CoCPF: Coordinate-based Continuous Projection Field for Ill-Posed Inverse Problem in Imaging

TL;DR

The Coordinate-based Continuous Projection Field (CoCPF) is proposed, which aims to build hole-free representation fields for SVCT reconstruction, achieving better reconstruction quality and outperforms state-of-the-art methods for 2D and 3D SVCT reconstructions under various projection numbers and geometries.

Abstract

Sparse-view computed tomography (SVCT) reconstruction aims to acquire CT images based on sparsely-sampled measurements. It allows the subjects exposed to less ionizing radiation, reducing the lifetime risk of developing cancers. Recent researches employ implicit neural representation (INR) techniques to reconstruct CT images from a single SV sinogram. However, due to ill-posedness, these INR-based methods may leave considerable ``holes'' (i.e., unmodeled spaces) in their fields, leading to sub-optimal results. In this paper, we propose the Coordinate-based Continuous Projection Field (CoCPF), which aims to build hole-free representation fields for SVCT reconstruction, achieving better reconstruction quality. Specifically, to fill the holes, CoCPF first employs the stripe-based volume sampling module to broaden the sampling regions of Radon transformation from rays (1D space) to stripes (2D space), which can well cover the internal regions between SV projections. Then, by feeding the sampling regions into the proposed differentiable rendering modules, the holes can be jointly optimized during training, reducing the ill-posed levels. As a result, CoCPF can accurately estimate the internal measurements between SV projections (i.e., DV sinograms), producing high-quality CT images after re-projection. Extensive experiments on simulated and real projection datasets demonstrate that CoCPF outperforms state-of-the-art methods for 2D and 3D SVCT reconstructions under various projection numbers and geometries, yielding fine-grained details and fewer artifacts. Our code will be publicly available.

Figures (7)

• Figure 1: Visual examples against state-of-the-art INRs: GRFF GRFF, NeRP NeRP, CoIL CoIL and SCOPE scope for 2D SVCT reconstruction on COVID-19 covid19 dataset. Each subfigure (bottom left) highlights the anatomic structures zoomed in the boxes, while the heatmap (bottom right) shows the difference related to the GT. Red text indicates the highest score.
• Figure 2: The overall architecture of the proposed CoCPF. In training stage, given a projection angle $\theta_t$ from sparse angle partitions, (a)CoCPF uniformly samples the points within a stripe $\mathcal{S}_{k}(\varpi, \rho, \theta_{t})$ (blue stripe). By feeding the stripe coordinate $\mathbf{z}_{i}$ and $\theta_{t}$ of each sampling point into the MLP $F_{\Theta}$, CoCPF produces the corresponding light intensity $I_i\in\mathbb{R}$ and attenuation coefficient $\sigma_i\in\mathbb{R}$. (b) As discussed in \ref{['eq:simple']}, the distribution of stripe is only related to the distance $\nu$, CoCPF predicts a coarse result (purple box) by employing the volume integral \ref{['eq:M_volume']} on the coarse estimations (pink stripe) of that stripe. Then, CoCPF resamples the points under the PDF of coarse estimation to acquire the fine one (orange stripe), and the fine result (orange box) can be generated by the former process. We use the fine result as the final output. Since the rendering functions are differentiable, CoCPF can optimize the MLPs by minimizing the loss \ref{['eq:loss']} between the dual outputs and sinogram pixels. (c) After optimization, CT images can be acquired by applying FBP FBP on the synthesized DV sinograms as \ref{['eq:view-syn-cnns']}.
• Figure 3: (a) Existing INR-based methods build the mappings between the positional coordinates and the sinogram pixels based on the Radon transformation. However, they struggle to reduce the ill-posed levels brought by the sparse sampler, thereby forming the holes (i.e., unmodeled spaces) in the fields, leading to blurry results and severe artifacts. (b)CoCPF broadens the sampling regions from rays (1D space) to stripes (2D space), thereby the holes can be jointly optimized in training, reducing the ill-posed levels for SVCT reconstruction.
• Figure 4: The architecture of our multi-layer perceptron (MLP) $F_{\Theta}$. For a given field coordinate $\mathbf{z}$ and projection angle $\theta$ encoded by the positional encoding $\gamma(\cdot)$ in \ref{['eq:PE']}, we first pass the coordinate vector $\gamma(\mathbf{z})$ through 9 fully-connected (FC) layers, each FC layer hasing $256$ channels. Then, to construct the residual connections, the feature of the 1st and 4th layers are added into the 4th and 7th layers, respectively. Since the attenuation coefficient $\sigma$ is only related to $\mathbf{z}$, $\sigma$ is predicted at the 7th layer with ReLU activations. Finally, we concatenate angle vector $\gamma(\theta)$ and downscale the feature channels into $128$, and the light intensity $I$ is predicted with Sigmoid activations.
• Figure 5: Comparisons against FBP FBP, FBPConv FBPConv, FreeSeed freeseed, GRFF GRFF, NeRP NeRP, CoIL CoIL and SCOPE scope for 2Dparallel- and fan-beam SVCT reconstructions on COVID-19 covid19 dataset. Subfigures (bottom left) show the anatomic structures zoomed in the boxes, and heatmaps (bottom right) display the difference related to the GT. Red text denotes the best score.