent illumination is the frequency response function of the system (see Born and Wolf [12], p. 527) de ned as MDIC (f ) = H (f )H  (?f ); (16) which relates the Fourier transforms Jo (f ; f ) and Ji (f ; f ) of the mutual intensities at the object and image plane respectively as follows: Ji (f ; f ) = MDIC (f ; f ) Jo (f ; f ): The eective frequency response function for DIC, is obtained by inserting Equation (2), in Equation (16) yielding: 0

0

0

0

0

0

MDIC (f ; f ) = sin(2f 0
x +
) sin(2(?f 00 )
x +
) M (f ; f ); (17) where M (f ; f ) = K (f )K  (?f ) is the frequency response function for transmitted optics. We note that under coherent illumination Js (f ) = (f ) (see [18], p. 34), and thus, Equation (14) becomes equivalent to MDIC (f ; ?f ) which is the same as the expression reported by Cogswell and Sheppard as Equation (13) in [5]. Thus, their expression is the coherent limit of Equation (14). Plots of MDIC (f ; ?f ) for various cases and a detailed analysis of frequency transfer through a 0

0

00

00

0

0

00

0

0

00

00

00

DIC microscope under coherent illumination can be found in [5].

3.4 Extension of the imaging model to three dimensions

Assuming an optically weak specimen, distortion of the wave eld due to multiple refraction through the specimen can be regarded as a secondary eect. With this assumption, the rst-order Born approximation (see Born and Wolf [12], p. 453) which permits the use of linear superposition is valid. To our knowledge, all existing 3-D models for transmitted-light optics are based on the rst-order Born approximation [19, 20, 21] because it simpli es the models. We have extended the 2-D model (Equation (5)) to three dimensions by assuming that an optically weak 3-D specimen with thickness t along the z -axis, is a set of N planar specimens of thickness
z = t=N , and that a wave eld that interacts with one plane in the specimen does not interact with the other planes. This allows the determination of the specimen's 3-D image from the linear superposition of 2-D images. A reasonable question here is what to superimpose: amplitudes or intensities? In order to answer this question, the issue of temporal coherence between two wave elds that interact with two dierent specimen planes that are a distance Dz apart must be examined. One way to do that is to compare the coherence length of the illuminating source, lc , to the smallest and largest possible optical path length dierence. The smallest and largest Dz of interest are the z?axis sampling distance (based on the Nyquist frequency required by the sampling theorem) and the thickness of the specimen, t, respectively. If lc  n t, where n is the average refractive index of the specimen, then temporal coherence holds among all waves interacting with the specimen and thus, wave elds can interfere at the detector; in this case the amplitudes of the dierent wave elds must be superimposed. On the other hand if lc  n
z then, temporal incoherence holds between any two waves interacting with dierent specimen planes, and thus, the intensities of the wave elds must be superimposed. For all other values of lc temporal partial coherence holds. Because of the complexity of the partial-coherence case, in practice assumptions are made in order to justify the use of temporal coherence or incoherence in the derivation of a model for a particular application. The choice of the best approximation depends on the various parameters used in an application. Instead of making any assumptions, we derived and tested (see Section 5) two models: one based on superposition of amplitudes and the other of intensities, in order to determine which one is a better approximation. When the image of a 3-D object can be obtained by the superposition of the intensity images of the planes that make up the object, then the 3-D image can be obtained by rst introducing the 13

specimen plane, zo , in Equation (5):

2 +1 Z izf (x; zf ? zo ) = () f (xo; zo ) h(x xo;zf ? zo ) hc( ; xo)dxo d; ?1 ?1

+1 Z

?

(18)

where zf is the plane at which the microscope is focused, and then by integrating this intensity over all the specimen planes,

i(x; z) =

+1 Z

?1

izf (x; z ? zo )dzo :

(19)

h(x; zf ? zo) is the 2-D PSF defocused by a distance
z = zf ? zo. Alternatively, Equation (5) can be extended to three dimensions by integrating amplitudes over zo , yielding a 3-D intensity: +1+1 2 Z Z f (xo;zo ) h(x xo;z ? zo ) hc(; xo )dxo dzo d: i(x; z) = () ?1?1 ?1

+1 Z

?

(20)

The coherent-illumination limit of this general 3-D models can be obtained by letting ( ) = a( ). The resulting simpli ed model is the extension of Equation (6) to three dimensions and, as in the case of the 2-D model, we refer to it as the 3-D point-aperture model.

4 Methods In order to test the DIC imaging model, several simple phantom specimens were constructed whose geometries and refractive-index distributions could be determined. Because our phantom fabrication facilities are limited to geometries of several microns, the models were primarily tested at low magni cations. These phantoms were imaged with conventional DIC microscopes using quasimonochromatic light (this is accomplished by placing a bandpass lter with a relatively narrow bandwidth in front of the light source) and dierent imaging conditions. The images were recorded using a cooled CCD camera. First, a 2-D phantom with shallow grooves was used for quantitative tests of the 2-D model. Second, a cross-shaped phantom and a bead phantom were used to test the 3-D models.

4.1 Phantom specimens

Two of the phantom specimens were constructed at the Photonics Research Laboratory (Electrical Engineering Department, Washington University, St. Louis, MO). The simple 2-D groove phantom was fabricated by etching two parallel grooves in a glass slide, each groove approximately 7 m (60 nm) wide and 60 nm (5 nm) deep. The grooves were measured with a stylus pro lometer (Alpha Step, Tencor Instruments) to be approximately 55-65 nm deep and 40m apart. Based on the etching process, the grooves are expected to have sharp edges. The 3-D cross phantom was constructed from resins with well known refractive indices [22] with the use of a computer-driven laser beam, and it consists of two perpendicular bars placed one on top of the other (see Figure 8). The two bars have width and height approximately 9 m, and refractive index slightly higher than the refractive index of the medium that surrounds them. This phantom was chosen because it has the ability to test the validity of intensity or amplitude superposition for weak specimens, an assumption that was used in extending the 2-D model to three dimensions (see Section 3.4). The bead phantom consists of a 4m in diameter polystyrene bead of refractive index n2 = 1:59 embedded in an optical cement of refractive index n1 = 1:524  0:0001 (Optical Adhesive No. 65, Norland Products, Inc., New Brunswick, N.J.). Optical cement was dropped on top of beads, dried on a coverslip, and was cured by a 10-second exposure to ultraviolet light from a 100-W mercury arc lamp at a distance of 5cm. 14

n1 y

n1

9µm

n2 n2

9µm

n1 = 1.494, n2 = 1.549 x

Figure 8: Schematic of the 3-D phantom specimen with crossed bars. n1 and n2 are the refractive index of the surrounding medium and of the bars respectively.

4.2 Data acquisition

A DIC image of the groove phantom was acquired at the Laboratory for Radiobiology, Academic Medical Center of the University of Amsterdam, the Netherlands, using an Ortholux II (Leitz, Germany) DIC microscope with a Leitz DIC 25x/0.5-NA dry objective lens and a 0.9-NA dry condenser lens with the condenser aperture open. This microscope has a calibrated DIC bias setting which is adjusted by rotating a polarizer (de Senarmont method [23]). The DIC bias was set to be approximately =2 radians. A band-pass lter (MAD8-1, Schott Glaswerke, Germany) with peak at 550 nm and bandwidth 8 nm around the peak was tted in front of the light source for quasi-monochromatic illumination. The images were acquired with a cooled CCD camera (Lambert Instruments, the Netherlands) equipped with a Kodak KAF0400 CCD chip (9-m well size). The eective pixel size in the recorded images is 0.36 m. A DIC image of small latex beads (460 nm in diameter, DOWlatex, 41984, Serva, Germany) was also measured at bias setting equal to =2 radians for the determination of the shear parameter. The shear for the 25x/0.5NA lens was determined to be approximately equal to 1.0m along the 45-degree axis by computing the coordinates of the center of the bright spot and the center of the dark spot in the image of the bead using a method similar to the one described by van Munster et al. [24]. Calibration images, dmin and dmax , were also acquired at bias of 0 and  radians respectively of an empty region on the slide using the same exposure time as for the other images. These images were used to correct the measured DIC images for CCD camera dark current and non-uniform at- eld response, and to normalize the DIC image values. For the correction, dmin was subtracted from each DIC image, then the result was divided by (dmax ? dmin ) (see [18], pp. 48-50). DIC images of the cross phantom and the bead phantom were acquired using an Olympus IMT2 inverted microscope (Olympus Corporation, Lake Success, N.Y.) with the IMT2-NIC attachement for DIC which has a 0.55-NA dry condenser lens. The images were recorded using a cooled (?45 C) CCD camera (Photometrics Ltd., Tucson, Ariz.) equipped with a Kodak KAF1400 CCD chip (6.8m well width). A band-pass lter with peak at 535 nm and full width at half maximum equal to 64.5 nm (IF550, Olympus) was placed in front of the light source (a tungsten halogen lamp, BRL, Ushio, Cypress, CA) for quasi-monochromatic illumination. The bias setting, controlled by sliding the Wollaston prism, is not calibrated on this microscope. The cross was imaged with an Olympus S Plan Achromat 10x/0.3-NA dry objective lens and the condenser aperture closed (for a closed aperture the eective condenser NA is 0.082) at four dierent orientations. The eective pixel size in each 2-D recorded image is 0.68 m. The spacing between focal planes along the z -axis (controlled by a stepping motor) was set to 3m. First, the two bars of the cross were aligned with the vertical and horizontal axes (Figure 8, right panel). The 15

cross was then rotated manually (by rotating the specimen slide) by approximately 13, 32 and 47 degrees clockwise. The bead phantom was imaged with an Olympus LWD CD Plan 40x/0.55-NA dry objective lens and the condenser aperture either closed or partially open. The pixel size in each 2-D image is 0.17 m and the spacing between focal planes was set to 1.8 m. A DIC image of small latex beads (210 nm in diameter, cat # 19391, Polysciences, Warrington, PA) with refractive index equal to 1.59 and air-dried on a slide was also acquired with the 10x/0.3NA objective lens for the determination of the shear parameter and for model testing. The shear determined from the bead image (using the method mentioned above) is approximately equal to 1.0m along the 135-degree axis (Figure 13).

4.3 Model predictions

Model predictions were obtained via simulations using theoretically-determined PSFs and computergenerated phantoms that approximate the phantom specimens described above. Synthetic 2-D and 3-D DIC images were generated using both the general models (Equation (5), Equation (19), and Equation (20)) and the point-aperture models (Equation (6) and its corresponding extensions to three dimensions). Theoretical PSFs were calculated using Equation (1) with R=0.5,  = 550nm, various bias values, and speci ed shear values for each of the objective lenses used: 10/0.3-NA, 25/0.5-NA, and 40/0.55-NA. The sampling rates used for the PSFs' calculations were based on the Nyquist condition (see Section 3.2). The shear value, 2
x, was determined to be equal to 1m for the three lenses. It is noted that the exact shear value used in the computation of the model predictions is aected by the pixel size used, and thus, some deviation from the above value was necessary in some cases. The bias, 2
, was estimated from the averagepvalue, Ib , of a region in the corrected DIC image with no specimen using the expression
 = sin?1 ( Ib ); it is easy to show that Equation (6) reduces to ik (x) = sin2 (
) = Ib for f (xo ) = 1 (see [18], pp. 176-178). Other methods for determining the bias can be found in [9, 25]. The computer generated phantoms represent the phase function of each phantom specimen which was calculated using  = 2(n2 ? n1 )
t=; where n2 is the uniform refractive index of a structure in the phantom (i.e. groove, bars or bead), n1 is the uniform refractive index of the surrounding medium,
t is the thickness of a thin section of a 3-D phantom (or the thickness of a 2-D phantom), and  the wavelength of light. This is a relative phase with respect to the background which is assumed to have zero phase. An implementation of Equation (5) and Equation (20) was developed based on the following approach. First, the illumination aperture is approximated by a lattice of points. Then, for each point in the aperture, a subimage is computed by performing a convolution operation between the PSF, h(x), and the product term f (x) hc (; x). The linear convolution operation is approximated with a circular convolution using fast Fourier transforms (FFT) [26]. To minimize aliasing errors, instead of padding with zeros, the phantom images are separated into two parts, background and structure, and superposition is used to determine the convolution result. This is done because our phantoms' transmission functions have a constant non-zero background and thus, the values in the phantom images do not fall near zero towards the edges of the images. The nal image is obtained by integrating over the aperture. The integration over the condenser aperture, Dc , is approximated with a summation. Implementation of the function hc (; x) requires evaluation ofpcomplex exponentials with frequencies (fcon ) and v = =(fcon ). For a given radius, rc =  2 + 2 , the highest frequency, p 2u = = 2 ur = u + v , is given by  ur = frc = r tan (21)  ; con

where r  rc =rmax , rmax = fcon tan  is the maximum radius of Dc ,  = sin?1 (NAcon=n) is the acceptance angle of the condenser lens, NAcon is the numerical aperture of the condenser lens, and n is the refractive index of the immersion medium used with the condenser lens. Thus, as the acceptance angle, , (or equivalently the NAcon ) or the normalized condenser aperture, r, increases, 16

complex exponentials with higher frequencies need to be computed. At some point, it becomes impossible to sample these functions adequately without decreasing the pixel size and at the same time increasing the number of points in the aperture lattice and thereby the computation time. This sampling problem gives rise to artifacts in the model predictions as the condenser aperture increases. In our computations, artifacts were sometimes observed for condenser apertures equal to 35% of the maximum radius although the exact size varies with the acceptance angle of the lens, the image size, and the pixel size. The point-aperture assumption simpli es the model equation, and thus, this model is easier to implement, and less time is required for its computation than the general model. For the point aperture model only one convolution operation is performed, and there is no need to compute complex exponentials as in the case of the general model. The program, written in the C programming language, was executed on an SGI R10000 180 MHz processor (Silicon Graphics, Inc., Mountain View, CA). Results from a performance analysis of the implementation using a commercially available program called \quantify" (Rational Software Corporation, Santa Clara, CA) showed that the implementation spends approximately 73% of the time executing FFTs. For the 2-D point-aperture model, the program takes approximately 6 seconds for a 512x512 image, 3while for the 3-D point-aperture model, it takes approximately 55 seconds for an image with 128 pixels. The time needed to compute the general model (2D or 3D) is equal to the time it takes to compute the point-aperture model scaled by the number of points in the aperture lattice.

5 Results In this section we present predictions obtained with our imaging models. Some of the model predictions are compared to measured images. In some cases exact values for imaging parameters such as bias, shear, and condenser aperture size were not available due to lack of system calibration. Furthermore, large aperture sizes could not be incorporated in the computation of the model predictions due to the required computational complexity (see also discussion in Section 4.3).

5.1 2-D Model

2-D model predictions were generated using a computer-generated groove phantom (Figure 9). As evident in Figure 9a, the model prediction obtained with the point-aperture model (r ! 0.0) which corresponds to coherent imaging shows oscillations that are characteristic of coherent imaging of a sharp step (Goodman [14], p. 131-132). These oscillations are not visible in partially coherent imaging, i.e., when the condenser aperture is partially open (Figure 9b, r = 0.3). Comparison of a model prediction (synthetic image) of the groove phantom to a measured DIC image of the phantom (Figure 10) shows good qualitative agreement. The two images appear to be similar and show a distance of approximately 7 m between the groove edges which is consistent with the groove's width. Quantitative dierences between model and data are evident from the pro les shown in Figure 10. The measured DIC image has lower and wider peaks than the synthetic DIC image. This mismatch could be attributed to several dierences between the actual imaging conditions and the ones assumed in the generation of the synthetic image: 1, condenser aperture size; 2, errors in the bias value and the shear value and direction; 3, PSF errors; and 4, de ciencies in the imaging model. It is noted that the condenser aperture used for the model prediction is smaller than the actual condenser aperture due to computation problems for larger apertures.

5.2 3-D Model

Due to the computational complexity of the 3-D general model (see discussion in Section 4.3, most of the model predictions presented in this section were computed using the point-aperture model. Although the point-aperture model is an approximation of the general model, evaluation of the point-aperture model provides a good starting point towards the evaluation of the general model. 3-D point-aperture model predictions were generated for the cross phantom and a small bead (210 17

(a)

y

shear x

(b)

0.12 r = 0.0 r = 0.3

0.1

Intensity

0.08 0.06 0.04 0.02 0 0

5

10

15 20 25 30 Distance [microns]

35

40

Figure 9: Comparison of synthetic DIC images of the groove phantom computed with the point aperture model (a), and the general model with an aperture of normalized radius, r, equal to 0.3 (b), assuming a 10=0:3-NA objective lens, a 0.55-NA condenser lens with focal distance, fc = 24mm, light wavelength  = 550nm, bias = -1.16 radians, and shear = 1.5 m. Horizontal pro les through the images are compared in (c) (r = 0.0 corresponds to the point-aperture model).

18

(a)

y

shear x

(b)

0.55 data model

0.5 0.45 Intensity

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0

5

10 15 Distance [microns]

20

25

Figure 10: Comparison of DIC images of the groove phantom: (a) measured DIC image from the physical phantom using a 25x/0.5-NA lens (shear = 1.0 m), 550 nm illumination wavelength, and an open condenser aperture; (b) synthetic DIC image generated using the general model with an aperture of normalized radius equal to 0.1 and an estimated bias equal to 1.178 radians; (c) Horizontal pro les through the measured image (data) and the synthetic image (model).

19

shear y

x

(a)

(b)

(c)

Figure 11: Measured and synthetic DIC images of a phantom with crossed bars. Measured DIC image acquired with a 10x/0.3-NA lens and a 0.55-NA condenser lens with the aperture closed (a). Model predictions generated with the 3-D model that assumes: superposition of amplitudes and a point condenser aperture (b); and superposition of intensities and a point condenser aperture (c). A DIC bias equal to zero radians was used for the model predictions. The section-images are cuts through the 3-D images, and they are approximately at the best focus of the objective lens. nm in diameter) while model predictions for various aperture sizes were generated for the bead (400 nm in diameter) phantom. For simplicity, 2-D cross-section images from the 3-D images are shown in the gures. Comparisons of model predictions of the cross-shaped phantom computed with the pointaperture model were compared to measured images acquired with a closed condenser aperture. Two speci c issues were tested in these comparisons: 1, the superposition assumption used to extend the 2-D model to three dimensions based on the temporal coherence of the light; and 2, the ability of the model to predict the direction sensitivity of DIC imaging. In what follows we describe results from these comparisons. To test the superposition assumption used to extend the 2-D model to three dimensions (see Section 3.4), model predictions were generated with both Equation 20 and Equation 19 (assuming a point condenser aperture for simplicity) and a computer generated crossed-bars phantom (Figure 11b and c, respectively). Measured images of the crossed-bars phantom specimen (Figure 8) acquired with a closed condenser aperture were compared with these model predictions. One such comparison is shown in Figure 11. As evident from the gure, the model prediction obtained by superimposing amplitudes (Figure 11b) shows good qualitative agreement with the measured image (Figure 11a), while the model prediction obtained by superimposing intensities (Figure 11c) fails to capture the distinct features of the measured DIC image. This result suggests that assuming temporal coherence between the interfering wave elds is a better approximation than temporal incoherence. We note that, for our expiremental parameters, the condition lc  n t that ensures temporal coherence (see discussion in Section 3.4) does not hold because the coherence length of the illuminating light, lc , is small; lc = 0:664 c=
 = 2:9 m (see [13], p. 168) where c is the speed of light in air, and
 = 6:8  107 MHz is the half-power bandwidth of the excitation lter (Olympus IF550) which was placed in front of the light source. We note that using a lter with a narrower bandwidth would yield a larger lc which could ensure temporal coherence between the interfering wave elds. In what follows we show model predictions obtained based on the temporal coherence assumption (i.e., with Equation 20) only. Another comparison of a measured image of the cross-shaped phantom to a model prediction is shown in Figure 12. Because the model prediction was obtained with the point-aperture model it shows oscillations characteristic of coherent imaging of a sharp step. As evident from Figure 12, the spread along the z axis in the model prediction is consistent with that observed in the measured 20

shear

x y

x z

(a)

(b)

Figure 12: Measured (a) and synthetic (b) DIC images of a cross phantom imaged with a 10x/0.3NA objective lens, and a 0.55-NA condenser lens, with the condenser aperture closed. The synthetic images were computed with the point-aperture model. The xy images are cuts through the 3-D image along the z axis, at z = 0m (i.e. near the best focus). The xz -section images are cuts through the top of the xy images. For the model prediction an estimated bias of 0.05 radians and a shear distance equal to 0.68m were used. The xz -section images were scaled (by linear interpolation) to a 1:1 aspect ratio. The scale bar is approximately 9m.

21

image. An apparent tilt along the z axis observed in the measured image (Figure 12a) is probably due to a misalignment in the microscope which is not accounted for by the model. A similar tilt along the z axis (o-axis aberration) is observed in the measured image of a small bead (Figure 13a) but not in the synthetic DIC image of the bead (Figure 13b). Another discrepancy between model and data evident in Figure 13 is due to the diraction rings observed in the synthetic image which are not visible in the measured image due to noise. To test how well the model accounts for the direction sensitivity of DIC imaging, measured images of the cross phantom acquired at dierent orientations of the cross are compared to model predictions. For simplicity, we show only xy-section images which correspond to a cut through the middle of a 3-D volume (Figure 14). Several similarities between the images are noted with respect to object features that are present in the images. At the rst orientation of the phantom (shown in Figure 8, right panel) the shear direction makes a 45-degree angle with both bars and, thus, the edges of both bars are visible (Figure 14a). As the phantom is rotated clockwise the phase-gradient information of one of the bars dominates making that bar more visible in the image. Finally, only the bar that is oriented perpendicular to the direction of shear is visible while the other one, along the direction of shear, disappears (Figure 14d). The asymmetry evident in the measured images (i.e., the dierence in the width of the bar edges in Figure 14b, c, and d, top row) suggests that the bars do not have a square cross section which was assumed for the computer-generated phantom used in the model predictions. 3-D model predictions of a computer-generated bead phantom were generated with the general model for dierent aperture sizes (Figure 15). Artifacts were observed in the model predictions for aperture sizes larger than 35% of the maximum aperture. One such artifact is the bright intensity in the center of the bead image in Figure 15d. Measured images of the bead phantom acquired with dierent aperture openings of the condenser lens do not show this artifact, but instead they show an asymmetry along the z -axis which seems to be due to spherical aberration (Figure 16). To con rm this, we generated a model prediction using a PSF with spherical aberrations (Figure 17). Comparison between data (Figure 16) and model (Figure 17) shows a qualitative agreement for the observed asymmetry along the z -axis. The aberrant PSF was computed using the Gibson and Lanni model [17] which incorporates spherical aberrations introduced by deviations of the thickness and refractive index of layers separating the object plane from the objective lens from the design conditions. We note that the amount of spherical aberration introduced in order to account for the observed asymmetry in the measured images is much larger than what would have been expected for the relatively thin bead phantom. Part of the observed spherical aberration may be due to residual aberrations reported for this objective lens by other users. Unfortunately, this was the only DIC lens available to us with an NA at least as large as the NA of the condenser lens.

22

shear

x z

Figure 13: xz -sections from a measured (a) and a synthetic (b) DIC image of a 210-nm in diameter bead imaged with a 10x/0.3-NA objective lens, and a 0.55-NA condenser lens, with the condenser aperture closed. The synthetic images were computed with the point-aperture model. For the (a) bias of 0.0015 radians and (b) a shear distance equal to 1.2m were model prediction an estimated used. The scale bar is approximately 4.3m.

23

shear

y

x

(a)

(b)

(c)

(d)

Figure 14: Measured DIC images of the crossed-bar phantom (top row); and model predictions (bottom row) of the DIC image of the phantom specimen shown in Figure 8 obtained with a 3-D model that assumes superposition of amplitudes and a point condenser aperture, with DIC bias equal to -0.001 radians, at dierent orientations of the phantom: (a) the phantom is oriented as shown in the right panel of Figure 8; (b) the phantom is rotated by 13 degrees clockwise from position (a); (c) the phantom is rotated by 32 degrees from position (a); and (d) the phantom is rotated by 47 degrees from position (a).

24

(a)

(b)

(c)

(d)

Figure 15: Meridional (xz ) sections from synthetic bead images computed with the general model for a 40/0.55-NA objective lens and a 0.55-NA condenser lens with the an aperture of normalized radius equal to: 0.06 (a); 0.12 (b); 0.25 (c); and 0.44 (d). A DIC bias equal to 0.0 radians and a 550nm illumination wavelength were used. The shear distance is 1.0 m and the direction is along the 135-degree axis as in Figure 13. The vertical and horizontal scale bars represent 25.7 m and 4.8 m respectively.

25

(a)

(b)

(c)

(d)

Figure 16: Meridional (xz ) sections of the bead phantom images measured with a 40/0.55-NA dry lens and a 0.55-NA condenser with the aperture: closed (a); 25% open (b); 50% open (c); 75% open (d); and open (e). Because the condenser aperture iris can be opened and closed with an uncalibrated slider, the aperture size reported here is the position of the slider with respect to its closed and open position. The vertical and horizontal scale bars represent 25.7 m and 4.8 m respectively. The shear direction is along the 135-degree axis as in Figure 13.

26

Figure 17: xz -section image from a synthetic bead image computed with the point-aperture model for a 40/0.55-NA objective lens with a spherically aberrant PSF. A DIC bias equal to 0.0 radians and a 550nm illumination wavelength were used. The shear distance is 1.0 m and the direction is along the 135-degree axis as in Figure 13. The vertical and horizontal scale bars represent 25.7 m and 4.8 m respectively.

27

6 Summary and conclusions The derivation of models for 2-D and 3-D DIC imaging based on rst principles has been presented. Evaluation of the models with three physical phantom specimens (a 2-D groove, a 3-D cross-shaped phantom, and a bead) has shown a number of similarities between model and data suggesting that although our models are not perfect, they do capture major features in the measured image. Currently, model predictions are limited to small condenser apertures. Computation of model predictions for large aperture sizes yield artifacts due to inadequate sampling (see discussion in Section 4.3). We expect that these sampling problems can be overcome with the increase of computing power (faster processors and more memory) available to us. By overcoming these current computational problems and by generating synthetic DIC images with larger condenser apertures, we expect that the match between model and data will improve. Several system parameters were estimated in order to best approximate DIC imaging formation with our model. Furthermore, in the case of the cross-shaped phantom, the exact phantom geometry was not known exactly and thus, estimated dimensions were used. In addition to errors in these estimated system and phantom parameters, aberrations in the optical system could be partly responsible for the observed dierences between model and data. Optical system aberrations due to incompletely corrected optical components or due to non-ideal set-up of the system can give rise to artifacts in the measured images (see detailed discussion in [27], pp. 405-412 and [17]) that are not accounted by the model. Thus, for further model testing a well calibrated DIC microscope equipped with high-NA optics and high resolution phantom specimens with known geometries will be used. The development and evaluation of the DIC image-formation models presented in this paper, is a necessary rst step for the development of model-based image-processing methods for DIC microscopy, and thus, it should facilitate future development of such methods. Although further evaluation of our DIC models is needed, our results have shown that the model has the potential to capture the main features of measured DIC images.

7 Appendix: DIC image of a point phase object

A point phase object has a phase function (x) = 1 (x), where 1 is a constant and (x) is a two-dimensional Dirac delta function and a complex amplitude transmission function, ( x = (0; 0) f (x) = exp(?j(x)) = cos(1 ) ?1 j sin(1 ) otherwise : (22) From Equation 6 the intensity in the DIC image of this object can be written as

i(x) =

+1 Z a1 (b1 + jb2 ) (xo ) h(x ?1

+1 Z

? xo) dxo + h(x ?

2 xo ) dxo

?1 2 = a1 jh(b1 + jb2 ) h(x) + H (0; 0)j i = a1 (b1 2 + b2 2 )j h(x)j2 + jH (0; 0)j2 + 2 Ref (b1 + jb2 ) h(x) H (0; 0) g ;

(23) where b1 = cos(1 ) ? 1 and b2 = sin(1 ), H (f ) is the Fourier transform of h(x), and Ref
g denotes the real part of a complex-valued function. The rst term in the right hand side of 2Equation 23 is the image of an intensity point object (i.e., a pinhole). The second term jH (0; 0)j = sin2
. However, the last term is not a constant, and thus, we conclude that the image of a phase point object is dierent from the image of an intensity point object.

8 Acknowledgements This research was supported in part by the National Institutes of Health (NIH) under research grants RR 01380, BTA-SIO-RR 10412, RO1 GM49798, and by Washington University. The authors 28

wish to thank Robert Krchnavek, Richard Livingston, and Brian Faircloth for the construction of the groove and cross phantoms and for the pro lometer measurement of the groove phantom, and James McNally (National Cancer Institute, NIH) for the construction of the bead phantom and for useful comments and discussions. Special thanks are extended to Erik van Munster (University of Amsterdam, the Netherlands) for acquiring the DIC image of the groove phantom specimen and the calibration images, and fruitful discussions, and to Frederick Lanni (Carnegie Mellon University) for suggesting the construction of a cross-shaped phantom, for his enthusiasm about this work, and for insightful discussions. Finally, the authors thank the anonymous reviewers for their constructive comments.

References [1] R. D. Allen, G. B. David, and G. Nomarski. \The Zeiss-Nomarski Dierential Interference Equipment for Transmitted-Light Microscopy". Zeitschrift fur Wissenschaftliche Mikroskopie und Mikroskopische Technik, 69(4):193{221, 1969. [2] W. Lang. \Nomarski Dierential Interference Contrast Microscopy". A Collection of Four Articles from Zeiss Information, Carl Zeiss, 7082 Oberkochen, West Germany, 1968. [3] H. Gundlach. \Phase Contrast and Dierential Interference Contrast Instrumentation and Applications in Cell, Developmental, and Marine Biology". Optical Engineering, 32(12):3223{ 3228, 1993. [4] M. Pluta. Advanced Light Microscopy: Specialized Methods , pages 146{197. Elsevier, Amsterdam, 1989. [5] C. J. Cogswell and C. J. R. Sheppard. \Confocal Dierential Interference Contrast (DIC) Microscopy: Including a Theoretical Analysis of Conventional and Confocal DIC Imaging". Journal of Microscopy, 165:81{101, 1992. [6] A. E. Dixon and C. J. Cogswell. \Confocal Microscopy with Transmitted Light". In J. B. Pawley, editor, Handbook of Biological Confocal Microscopy, pages 479{490. Plenum Press, New York, 1995. [7] C. Preza, D. L. Snyder, and J.-A. Conchello. \Imaging Models for Three-Dimensional Transmitted-Light DIC Microscopy". In C. J. Cogswell, G. S. Kino, and T. Wilson, editors, Proceedings of the IS&T/SPIE symposium on Electronic Imaging, Science and Technology, volume 2655, pages 245{257, 1996. [8] C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, and J.-A. Conchello. \Phase Estimation from Transmitted-Light DIC Images Using Rotational Diversity". In T. Schulz, editor, Image Reconstruction and Restoration II, volume Proc. SPIE 3170, pages 97{107, 1997. [9] K. Dana. \Three Dimensional Reconstruction of the Tectorial Membrane: An Image Processing Method Using Nomarski Dierential Interference Contrast Microscopy". Master's thesis, Massachusetts Institute of Technology, Massachusetts, 1992. [10] T. J. Holmes and W. J. Levy. \Signal-Processing Characteristics of Dierential-InterferenceContrast Microscopy". Applied Optics, 26(18):3929{3939, 1987. [11] W. Galbraith. \The image of a Point of Light in Dierential Interference Contrast Microscopy: Computer Simulation". Microscopica Acta, 85(3):233{254, 1982. [12] M. Born and E. Wolf. Principles of Optics. Pergamon Press, Oxford, 1964. [13] J. W. Goodman. Statistical Optics. John Wiley and Sons, New York, 1984. [14] J. W. Goodman. Introduction to Fourier Optics. McGraw-Hill Book Company, New York, 1968. 29

[15] S. F. Gibson and F. Lanni. \Diraction by a Circular Aperture as a Model for Threedimensional Optical Microscopy". J. Opt. Soc. Am. A, 6(9):1357 { 1367, September 1989. [16] H.H. Hopkins. \The Frequency Response of a Defocused Optical System". Proceedings of the Royal Society of London A, 231:91{103, 1955. [17] S. F. Gibson and F. Lanni. \Experimental Test of an Analytical Model of Aberration in an Oil-Immersion Objective Lens Used in Three-Dimensional Light Microscopy". J. Opt. Soc. Am. A, 9(1):154{66, 1992. [18] C. Preza. \Phase Estimation Using Rotational Diversity for Dierential Interference Contrast Microscopy". DSc thesis, Washington University, Sever Institute of Technology, St. Louis, MO, August 1998. [19] N. Streibl. \Three-Dimensional Imaging by a Microscope". Journal of the Optical Society of America A, 2(2):121{127, 1985. [20] I. Nemoto. \Three-Dimensional Imaging in Microscopy as an Extension of the Theory of two-Dimensional Imaging". Journal of the Optical Society of America A, 5(11):1848{1851, 1988. [21] C. J. R. Sheppard and X. Q. Mao. \Three-Dimensional Imaging in a Microscope". Journal of the Optical Society of America A, 6(9):1260{1269, 1989. [22] K. Nakagawa. \Cross-linked Acrylic Polymers for Optical Waveguide Applications ". Master's thesis, Washington University, Sever Institute of Technology, St. Louis, Missouri, 1994. [23] C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan. \Quantitative DIC Microscopy Using a Geometric Phase Shifter". In C. J. Cogswell, J.-A. Conchello, and T. Wilson, editors, Three-Dimensional Microscopy: Image Acquisition and Processing IV, volume Proc. SPIE 2984, pages 72{81, 1997. [24] E. B. van Munster, L. J. van Vliet, and J. A. Aten. \Reconstruction of Optical Pathlength Distributions From Images Obtained by a Wide-Field Dierential Interference Contrast Microscope". Journal of Microscopy, 188(2):149{157, 1997. [25] J. S. Hartman, R. L. Gordon, and L.L. Delbert. \Quantitative Surface Topography Determination by Nomarski Re ection Microscopy". Applied Optics, 19(17):2998, 1980. [26] A. V. Oppenheim and R. W. Schafer. Digital Signal Processing. Printice-Hall, New Jersey, 1975. [27] S. Inoue. \Ultrathin Optical Sectioning and Dynamic Volume Investigation with Conventional Light Microscopy". In J. Stevens, L. Mills, and J. Trogadis, editors, Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems, pages 397{419. Academic Press, San Diego, 1994. [28] M. Pluta. Advanced Light Microscopy: Specialized Methods . Elsevier, Amsterdam, 1989.

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