Interpolation of histological slices

Motivation

The quality of 3-D reconstructions from histological slice sequences strongly depends on the completeness of the data set. Unfortunately, due to the sever mechanical stress imposed on tissue slices during the preparation and staining stages, the possibility that tissue parts become missing or cannot be used for reconstruction is high. Depending on the specific case, a loss of 5-30% of images is possible.

In this case, the final reconstructed volume is of reduced quality, with structures looking distorted or discontinuous due to the lack of information in between. An examplary reconstruction of a hair follicle from a mouse's eye lid showing holes where slices are missing can be seen below. In addition, the reversal of tissue deformations is an essential part of histological image reconstruction, and relies on the completeness of the data set.

We propose a method to eliminate this problem. We suggest to interpolate the missing images using the adjacent slices and a variational non-rigid registration approach.

 Method

After some preprocessing steps, the images adjacent to a missing slice are used as reference R and template image T, and a deformation field u transforming T to R is calculated, i.e. R = Tu = T(x - u(x)). The parameters of the previously performed
rigid registration are used as initialization, and the following registration is performed on an overlapping region of both images, to prevent unrealistic deformations of the edges. The non-rigid registration method of choice is the variational formulation of the problem, introduced in [3]. 
The objective function to be minimized is given by

where D[R; T; u] is a distance metric used to evaluate the similarity between the reference image and the transformed template image. We choose mutual information (MI), measuring the amount of information one image contains about the other image. The second term S[u], called regularizer, is weighted by an adjustable parameter, and
keeps the deformation field smooth.

Then, the missing slices are eventually interpolated. This avoid holes or discontinuities in the final reconstruction, as in the figures above. The deformation field u resulting from the non-rigid registration is divided into several parts, depending on the number of missing slices k in the respective gap. The final deformation field that has to be applied to the template image T to approximate the i-th intermediate slice S_i is then
given by

and the slice itself is given by application on the template image,

Results

To achieve histological reconstructions with high quality, missing slices have to be restored to prevent distortions and discontinuities. The quality of the interpolated images is good, and can effectively restore spatial connectivity between original slices. You can see the resulting reconstructions below.

The BVM publication about this topic can be found here.