A Gauss-Seidel Iteration Scheme for Reference-Free 3-D Histological Image Reconstruction

Abstract

Three-dimensional (3-D) reconstruction of histological slice sequences offers great benefits in the investigation of different morphologies. It features very high-resolution which is still unmatched by in-vivo 3-D imaging modalities, and tissue staining further enhances visibility and contrast. One important step during reconstruction is the reversal of slice deformations introduced during histological slice preparation, a process also called image unwarping. Most methods use an external reference, or rely on conservative stopping criteria during the unwarping optimization to prevent straightening of naturally curved morphology. Our approach shows that the problem of unwarping is based on the superposition of low-frequency anatomy and high-frequency errors. We present an iterative scheme that transfers the ideas of the Gauss-Seidel method to image stacks to separate the anatomy from the deformation. In particular, the scheme is universally applicable without restriction to a specific unwarping method, and uses no external reference. The deformation artifacts are effectively reduced in the resulting histology volumes, while the natural curvature of the anatomy is preserved. The validity of our method is shown on synthetic data, simulated histology data using a CT data set and real histology data. In the case of the simulated histology where the ground truth was known, the mean Target Registration Error (TRE) between the unwarped and original volume could be reduced to less than 1 pixel on average after 6 iterations of our proposed method.

Transformation diffusion reconstruction of three-dimensional histology volumes from two-dimensional image stacks

“…For the $N-2$ inner slices, using the Forward-Time Central-Space (FTCS) method (Roache, 1972) produces $\frac{{\varphi }_i^{m+1}-{\varphi }_i^m}{\Delta t}=D\frac{{\varphi }_{i+1}^m-2{\varphi }_i^m+{\varphi }_{i-1}^m}{\Delta s^2},i=1,\dots ,N-2,$with m denoting the iteration number, Δs the spatial increment, Δt the time increment, and ${\varphi }_i^m=\varphi \left(i\Delta s,m\Delta t\right)$. For the two end slices, $i=0$ and $i=N-1$, we impose Neumann boundary conditions as Gaffling et al (2015), as these do not fix their position, which is convenient for reconstruction $\begin{array}{c}\nabla \varphi \left(0,t\right)=0\\ \\ \nabla \varphi \left(L,t\right)=0\end{array}$…”

“…(e) After 7000 diffusion iterations, reconstruction converges to maximum alignment of slices, but far from their true shape. (f) Difference between reconstruction and true shape, measured as mean square error $=\parallel y_0-{\widehat{y}}_0,\dots ,y_{N-1}-{\widehat{y}}_{N-1}\parallel /N$ for 7000 stack sweeps of our TDR method (solid line) and Gaffling et al's (2015) Gauss Seidel approach (dashed line). The minima corresponding to the best reconstructions are obtained after 42 (TDR) and 25 (Gaffling) diffusion iterations, and are $7.42·{10}^{-3}$ and $5.16·{10}^{-3}$, respectively.…”

“…The methods in Adler et al, 2014, Adler et al, 2012), Yushkevich et al (2006) are also causal sequential, but the neighbor $I_{i-k}$ that I i is registered to is not necessarily $I_{i-1}$. Non-causal algorithms that align I i to a bilateral neighborhood $I_{i-d},\dots ,I_{i-1},I_{i+1},\dots ,I_{i+d}$ (Gaffling et al, 2015, Guest and Baldock, 1995, Saalfeld et al, 2012, Wirtz et al, 2004) cannot converge in one sweep, because after I i is registered to $I_{i+1},\dots ,I_{i+d}$, those neighbors get transformed too. Instead, repeat sweeps (forwards or back and forth) can be used to increasingly smooth the stack, but this increases the computational cost, particularly if the algorithm depends on expensive registration operations.…”

“…Gaffling et al (2015) pose the problem of histology smoothing as intensity curvature minimization using Laplace's partial differential equation (PDE) $0={\nabla }^{2}Itrue(rtrue),$where ∇ 2 is the Laplacian along stack direction r . Discretization of this equation produces a sequential algorithm with a radius-1 neighborhood where back and forth sweeps increasingly smooth the stack.…”

Towards a comprehensive atlas of cortical connections in a primate brain: Mapping tracer injection studies of the common marmoset into a reference digital template

“…The deformable reconstruction method stems from an assumption that variability of the shape of the brain structures is larger (i.e. it extends further) than the section thickness itself, thus the neighboring images are similar to one another in a formal sense ( Adler et al, 2014 ;Gaffling et al, 2015 ). Therefore, the first stage of this process relies on warping each given section image towards an average image of its immediate neighbors in either direction.…”