Synopsis

Constrained image reconstruction incorporating prior information has been widely used to overcome the ill-posedness of reconstruction problems. In this work, we propose a novel "kernel+sparse" model for constrained image reconstruction. This model represents the desired image as a function of features "learned" from prior images plus a sparse component that captures localized novel features. The proposed method has been validated using multiple MR applications as example. It may prove useful for solving a range of image reconstruction problems in various MR applications where both prior information and localized novel features exist.

Introduction

Image reconstruction is known to be an ill-posed mathematical problem because most imaging operators are ill-conditioned and its feasible solutions are not unique due to finite sampling. To address this issue, constrained reconstruction incorporating prior information has been widely used. A popular approach to constrained reconstruction is to use regularization in which priori information is incorporated implicitly in a regularization functional. In this work, we propose a novel “kernel+sparse” model for constrained reconstruction. This model represents the desired image as a function of features “learned” from prior images plus a sparse component that captures localized novel features. The proposed representation has been validated using multiple MR applications as a testbed.

Method

Kernel+Sparse Model

We decompose the spatial variations of a desired image function into two terms, one absorbing prior information (using a kernel model) and the other capturing localized sparse features:

$$\hspace{14em}\rho(\boldsymbol{x}_n)=\sum_{i=1}^N\alpha_{i}k(i,n)+\tilde{\rho}(\boldsymbol{x}_n).\hspace{14em}(1)$$

The kernel component was motivated by the success of kernel models in machine learning. More specifically, this component models the desired image value at spatial location $\boldsymbol{x}_n$ as a function of a set of low-dimensional features $\boldsymbol{f}_n\in\mathbb{R}^m$:

$$\hspace{16.5em}\rho(\boldsymbol{x}_n)=\Omega(\boldsymbol{f}_n).\hspace{16.5em}(2)$$

The features $\{\boldsymbol{f}_n\}_{n=1}^N$ are learned/extracted from prior images, which leads to implicit incorporation of priori information. However, the function $\Omega(\cdot)$ is often highly complex in practice and cannot be accurately described as a linear operator in the original feature space^1-2. Inspired by the "kernel trick" in machine learning, we linearize $\Omega(\cdot)$ in a high-dimensional transformed space spanned by $\{\phi(\boldsymbol{f}_n):\boldsymbol{f}_n\in\mathbb{R}^m\}$:

$$\hspace{16.0em}\Omega(\boldsymbol{f}_n)=\omega^T\phi(\boldsymbol{f}_n).\hspace{16.0em}(3)$$

In the sense of empirical risk minimization (ERM), the optimal $\omega$ should minimize the empirical risk:

$$\hspace{12.8em}r_{amp}(\omega)=\frac{1}{N}\sum_{n=1}^{N}l(\omega^T\phi(\boldsymbol{f}_n),\rho(\boldsymbol{x}_n)),\hspace{12.8em}(4)$$

where $l(\cdot)$ is some loss function (e.g., square-error loss). The well-known representer theorem ensures that this optimal $\omega$ takes the following form³:

$$\hspace{16.2em}\omega=\sum_{n=1}^N\alpha_i\phi(\boldsymbol{f}_i).\hspace{16.2em}(5)$$

Hence we obtain the kernel-based representation for $\rho(\boldsymbol{x}_n)$ as:

$$\hspace{11.7em}\rho(\boldsymbol{x}_n)=\sum_{i=1}^N\alpha_i\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f}_n)=\sum_{i=1}^N\alpha_ik(i,n),\hspace{11.7em}(6)$$

where $k(i,n)=\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f}_n)$ is a kernel function. However, Eq. (6) alone may bias the model towards prior information. To avoid this potential problem, we introduce a sparsity term into Eq. (6) to capture localized novel features as described in Eq. (1) with the requirement that $||M\{\tilde{\rho}(\boldsymbol{x}_n)\}||_0\leq{\epsilon}$ where $M(\cdot)$ is some sparsifying transform.

Image Reconstruction

Image reconstruction using the proposed model requires specification of the kernel function and features. In this work, we choose the radial Gaussian kernel function:

$$\hspace{13.9em}k(\boldsymbol{f}_i,\boldsymbol{f}_n)=\exp(-\frac{||\boldsymbol{f}_i-\boldsymbol{f}_n||_2^2}{2\sigma^2})\hspace{13.9em}(7)$$

which corresponds to an infinite-dimensional mapping function². Choices of features are rather flexible, such as image intensities and edge information, making the proposed model even more powerful in absorbing a large range of priors.

The proposed kernel-based signal model results in maximum likelihood reconstruction by solving:

$$\hspace{5.7em}\{\alpha_i^*,\tilde{\rho}^*\}=\arg\max_{\{\alpha,\tilde{\rho}\}}L(d,I(\{\sum_{i=1}^{N}\alpha_ik(i,n)+\tilde{\rho}(\boldsymbol{x}_n)\})),\mathrm{s.t.}||M\{\tilde{\rho}(\boldsymbol{x}_n)\}||_0\leq{\epsilon}.\hspace{5.7em}(8)$$

where $d$ denotes the measured data, $I(\cdot)$ the imaging operator, and $L(\cdot,\cdot)$ the likelihood function.

Results

Conclusions

Acknowledgements

References

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