Interpolative Model on Hueristic Projection Transform for Image Reconstruction

The Foundations of 3D Reconstruction from Projections

Three-dimensional (3D) reconstruction of an object’s mass density from a set of its two-dimensional (2D) line projections lies at the core of both single-particle reconstruction (SPR) techniques and electron tomography (ET) in the field of macromolecular electron microscopy. These techniques utilize an electron microscope to collect a set of projections of either multiple objects representing the same macromolecular complex in an isolated form or a subcellular structure isolated in situ.

The problem of inverting the projection transformation to recover the distribution of the mass density of the original object is an interesting one, as in its discrete form it is ill-posed and not invertible. Various algorithms have been proposed to cope with the practical difficulties of this inversion problem, differing widely in terms of their robustness with respect to noise in the data, completeness of the collected projection data set, errors in projections’ orientation parameters, and the ability to efficiently handle large data sets.

In this article, we will review the theoretical foundations of 3D reconstruction from line projections, followed by an overview of the reconstruction algorithms routinely used in the practice of electron microscopy. We will also discuss the unique challenges faced in EM applications and how they impact the design and performance of reconstruction algorithms.

Theoretical Foundations of 3D Reconstruction from Projections

The relationship between an object and its projections can be described mathematically using the ray transform, also known as the projection transform. For a continuous distribution of projection directions τ, the 2D projection g(x) of a 3D specimen d(r) is given by the integral transformation:

g(x) = ∫ d(r) δ(x - r⊥τ) dr

where x is the coordinate on the projection plane perpendicular to the projection direction τ, and r⊥τ denotes the component of r perpendicular to τ.

The distribution of projection directions τ is typically parametrized using two Eulerian angles (ϕ, θ), and the in-plane rotation ψ. In electron microscopy, the ZYZ convention is commonly used, where the projection direction τ is defined as:

τ(ϕ, θ) = (cos(ϕ)cos(θ), sin(ϕ)cos(θ), sin(θ))

The relationship between the Fourier transform G of a projection g and the Fourier transform D of the 3D object d is known as the central section theorem:

G(s) = D(s⊥τ)

where s is the vector of spatial frequencies. This theorem states that the 2D Fourier transform of a projection is a central 2D plane cross-section of the 3D Fourier transform of the object, perpendicular to the projection direction τ.

The inversion of the 3D ray transform is possible if there is a continuous distribution of projections such that their 2D Fourier transforms fill the 3D Fourier transform of the object without gaps. This requirement is known as Orlov’s condition, which states that the minimum distribution of projections necessary for inversion is given by the motion of the unit vector over any continuous line connecting the opposite points on the unit sphere.

However, in the practice of electron microscopy, the distribution of object orientations is often random and non-uniform, making the Orlov’s condition difficult to satisfy. Moreover, the high noise level of the data compounds the problem, leading to a highly non-uniform distribution of the spectral signal-to-noise ratio (SSNR) in the reconstructed 3D object.

Reconstruction Algorithms in Electron Microscopy

The two main groups of reconstruction methods are the algebraic and the transform methods. The algebraic methods begin by discretizing the integral transformation, leading to a system of linear equations that are typically overdetermined. The advantages of this approach include the ability to handle particular data collection geometries and experimental limitations, such as limited or truncated views of the object. However, these methods are computationally intensive, especially when applied to EM data where the number of projections is very large.

The transform methods are based on the central section theorem and tend to rely on Fourier space analysis of the problem. These methods can be further divided into two subgroups: those that construct a Fourier space linear filter to account for the particular distribution of projections, and those that directly operate in Fourier space by casting the problem as one of Fourier space interpolation between polar and Cartesian coordinate systems.

The filtered backprojection (FBP) algorithm is a representative of the first subgroup of transform methods. It consists of the following steps:

1. Compute the 1D Fourier transform of each projection.
2. Multiply the Fourier transform of each projection by a weighting filter (e.g., a ramp filter).
3. Compute the inverse 1D Fourier transforms of the filtered projections.
4. Apply backprojection to the processed projections in real space to obtain the reconstructed 3D object.

The design of a good weighting function for the FBP algorithm is challenging, as it needs to account for the uneven distribution of projection directions, which is often irregular and cannot be approximated by an analytical function.

The direct Fourier inversion methods, on the other hand, directly operate in Fourier space. The most accurate of these methods employ non-uniform Fourier transforms, particularly the 3D gridding method. The Gridding-based Direct Fourier Reconstruction (GDFR) algorithm is a representative of this approach, which involves the following steps:

1. Resample the 2D Fourier transform of each projection into 2D polar coordinates using a reverse gridding method.
2. Calculate “gridding weights” as the areas of 2D Voronoi cells on a unit sphere to compensate for the non-uniform distribution of the grid points.
3. Apply gridding using a convolution kernel, followed by a 3D inverse FFT to obtain samples on a 3D Cartesian grid.
4. Remove the weights in real space to yield the reconstructed 3D object.

The GDFR algorithm has been shown to provide virtually perfect 3D reconstruction within the full frequency range in a noise-free case, while being significantly faster than other reconstruction methods.

Unique Challenges in EM Reconstruction

The reconstruction of 3D structures from 2D projections in electron microscopy faces several unique challenges that are not commonly encountered in other fields of tomographic reconstruction:

1. Incorporating CTF Correction: Accounting for the contrast transfer function (CTF) of the electron microscope and the non-uniform distribution of the spectral signal-to-noise ratio (SSNR) in the projection data is crucial for obtaining high-quality reconstructions. While iterative algebraic methods can relatively easily incorporate CTF correction, transform-based methods require more sophisticated approaches, such as the implementation of a Wiener filter.

2. Handling Uneven Distribution of Projections: The distribution of projection directions in electron microscopy is often highly irregular and cannot be approximated by an analytical function. Designing appropriate weighting functions to account for this uneven distribution is a significant challenge, especially in the presence of gaps in the coverage of Fourier space.

3. Dealing with Incomplete or Truncated Views: In electron tomography, the object may not be fully accessible due to the limited range of tilt angles, resulting in incomplete or truncated views. Reconstruction algorithms need to be able to handle such scenarios, which is not a trivial problem.

4. Accounting for Errors in Orientation Parameters: The orientation parameters of the projections are known only approximately, with both random and systematic errors. Reconstruction algorithms need to be considered within the context of the structure refinement procedure to address this issue.

5. Computational Efficiency: The large number of projections in EM data (often exceeding 100,000) places extreme computational demands on the reconstruction algorithms. This requirement eliminates certain methods from consideration and necessitates the development of highly efficient algorithms, such as the GDFR method.

These unique challenges have led to the continuous development and refinement of reconstruction algorithms in the field of electron microscopy, ensuring that it remains a vibrant research area with significant room for further progress.

Conclusion

The field of 3D reconstruction from projections has a rich history and a well-developed theoretical foundation. However, the unique challenges posed by electron microscopy applications have driven the development of specialized reconstruction algorithms that can effectively handle the high noise levels, uneven distribution of projections, and computational demands of EM data.

While the incorporation of contrast transfer function (CTF) correction and the ability to deal with incomplete or truncated views remain active areas of research, the gridding-based direct Fourier inversion methods, such as the GDFR algorithm, have emerged as highly accurate and computationally efficient solutions for EM reconstruction. As the field of electron microscopy continues to advance, further refinements and innovations in reconstruction algorithms are expected to play a crucial role in unlocking the full potential of this powerful imaging technique.