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.
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.
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 space1-2. 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.
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