Reconstructing Thin 3D Structures from Optical Blur Using Optical Sectioning Microscopy

Modeling the electrical properties of neurons (e.g. as a distributed transmission line) requires extensive knowledge of dendritic spatial organization and trajectories in three dimensions (3D). Confocal microscopy provides sharp contrast for 3D histological examination and is a high resolution means for reconstructing biological micro-scale structures. However, confocal microscopy is very expensive, which prompts many researchers to resort to the much more accessible option of optical sectioning microscopy (OSM) for 3D histology. However, the major difficulty associated with using OSM for 3D reconstruction is that depth information must be estimated from planar transmission light imagery. The technique for 3D reconstruction of OSM data is camera lucida. In camera lucida, the plane of focus is moved through the slide thickness and the in-focus (sharp-edged) parts of each image are traced. Ideally, each tracing is a set of contours representing the parts of the neuron existing at the corresponding plane of focus which, when stacked, forms a 3D representation of the neuron. The intensive human intervention required for tracing makes this option slow and costly. Camera lucida extracts depth information from imagery based on the perceived sharpness of image regions. Depth from defocus methods exploit the same relationship between depth and focus [1]. These methods have been used to find distance to a point in a scene in visual imagery given a parameterization of the point spread as a function of depth. To date, this approach has not been applied to recovering depth information in microscopic imagery. Here we develop a method in the spirit of the depth from defocus literature for recovering depth from microscopic imagery. Unlike depth from defocus methods, our proposed method does not require detailed knowledge of the form of the point spread nor its identification. Structures which are closer to the plane of focus for an image will appear sharper and hence have more high frequency energy in the image. Conversely, structures farther from the plane of focus will appear more blurred, and contribute less high frequency energy to the image. Thus distance from the depth of focus is inversely related to the amount of high frequency energy in the image. Our approach is to interpret the amount of high frequency energy in an image patch as the probability that it comes from the depth of focus it was acquired at. This interpretation of the high frequency energy as a probability leads to a definition of depth estimation without specifically parameterizing point spread, requiring only the weaker assumption that the point spread is symmetric as a function of depth. The original contribution of this work is applying this new probabilistic interpretation of depth estimation so that a 3D reconstruction of thin microscopic structures can be made without detailed knowledge of the point spread function. To understand our method, suppose we want to estimate the depth of a thin microscopic structure that exists at a depth, z = D, in the slide2. We denote this structure gD(r, θ). We