The imaging performances of multiphoton excitation and confocal laser scanning microscopy are herby considered: in typical experimental imaging conditions, a small finite amount of photon reaches the detector giving shot-noise fluctuations which affects the signal acquired. A significant detriment in the high frequencies transmission capability is obtained. In order to partially recover the high frequencies information lost, the insertion of a pupil plane filter in the microscope illumination light pathway on the objective lens is proposed. We demonstrate high-frequency and resolution enhancement in the case of linear and non linear fluorescence microscope approach under shot-noise condition.
© 2009 Optical Society of America
Confocal laser scanning microscopy (CLSM) and two photon excitation (2PE) microscopy  are crucial optical techniques, commonly used in several research areas, especially in the biomedical field. CLSM and 2PE performances have been extensively analyzed both in the spatial and in the frequency domain [4–7] providing a full description of their optical capabilities and an exhaustive evaluation of their theoretical resolution limits. These analysis, which generally assumes ideal imaging conditions, i.e. detection without perturbations of a diffraction-limited illumination spot, have been combined with detailed analysis taking into account also the noise contribution and evaluating the overall Signal to Noise Ratio (S/N) in the image formation [8–10]. These further characterizations are essential as in several live-cell confocal and two photon imaging applications, extremely low signal levels are acquired and noise contributions can not be neglected, significantly hampering the imaging process formation and hence deteriorating the image quality. In particular, such a low photon influx to the detector determines an uncertainty in the detection photon counting known as shot-noise, extremely relevant for CLSM and 2PE microscopy.
Therefore, in the present work a characterization of the imaging performances of CLSM and 2PE microscopy affected by shot-noise is presented. In particular, a detailed analysis in the frequency domain will be adopted to emphasize shot-noise influence in the high frequencies range with a substantial reduction of the transfer function bandwidth which produces a remarkable decrease of the spatial resolution.
In order to improve the imaging quality, the insertion of an annular amplitude filter juxtaposed to the microscope objective lens in the illumination arm is proposed using a consolidated approach [13–18]. In fact, the interference effect induced by the proposed filter results in a redistribution of the optical transfer function, mostly prominent along the axial direction and particularly displaying an improvement of the high spatial frequency response of the system at the expense of a slight low-frequency information loss [19–22]. Consequently, such a pupil filter can be regarded as a filter able to improve the S/N at high frequencies and, hence, to partially recover the high frequencies information loss because of shot-noise.
2.1 PSF and OTF characterization
The imaging capability of an optical system could be fully characterized by the system response in terms of 3D intensity Point Spread function (3D-PSF) and 3D Optical Transfer Function (3D-OTF) .
where a point-like illumination source is assumed; the first factor represents the illumination, while the second square brackets describes the detection intensity PSF; hex(u,v,ϕp) and represent respectively the amplitude PSF of the objective lens for the excitation and the emission wavelength; β is a scale factor taking into account the excitation and the emission wavelength β = λem/λex ; D(v) characterizes the pinhole aperture; m is the number of photon absorbed during the excitation process; u,v,ϕP specifies the position of a typical point P in the focal region of the objective lens; u,v are the radial and axial optical coordinates which properly take into account the wavelength of the incident light on the objective lens:
with k = 2π/λex,λex, is the wavelength of excitation light, α is the angular aperture of the lens, θP, ϕP, rP are the spherical coordinates of P.
In the present investigation, we refer to two widely diffused fluorescence microscopy systems: a confocal laser scanning microscope (CLSM) and a two-photon excitation (2PE) scheme. The CLSM scheme has been modelled considering a single photon excitation and an ideal point-like pinhole aperture, approximating D(v) as a Dirac’s delta function. On the contrary, the TPE scheme adopted is considered with no pinhole detection, approximating D(v) as infinitely extensive with unitary response. With this usual assumptions from Eq. (1) we evaluate our working confocal and two photon intensity PSF [11,24] as:
Here, we consider a circularly polarized illumination light, hence:
The insertion of a pupil-plane filter juxtaposed to the objective lens in the incident light pathway induces an interference effect which modifies the components of the electromagnetic field in P by a factor K(θ,ϕ) characterizing the pupil-plane function:
We assume an obscuring annular amplitude filter centred on the lens optical axis and completely absorbing the incident light over its extension (Fig. 1), hence only depending on θ:
where C takes into account the annular covering portion of the overall angular aperture.
The effect of the filter insertion on the intensity PSF, either for CLSM or for 2PE, has been previously analyzed and it results in a double effect: sharpening up of the PSF central lobe in despite of an increase in the side-lobe levels, both more prominent as the covering portion C parameter increases .
2.2 Shot-noise effect
One of the most significant source of imaging noise in fluorescence microscopy is known as photon shot-noise or poissonian noise. Due to the statistical nature of photon production, the probability distribution to detect n photons induced by a photons per second influx γ in a temporal observation window t obeys a poissonian distribution :
where the mean of the poissonian distribution γt represents the expected number of photons np contained in the light beam in the time t. Due to this probabilistic behaviour, the photons detection counting is affected by a remarkable uncertainty more prominent as the number of expected photons decreases. As noise is given by the square root of the signal variance, shot-noise over a certain temporal detection interval is assumed as the square root of the expected number of photons contained in the beam. Assuming only shot-noise, this results in a finite S/N equal to square root of np.
Clearly, shot noise contribution to the detected signal becomes relevant as the photons rate to the detector is reduced. For standard confocal and 2PE scanning applications, the amount of photons reaching the detector has been estimated nearly 10-1000photons/voxel, depending on the imaging conditions used [2,5,29–31]. The slight quantity of photons acquired is mainly due to the huge portion of the emitted photons lost in the detection pathway , as losses due to the angular aperture lens which limits signal collection or the several signal losses during the transmission inside the microscope. For such an estimated photon amount, shot-noise contribution ranges from 1% up to a 30% of the signal. It follows that, in typical CLSM and 2PE investigations the uncertainty in the detected photon counting can become quantitatively remarkable. Apparently, a proper rearrangement of the imaging microscope conditions could minimize shot-noise effect: oversampling, increasing the illumination photon rate or the exposition time, can induce a high np and thus a negligible shot noise contribution. However, these solutions can be significantly limited by saturation, photobleaching and light-matter damaging interactions [33,34].
Since shot-noise is a subtle intrinsic limit of several confocal and 2PE scanning microscopy experimental applications, an evaluation of its influence on the spatial resolution is essential. Hence, in order to establish the effective practical resolution power, a simulation of the shot-noise effect on the 2PE and CLSM imaging process formation has been performed.
Simulation algorithm perturbs the theoretical PSF of the system (Fig. 2(a), Fig. 2(c)) by a shot-noise contribution corresponding to a finite amount of photons influx to the detector: in particular each pixel is perturbed by a poissonian probabilistic distribution [35–37]. In order to take into account the differences of photons collected in the central lobe of the PSF with respect to the side lobe, we properly scaled the mean of the poissonian fluctuations depending on the pixel position in the PSF. Under these assumptions, the new PSF represents one of the possible PSF resulting by taking into account the statistical fluctuations in the amount of photons detected (Fig. 2(b), Fig. 2(d)). This allows to evaluate the real diffractive intensity distribution associated with a finite amount of photons to the detector.
We have to mention that the imaging process is affected also by several other background signals not treated in the present work in order to emphasize the role of shot-noise.
2. Data and results
In order to investigate the effect of shot-noise on the CLSM and 2PE imaging, an analysis of the optical system response in the frequency domain has been carried out (Fig. 3). Graphs of Fig. 4 and Fig. 5 show the 2PE and CLSM OTF radial and axial profile performed in presence of shot-noise contributions.
Noise slightly influences the low-frequency transmission, but it significantly affects the information capability at the high frequencies. In particular, it is possible to outline a noise level threshold below which the real object frequencies is merged to noise contribution (Fig. 4, Fig. 5 insets).
Hence, this noise level limit redefines the baseline OTF level, resulting in some new effective cut-off frequency values, lower than the theoretical one. The threshold limit has been obtained by the simulations results assuming a noise band equal to μ±3σ , with μ the mean of the OTF values above the theoretical cut-off frequency, hence due only to noise signal, and σ the relative standard deviation. The ±3σ interval has been set in order to have a 99% confidential interval for the noise statistic.
Such a noise level has been estimated in a range of 2%–3% of the OTF peak value for an amount of tens photons reaching the detector and it decreases tending towards the theoretical zero value increasing the quantitative of expected photons.
Because of the extremely weak high-frequencies response capability, even if the threshold noise limit calculated is relatively small respect to the OTF peak, for slight amount of photons it appreciably deteriorates high frequencies capability reducing the theoretical cutoff frequencies of the system and thus the effective resolution power of the optical system. Table1 shows the new 2PE and CLSM cut-off frequencies obtained simulating the shot-noise effect on a theoretical PSF assuming a finite amount of photons detected. Each value represents the averaged cut-off frequency obtained by 20cut-off frequencies evaluated by different simulations with the same amount of photons at different noise realizations. The errors have been evaluated as the standard error. Theoretical values have been calculated assuming ideal conditions. As notable, the transfer bandwidth is significantly reduced.
In order to recover those high-frequency details lost, we propose to use an amplitude ring pupil plane filter. We compare the characteristics of the proposed annular filters, simulating the diffractive pattern obtained in CLSM and 2PE scheme in the frequency domain for differently covering portion of the whole plane of the entrance pupil (C=0;15;25;30).
As expected, the theoretical bandwidth extension is not affected by the pupil filter, keeping unchanged the cut-off frequencies . Nonetheless, a different distribution of the frequency information has been obtained: a substantial increase of the high-frequencies transmission is gained at the expense of a low-frequencies loss. As shown by graphs in Fig. 6 and Fig. 7, this effect could be tuned by a proper choice of the obscured pupil surface covered by the designed filter. Moreover, it is notable that such a filtering effect is more remarkable in the axial direction and slightly affects the radial profile.
As this new OTF distribution exhibits high-frequency response enhancements, it can be exploited to recover the high frequencies details lost for the shot-noise effect explained in the previous section.
Due to the enhanced high frequency response obtained in the annular filtering configuration, the high frequencies removed by the noise effect can be remarkably recovered in the axial direction, less significantly in the radial profile. Furthermore, the filtering high-frequencies recovery effect in presence of noise, is accentuated in 2PE.
Table 2 and Table 3 show the new cut-off frequencies estimated inserting a ring filter on objective lens and assuming different photons influxes to the detector. Each value represents the averaged cut-off frequency obtained by 20cut-off frequencies evaluated by different simulations with the same amount of photons at different noise realizations. The errors have been evaluated as the standard error.
For a low photons influx to the detector, in the 2PE scheme the filtering effect shows significant improvements in the axial direction, where the high frequencies lost because of shot-noise are significantly recovered. Such effects are more prominent increasing the obscuring portion of the annular filter. Even if it reaches a weaker improvements, similar effects are obtained along the axial direction for a CLSM scheme. In both cases, increasing the expected number of photons the filter benefits become less important.
On the other hand, in the radial direction the effect of the filter insertion exhibits a more complex behavior. In particular, in the confocal scheme the filter induces slight extensions of the effective transfer bandwidth, while in 2PE scheme slight improvements seem to be joined with some worsening. A detailed observation of the 2PE and CLSM filtered profiles (Fig. 6, Fig. 7) shows how each filtered OTF transmits the spatial frequencies better than the unfiltered case in a region from the theoretical cut-off frequency until an inversion point above which an opposite effect is obtained. Fig. 7(c) shows how in the radial direction of 2PE case, this inversion takes place at an OTF level comparable with the estimated shot-noise threshold levels. This behavior, combined with the extremely small gaps between the filtered and unfiltered OTF in the radial profile can explain the slight improvement or worsening obtained in the new radial cut-off frequencies displacement. Therefore, the employment of such a filter has to be carefully evaluated and it is particularly recommended in order to improve the imaging quality along the optical axis, especially in 2PE applications.
It is worth mentioning that the low-frequency loss can be neglected and overcome by linear imaging restoration until it does not reach the threshold noise level, where the signal frequencies are irreversibly blurred with noise. However, since the filter effect can be tuned varying the ring covering portion, a proper choice of the ring extension on the basis of the expected signal level, can prevent this eventuality. For the shot-noise threshold level evaluated in the present investigation, the ring covering portion is limited up to C=25.
As reported in a previous work , it is of pivotal relevance to mention that the insertion of such a ring filter in the pathway of the illumination light can induce a remarkable increase of the secondary side lobes of the system point spread function. This effect can limit the applicability of the filter since it can lead to unacceptable artifacts. However, linear deconvolution can be applied to remove the side lobes artifacts obtaining a genuine high frequency recovery at a straightforward and fast computational level .
The application of such ring filter, partially obstructing the light optical pathway, could be limited by the laser intensity on the sample, considerably low respect to the unobstructed case. However, since the filter is inserted in the microscope illumination arm, these limits has been overcome in confocal and 2PE scanning microscopy as the scanning laser illumination could be adjusted to have adequate intensity on the sample . In the analysis carried out, the excitation source power has been set in order to have the same incident power on the sample in each configurations.
In the present treatment we consider circularly polarized illumination light. Analogous filter effects both in the spatial and in the frequency domain can be obtained assuming linearly polarized light. Clearly, in this case the high frequencies recovery induced by the filter has to be evaluated for each different radial directions as the radial section of the OTF induced by a linearly polarized illumination is not symmetric.
In order to deepen in the characterization of signal to noise performances of the proposed optical system scheme, a further study of the pinhole size should be performed. As the pinhole size can modulate the signal to noise ratio of the imaging process, such an analysis would allow an estimation of the optimum pinhole size for our situation depending on the shot-noise effect and on the background signal level originating from non-specific staining or autofluorescence, from either the specimen or optical elements. Such a study would be particularly relevant within an experimental perspective of the proposed scheme.
4. Conclusion and perspectives
In this work a detailed analysis of the shot-noise effect in CLSM and 2PE microscopy has been discussed. In particular, the loss of high frequencies information in presence of shot-noise has been studied and the new effective CLSM and 2PE cut-off frequencies have been estimated. In order to retrieve the high frequencies information loss because of shot-noise, a ring filter configuration able to enhance the high spatial frequencies response has been presented. The proposed annular filter allows to recover the high frequencies loss in presence of noise especially in the axial direction, improving the practical optical resolution deteriorated by the noise. This is particularly evident for the 2PE case.
Furthermore, the utilization of tuneable white light  sources coupled to the spectral characteristic of the emission of fluorescent molecules being used, can allow a sort of tuning of such enhanced high frequencies within the cut-off frequencies domain. This tuning could allow to select specific spatial object features. It is worth noting that such a tuneable feature in illumination can be also effectively used in those applications utilizing photoactivatable or photoswitchable fluorescent proteins[43–45].
Authors are indebted with Paolo Solinas for the useful discussions on the poissonian photon statistic characterization and the shot-noise evaluation. This work is supported by University of Genoa, IFOM-IEO grants and PRIN2006-MIUR(2006028909)(Italian Minister of University and Research).
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