Single particle imaging approaches have become powerful tools for studying cellular processes due to several advantages they offer over conventional bulk imaging techniques. One of the important factors in conducting single particle imaging experiments is that a sufficient number of photons be detected from the fluorescent label. This is important not only for visually identifying and tracking the labeled entity, but also has implications in the quantitative analysis of the data. We have introduced a new microscope imaging configuration dMUM for single particle imaging in 2D and 3D. The dMUM configuration uses two objective lenses and collects photons from above and below the sample. That the dMUM has improved light collection capability is evident from its optical configuration and, here, we have verified this experimentally. However, an important question arises as to how one can exploit the additional photon counts available in the dMUM images to obtain improved performance. Equally important is the question of how to quantitatively assess the level of improvement that can be attained from the dMUM images. That latter is especially relevant, for example, for designing experimental setups and for evaluating the feasibility of carrying out a particular experiment. In this manuscript, we have addressed these issues by using newly developed tools as well as previously established methodologies and have demonstrated the improved performance of dMUM through experimental data, simulations and analytical calculations.

We have considered two imaging configurations of dMUM, namely the 2D infocus imaging and the 3D imaging configurations. The 2D infocus imaging configuration can be used in place of the classical 2D conventional microscope to image processes that are typically confined to two dimenions. The 3D imaging configuration exploits the multifocal plane imaging capability of dMUM in which the sample is simultaneously imaged at two different focus levels and can be used for 3D tracking applications.

We have developed a new algorithm to determine the X-Y coordinates of nanoprobes from images acquired in the 2D infocus imaging configuration. Using experimental data we have showed that the X-Y coordinates of nanoprobes from their infocus images can be determined with better accuracy with dMUM than with a standard microscope. We have presented analytical calculations to compute the 2D localization measure, which provides a limit to the 2D localization accuracy of an infocus nanoprobe for a given microscope setup. Our calculations show that the new algorithm attains this limit thereby validating that this algorithm indeed provides the best possible accuracy.

While the 2D infocus imaging configuration provides improved x-y localization accuracy, it is not well suited for z-localization. This is due to the poor depth discrimination property that is intrinsic to this imaging configuration. Hence we considered the 3D imaging configuration of dMUM for z-location estimation. We have reported analytical calculations to compute the 3D localization measure of *z*_{0}, which provides a limit to the z-localization accuracy of a nanoprobe for a given microscope setup. Using this, we have compared the z-localization capabilities of three different microscope setups, i.e., dMUM, MUM and a standard microscope. Our results showed that for the 3D imaging configuration dMUM provides uniformly better z-localization accuracy when compared to a standard microscope and MUM. Further, we validated the improved z-localization capability of dMUM over MUM through simulated data, where we showed that the accuracy of the z-estimates from dMUM is consistently better than the accuracy of the z-estimates from MUM.

In [

9] we showed that the z-localization algorithm MUMLA is optimal for MUM data in the sense of attaining the lowest possible standard deviation as specified by the corresponding 3D localization measure. Here, we have shown that with minor modifications the same algorithm can also be applied to dMUM data from the 3D imaging configuration for z-position estimation. A comparison of standard deviations of z-estimates from a simulation study with the 3D localization measures for this dMUM configuration confirmed that MUMLA is also optimal for this setting. Although not discussed here in detail, in the 3D imaging configuration x-y location estimation can be carried out by fitting an appropriate profile to suitably registered images of either the top scope or the bottom scope, analogous to the approach proposed for MUM data analysis in [

9]. With this approach we can expect the x-y localization performance to be similar to what can be achieved with a standard microscope. Alternatively, x-y location estimation could be carried out by suitably modifying the x-y localization algorithm for the 2D infocus imaging configuration, by fitting 3D PSF models rather than infocus Airy profiles. In this case we can expect improved x-y localization performance over imaging with a standard microscope in situations when the image of the point source can be detected in the images of both focal planes.

In the 2D infocus imaging configuration, we have used the Airy profile ([

44]) to describe the image of the nanoprobe. In some situations, the use of an Airy profile may not be suitable, for example, in the case of polarized illumination and/or detection. In such cases, the Airy profile needs to be replaced by the appropriate image profile. In the same way, for the 3D imaging configuration we have made use of the diffraction limited 3D PSF profile that is based on the Born and Wolf model. Depending upon the specific imaging conditions, other 3D PSF image profiles may need to be used. Similar considerations also apply to other aspects of our data analysis, for example, in the use of a background estimation algorithm. It can be deduced from the analytical formula given in

Eq. 3 that indeed the improved localization accuracy of dMUM will hold even if other image profiles and/or background estimation procedures need to be used.

The dMUM setup was built using commercially available, off-the-shelf components and is straightforward to implement with little or no customization. The dMUM configuration reported here supports simultaneous imaging of two focal planes. In general, more than two focal planes can be simultaneously imaged with dMUM. This can be achieved by implementing the MUM imaging configuration in the top and bottom scopes of dMUM. In our implementation of dMUM, we have placed one of the microscopes on top of the other in an upside down orientation. However, other configurations are also possible and our algorithms and analytical calculations can be used with little or no modifications. For example, an alternative configuration for dMUM is to place two inverted microscopes next to each other. The sample is placed in one of the microscopes (bottom scope) and the objective lens in that microscope illuminates the sample and collects the light from the bottom side of the sample. In the other microscope (top scope), the objective lens is attached to the nose piece through an extension arm which positions the objective lens on top of the sample.

The dMUM imaging configuration is not limited to single particle imaging applications. For example, it can be used to improve the photon collection efficiency in a wide variety of low-light level cellular imaging applications including but not limited to the tracking of single molecules, vesicles and viruses in a live cell environment. For the fitting of larger objects, the use of point-source image profiles (Airy profile and 3D point spread function profile) may not be appropriate but can be replaced by a profile that describes the image of the object being tracked. In conclusion, the dMUM imaging configuration permits high accuracy localization of individual nanoprobes in 2D and 3D.