A statistical approach for intensity loss compensation of confocal microscopy images

Transcription

1 Journal of Mcroscopy, Vol. 230, Pt , pp Receved 4 Aprl 2007; accepted 2 October 2007 A statstcal approach for ntensty loss compensaton of confocal mcroscopy mages S. GOPINATH, Q. WEN, N. THAKOOR, K. LUBY-PHELPS & J. X. GAO Computer Scence and Engneerng Department, Unversty of Texas at Arlngton, U.S.A. Electrcal Engneerng Department, Unversty of Texas at Arlngton, U.S.A. Cell Bology Department, Unversty of Texas Southwestern Medcal Center, Dallas, TX, U.S.A. Computer Scence and Engneerng Department, Unversty of Texas at Arlngton, U.S.A. Key words. Compensaton, confocal mcroscopy, expectaton maxmzaton, ntensty loss, photobleachng. Summary In ths paper, a probablstc technque for compensaton of ntensty loss n confocal mcroscopy mages s presented. For sngle-colour-labelled specmen, confocal mcroscopy mages are modelled as a mxture of two Gaussan probablty dstrbuton functons, one representng the background and another correspondng to the foreground. Images are segmented nto foreground and background by applyng Expectaton Maxmzaton algorthm to the mxture. Fnal ntensty compensaton s carred out by scalng and shftng the orgnal ntenstes wth the help of parameters estmated for the foreground. Snce foreground s separated to calculate the compensaton parameters, the method s effectve even when mage structure changes from frame to frame. As ntensty decay functon s not used, complexty assocated wth estmaton of the ntensty decay functon parameters s elmnated. In addton, mages can be compensated out of order, as only nformaton from the reference mage s requred for the compensaton of any mage. These propertes make our method an deal tool for ntensty compensaton of confocal mcroscopy mages that suffer ntensty loss due to absorpton/scatterng of lght as well as photobleachng and the mage can change structure from optcal/temporal sectonto-secton due to changes n the depth of specmen or due to a lve specmen. The proposed method was tested wth a number of confocal mcroscopy mage stacks and results are presented to demonstrate the effectveness of the method. Correspondence to: Jean X. Gao. e-mal: gao@uta.edu Ths work s supported n part by Natonal Scence Foundaton NSF IIS , NSF IIS , NSF IIS Introducton Images produced by confocal mcroscope tend to decrease n ntensty wth tme as an effect of photobleachng when conventonal fluorescence tags are used or wth depth due to absorpton or scatterng of exctaton and fluorescence. These effects make analyss of the mages wthout ntensty correcton a complcated problem. Methods used to compensate the ntensty loss can be categorzed nto two types, Pre-processng methods: Ones that correct the ntensty loss wth modfed optcs as the mages are beng captured (Atkns & De Paula, 1994; Chen et al., 1995; Becker, 1996; Song et al., 1996). More recently, a method, whch manpulates the photomultpler gan, was ntroduced to counter the hgh ntensty losses n the deep layers of the specmen by Ĉapek et al. (2006). Post-processng methods: These methods compensate the mages after they are captured (Rgaut & Vassy, 1991; Oostveldt et al., 1998; Ortz et al., 1999; Kervrann et al., 2004; Ĉapek et al., 2005). Intensty decay functon (IDF) based methods model ntensty loss n the mages as a parametrc decay functon of depth or tme. The decay parameters are estmated and compensated for n these methods. Another famly of methods reles on matchng hstogram profles of mage stacks. These methods however cannot handle change n mage structure along the optcal axs. Optcs-based methods assume that the majorty of the ntensty loss s due to absorpton and scatterng of lght as t travels through the specmen. As the rate of photobleachng can vary for dfferent types of specmens, ntensty loss cannot be deally compensated by optcs alone. For ths reason, we concentrate on post-processng methods to correct the ntensty loss. Journal complaton C 2008 The Royal Mcroscopcal Socety

2 144 S. GOPINATH ET AL. As factors contrbutng to the ntensty loss cannot be modelled accurately for practcal mages accordng to Wu & J (2005), t poses a problem when IDF s used for ntensty compensaton. In addton, the combnaton of ntensty loss due to photobleachng and depth can gve rse to a complcated IDF. Our method s motvated by hstogram matchng; however, t deals wth a contnuous doman by modellng an mage as a mxture of two Gaussan probablty dstrbuton functons (PDFs) and matchng the profles of foreground probablty dstrbutons. By matchng foreground and not the entre mage, our method avods the problems arsng due to change n structure of the mage. Dfferent postprocessng approaches to correct the ntensty varatons can be found n the lterature. Negahdarpour & Yu (1993) apply a general model n whch the horzontal and vertcal flow felds as well as addtve and multplcatve ntensty relatonshps are estmated for every pxel. Accordng to Ĉapek et al. (2005), ths approach s computatonally expensve. A least-squares optmzaton-based approach, whch optmzes brghtness and contrast, s proposed by Peraswamy & Fard (2003) and Kervrann et al. (2004). These technques are hghly senstve to outlers. The reweghed least-squares method s used by Kervrann et al. (2004) to correct the dsadvantage of Peraswamy & Fard (2003). The method dscussed n Kervrann et al. (2004) s not only senstve to nose whch can be elmnated by medan flterng but also to the dynamc movement of objects n neghbourng optcal sectons. Accordng to Ĉapek et al. (2005), ths gves erroneous and unstable results even n the presence of a very few outlers n optcal sectons. In Cox et al. (1995), ntensty varatons are corrected based on hstogram warpng, but t s restrcted to the case where a global, spatally nvarant, nonlnear, monotoncally ncreasng relatonshp exsts between the ntenstes of the two mages. Ĉapek et al. (2005) extend the approach of Cox et al. (1995) and attempt to gve a general and fully automatc method of correctng ntensty loss n confocal mcroscopy mages. The proposed method manpulates the mage hstogram as n Ĉapek et al. (2005), but t focusses on the contnuous doman of probabltes to flter the foreground nformaton to calculate the correcton parameters. Before we present our approach, we wll dscuss the approach by Ĉapek et al. (2005) n short. The approach proposed n Ĉapek et al. (2005) conssts of two stages. In the frst stage, a standard hstogram s constructed wth the help of hstograms of all the optcal sectons n the mage stack. In the second stage, ndvdual hstograms are warped accordng to the standard hstogram to acheve the brghtness and contrast of the standard hstogram. The constructon of standard hstogram s adopted from Nyul et al. (2000). The approach s based on landmarks chosen n the mage hstogram. The landmarks chosen are the mnmum and maxmum ntenstes and percentles of the ntenstes of the mage. However, mnmum and maxmum ntensty of mage are hghly senstve to nose. For the mages that change the structure from optcal secton-to-secton, proporton of the foreground to the background vares. Ths causes substantal changes n hstograms, makng hstogram-based methods less effectve. The man dea of the proposed method s to flter the foreground nformaton from a gven mage by modellng t as a mxture of Gaussan PDFs and use ths nformaton to compensate mage ntensty loss. The foreground mean and standard devaton are used to transform the pxel ntenstes of the orgnal mage relatve to the ntensty parameters of a reference mage. The paper s organzed as follows: Secton 2 explans the proposed approach n detal. Secton 3 presents expermental results. Paper concludes n Secton 4. Proposed approach In many statstcal applcatons, Gaussan mxture modellng (GMM) s used as a general tool for modellng a large heterogeneous populaton. Detaled ntroducton to GMM can be found n Theodords & Koutroumbas (1999). GMM s a sem-parametrc estmaton approach that provdes good flexblty and precson n modellng the statstcs of unlabelled sample data. In our case, the mage data can be assumed to be generated from two components, one formng the Fg. 1. Flowchart for the proposed compensaton algorthm, numbers n the bracket ndcate the correspondng equatons.

3 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 145 Fg. 2. Sequence 1: Orgnal optcal sectons at tme t = 1 from (a) top (z = 1) to (l) bottom (z = 12) (bottom frame s used as the reference). background of the mage and the other pertanng to the foreground of the mage. However, t s not known that whch pxel belongs to whch component. Because of ths, the problem can be consdered to have mssng data, that s, background/foreground membershp nformaton. Each component can be consdered to have ts own parameters θ, whch defne the probablty densty functon P (x;θ). These parameters can be estmated through the expectaton maxmzaton (EM) algorthm, whch s the wdely used approach to solve the mssng data problem. It devses approprate parameters for the chosen model wth respect to the data ponts generated by ndvdual components. In the EM algorthm, ntal estmates for the parameters are chosen arbtrarly. As the selecton of ntal estmates affects the result, they must be chosen carefully. The teratve parameter estmaton process conssts of two steps, the expectaton (E) step and the maxmzaton (M) step. In the E step, the expected value of the mssng data s calculated. In the M step, the resultng value of the expectaton s maxmzed by selectng new set of parameters. The E and M steps are terated untl a stoppng crteron such as a number of teratons s met or untl there s no change n the mxture model parameters. Most of the mages captured wth confocal mcroscopy are bmodal, one mode each for background and foreground. Hence, the

4 146 S. GOPINATH ET AL. Fg. 3. Sequence 1: Foreground membershp probablty for optcal sectons at tme t = 1 (Brght regons denote hgher foreground membershp probablty and dark regons denote hgher background membershp probablty). mage data s modelled as a two-component GMM. Based on the assumpton that the loss of ntensty ncreases relatvely wth tme or depth or both, the frst mage of the tme seres or the frst z-slce of the stack wll have mnmal loss of ntensty and can be consdered as the reference mage. The reference mage should have good vsual nformaton of the object or specmen to be studed. Intally, mean ntensty and standard devaton for the foreground and background are estmated wth EM algorthm. Then the parameters of the foreground component are used to warp each pxel of the mage to ts relatve reference ntensty. Followng subsectons explan ndvdual steps taken durng ths process n detal. Parameter estmaton For a two component GMM of the j th mage n a stack, there are sx unknown parameters, { (w θ j j = 1,μj 1,σ j ) ( j 1, w 2,μj 2,σ j ) } 2, (1)

5 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 147 (a) Foreground mean nrensty Tme Depth (b) Foreground mean nrensty Tme Depth Fg. 4. Sequence 1: Mean ntensty of foreground regon n (a) orgnal mage stack and (b) restored mage stack.

6 148 S. GOPINATH ET AL. (a) Foreground mean ntensty Before, t=10 Before, t=20 60 Before, t=30 After, t=10 After, t=20 After, t= Depth (b) Foreground mean ntensty Before, z=3 Before, z=6 Before, z=12 After, z=3 After, z=6 After, z= Tme Fg. 5. Sequence 1: Varaton n the mean ntensty of foreground regon of orgnal mage stack and restored mage stack (a) wth depth (b) wth tme. where w j 1, w j 2 are mxture weght constants, μ j 1, μ j 2 represent mean ntenstes and σ j 1, σ j 2 gve the standard devatons correspondng to background and foreground Gaussan PDF, respectvely, for the j th mage. The frst step s to estmate the membershp probablty for each nth pxel of j th confocal mcroscopy mage. Gven ntensty for ths pxel s xn, j the membershp probablty can be calculated as, where P j F j (n) = w j 1 P j 1 P j ( j ) xn ( j ) j xn + w 2 P j 2 w j ( x j n ), (2) { ( j ( ) x j 1 xn μ j ) 2 } n = σ j exp 2π 2 ( σ j ) 2. (3)

7 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 149 In the above equatons, = 1, 2 and n = 1, 2,..., N M, where N M s the dmenson of the mage and j = 1, 2,..., K wth K beng the number of the mage slces. In the second step, the Gaussan PDF mxture parameter values are estmated from the above membershp probablty: σ j = w j = M N 1 F j (n), (4) M N n=1 M N μ j n=1 = F j (n) xn j M N n=1 F j (n) M N n=1 F j (n) (xn j μ j M N n=1 F j (n), (5) )2. (6) The above two steps are terated untl all the parameters converge. The teratve process s repeated for each mage n the stack. Image warpng Once the mxture parameters for the mage are known, ntensty can be compensated by compensatng these parameters to match some reference. The compensated ntensty for the nth pxel n the j th mage can be calculated from orgnal ntensty x j n as: { ( j x (x ) n j = F j 2 (n) n μ j ) } 2 σ r + μ r + F j 1 (n) x n j. (7) σ j 2 Here, μ r and σ r represent mean and standard devaton references for the foreground. Mean and standard devaton of the frst mage n the stack can be set as the reference parameters for the mage stack restoraton. However, n the case where the frst mage n the stack does not have enough detals or s not the brghtest, one of the other mages can be chosen to be the reference. Fgure 1 summarzes the entre compensaton algorthm. Expermental results The proposed approach was mplemented n MATLAB and was tested on several sets of mages n the Bovson lab database at Unversty of Texas at Arlngton. Before proceedng to the expermental results, we wll brefly talk about the ntalzaton used for the experment. Intalzaton s crucal for the EM algorthm. As parameters from the reference frame are needed for mage restoraton, EM s carred out on the reference mage frst. The mxture weghts for the reference frame m can be ntalzed as w m 1 = wm 1 = 0.5. Snce the background mean s lower than the overall mage mean and foreground mean s hgher, one can select the ntal values arbtrarly to follow ths restrcton, M N 1 μ = xn m M N,μm 1 = μ 2,μm 2 = 3μ 2. n=1 A good ntal value for the mxture standard devaton s the overall standard devaton of the mage. σ1 m = σ 2 m = 1 (M N) 1 M N n=1 (x m n μ)2. These values can be mproved upon by randomly usng varous ntalzatons and then choosng the one that maxmzes the membershp probabltes. However, reasonable fxed values as stated above were used for the repeatablty of the experment. After successful completon of the EM procedure for reference frame m, reference parameters are set as μ r = μ r 2,σ r = σ r 2. As any mage n the sequence s very smlar to ts prevous mage, the parameters of the prevous mage after EM are used to ntalze the EM procedure for the next mage. w j 1 = w j 1 1,μ j 1 = μ j 1 1,σ j 1 = σ j 1 1, w j 2 = w j 1 2,μ j 2 = μ j 1 2,σ j 2 = σ j 1 2. Ths ntalzaton also helps to reduce the computatonal burden by reducng the number of EM teratons. MATLAB mplementaton of the proposed method wth above ntalzaton takes less than 1 s per optcal secton on an average on a 2.53 GHz Pentum 4 computer. Ths s faster compared to Ĉapek et al. (2006), who report that C++ mplementaton of ther approach takes approxmately 2 s per optcal secton. A C++ mplementaton of our approach can provde further speed-up f needed. Contrast to Nose rato Before, z=3 Before, z=6 Before, z=12 After, z=3 After, z=6 After, z= Tme Fg. 6. Sequence 1: Contrast-to-nose rato.

8 150 S. GOPINATH ET AL. Fg. 7. Sequence 1: Restored optcal sectons at tme t = 1 from (a) top (z = 1) to (l) bottom (z = 12). Sequence 1 The sequence tested here s a 4D-xyzt sequence wth resoluton acqured by spnnng dsk confocal mcroscopy, and showng the traffckng of caveoln1-gfp n a CHO (Chnese hamster ovary) cell. Fgures 2(a) to (l) show all the 12 optcal sectons at tme t = 1. It can be observed from the mages that the ntensty of the optcal sectons vares sgnfcantly from one secton to the other. The ntensty rses from depth z = 1toz = 5 and drops agan tll z = 11 before t rses n the fnal optcal secton at z = 12. These ntensty changes prmarly result from a combnaton of ncreasng z depth, changes n cross-sectonal area of the cell, and actual changes n the dstrbuton of caveoln1-gfp, whch s concentrated on the cell surface relatve to the cell nteror. Secton 12 shows the bottom surface of the cell where t s spread out on the glass cover slp. The top surface of the cell was not ncluded n the z stack. In addton, the morphology of dscrete fluorescent objects changes wth the depth. GMM parameters were calculated wth EM algorthm. For each frame, the teratve process was termnated when foreground and background mean values changed by less than 0.01.

9 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 151 Fg. 8. Sequence 2: Orgnal mage stack: optcal sectons from z = 1to18,att =30. Fgure 3 shows the foreground membershp probablty after the convergence of EM algorthm. Snce these are membershp probabltes and not membershps, these can take any value from 0 to 1. The brghtness of a pxel s hgher, that s the probablty s close to one, f t belongs to the foreground. On the other hand, the darkness of a pxel ndcates that t belongs to background (ths means the probablty s close to 0). Despte the structural changes and changes n ntensty, Fg. 9. Sequence 2: Orgnal mage stack: optcal sectons at z = 15 for t = 1, 12, 24, 36, 48, 60.

10 152 S. GOPINATH ET AL. (a) Foreground mean ntensty (b) Foreground mean ntensty Before, t=1 Before, t=20 Before, t=40 Before, t=60 After, t=1 After, t=20 After, t=40 After, t= Depth Before, z=1 Before, z=6 Before, z=12 Before, z=18 After, z=1 After, z=6 After, z=12 After, z= Tme Fg. 10. Sequence 2: Varaton n the mean ntensty of foreground regon of orgnal mage stack and restored mage stack (a) wth depth (b) wth tme. the foreground regons are consstently detected. The success of the proposed method can be attrbuted to ths consstency. Fgure 4 shows foreground mean ntenstes for the entre mage stack before and after compensaton. A few of the curves are extracted n Fg. 5 to observe the ntensty loss trends closely. Fgure 5(a) shows the plot for the varaton n the foreground mean wth depth at tme t =10, 20, 30, whch s n agreement wth the vsual observatons made. However, varaton of mean ntensty wth tme plotted n Fg. 5(b) for depth z = 3, 6, 12 reveal facts that are dffcult to observe vsually. Intensty of the foreground drops as the tme progresses as expected owng to effects of photobleachng. However, rate of the decay s dfferent at dfferent depth levels. At depth z = 6, the mean ntensty drops from 77.3 to 75.3, whereas at depth z = 12 t drops from 82.2 to Decay rate at z = 12 s more than two tmes the decay rate at z = 6. As Contrast to Nose rato Before, z=1 Before, z=6 Before, z=12 Before, z=18 After, z=1 After, z=6 After, z=12 After, z= Tme Fg. 11. Sequence 2: Contrast-to-nose rato. our method does not use IDF, estmaton of complcated IDF requred to model ths mage sequence becomes unnecessary. The mage sequence was restored wth reference values μ r = 82.2 and σ r = 50.9, whch were estmated from frame at t = 1 and z = 12. Ths frame was chosen as t has the maxmum mean ntensty. Restored mages at z = 1 are shown n Fg. 7. Foreground ntenstes of the restored mage stacks are plotted n Fg. 4(b). Steady values of the ntenstes can be observed n the plot. The steady values are also reflected n the restored mages n Fg. 7. Contrast-to-nose rato (CNR) was computed for the sequence for quanttatve analyss. CNR = μ j 2 μ j 1 σ j. (8) 1 Fgure 6 shows CNR curves before and after compensaton for depths z = 3, 6, 12. A substantal ncrease n the CNR after the compensaton can be seen for all the depths. Sequence 2 The next sequence s a 4D-xyzt sequence smlar to the frst sequence wth resoluton Compared to the frst sequence, the second sequence shows more structural changes as seen n Fg. 8. The shape of the foreground object changes from a sngle round object n the top optcal secton to the two elongated objects n the deeper optcal sectons. It also exhbts severe photobleachng wth tme (Fg. 9). Foreground mean ntenstes are droppng drastcally wth tme as seen n Fg. 10(b). Worst drop s from 70.3 to 31.4 at depth z = 18. Although, the least drop s experenced by z = 6, t has the least foreground mean ntensty to start wth. The sequence was restored by selectng optcal secton at z = 18 at tme t = 1 wth μ r = 70.3 and σ r = Steady ntenstes can be seen after the restoraton n Fgs 12, 13 and also n 10 (a) and 10 (b). CNR for ths sequence s shown n Fg. 11.

11 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 153 Fg. 12. Sequence 2: Restored mage stack: optcal sectons from z = 1 18, at t =30. Fg. 13. Sequence 2: Restored mage stack: optcal sectons at z = 15 for t =1, 12, 24, 36, 48, 60. Sequence 3 The thrd mage sequence s a 3D-xyt sequence of dmensons showng traffckng of caveoln1-gfp n a sngle optcal secton. For ths long sequence, ntensty loss due to photobleachng s promnent as frame number ncreases. Mean ntensty of the foreground drops from 35 to 21.5 from frame at tme t = 1tot = 220. Changes of object shape

12 154 S. GOPINATH ET AL. Fg. 14. Sequence 3: Reference secton t = 1 and some sectons from an orgnal mage stack. Sectons: (a) t = 1, (b) t = 20, (c) t = 40, (d) t = 60, (e) t = 80, (f) t = 100, (g) t = 120, (h) t = 140, () t = 160, (j) t = 180, (k) t = 200, (l) t = 220. from frame to frame are manly due to object moton, and are mnmal compared wth those resultng from changng optcal secton as n the frst two sequences. In Fg. 14, we show the few mages from the orgnal sequence of confocal mcroscopy mages. The entre sequence was processed by our algorthm usng frst frame (Fg. 14(a)) as the reference mage. Reference values were μ r = 34.7 and σ r = From Fg. 16, one can see that the ntensty drops steadly wth ncreasng tme-pont. Generally, ths decay s modelled wth an IDF. A smple photophyscal model for photobleachng s a sngle exponental decay, but the actual IDF may be much more complcated and mpossble to estmate a pror. The proposed

13 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 155 Fg. 15. Sequence 3: Reference secton t = 1 and some sectons from restored mage stack. Sectons: (a) t = 1, (b) t = 20, (c) t = 40, (d) t = 60 (e) t = 80, (f) t = 100, (g) t = 120, (h) t = 140, () t = 160, (j) t = 180, (k) t = 200, (l) t = 220. method has helped to mantan a constant ntensty for the entre stack after the compensaton wthout any IDF. In Fgs 15 and 16, t can be observed that the ntenstes of the foreground object of the restored seres are unform wth very lttle varaton. For the frst frame, the number of teratons taken for EM to converge was 63. For the rest of the frames, t requred only 1.36 teratons on average, wth a mnmum of 1 and a maxmum of 11 teratons. Thus, usng prevous frames estmated parameters to ntalze the next frames parameters helps to reduce computatonal load sgnfcantly. Fg. 17 shows CNR before and after compensaton.

14 156 S. GOPINATH ET AL. Mean ntensty Orgnal foreground Restored foreground Orgnal background Restored background Tme Fg. 16. Sequence 3: Mean ntensty of background and foreground. Contrast to Nose rato Orgnal mage Restored mage Tme Fg. 17. Sequence 3: Contrast-to-nose rato. Sequence 4 Sequence 4 s a 3D-xyt sequence smlar to Sequence 3 wth resoluton For ths mage sequence, t s dffcult to separate the foreground and background vsually. Foreground membershp probablty for the optcal sectons shown n Fg. 18 are depcted n Fg. 19 as estmated by the proposed algorthm. Smlar to sequence 3, Fg. 21 shows the decay of foreground ntensty over tme. The frst frame of the sequence was chosen as the reference frame. Reference mean μ r was calculated to be 35.4 and reference standard devaton σ r was Restored mages are shown n Fg. 20 and correspondng contrast-to-nose ratos are plotted n Fg. 22. Concluson For relable analyss as well as vsualzaton of cell dynamcs, t s essental that the acqured mages reflect the exact nformaton of the specmen. The objectve of the proposed method was to help regan the vsual nformaton lost due to varous deteroratng factors such as scatterng and absorpton of the exctaton, photobleachng of fluorescent mages etc. Majorty of the current approaches to solve ths problem are ether computatonally complex, tme-consumng, restrcted to parametrc decay models (IDF) or are hghly senstve to nose. The proposed method provdes a smple yet effectve statstcal approach to solve ths problem. It overcame the dsadvantages of current methods and at the same tme ncreased the vsual value of confocal mcroscopy mages. The man dea was to flter the foreground nformaton from a gven mage by modellng t as a mxture of Gaussan PDFs and use ths nformaton to compensate the ntensty loss of the confocal mcroscopy mages. When multple fluorescence tags are used n a specmen, the proposed method can be smply appled to the ndvdual tags or a multple Gaussan PDF mxture model can be used to handle the scenaro. References Atkns, P. & De Paula, J. (1994) Spectroscopy 2: electronc transtons. Physcal Chemstry, pp W.H. Freeman and Company, New York. Becker, P.L. (1996) Quanttatve fluorescence measurements. Fluorescence Imagng Spectroscopy and Mcroscopy (ed. by Wang, X.F. & Herman, B.), pp Wley, New York. Ĉapek, M., Janàĉek, J. & Kubínová, L. (2006) Methods for compensaton of lght attenuaton wth depth of mages captured by a confocal mcroscope. Mcros. Res. Tech. 69, Ĉapek, M., Kubínová, L., Hána, K. & Smrĉka, P. (2005) Compensaton of the contrast and brghtness attenuaton wth depth n confocal mcroscopy. In Proceedngs of the Sprng Conference on Computer Graphcs 2005, Chen, H., Swedlow, J.R., Grote, M., Sedat, J.W. & Agard, D.A. (1995) The collecton, processng, and dsplay of dgtal three-dmensonal mages of bologcal specmens. Handbook of Bologcal Confocal Mcroscopy (ed. by Pawley, J.B.), pp Plenum Press, New York. Cox, I., Roy, S. & Hngoran, S. (1995) Dynamc hstogram warpng of mage pars for constant mage brghtness. In Proceedngs of the IEEE Internatonal Conference on Image Processng 1995, 2, Kervrann, C., Legland, D. & Pardn, L. (2004) Robust ncremental compensaton of the lght attenuaton wth depth n 3D fluorescence mcroscopy. J. Mcrosc. 214(3), Negahdarpour, S. & Yu, C.-H. (1993) A generalzed brghtness change model for computng optcal flow. In Proceedngs of the IEEE Internatonal Conference on Computer Vson 1993, Nyul, L., Udupa, J. & Zhang, X. (2000) New varants of a method of MRI scale standardzaton. IEEE Trans. Med. Img. 19(2), Oostveldt, P.V., Verhaegen, F. & Messens, K. (1998) Heterogeneous photobleachng n confocal mcroscopy caused by dfferences n refractve ndex and exctaton mode. Cytometry 32, Ortz, S., Garca, E., Jones, A., Pnkel, D., Gray, J., Sudar, D. & Lockett, S. (1999) Segmentaton of confocal mcroscope mages of cell nucle n thck tssue sectons. J. Mcrosc. 193,

15 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 157 Peraswamy, S. & Fard, H. (2003) Elastc regstraton n the presence of ntensty varatons. IEEE Trans. Med. Imag. 22(7), Rgaut, J. & Vassy, J. (1991) Hgh-resoluton 3D mages from confocal scannng laser mcroscopy: quanttatve study and mathematcal correcton of the effects from bleachng and fluorescence attenuaton n depth. Anal. Quant. Cytol. 13, Song, L., Varma, C.A., Verhoeven, J.W. & Tanke, H.J. (1996) Influence of the trplet excted state on the photobleachng knetcs of fluorescen n mcroscopy. Bophyscs 70, Theodords, S. & Koutroumbas, K. (1999) Pattern Recognton. Academc press, San Dego. Wu, H.-X. & J, L. (2005) Fully automated ntensty compensaton for confocal mcroscopc mages. J. Mcrosc. 220(1), Fg. 18. Sequence 4: Reference secton t = 1 and some sectons from orgnal mage stack. Sectons: (a) t = 1, (b) t = 20, (c) t = 40, (d) t = 60, (e) t = 80, (f) t = 100, (g) t = 115, (h) t = 130, () t = 166.

16 158 S. GOPINATH ET AL. Fg. 19. Sequence 4: Foreground membershp probablty for optcal sectons at tme (a) t = 1, (b) t = 20, (c) t = 40, (d) t = 60, (e) t = 80, (f) t = 100, (g) t = 115, (h) t = 130, () t = 166 (Brght regons denote hgher foreground membershp probablty and dark regons denote hgher background membershp probablty).

17 STATISTICAL APPROACH FOR INTENSITY LOSS COMPENSATION 159 Fg. 20. Sequence 4: Reference secton t = 1 and some sectons from restored mage stack. Sectons: (a) t = 1, (b) t = 20, (c) t = 40, (d) t = 60, (e) t = 80, (f) t = 100, (g) t = 115, (h) t = 130, () t = 166.