A number of dynamic MRI applications have seen the adaption of data-driven models for efficient de-noising, and reconstruction from under-sampled data [1-6]. Data-driven models have shown to be advantageous over models that are based on pre-determined transforms. In this work, we develop a novel temporal point spread function interpretation of the low rank [1-5], and dictionary-learning models [6] that are used for de-noising dynamic MRI. Through this interpretation, we show (a) the low rank model to perform spatially invariant non-local view-sharing, and (b) the dictionary-learning model to perform spatially varying non-local view-sharing. Both the models can be viewed as efficient data-driven retrospective binning techniques.

Dynamic signal model: The dynamic pixel time profile $$$\gamma(\mathbf x,t)$$$ is modeled as a linear combination of temporal basis functions $$$v_{i}(t)$$$ derived from the data [1-2]: $$\mathbf \Gamma_{p\times m} = \mathbf V_{p \times r} \mathbf U_{r \times m};$$where $$$\mathbf \Gamma$$$ is the Casorati matrix representation of the dynamic data, and $$$\mathbf U$$$ is the spatial weight matrix; $$$p, r,m$$$ are respectively the number of time frames, number of basis functions, and number of pixels in a time frame. The low rank model assumes linear combination with a few number of orthogonal bases (i.e, $$$r<p$$$), while dictionary learning assumes a sparse linear combination of a large number of bases that are not necessary orthogonal (i.e $$$r>p$$$; $$$\|u_i\|_{0}<=k; k<r$$$), where $$$k$$$ is the sparsity level.

Denoising using the low-rank model: Denoising with the low rank model can be formulated as $$\min_{\mathbf U}\|\mathbf V\mathbf U-\mathbf \Gamma_{n}\|_{F}^{2};$$ where $$$\mathbf \Gamma_{n}$$$ is the noisy dynamic data. $$$\mathbf V$$$ is estimated from the data itself via SVD decomposition of rank $$$r$$$ approximation of $$$\Gamma_n$$$; $$$(r<p)$$$. The denoised solution is given as: $$\hat{\mathbf \Gamma}=\underbrace{\mathbf V(\mathbf V^{T}\mathbf V)^{-1}\mathbf V^{T}}_{\mathbf Q_{p\times p}}\mathbf \Gamma_n;$$The above suggests that every spatial frame in $$$\hat{\mathbf \Gamma}$$$ is a weighted linear combination of spatial frames from $$$\mathbf \Gamma_n$$$. The weights are determined by columns (or rows) of the symmetric $$$\mathbf Q$$$ matrix. We term the columns of this matrix as temporal point spread functions (TPSF) as it characterizes averaging across time.

Denoising using the dictionary-learning model: The problem is formulated as joint estimation of $$$\mathbf U$$$ and $$$\mathbf V$$$: $$\min_{\mathbf U, \mathbf V}\|\mathbf V \mathbf U-\mathbf \Gamma_n\|_{F}^{2}; \mbox{such that}, \|u_{i}\|_{0}<=k; \|v_i\|_{2}^{2}<=1.$$ The above can be solved by dictionary learning algorithms such as k-SVD [7], with the resulting solution: $$\hat{\mathbf \Gamma}=\underbrace{\mathbf V_{red}(\mathbf V_{red}^{T}\mathbf V_{red})^{-1}\mathbf V_{red}^{T}}_{\mathbf Q_{p\times p}}\mathbf \Gamma_n;$$where the rows of the matrix $$$\mathbf V_{red}$$$ are the temporal basis functions that are active at a specified pixel. Note, $$$\mathbf V_{red}$$$ is the reduced subset from the dictionary $$$\mathbf V$$$ and will vary for different spatial pixels. This implies the TPSF is spatially varying.

Dynamic data of the upper-airway in the mid-sagittal plane was acquired on a GE 3T scanner with the head coil, using a fast gradient echo sequence (Flip angle: 15⁰, TR= 3.28 ms; time-resolution: 420ms). The subject produced repeated utterance of the sound: "za-na-za-na". The head coil has low sensitivity in the dynamic articulatory regions of interest. The low rank, and dictionary-learning algorithms are applied to improve the signal to noise in these regions. Rank, and sparsity in the respective algorithms were chosen such that $$$\| \hat{\mathbf \Gamma} - \mathbf \Gamma_n \|_{F}^{2} $$$ lied within the noise level.Figure 1 shows representative TPSFs during low rank denoising. Two frames are picked which correspond to different motion states: velum touching the pharyngeal wall, and velum in resting position. The peaks of the TPSF correspond to the frames that have the most weight in the linear combination, and correspond to instances of similar motion states, which suggests implicit non-local view-sharing. Figure 2 show representative spatially varying TPSFs during dictionary learning de-noising. The spatial locations are chosen at three different air-tissue interfaces, where the articulators have different rates of movements. Note how the characteristics of the TPSFs are correlated with underlying denoised dynamic pixel time profiles, suggesting spatial adaption of non-local view-sharing. Figure 3 finally shows the denoising results using the two algorithms. We provide a novel interpretation of low rank and dictionary learning algorithms as retrospective data-rebinning techniques. The low rank model can be interpreted as a spatial invariant non-local view-sharing method. The dictionary learning model can be interpreted as a spatially variant non-local view-sharing method. The temporal point spread functions can be used as a means to characterize blurring with these methods. Our analysis suggests the data-driven models to be very efficient in DMRI applications with quasi-periodicity such as dynamic lung imaging, free breathing cardiac imaging, and repeated speech utterances.