Learned Multi-layer Residual Sparsifying Transform Model for Low-dose CT Reconstruction

Signal models exploiting sparsity have been shown to be useful in a variety of applications such as compression, restoration, denoising, reconstruction, etc. Natural signals can be modeled as sparse in a synthesis dictionary (i.e., represented as linear combinations of few dictionary atoms or columns) or in a sparsifying transform domain.

Transforms such as wavelets [1] and the discrete cosine transform (DCT) are well-known to sparsify images. Methods such as synthesis dictionary learning [2] and analysis dictionary learning [3] allow adapting the model to the data. These dictionary learning problems are NP-hard in general and involve computationally expensive alternating-type algorithms that limit applicability to large-scale data. In contrast, the recently proposed sparsifying transform learning approaches [4] involve exact and highly efficient updates. In particular, the transform model approximates the signal as sparse in a transformed domain. Adopting a multi-layer sparsifying transform model enables sparsifying an input image successively over layers [5], creating a rich and more complete sparsity model, which forms the core of this work.

One of the most important application of such image models is for medical image reconstruction. In particular, an important problem in computed tomography (CT) is reducing the X-ray exposure to patients while maintaining good image reconstruction quality. A conventional method for CT reconstruction is the analytical filtered back-projection (FBP) [6]. However, image quality degrades severely for FBP when the radiation dose is reduced. In contrast, model-based image reconstruction (MBIR) achieves comparable or better image quality [7].

A typical MBIR method for low-dose CT (LDCT) is the penalized weighted least squares (PWLS) approach. The cost function for PWLS includes a weighted quadratic data-fidelity term and a penalty term or regularizer capturing prior information or model of the object [8, 9]. Recent works have shown promising LDCT reconstruction quality by incorporating data-driven models into the regularizer, where the models are learned from datasets of images or image patches. In particular, PWLS reconstruction with adaptive sparsifying transform-based regularization has shown promise for tomographic reconstruction [10, 11, 12]. Recent work has also shown that they may generalize better to new data than supervised deep learning schemes [13]. The adaptive transform-based image reconstruction algorithms can exploit a variety of image models [10, 11, 14] learned in an unsupervised manner, and involve efficient closed-form solutions for sparse coding.

In this work, we propose a new formulation and algorithm for learning a multi-layer transform model [5], where the transform domain residuals (the difference between transformed data and their sparse approximations) are successively sparsified over several layers. We refer to the model as a multi-layer residual sparsifying transform (MRST) model. The transforms are learned over several layers from images to jointly minimize the transform domain residuals across layers, while enforcing sparsity conditions in each layer. Importantly, the filters in deeper layers can help better exploit finer features (e.g., edges and correlations) in the residual maps. We investigate the performance of the (unsupervised) learned MRST model for LDCT reconstruction using PWLS. We propose efficient alternating minimization algorithms for both learning and reconstruction. Experimental results on Mayo Clinic data illustrate that the learned MRST model outperforms the conventional FBP as well as PWLS methods based on the non-adaptive edge-preserving (EP) regularizer and learned single-layer transform model, especially for maintaining some subtle details. In the following sections, we will first discuss the proposed model and formulation, followed by our algorithms and experimental results on Mayo Clinic data.

Here, we will introduce the proposed general multi-layer learning framework and the formulation for LDCT image reconstruction. The MRST learning cost and constraints are shown in Problem (P0), which is an extension of simple single-layer transform learning.

Here, $\{\mathbf{\Omega}_{l}\in\mathbb{R}^{p\times p}\}$ and $\{\mathbf{Z}_{l}\in\mathbb{R}^{p\times N}\}$ denote the learned transforms and sparse coefficient maps for the $1\leq l\leq L$ layers. Parameter $\eta_{l}$ controls the maximum allowed sparsity level (computed using the $\ell_{0}$ “norm” penalty) of $\mathbf{Z}_{l}$. The residual maps $\{\mathbf{R}_{l}\in\mathbb{R}^{p\times N}\}$ are defined in recursive form over layers, with $\mathbf{R}_{1}$ denoting the input training data. Here, we assume $\mathbf{R}_{1}$ to be a matrix, whose columns are (vectorized) patches drawn from image data sets. The unitary constraints for $\{\mathbf{\Omega}_{l}\}$ enable closed-form solutions for the sparse coefficient and transform update steps in our algorithms. The learned MRST model can then be used to construct a data-driven regularizer in PWLS as shown in Problem (P1).

In particular, we reconstruct the image $\mathbf{x}\in\mathbb{R}^{N_{p}}$ from noisy sinogram data $\mathbf{y}\in\mathbb{R}^{N_{d}}$ by solving (P1), where $\mathbf{A}\in\mathbb{R}^{N_{d}\times N_{p}}$ is the system matrix of the CT scan and $\mathbf{W}=\mathsf{diag}\{w_{i}\}\in\mathbb{R}^{N_{d}\times N_{d}}$ is the diagonal weighting matrix with elements being the estimated inverse variance of $y_{i}$. Operator $\mathbf{P}^{j}\in\mathbb{R}^{p\times N_{p}}$ extracts the $j$th patch of $\mathbf{x}$ as $\mathbf{P}^{j}\mathbf{x}$. The $j$th colums of $\mathbf{R}_{l}$ and $\mathbf{Z}_{l}$ are denoted $\mathbf{Z}_{l}^{j}$ and $\mathbf{R}_{l}^{j}$. The non-negative parameters $\{\gamma_{l}\}$ control the sparsity of the coefficient maps in different layers, and $\beta>0$ captures the relative trade-off between the data-fidelity term and regularizer.

In this section, we evaluate the image reconstruction quality for the proposed PWLS-MRST algorithm and compare it with conventional FBP and the PWLS-EP (Edge Preserving) method [16].

This paper proposes a general framework for Multi-layer Residual Sparsifying Transform (MRST) learning wherein the transform domain residual maps over several layers are jointly sparsified. Our work then applies MRST learning to low-dose CT (LDCT) image reconstruction by using a PWLS approach with a learned MRST-based regularizer. Experimental results illustrate the promising performance of the multi-layer scheme over single-layer learned sparsifying transforms. Learned MRST models also offer significant improvement over typical nonadaptive methods. In the future, we will investigate additional strategies such as pooling operations to reduce noise over layers.

The authors thank Dr. Cynthia McCollough, the Mayo Clinic, the American Association of Physicists in Medicine, and the National Institute of Biomedical Imaging and Bioengineering for providing the Mayo Clinic data.