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Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy José-Angel Conchello* and James G. McNally*+ Institute for Biomedical Computing* and Biology Department+, Washington University, St. Louis, MO 63110 ABSTRACT Maximum likelihood image restoration is a powerful method for three-dimensional (3D) computational optical sectioning microscopy of extended objects. With punctate specimens, however, this method produces a few very bright isolated spots and dim detail around them is lost. The commonly used regularization methods (sieves and roughness penalty) decrease the amplitude of the bright spots, but do not avoid loosing dim detail. We derived an intensity regularization that decreases the amplitude of bright spots without loosing dim detail. In contrast with other regularization methods, this method does not increase significanlty the computational complextity of the estimation algorithm. Keywords: three-dimensional microscopy, deconvolution, regularization. 1. INTRODUCTION Optical sectioning microscopy is a powerful tool for the non-invasive visualization of fluorescentlystained three-dimensional (3D) biological specimens. In optical sectioning microscopy a series of two-dimensional (2D) images is collected focusing the microscope at different focal planes through the specimen. Each 2D image in this through-focus series contains the in-focus plane plus contributions from out-of-focus material that obscures the image. The effect of out-of-focus contributions may be reduced optically or computationally. In the optical approach, a microscope is used that collects less out-of-focus light than the conventional fluorescence microscope and thus has better optical axis resolution. Several approaches exist to increase the optical axis resolution of fluorescence microscopes, such as confocal scanning microscopes [see 1,2 and references therein], twophoton fluorescence excitation3-5, and standing-wave or synthetic excitation-wave fluorescence microscopes6,7. Although all the optical methods do improve the optical axis resolution of the microscope, there is some residual axial blur that can be removed by computational methods8-11. In the computational approach the recorded image is processed by a computer to undo the degradations introduced by the process of image formation and recording. Several methods have been devised to reduce the out-of-focus contributions8-17. From these methods, we have found that those based on maximum-likelihood (ML) image estimation10-17 assuming that the image follows Poisson statistics are very successful at restoring images of extended objects. In fact, we have shown [16] that these methods are capable of restoring a significant amount of frequency components that are lost during the imaging process, and thus are capable of meaningful band extrapolation. Unfortunately, the application of ML image estimation to microscopical images of mesh-like and punctate biological specimens has been less successful. The ML image estimate is dominated by a few very bright spots and dimmer structures near these bright spots are lost in the estimation process. The presence of overly bright spots in the estimates is a known problem of ML estimation. For example, given a series of observations, the ML estimate of a probability density function is a series of impulses at the observation points [30]. The common approach to avoid these overly bright isolated spots is to regularize the estimation process using any of the existing regularization approaches, such as the method of sieves31 or the maximum penalized likelihood (MPL) approach32. The existing regularization methods successfully ameliorate the presence of overly bright pixels. Unfortunately, these methods still loose many dim structures around bright spots. In this article we present a new regularization method based on MPL that not only reduces the brightness of the bright isolated spots, but also preserves the dim structures around these spots. The method presented here, furthermore, has lower computational complexity than the existing methods. The article is organized as follows. Section 2 presents a brief overview of the computational approach to optical sec-

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tioning microscopy. Section 3 presents an overview of maximum likelihood image estimation, the expectation maximization (EM) algorithm, and the most commonly used regularization methods. In Section 4 we present the derivation of our new regularization method and in Section 5 the results of applying it to microscopical images of punctate and mesh-like biological specimens are presented. Finally, Section 6 summarizes our conclusions and lists some problems that remain to be addressed. 2. COMPUTATIONAL OPTICAL SECTIONING MICROSCOPY As mentioned in the introduction, in computational optical sectioning microscopy (COSM) a 3D microscopical image is obtained by recording a through-focus series of 2D images. A computational method is then used to remove the contributions from out-of-focus fluorescence that obscure the image and reduce contrast. Several deconvolution algorithms have been devised to reduce these out-of-focus contributions8-17. In mathematical terms, the purpose of these algorithms is summarized as follows. Given one or more recorded images g(xi), where xi=(xi, yi, zi) is a 3D point in image space I, these methods aim to find the specimen function s(xo) –the distribution of fluorescent intensity at pixel xo = (xo, yo, zo) in object space O, that best matches the imaging equation g ( xi) =

∫ h ( x i – x o ) s ( x o ) dx o

= h ( xi) * s ( xi)

(1)

O

where h(•) is the microscope’s point spread function (PSF), the image of a point source of light, and * denotes convolution. The PSF may or may not be known and, in the cases where it is unknown, the PSF must be estimated together with the specimen function in a process called blind deconvolution [see for example27-29, and references therein]. The main difference between the many existing methods is the definition of a best match, commonly used approaches include least squares22-25,36, projection into convex sets26,33-35 and statistical image restoration (SIR). In SIR each recorded image is assumed to be a realization of a random process whose probability distribution depends on the specimen function and the PSF. The specimen function and, if necessary, the PSF, are then estimated using any of the well-known approaches to statistical parameter estimation37, such as maximum likelihood, maximum a posteriori probability, or maximum entropy, to mention a few. We have found that the ML estimation based on Poisson image statistics –a reasonable assumption for fluorescence microscopy10,11, is a successful approach to SIR10,11,16. In the next section we present a brief description of the ML approach and present its drawbacks and possible ways to overcome them. 3. EXPECTATION MAXIMIZATION ALGORITHM FOR MAXIMUM-LIKELIHOOD IMAGE RESTORATION 3.1. Introduction In the maximum likelihood approach to SIR the solution to Equation (1) is the non-negative function sˆ ML ( • ) that maximizes the log-likelihood functional38 L [ s ( •) g ( •) ] = –∫

I

∫O h ( xi – xo ) s ( xo ) dxo – g ( xi ) log { ∫O h ( xi – xo ) s ( xo ) dxo }

dx i .

(2)

For the non-trivial task of maximizing the log-likelihood functional, we use the expectation-maximization formalism of Dempster, Laird, and Rubin39 which leads to the iterative algorithm ( k) ( x ) sˆ EM g ( xi) o ( k + 1 ) ( x ) = ---------------------- × ∫ h ( x i – x o ) ---------------------- dx i , sˆ EM o H0 gˆ ( k ) ( x ) I

(3)

i

where the superscript (k) denotes iteration number, gˆ ( k ) ( x i ) =

( k ) ( x ) dx ∫ h ( xi – xo ) sˆ EM o o

(4)

O

is the image of the specimen function estimate at iteration (k), and H0 =

∫ h ( x o ) dx o .

(5)

O

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In Equations (3) and (4) wa have changed the subscript ML to EM to denote the choice of algorithm for the maximization of the likelihood function.This algorithm has proved a powerful method for three-dimensional (3D) restoration of fluorescent microscopical images of extended objects16. Remarkably, this method is capable restoring spatial-frequency components within the missing cone of frequencies intrinsic to non-confocal microscopy30, but to avoid these overly bright isolated spots, the estimation process must be regularized. 3.2. Common regularization techniques. To regularize the ML estimation problem, several methods have been proposed . A regularization method is including the method of sieves31, widely used for nuclear medical images. In this method, the specimen function estimate is constrained to be a low-pass object, i.e. (6) sˆ s ( x ) = h s ( x ) * bˆ ML ( x ) where hs(•) is the sieve kernel, a non-negative smoothing kernel, bˆ ML ( • ) is the ML solution of g ( x) = hr ( x) * b ( x) ,

(7)

hr ( x) = h ( x) * hs ( x)

(8)

and is the PSF for the sieve-regularized estimation problem. The method of sieves effectively reduces the brightness of isolated bright spots by spreading their intensity over the support of the sieve kernel. A drawback of this algo( • ) suffers at least the same loss of dim detail as the unregularized ML estimate sˆ ( ˙• ) . This is rithm is that bˆ ML

ML

easily explained from Equations (7) and (8) as follows. Because hs(•) is a low-pass filter, hr(•) causes more blur than the microscope PSF h(•), thus bˆ ML ( • ) is sharper than the unregularized ML estimate (except at places where sˆ ML ( • ) approaches an impulse). Therefore, at the pixels where sˆ ML ( • ) has isolated bright spots, bˆ ML ( • ) will also have isolated spots, potentially brighter than those in sˆ ML ( • ) , with the corresponding loss of dim structures around these spots. Application of Equation (6) will not recover these dim structures, on the contrary, any low-contrast structure in bˆ ( • ) will be lost because of the smoothing effect of the low-pass filter hs(•). ML

Another method commonly used to regularize the ML estimation problem is maximum penalized likelihood (MPL). In MPL a new merit function is defined by adding a term to the likelihood or log-likelihood functional to penalize any undesired behavior of the ML estimate, i.e. M ( s g ) = L ( s g ) – αR ( s ) (9) Where R[•] is the penalty functional and α is a constant factor that weights the contribution of the penalty term to the merit functional M(s|g). The MPL estimate of s(•) is the function that maximizes this functional. Clearly, in the limit as α→0, the MPL estimate approaches the ML estimate. A widely used penalty term R(•) is Good’s Roughness32,13 R [ s ( •) ] = RR [ s ( •) ] =

∫ ∇s ( x )

2 dx

(10)

O

which penalizes rapid changes in the specimen function. This penalty term has an effect similar to the method of sieves, in that the specimen function estimate is forced to be smooth, and this smoothness sacrifices low-contrast structures in the restoration. In the next section we present an MPL regularization method that does not sacrifice dim structures of low contrast. 4. INTENSITY PENALTY Instead of using Good’s roughness penalty we propose a penalty functional that penalizes only the overly bright spots without enforcing smoothness on the restoration, this penalty functional is

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R [ s ( •) ] = R0 [ s ( •) ] =

∫O [ s ( x o ) ]

2

dx o .

(11)

To derive an algorithm to maximize the penalized likelihood (Equation (9)) with R given by Equation (11) we use the EM formalism as presented in Sect 3.3 of38. In the EM algorithm for maximum likelihood (ML) estimation two data sets are defined, the unobservable complete data set and the observable incomplete data set. In 3D fluorescence microscopy the complete data set has commonly been defined as the specimen function s(xo) to be estimated and the incomplete data set as the recorded image g(xi), the relation between image and object is given by the imaging equation (Equation (1)). The log-likelihood functional of the complete data set is then defined Lcd[s]. For computational optical-sectioning fluorescence microscopy this functional has the form L cd [ s ( • ) ] = – ∫ s ( x o ) dx o + ∫ log [ s ( x o ) ] N ( dx o ) O

O

,

where N(dxo) is the number of photons emitted from a small region [xo, xo+dxo] in object space and is assumed to be a Poisson-distributed random variable with mean s(xo). Each iteration of the EM algorithm consists of two steps. In the expectation step (or E-step), the expectation of the complete-data log-likelihood is calculated, conditioned on the current estimate of the complete data set and on the recorded realization of the incomplete data set Q s ;sˆ

( k)

( k)

= E { L cd [ s ] |g, sˆ

}

(12)

( k)

( • ) is the estimate of the specimen function at iteration k. The where E{} denotes statistical expectation and sˆ maximization step (or M-step) consists of finding sˆ

( k + 1)

( • ) = argmax Q s, sˆ

( k)

.

s≥0

(13)

The EM algorithm may also be applied to MPL estimation, i.e. to the maximization of the functional M[s|g] in Equation (9). For the EM algorithm the penalty term is applied to the log-likelihood of the complete data set. The E-step thus consists of finding the conditional expectation of the penalized log-likelihood of the complete data set, i.e. Q s ;sˆ

( k)

= E { L cd [ s ] – α R [ s ] |g, sˆ

( k)

} .

(14)

The M-step remains unchanged. It has been shown40 that repeated application of Equations (12) and (13) produces a sequence of estimates sˆ

( k)

( • ) for k = 0, 1, 2, … such that the sequence of log-likelihood values L sˆ

is non-decreasing, and that if the iteration reaches a stationary point sˆ

( k + 1)

( x o ) = sˆ

( k)

( k)

g

( x o ) for all xo, then

( k)

sˆ ( • ) is a maximizer of the log-likelihood functional, and thus a maximum-likelihood estimate of the specimen function s(•). Likewise, repeated application of Equations (12) and (14) lead to the MPL estimate of the specimen function s(•). Substituting the intensity penalty (Equation (11)) into Equation (14), and substituting the complete data log-likelihood from reference [38] the E-stetp gives Q s ;sˆ

( k)

= – ∫ s ( x o ) dx o + ∫ log [ s ( x o ) ] E { N ( x o ) sˆ O

( k)

O

, g } dx o – α ∫ [ s ( x o ) ] 2 dx o ,

(15)

O

where we substituted E { N ( dx o ) sˆ

( k)

, g } = E { N ( x o ) sˆ

( k)

, g } dx o

.

The M-step is easily found applying the Euler Equation for variational calculus42 to Equation (15). This gives

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( k)

E { N ( x o ) sˆ , g } ------------------------------------------------ – 2αsˆ ( k + 1 ) ( x o ) – 1 = 0 . sˆ ( k + 1 ) ( x o )

(16)

Equation (16) may be written as a quadratic equation in sˆ ( k + 1 ) ( x o ) . Then using E { N ( x o ) sˆ

( k)

, g } as defined

by Equation (3.17) in reference38 we get sˆ MPL where α0 = 4α, and sˆ EM

( k + 1)

( k + 1)

( k + 1)

– 1 + 1 + 2α 0 sˆ EM ( xo) ( x o ) = --------------------------------------------------------------------------- , α0

(17)

( x o ) is given by Equations (3) and (4) above (using sˆ MPL

( k)

( x o ) on the righ-hand

side). Straightforward application of L’Hospital’s rule shows that when α0 → 0, sˆ MPL sˆ MPL

( k + 1) ( k + 1)

( x o ) → sˆ EM

( k + 1)

( x o ) for all xo as expected. From Equation (17), it is clear first that

( x o ) ≥ 0 for all xo and k, and second that, because the intensity penalty does not couple the samples of

the specimen function s(•), the M-step of the MPL-EM algorithm may be carried out in closed form. In contrast, Good’s roughness penalty (Equation (10)) couples each pixel in the restoration with its adjacent neighbors in such a way that a closed form for the M-step is not possible. In fact, the M-step for Good’s roughness requires the solution of a non-linear partial differential equation, greatly increasing the computational complexity of the algroithm. To understand how the intensity penalty works, it is useful to analyze Equation (17) in two regimes. First, for bright pixels where α 0 sˆ EM

( k + 1)

( xo) » 1 ,

(18)

Equation (17) may be approximated as α 0 sˆ MPL

( k + 1)

( x o ) ≈ 2α 0 sˆ EM

( k + 1)

( x o ) < α 0 sˆ EM

( k + 1)

( xo) ,

(19)

and thus Equation (17) effectively penalizes bright pixels (note that the inequality in Equation (19) does not hold in general, but is true if Equation (18) holds). For dim pixels such that α 0 sˆ EM

( k + 1)

( x o ) « 1 , the square root in

Equation (17) may be approximated as (1+2x)1/2 ≈ 1+x, so sˆ MPL

( k + 1)

( x o ) ≈ sˆ EM

( k + 1)

( xo)

,

and thus, as desired, dim detail in the restored image is preserved by the MPL-EM algorithm with an intensity penalty. The preceeding analysis may also be useful to determine an approprite value of α0 to use, namelly 1/α0 should be set to approximately the largest pixel value expected in the estimated specimen funciton. For images where the bright pixels occur in isolated regions comparable in size to the diffraction limited spot, the maximum value of the specimen function may be approximately estimated as the maximum pixel value in the image divided by the maximum value in the PSF. Likewise, for images where the bright pixels occur along line-structures, the maximum value of the specimen function may be approximately estimated as the maximum image value divided by the maximum value of the line-spread function (the integral of the PSF with respect to either x or y). 5. RESULTS 5.1. Experimental methods The images presented here were collected using an Olympus IMT2 inverted microscope using a 100x/1.3 NA oil-immersion objective. Pixel sizes are 0.068 µm laterally. The raw image has 64 planes with 256 rows and 256 columns. A PSF was calculated over the same grid using the theoretical model of Gibson and Lanni43. After

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data collection, the image was transmitted via FTP to either a DEC/5000-400 computer (Digital Equipment Corp. Nashua, NH) with 160 MBytes RAM and a 175 MHz alpha CPU or to a SGI Challange-L computer with four R8000 processors and 712 Mbytes RAM. The algorithms were implemented in Fortran77 and compiled with the f77 compiler with the highest level of optimization available on either computer. The convolutions in Equations (3) and (4) were carried out using discrete Fourier transforms (DFT) implemented with 3D fast Fourier transforms (FFT) using the row/column-decomposition method41 using the 1D FFT routines from netlib’s fftpack.a library (http://netlib.att.com/netlib/fftpack). Convolutions carried out by DFT methods are cyclic. The PSF is compact in the lateral direction and thus the cyclic convolution is not a serious problem in this direction. The axial extent of the PSF, however, is more significant and thus cyclic convolution presents a problem in the axial direction. The common approach of zero-padding the two arrays to twice their axial support, however introduces an edge in the two arrays. This problem is not severe for the PSF whose amplitude decreases as 1/z2. For the specimen function s(•) and for the ratio g ⁄ gˆ , however, this edge causes large oscillations in the estimated specimen function at the location of these edges. To avoid introducing an edge in the specimen function s(•) and in the ratio g ⁄ gˆ in Equation (3), instead of zero padding these arrays, we extend their axial extent to twice the original support by forcing them to have even symmetry in z. The advantage of the approach we use is that it changes little the frequency content of the extended arrays relative to the original arrays. With these extended convolutions one iteration of Equations (3) and (4) takes about one minute of CPU in either computer (DEC alpha or Challange-L). The CPU time added to evaluate Equation (17) is negligible. 5.2. Experimental results The algorithm given by Equations (3), (4), and (17) was applied to the image described above. Figure 1 shows the raw image as well as the unregularized restoration (α0=0). In these images a maximum intensity projection through the 3D stack is shown, that is, for a given pixel (xp, yp) in the projection, the largest pixel value for all z is displayed This image illustrates the problem of the unregularized restoration, in the area marked by the circle, a pixel becomes very bright in the restoration and a dimmer structure above and to the left disappears. For comparison, the restoration using Good’s roughness penalty for α=10-3 and α=10-4 is shown in Figure 2. From this figure we see that for α=10-3 the estimate of the specimen function is very smooth. For α=10-4 the estimated specimen function is less smooth, but the dim structure above and to the left of the bright pixel in the circle is missing in both estimates. For α=10-5, the specimen funciton estimate is very similar to the unregularized ML

unregularized restoration

raw

Figure 1. Raw image (left) and unregularized ML restoration after 3000 iterations of the EM algorithm (right). In the area marked by the circle a pixel becomes very bright in the restoration and the structure at the above and to the left, which is present in the raw image, disappears

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α1=10-3

α1=10-4

Figure 2. Restorations regularized by Good’s roughness penalty after 3000 iterations for two different weights of the penalty functional, α1=10-3(left) and α1=10-4(right). Notice that the dim structure above and to the left of the bright pixel in the circle is still lost from the restoration estimate for the same number of iterations, this is a sign that for this value of α1the penalty term has little or no effect on the merit function. We applied the intensity regularized MPL-EM algorithm to the same image. The value of α0 obtained using the criterion described in Seciton 4 above (namely α0=max{h}/max{g}) is α0=4×10-5. The results for this value of α0 are shown in Figure 3. From this figure it is clear that the MPL-EM algorithm with an intensity penalty successfully preserves the dim structure that is not present in the unregularized and roughness-regularized restorations. When the weight factor is larger than this recomended value (see Figure 4 (left)), the restoration is less sharp, but the dim detail is still preserved, on the other hand, a smaller α0=10-5(Figure 4 (right)) reduces the

Figure 3. Specimen function estimate after 1000 iteraitons of the MPL-EM algorithm with intensity penalty using α0=4×10-5, the value determined with the criterion described in Section 4. Notice that the dim detail above and to the left of the bright pixel in the circle shows clearly.

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α0=10-4

α0=10-5

Figure 4. Specimen function estimate after 1000 iterations of the MPL-EM algorithm with intensity penalty using α0=10-4 (left) and α0=10-5 (right). The larger α0 the less sharp the estimated specimen function, the smaller this value, the less visible dim detail is. weight of the penalty term in the merit function and dim detail starts to disappear. For α0 smaller than this value, the estimate of the specimen function becomes very similar to the unregularized estimate. 6. CONCLUSIONS AND FUTURE WORK We have derived a regularization method that succesfully avoids loosing dim detail around bright pixels in the specimen fiunction estimate. This method is based on the maximum penalized likelihood approach to statistical image restoration. Each iteration of the algorithm we derived consists of one iteration of the unregularized EM algorithm, followed by a step that penalizes bright pixels. The computational complexity added by this step is negligible compared to the complexity of the unregularized EM. The method we derived, however, presents at least one problem that we are currently addressing. This is, the weight of the penalty is the same over all the specimen estimate. Thus if two areas of the image have different overall intensitites, both with punctate structures, the intensity penalty will act on the area of largest intensity, but will have little or no effect on the area of lower intensity. We are currently investigating the possibility of a space varying penalty term. 7. ACKNOWLEDGMENTS The images used for this article were provided by Dr. B. Pickard and Dr. C. Reuzeau. We thank Joanne Markham for the programs for Good’s roughness MPL-EM. The support of NIH grants RR01380 and R01GM9798 and NASA/NSF grant IBN 941601 is greatfully acknowledged. 1

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Conchello and McNally

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Fast regularization technique… SPIE 2655-27