The spatial resolution and definition of the cellular protein matrix is fundamental to the characterization and analysis of cellular function. The accurate resolution of sub-organelle protein localization, in tissue, on a proteomic scale is immensely useful. It is with this in mind that we developed Array Tomography (AT), a proteomic imaging technique. AT uses ribbon arrays of ultrathin (50–100 nm) physical sections of resin-embedded, fixed tissue for multiple rounds of immunohistological detection, which produces a rich, high-dimensional matrix of protein information in an ex-vivo context 
. AT allows the collection of 30+ channels of protein information in a cubic millimeter volume of brain tissue 
. This information is only useful if we can, with spatial accuracy, localize spatially aggregated protein units within cellular structures and in relation to all other imaged protein channels. This places a premium on the computational segmentation of objects in the image volume, and is highly dependent on resolution and contrast.
The axial resolution of AT image volumes is limited only by the physical sectioning, which is 50–100 nm and is far smaller than the diffraction limited axial resolution of most microsocopes (~385 nm). However, the lateral resolution of AT image volumes is still limited by the Abbe diffraction limit (~200 nm for visible wavelengths) 
. At that lateral resolution, the segmentation of densely packed proteins, such as Synapsin (a highly abundant presynaptic protein in the brain), is unreliable and difficult. Recently, AT was combined with direct stochastical optical reconstruction microscopy (dSTORM) to achieve lateral resolution of ~40 nm 
. However, dSTORM imaging is time consuming and requires specialized microscopes. Thus, we investigated deconvolution as a simple and efficient method to improve our resolution in AT. The reason for considering deconvolution is that the physical sectioning of AT provides full removal of out of focus light, and the ideal correction of refractive index, astigmatism, coma, spherical aberration and curvature of field 
. Moreover, the thinness of the tissue coupled with the direct placement of the sample onto glass also means that the heterogeneity of refractive indexes in normal biological samples is not present, which further eliminates sources of aberration and wave-front distortions. These properties, which are not present in most imaging techniques, allow AT to produces image volumes where the point spread function (PSF) is truly spatially invariant throughout, which makes these images an ideal substrate for deconvolution.
Deconvolution is a method by which the diffracted light is computationally returned back into its actual source using either an idealized or empirically measured PSF 
. The PSF describes the diffraction of light from a point source. Specimens in the image are blurred by the PSF at a point by point basis. This blurring can be considered a convolution operation on the image 
, if it is linear (each point source in the image sums their intensity linearly) and shift invariant (the PSF is the same for the entire field of view). Wide-field is such an imaging systems 
, although in actual biological tissue the heterogeneity and depth of the tissue volume does introduce aberrations, wave-front distortions and out of focus light contributions that can cause significant deviations in the PSF across the image volume, which adversely affect the quality of deconvolution. This is not the case for AT thin sections where the PSFs are truly spatially invariant. Moreover, it might be easier to appreciate the advantages of thin physical sections by thinking about the analogy to conventional optical sectioning microscopes such as confocals. Confocals achieve optical sectioning by using a pinhole to reject out of focus light. This improves image quality by increasing the collection of high spatial frequency information in the image, but this comes at a cost of reduced signal to noise, due to the rejection of in focus light by the pinhole. AT physically removes all out of focus light sources, which means that AT does not need to use a pinhole for optical sectioning thus allowing it to provide both high signal to noise (which, in normal confocal microscopy, would be maximized by a large-diameter pinhole) and measurement of high-frequency spatial information (which would be maximized by a small-diameter pinhole) 
The content of high-frequency information in the image is reflected in the bandwidth of the Optical Transfer Function (OTF), which is the Fourier Transform (FT) of the PSF. In confocal the OTF bandwidth varies inversely with pinhole diameter 
. The OTF determines the actual spatial frequencies transferred to the recorded image. Thus, if the OTF were small at high spatial frequencies (as is the case for an expanded confocal pinhole or a conventional wide-field setup), the high-frequency components of the specimen would be greatly attenuated, causing blurring and decreased resolution. Interestingly, the OTF of a theoretical infinitely-small pinhole would have twice the bandwidth of a standard wide-field OTF 
. In AT, we approximate this ideal pinhole with physical sectioning, and combined with the spatially invariant PSF, allow us to perform deconvolution at its mathematical optimum, which should, with the correct algorithm, allow us to greatly increase the magnitude of recovery for high spatial frequency information in the OTF up to the physical bandwidth limit, which is defined by diffraction.
Richardson-Lucy deconvolution (RL) is a Bayesian based expectation maximizing deconvolution method originally developed for the restoration of images in astronomy 
. RL has several advantages for AT images. It assumes the non-negativity of the observations and that the statistic of the associated noise follows a Poisson distribution, which is appropriate for fluorescent images 
. RL is globally and locally intensity-conserving at each iteration 
, thus ensuring that intensity data remain quantifiable after deconvolution 
. RL is computationally efficient, and the restored images are robust against small errors in the image and the point-spread function (PSF) 
, which makes its real world implementation realistic. Finally, in our tests on AT images, RL significantly out performs other non-Bayesian based deconvolution methods, and has demonstrated a greater than 8 fold increase in the magnitude of spatial frequency recovery up to the diffraction limit, without any measurable introduction of artifact or noise into the images. Moreover, RL in our application demonstrated mathematically a potential for the recovery of spatial frequencies beyond the diffraction limit, which likely contributes to the analytical improvements seen in the analysis of the deconvolved tissue volumes.
Thus, the confluence, in AT, of an essentially two-dimensional sample imaged at the optical optimum of the imaging system (e.g., minimal spherical aberration, optimal refractive index correction, ideal flatness of field, high signal to noise and a spatially invariant PSF) 
allows AT in combination with RL to achieve volumetric resolution significantly better than the diffraction limit. Using this technique, we demonstrate accurate and clean computational separation of objects in densely labeled tissue volumes.