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A Parallel Product-Convolution

approach for representing depth varying

Point Spread Functions in 3D widefield

microscopy based on principal

component analysis

Muthuvel Arigovindan1*, Joshua Shaevitz3, John McGowan1, John W.

Sedat1and David A. Agard1,2

1Keck Advanced Microscopy Center and the Dept. of Biochem. and Biophys., University of

California at San Francisco, San Francisco, CA-94158, USA

2Howard Hughes Medical Institute

3Department of Physics and Lewis-Sigler Institute for Integrative Genomics, Princeton

University, Princeton, NJ-08544, USA

*mvel@msg.ucsf.edu

Abstract:

of image formation in 3D widefield fluorescence microscopy with depth

varying spherical aberrations. We first represent 3D depth-dependent point

spread functions (PSFs) as a weighted sum of basis functions that are

obtained by principal component analysis (PCA) of experimental data.

This representation is then used to derive an approximating structure that

compactly expresses the depth variant response as a sum of few depth

invariant convolutions pre-multiplied by a set of 1D depth functions,

where the convolving functions are the PCA-derived basis functions. The

model offers an efficient and convenient trade-off between complexity and

accuracy. For a given number of approximating PSFs, the proposed method

results in a much better accuracy than the strata based approximation

scheme that is currently used in the literature. In addition to yielding better

accuracy, the proposed methods automatically eliminate the noise in the

measured PSFs.

We address the problem of computational representation

© 2010 Optical Society of America

OCIS codes: (110.0110) Imaging systems; (110.0180) Microscopy; (110.6880) Three-

dimensional image acquisition.

References and links

1. D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol.

30, 353–377 (1989).

2. P. Sarder and A. Nehorai,“Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Sig. Proc.

Mag. 23, 32–45 (2006).

3. S. 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 8, 1601–1613 (1991).

4. B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, “Phase-retrieved pupil functions in wide-field fluorescent

microscopy,” J. Microsc. 216, 32–48 (2004).

5. J. Shaevitz and D. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-

varying point-spread function,” J. Opt. Soc. Am. A 24, 2622–2627 (2007).

(C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6461

#121941 - $15.00 USDReceived 8 Jan 2010; revised 5 Mar 2010; accepted 5 Mar 2010; published 15 Mar 2010

Page 2

6. C. Preza and J. Conchello, “Image estimation account for point-spread function depth variation in three-

dimensional fluorescence microscopy,” Proc. SPIE (2003), pp. 1–8.

7. C. Preza and J. Conchello, “Depth-variant maximum likelihood restoration for three-dimensional fluorescence

microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

8. E. Hom, F. Marchis, T. Lee, S. Haase, D. Agard, and J. Sedat, “AIDA: an adaptive image deconvolution algorithm

with application to multi-frame and three-dimensional data,” J. Opt. Soc. Am. A 24, 1580-1600 (2007).

9. C.VoneschandM.Unser,“AFastThresholdedLandweberAlgorithmforWavelet-RegularizedMultidimensional

Deconvolution,” IEEE Tran. Imag. Proc. 17, 539–549 (2008).

10. C. Vonesch and M. Unser, “A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration,” IEEE

Tran. Imag. Proc. 18, 509–523 (2009).

11. I. Jolliffe, Principal component analysis (Springer, 2002).

12. S. Wiersma, P. Torok, T. Visser, and P. Varga, “Comparison of different theories for focusing through a plane

interface,” J. Opt. Soc. Am. B 14, 1482–1490 (1997).

1.Introduction

Deconvolution plays a crucial role in 3D wide-field microscopy [1]. It attempts to estimate the

underlying image structure from the acquired data by inverting the imaging response (forward

model) of the microscope, which leads to an improved resolution and contrast. Undeniably,

accuracy of the forward model has a significant impact on the accuracy of the estimated struc-

ture. In the standard space-invariant assumption, the forward model is represented by a point

spread function (PSF) which is the image resulting from placing a point source (sub-diffraction

fluorescent bead) under the microscope. Most of the deconvolution methods rely on this as-

sumption [2]. However, in reality, this assumption is invalid.

In any realistic widefield microscope with high NA objective lens observing biological sam-

ples, the actual PSF varies as a function of depth from the coverslip. The dominant factor

causing this variation is the depth dependent spherical aberration resulting from the mismatch

between the refractive indices of mounting and immersion mediums [3]. Hence, if Nx×Ny×Nz

is the image size with Nzbeing the axial dimension, then a rigorous representation amounts

to using an Nznumber of 3D depth-varying PSFs (DV-PSFs). For a particular imaging set-up,

if one has measured the PSF at zero depth, all DV-PSFs are predictable with a reasonable ap-

proximation by modifying the phase-retrieved pupil function [4]. It can also be measured with

sufficiently fine sampling using specialized hardware [5].

Observation of such PSFs indicate that the magnitude of depth variation is significant, sug-

gesting that including depth-dependent effects in deconvolution will lead to a substantial im-

provement in the results. This has indeed been demonstrated experimentally in [5] where both

the data and PSFs used were from real measurements. The emphasis therein was on showing

the advantage in using the depth variant forward model against the invariant one, but not on the

computational complexity. Since, it is impractical to use Nznumber of PSFs, the authors use

an approximation method where the depth-dependent effects are handled as discrete layers or

strata in Z (strata method) [6,7].

Itshouldbenotedthatmostofthehighperformancenonlineardeconvolutionmethods[8–10]

rely on repeated computation of the forward model. Hence, an improved approximation method

will results in a significant improvement in the deconvolution process.

Our goal in this paper is to develop a better approximation scheme based on principal com-

ponent analysis. We restrict ourselves in developing the forward model only and we do not

consider the deconvolution problem here. With the use of theoretical and measured PSFs, we

demonstrate how our model provides a better alternative for constructing deconvolution algo-

rithms.

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2.Developing the model

Our model is based on expressing the DV-PSFs as weighted sum of a fixed set of basis functions

derived from the the principal components of experimental observations. Since, PCA of 3D

images is extremely complex and computationally impractical (with no reported applications

so far), we first derive an approximate two-stage method, which we call the tensor product

PCA (TP-PCA). We then derive the final computational structure. Thus, the depth variation

is accounted for by premultiplication with a set of 1D depth functions, followed by a set of

standard convolutions.

2.1.

Principal componentanalysis [11]allows acompact representationof imagesof a givenspecific

class. It represents a given set of images as a weighted sum of a small number of 2D basis

functions, These basis functions are known as the principal components which are computed

using Singular Value Decomposition (SVD) as described below.

Let {X1,X2,...,XN} be the given set of images. Let¯X be the mean image given by

Tensor product PCA of 3D images

¯X = (1/N)

N

∑

i=1

Xi.

Let vibe the 1D vector obtained by scanning the mean removed image X�

the matrix defined by

i= Xi−¯X. Let R be

R = [v1 v2···vN]

R is an M×N matrix where M is the total number of pixels in each of the images

{X1,X2,...,XN}. Let the following be the singular value decomposition of R.

R = UDVT,

where U and V are orthogonal matrices and D is a diagonal matrix. Without the loss of gener-

ality, let us assume that the diagonal elements of D are ordered with decreasing magnitude. Let

uibe the ith column of U, and Pibe the corresponding image obtained by reverse scanning ui.

Then in PCA, one choses the first B column vectors of U and expresses an input image Xias

Xi=¯X+

B

∑

j=1

ci,jPj,

where ci,j=�Pj,X�

ber of basis functions, which is chosen according to the required level of approximation. The

diagonal elements of D provide an estimate of approximation error. Specifically, the ratio,

i

�, with �, � representing the inner product, i.e., point-wise multiplication

and summation. The user-defined parameter, B, in the above expression represents the num-

rB=dB

d1,

gives relative mean square approximation error, where the dis represent the diagonal elements

of D. Typically, for an adequate level of approximation, B is much smaller than N.

Almost all of the imaging applications that use PCA are on 2D images. A direct applica-

tion of the above procedure is impractical for 3D images because SVD becomes prohibitively

expensive. Here we develop a novel two-stage method, which we call the tensor product PCA

(TP-PCA) that can be applied to the 3D PSF images with an affordable complexity.

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Let S = {S1(x,y,z),...,SN(x,y,z)} be the set of 3D images. Let Nz be the num-

ber of z sections. For each value of z, we form a set of 2D images given by S(z)=

{S(z)

sets to get basis functions P(z)

expressed as

1(x,y),...,S(z)

N(x,y)}, where S(z)

i(x,y) = Si(x,y,z). We then apply PCA on each of the sub-

1(x,y),...,P(z)

B1(x,y), where B1is a user parameter. The images are

S(z)

i(x,y) =¯S(z)(x,y)+

B1

∑

j=1

c(z)

i,jP(z)

j(x,y),

(1)

where¯S(z)(x,y) is the mean image of the set S(z)and

c(z)

i,j=

�

S(z)

i(x,y)−¯S(z)(x,y),P(z)

j(x,y)

�

.

We now form a set of 2D images given by

S�=

�

Ci(z, j) = c(z)

i,j,i = 1,...,N; j = 1,...,B1

�

.

(2)

Finally, we apply a PCA again on the set S�to get basis functions P�

B2is the second user parameter. The images are approximated as

1(z, j),...,P�

B2(z, j), where

Ci(z, j) =

B2

∑

k=1

c�

i,kP�

k(z, j),

(3)

where c�

i,k=�Ci(z, j),P�

k(z, j)�. Substituting Eqs. (3) and (2) in Eq. (1) gives

S(z)

i(x,y) =¯S(z)(x,y)+

B2

∑

k=1

c�

i,kQk(x,y,z),

(4)

where

Qk(x,y,z) =

B1

∑

j=1

P�

k(z, j)P(z)

j(x,y)

Replacing S(z)

i(x,y) and¯S(z)(x,y) by Si(x,y,z) and¯S(x,y,z) in Eq. (4) yields

Si(x,y,z) =¯S(x,y,z)+

B2

∑

k=1

c�

i,kQk(x,y,z),

(5)

The above expression represents our final TP-PCA approximation. In the above procedure, the

first user parameter B1is chosen such that the approximation error is negligible or zero in the

first stage, and the second user parameter is chosen according to the desired final approximation

error. By this way, the overall approximation error is governed only by the z-dependent second

stage.

We have described so far a procedure that gives orthogonal basis functions for 3D images by

performing Nz+1 2D PCA decompositions. Even though the resulting basis function are not

as optimal as the ones that might be obtained by direct 3D PCA (the impractical method), we

will show experimentally that this approach works well for approximating 3D PSFs.

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2.2.

We are now ready to derive the Parallel Product-Convolution structure. Consider the scheme

represented in the Fig. 1(a). The fluorescent object to be studied is mounted under a coverslip

and placed below the objective lens. The space between the objective lens and the coverslip is

filled with the immersion medium (oil) whose refractive index matches with that of the cover-

slip. The refractive index of the object differs from that of the coverslip and oil, and is typically

assumed to be uniform. In essence, there is only one discontinuity along the optical axis that is

represented by a plane, which results in the depth varying response of the system.

Typical 3D imaging involves acquiring a series of 2D images for different values of z or

equivalently different positions of the focal plane [Fig. 1(a)]. Note that z represents the position

ofthefocalplanewithrespecttotheinterfaceplane.Ifthereisnorefractiveindexmismatch,the

3D image is expressed in terms of the object by a simple convolution with a 3D PSF. However,

to express the image in terms of the object under the coverslip when there is a mismatch, we

need the 4D function, h�(x,y,z,z�), which represents the series of 2D image resulting from

positioning the focal plane at z and a point source at z�[Fig. 1(b)].

Let f(x,y,z) be the object (fluorescent dye structure) and g(x,y,z) be the image acquired by

the microscope. The forward model is given by

�

In the above expression, z is the axial image coordinate and z�is the axial object coordinate and

the system response is a function of both.

Under this notion, when the system is depth independent, the response is a function of only

the difference z−z�. In other words, the depth invariance corresponds to following relation:

h�(x,y,z,z�) = h�(x,y,z−z�,0),

and Eq. (6) becomes a simple convolution given by

The PCA based imaging model

g(x,y,z) =

x�,y�,z�h�(x−x�,y−y�,z,z�)f(x�,y�,z�)dx�dy�dz�,

(6)

g(x,y,z) = h0(x,y,z)∗ f(x,y,z),

where h0(x,y,z) is the standard depth independent PSF given by h0(x,y,z) = h�(x,y,z,0).

The next step is to account for the focus shift [12] in the representation of h�. The term focus

shift signifies the fact that, for a given value of object depth variable z�, the maximum intensity

is observed when the image depth variable z is equal to az�, where a is a constant that depends

on the refractive indices of the mounting and immersion mediums. To account for the focus

shift, we consider the followed transformation [Fig. 1(c)]:

h(x,y,z,z�) = h�(x,y,z+az�,z�).

(7)

Note that now in the new representation, the reference plane for z is the plane containing the

bead with focus shift. This representation is advantageous in the sense that, for a fixed value

of x,y, and z, the variation with respect to axial object coordinate z�is lower compared to the

former representation.

The forward model with new representation becomes

�

The above equation can be equivalently represented by

�

g(x,y,z) =

x�,y�,z�h(x−x�,y−y�,z−az�,z�)f(x�,y�,z�)dx�dy�dz�,

g(x,y,z/a) =

x�,y�,z�h(x−x�,y−y�,z/a−z�,z�)f(x�,y�,z�)dx�dy�dz�,

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(a) Schematic of 3D acquisition

(b) Axial variables of h�

(c) Axial variables of h (Eq. (7))

Fig. 1. Schematic representation 3D depth dependent system

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The factor a in the above expression disappears if it is translated into discrete summations

assuming that discretization step size of z is a times larger than that of z�. Assuming now that

the variables represent discrete indices, the above model becomes

g(x,y,z) = ∑

x�,y�,z�h(x−x�,y−y�,z−z�,z�)f(x�,y�,z�),

(8)

Now, define the set of 3D images given by

S = {hz�(x,y,z) = h(x,y,z,z�),z�= 0,...,Nz−1}

We then apply a TP-PCA as described in the previous subsection. Let P1(x,y,z),...,PB(x,y,z)

be the selected principal components [Qk’s in Eq. (5)]. Then, the function h(·,·,·,·) can be

approximated as

h(x,y,z,z�) =¯h(x,y,z)+

B

∑

j=1

cj(z�)Pj(x,y,z),

(9)

where¯h(x,y,z) is the function obtained by averaging h(x,y,z,z�) along z�, and

�h(x,y,z,z�)−¯h(x,y,z)�)Pj(x,y,z)

Substituting Eq. (9) in Eq. (8) yields the final structure:

cj(z�) =∑

x,y,z

g(x,y,z) =¯h(x,y,z)∗ f(x,y,z)+

B

∑

j=1

Pj(x,y,z)∗[cj(z)f(x,y,z)],

(10)

where ∗ represents convolution. The equation [Eq. (10)] is our parallel product-convolution

(PPC) model. The complicated integral in Eq. (6) is approximated by simple expression in Eq.

(10) involving a set of standard convolutions with the input pre-multiplied by 1D functions. The

above approximation model can be re-expressed as in the equation [Eq. (6)] with h replaced by

h(B)

a , where the latter is given by

h(B)

a (x,y,z,z�) =¯h(x,y,z)+

B

∑

j=1

Pj(x,y,z)cj(z�).

(11)

Let Nx×Ny×Nzbe the image size. A quick examination of Eq. (10) might indicate that its

implementation will require B+1 number of 3D FFTs and a 3D IFFT. However, observing the

fact that input to B+1 convolutions differ only axially, it can be verified that B+1 number of

3D FFTs can be replaced by Nztimes 2D FFTs and then (B+1)NxNytimes 1D FFTs, providing

additional computational efficiency.

3.Results

We apply our method on both theoretical and measured PSFs and compare with the bilinear

interpolation method reported in [7]. In this method, the z�-axis is divided into some number

of intervals (strata) of thickness T and the DV-PSFs at the edge of the intervals are used to

approximate all the DV-PSFs using bilinear interpolation.

We used the same experimental set-up as in [5] to obtain measured DV-PSFs. DV-PSFs were

measured by positioning fluorescent beads at various depths (z�) using an optical trap (refer

to [5] for experimental set up). Imaging wavelength was 532 nm. The discretization step size

for x and y was 42 nm and that of z�was 75 nm. z step size was chosen appropriately to

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compensate for the focus shift. The objective numerical aperture was 1.49 NA. We also used

the same specifications to generate theoretical DV-PSFs using the method described in [4].

To visualize the DV-PSFs we utilize both xz sections for various values z�with y chosen to

be at the center of the PSFs, and xz�sections for various values z with y chosen similarly. In

all the visualizations, the top left corner corresponds to the origin of the axes. To make low

intensities visible, we display the fourth root of the intensity whenever the images are positive.

The exceptions are the approximation error images, and the PCA basis functions.

For a given B, we define the normalized approximation error as

���h−h(B)

where h(B)

a

is as defined in Eq. (11).

E =

a

���

2

�h�2

,

3.1.

We used 41 theoretical 3D PSFs of size 100×100×40 for {z�= i×75nm,i = 0,...,40}. Fig-

ure 2 displays the xz and xz�sections and Figure 3 shows few first PCA basis functions.

Application to theoretical PSFs

(a) xz section for various values of z�(b) xz�sections for various values of z

Fig. 2. Theoretical DV-PSF sections. Horizontal axis represent x. Image size is 2925 nm×

4200 nm.

Figure 4 compares the approximation error for fitting a set of 41 3D PSFs using either the

PCA method or the strata based method given in [7] for various number of PSF basis func-

tions. Note that the basis PSFs for the strata method are selected from the DV-PSFs themselves,

whereas for our method, the basis PSF functions are the mean PSF and the principal compo-

nents. The advantage of using the PCA method over the strata method is evident from the plot;

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Fig. 3. Basis functions for theoretical PSFs (xz sections). Image at the top left is the mean

PSF¯h. The remaining are the first 5 principal components.

2 PCA functions out perform 5 strata functions and 3 PCA functions are comparable to 8 strata

functions.

23456789 10 11

?7

?6

?5

?4

?3

?2

?1

No. of basis PSFs

Log of normalized MSE

Strata approx. error

PCA approx. error

Fig. 4. Approximation error for various values of number of basis PSFs. x-axis represents

number of PSFs used to approximate 41 PSFs spanning the range z�= [0,3000 nm]. PCA

method yields a significant advantage over the strata based method.

We now consider the special case of approximating the DV-PSFs with two bases PSFs. This

amountsto,forPCAmethod,usingthemeanPSFs¯handthefirstprincipalcomponent;whereas,

for the strata method, this means using the PSF corresponding to values of z�equal to zero and

bottom of the total range. Figure 5 compares the approximated PSFs with the originals for both

methods. In the strata method, for the PSFs at the top and the bottom, the approximated PSFs

are identical to the originals since these PSFs are themselves the basis functions. However, for

the total range of z�, the mean approximation error is much higher than that of PCA method

(about 17 times). This is evident from the approximated images in the middle (for z�= 750 nm,

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1500 nm, 2250 nm). Figure 6 shows the xz�sections of the same results. While visually less

obvious, comparable behavior extends to any number of basis functions.

3.2.

We used a set of 41 measured 3D PSFs obtained using the set-up described at the beginning of

this section. The results are analogous to that of the previous subsection; the only difference is

that the PSFs used here are real measured PSFs. Figure 7 displays the xz and xz�sections and

Fig. 8 shows first two PCA basis functions. We observed that, subsequent basis functions con-

tain background noise as well as high frequency structures that lie well beyond the theoretical

frequency support of the microscope, which possibly originate from the Poisson noise. This

indicates that two bases function are sufficient to represent the essential details of DV-PSFs for

the range of depth considered in this experiment.

Figure 9 compares the approximation error of the PCA method with that of the strata based

method for various number of bases PSFs. As with the theoretical PSFs, the PCA method

provides an improved approximation. However, the error does not fall off as fast as in the case

of theoretical PSF approximation, since the measured PSFs contain noise.

We now again consider the special case of approximating the DV-PSFs with two basis PSFs.

Figure 10 displays the results. It should be noted that, for the strata method, the calculated

PSFs at the top and the bottom are identical to the originals (including the background noise)

since these PSFs are themselves used as the basis functions. Whereas, in PCA method, the

approximations differ from the original both in terms of the noise and signal for these two

values of z�. However, for the other values of z�, the PCA method yields a smaller error, and all

of the approximated PSFs are nearly noise-free. This result is also evident from the displayed

error images [Fig. 10(b)]. In contrast to the case of theoretical PSFs, here the PCA method

provides only a two fold improvement in the error, due to the inclusion noise. Figure 11 displays

the xz�section of the same result for z = 0. We can see significant line artifacts in the strata

approximated PSFs which originates from interpolated noise.

Based on the above experimental observations, we now summarize the factors that render

the proposed method an attractive tool for constructing robust depth dependent deconvolution

algorithms. We first note that a robust deconvolution method typically uses a nonquadratic

regularization. It computes the required solution as a minimizer of a weighted sum of this reg-

ularization and a quadratic data fidelity term [8–10]. Such algorithms typically take about 300

seconds for computing a 3D stack of standard size. If these algorithms are extended for depth

dependent deconvolution, the computational time will be multiplied by a factor proportional to

the number of PSFs used. Hence using a large number of PSFs is not practical. Since the pro-

posed method approximates the DV-PSFs with a very few basis PSFs, it will be a promising tool

for solving depth dependent deconvolution problems using general purpose computers with a

moderate computational power. For example, a 5 PSF model, which approximates both exper-

imental and theoretical PSFs with more than 95% accuracy, offers a good trade-off between

computational complexity and accuracy.

Second, for a given number of basis PSFs, the proposed method has a lower computational

complexity than that of the strata method as explained at the end of section 2. For example,

when a 5-PSF model is used, the strata method will cause a 5 fold increase in the compu-

tational complexity when compared to depth invariant deconvolution, whereas the proposed

method will cause only about a 3 fold increase. This gain originates from the compactness of

the forward model expressed in the equation [Eq. (10)].

Another interesting point is that, since the proposed model automatically eliminates noise

without losing low intensity features in the DV-PSFs, the PCA-approach will reduce the poten-

tial for artifacts in the deconvolution results.

Application to measured PSFs

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(a) Approximated xz sections

(b) xz sections of normalized approximation error

Fig. 5. Results of approximating using two basis functions only. Displayed are the approx-

imated xz sections and error images for various values of z�. For z�= 0 nm, 3000 nm, strata

approximation is identical to the originals because the basis function therein are originals

themselves. However, for the other values of z�, PCA method gives better approximation

resulting in a 17 fold improvement in overall approximation error. The displayed error im-

ages are normalized to the overall maximum of the original PSFs.

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(a) Approximated xz�sections

(b) xz�sections of normalized approximation error

Fig. 6. xz�sections of the same results in Figure 5 for various values of z

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(a) xz section for various values of z�(b) xz�sections for various values of z

Fig. 7. Measured DV-PSF sections. Image size is 2925nm×4200nm.

Fig. 8. Basis functions for measured PSFs (xz sections). Image at the left is the mean PSF

¯h. The right one is the first principal component.

(C) 2010 OSA29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6473

#121941 - $15.00 USD Received 8 Jan 2010; revised 5 Mar 2010; accepted 5 Mar 2010; published 15 Mar 2010

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?1.35

?1.3

?1.25

?1.2

?1.15

?1.1

?1.05

?1

?0.95

No. of basis PSFs

Log of normalized MSE

Strata approx. error

PCA approx. error

Fig. 9. Approximation error for various values of number of basis PSFs. x-axis represents

number of PSFs used to approximate 41 PSFs spanning the range z�= [0,3000 nm]. PCA

method yields a better approximation.

4. Conclusions

We have developed a PCA based approximation model for use in deconvolving the depth vary-

ing response of widefield microscopes. When compared to the only available method in liter-

ature, namely the strata based interpolation method, the PCA method results in a much lower

approximation error for any given number of model PSFs. Simultaneously, the PCA method

automatically eliminates the noise in the measured PSFs, in contrast to the strata based method.

Using this new model in any iterative deconvolution algorithm is thus expected to result in a

greatly improved restoration of biological structures.

(C) 2010 OSA29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6474

#121941 - $15.00 USDReceived 8 Jan 2010; revised 5 Mar 2010; accepted 5 Mar 2010; published 15 Mar 2010

Page 15

(a) Approximated xz sections

(b) xz sections of normalized approximation error

Fig. 10. Results of approximating using two basis functions only. Displayed are the approx-

imated xz sections and error images for various values of z�. For z�= 0 nm, 3000 nm, strata

approximation is identical to the originals because the basis function therein are originals

themselves. However, for the other values of z�, PCA method gives better approximation.

(C) 2010 OSA29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6475

#121941 - $15.00 USD Received 8 Jan 2010; revised 5 Mar 2010; accepted 5 Mar 2010; published 15 Mar 2010